Talk:Equal temperament

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Measure of Purity[edit]

I've removed this recent addition to the page for the moment because its meaning is quite vague. How is this measurement of "purity" of an equal temperament derived? In what way does this measure "closeness" to just intervals? Is there are source for the derivation and use of this function? What does it mean, and why? (Is this original research?) - Rainwarrior (talk) 05:11, 6 December 2007 (UTC)

The musical purity of an equal temperament can be measured mathematically:
where r is the ratio and is Riemann zeta function.
The larger is purity, the closer are temperated intervals to just ones. Some equal temperaments are compared by their purity in the following table:
Scale Ratio Purity
5-EDO 1.148698355 2.33814509
7-EDO 1.104089514 2.43157737
BP scale 1.088182243 1.90437042
12-EDO 1.059463094 2.81128026
15-EDO 1.047294123 2.39794095
19-EDO 1.037155044 2.70707630
22-EDO 1.032008280 2.91094249
24-EDO 1.029302237 2.82078484
31-EDO 1.022611436 3.15408628
34-EDO 1.020595910 2.77166598
41-EDO 1.017049744 3.30355852
46-EDO 1.015182518 3.22978190
48-EDO 1.014545335 2.62243154
53-EDO 1.013164143 3.46211565
55-EDO 1.012682424 1.49969344
72-EDO 1.009673533 3.09900534
99-EDO 1.007026054 3.07429123
118-EDO 1.005891415 3.41236220
171-EDO 1.004061719 3.81529362

Shouldn't the term 'musical' purity be removed if the above table is to feature in the article... and maybe replaced? 'Mathematical' purity seems a more appropriate concept for the data provided given that maths is used to determine this purity and also that simple intervals such as a conventional 5th are arguably more 'musically' pure than say a major 7th... even when both are mathematically 100% pure using just intonation. (17:11 can be mathematically pure but musically it's pretty discordant). Also note that just intonation in principle presents an infinitude of pitches and so it is nothing more than custom that makes us compare any given pitch to one of the traditional notes that are derived by taking simpler ratios to the root. (e.g. 125:64 is a just intonation pitch, if not a common one). AnotherPath (talk) 14:15, 25 October 2013 (UTC)

This could be used as a reference I think: Droog Andrey (talk) 13:12, 25 July 2014 (UTC)

Mention of Pythagorean Comma[edit]

Maybe it should be mentioned that this system is the ultimate outcome of searching for a solution to the initial problem of the Pythagorean Comma, where the so-called Pythagorean scale represents the earliest attempts to provide a mathematical understanding of pitch scale and the equal tempered system provide the final solution to all its problems. A brief paragraph about the history of the difficulty in combining mathematics with human hearing and harmony might provide a (more) useful introduction. (talk) 15:22, 4 March 2008 (UTC)


It is now well-accepted that of the two primary tuning systems in gamelan music, slendro and pelog, only slendro somewhat resembles five-tone equal temperament while pelog is highly unequal; however, Surjodiningrat et al. (1972) has analyzed pelog as a seven-note subset of nine-tone equal temperament.

What does "unequal" mean? (talk) 11:02, 3 June 2008 (UTC)

Figure out what's "equal" about equal temperament, and you've got your answer. (talk) 15:56, 3 June 2008 (UTC)

Graphical ET comparison[edit]

File:Comparison of some tet-scales against M3P5P7.jpg
A graphical comparison of a few equal temperament scales with the first few harmonics. [dubious ]

This image is nice and colorful but it's not clear what the X-axis numeric values are measuring (e.g. 0.322, 0.585, etc.) Proportion of the octave? logarithmic? Maybe cents would make more sense. :) Any unit label would probably be an improvement. I'm inclined to think this diagram might actually be in error, though, because if M3 and P5 refer to a Major Third and Perfect fifth... then what does P7 refer to? I've tagged this image as dubious but really, just "relabel and fact-check" is what I'm suggesting. --Ds13 (talk) 07:40, 9 July 2008 (UTC)

Thanks for your comments. There is more about the image in Music and mathematics.
The numbers seem to be interval ratios log2:
  • 20.322=1.250
  • 20.585=1.500
  • 20.807=1.750
I agree that a more standard scale (e.g. cents) would be preferable, so that the values would correspond to the values in the table comparing equal temperament and just intonation.
The octave is labeled "U1". ISTM, this should be "P8".
Also, the very informative comparision table shows the minor seventh as having an interval of 7/4=1.75 in just tuning. However, the articles on the minor seventh and the harmonic seventh say 7/4 is distinct from any interval in just intonation.
--Jtir (talk) 21:31, 9 July 2008 (UTC)
7:4 is a just intonation interval, by definition: it's expressed as a ratio between frequencies. — Gwalla | Talk 18:33, 10 July 2008 (UTC)
OK. Just intonation says so in the first sentence. What I meant was that calling 7:4 a minor seventh in the table is confusing or misleading. (I now see a note below the table that partly addresses this problem.) --Jtir (talk) 23:18, 11 July 2008 (UTC)

If you compare the "P7" with the 12tet lines it is a flat minor seventh or "m7". Hyacinth (talk) 21:47, 9 July 2008 (UTC)

However, since Wikipedia is not a textbook nor the place for original research perhaps we should lay this aside until the person who made it can answer our questions. Hyacinth (talk) 18:40, 10 July 2008 (UTC)

The lines are pretty much what they say: the first few harmonics. The first few harmonics have frequency ratios of 2, 3, 4, 5, 6, 7, 8 relative to the fundamental. From the fundamental, those intervals are: octave, octave+fifth, 2 octaves, 2 octaves plus third, 2 octaves plus fifth, 2 octaves plus "minor seventh", 3 octaves. The lines on the diagram show these intervals, with the whole octaves subtracted. This leaves: just third (5/4), just fifth (3/2), and "just minor seventh" (7/4). Rracecarr (talk) 19:10, 10 July 2008 (UTC)
My point above is that after spending all this time to answer one simple question, one still may be left with so many other questions. For example: What is .807? Why does it matter? What is the source for this information? Why is it notable? Why make the comparison in the first place? It may be more worthwhile to go create your own solid work than to try to figure this out. Hyacinth (talk) 19:19, 10 July 2008 (UTC)
I don't understand the problem with the figure. I actually found pretty much everything GaulArmstrong wrote to be incomprehensible, and that's who made the figure, but I actually think it's very good. It shows in an obvious way how well various equal tempered scales coincide with simple frequency ratios. Why take it out of the article? Presumably you have the answers to your questions (is that why you crossed them out?) Rracecarr (talk) 19:50, 10 July 2008 (UTC)
No, I crossed them out because I wasn't asking you them directly.
The problem with this figure is that it contains error(s) ("P7") and needs explanation.
Regarding the error, would you recreate the image with corrections and find citations? (note that the text and the graph in the image do not overlap!)
Regarding the explinations, what is .807?
Hyacinth (talk) 20:27, 10 July 2008 (UTC)

I might be able to photoshop the image, but I do not really feel like hunting down references for this. .807 is log base 2 of 7/4. 7/4 is the ratio of frequencies of the 6th harmonic to the 3rd, the 3rd harmonic being 2 octaves up from the fundamental. So the 6th harmonic is .807 of the way (969 cents) from one octave to the next. This could be explained better, maybe in the caption.Rracecarr (talk) 01:41, 11 July 2008 (UTC)

I agree that the image is informative (and colorful). Photoshopping is a good idea. IMO, the simplest "fix" would be to replace the numbers with the whole number ratios: 1, 5/4, 3/2, 7/4, 2. The octave label could be replaced with "P8". I'm not sure how to fix "P7" except by replacing it with a more standard interval. A general approach would be to label more intervals. There is an excellent example in this image of a Blackman Spectral Analysis of two sinusoidal tones diverging from unison to octave. A more ambitious project would be to reimplement the image as an SVG.
I'm not sure why any of this would need to be sourced beyond what the articles on musical intervals already provide. Hyacinth?
--Jtir (talk) 18:39, 11 July 2008 (UTC)
A rationale for the contrast and comparison, and then an explanation of it. Hyacinth (talk) 19:25, 11 July 2008 (UTC)
OK. Does the article already have a rationale that could be adapted for a caption, IYO? --Jtir (talk) 20:32, 11 July 2008 (UTC)
That temperaments are meant to approximate just intervals is rationalization enough, IMO. A brief explanation or legend would be more important than a rationale. — Gwalla | Talk 22:35, 11 July 2008 (UTC)

OK, I failed at the svg thing. Here is a png. Comments? Rracecarr (talk) 19:44, 11 July 2008 (UTC)


Thanks. Your relabeling looks good. Simply omitting "P7" is fine with me. Hyacinth?
Your SVG file had the JPEG image embedded in it, so it wouldn't have been truly scalable. Converting to PNG is a good idea. To create a truly scalable SVG image would require redrawing the image, I believe. --Jtir (talk) 19:52, 11 July 2008 (UTC)
Unless there's a slicker way I don't know about, redrawing is the way to do it. See Image_talk:Equaltemper.svg __Just plain Bill (talk) 19:59, 11 July 2008 (UTC)
Thanks for your reply. More at Image_talk:Equaltemper.svg. --Jtir (talk) 20:34, 11 July 2008 (UTC)
Hey that's looking good. This is definitely an improvement. --Ds13 (talk) 22:05, 11 July 2008 (UTC)

Here is how far I got with it tonight: work in progress: Equaltemper.svg

I added a scale from 0 to 1 at the bottom, representing the base-2 log of the ratios, and tweaked the placement of the lines. Log of 1.25 is 0.32193, log of 1.5 is 0.58496, and log of 1.75 is 0.80735 to five places, according to my open office spreadsheet.

Remaining issues that I see are two: the "19, 21, and 22" bars each have an extra box, making them actually represent 20, 22, and 23-tet. Second, the boxes in the 31-tet row were not all exactly the same size in the original JPG, and still aren't in this early "tracing" of it. I'll get around to fixing them soon enough, I hope, and then reload it without the "PRELIMINARY" stamp. In the meantime, comment is invited. Any improvements you see that could be made? This may actually turn out to be a useful presentation, once the dimensions of it are adjusted better. In my opinion, the colors are chartjunk, and could just as well be all white. 'Night for now, __Just plain Bill (talk) 04:10, 13 July 2008 (UTC)

Further adjustments: boxes drawn to better accuracy, limits imposed by manual positioning of bar ends at unison and octave lines, & perhaps vanishingly minor artifacts of Inkscape's copy, paste, and scale functions. Interesting that none of the equal temperaments renders a perfect fifth perfectly. __Just plain Bill (talk) 07:29, 13 July 2008 (UTC)
Thanks! That is much crisper (the text especially) and is more subdued with shades of grey. I appreciate your interest in making the image accurate. Adding the decimal scale along the bottom is a good idea. Here are some suggestions and comments (ordered by priority, IMO):
  1. Add vertical bars for more just intervals, such as the ones in the table comparing 12-TET and JI or in this image.
  2. Use cents for the scale along the bottom (x cents = 1200 log2 m/n) and label the scale (from my limited knowledge, log2 isn't used much in music, so it would need to be explained). The log2 scale could be retained as a hidden layer, however.
  3. Add 24-TET (or replace one of the others with it). The importance of this scale is a matter of opinion, and I am not qualified to say more than that it has been used by Western composers and is used in Arab music.
  4. Add a title so that the graph (chart? diagram?) is self-documenting.
  5. Use "TET" instead of "tet". The article consistently uses upper case for this abbreviation.
  6. Unison says its abbreviation is "P1".
  7. The grey-shaded horizontal bars could be [mis]interpreted as keys on hypothetical keyboards. (I like this interpretation, because it makes the image much more concrete.)
--Jtir (talk) 20:32, 13 July 2008 (UTC)
You'll now see most of those items done; if not, a screen refresh should take care of things. I'm not actually that speedy, but was working on the cents scale this morning, as well as adding a 24-TET bar at that time. More just intervals coming up in a bit... thanks for the quick feedback! __Just plain Bill (talk) 21:25, 13 July 2008 (UTC)
Nice! Can we get the 7/4 line labeled a minor seventh (m7)? Hyacinth (talk) 21:56, 13 July 2008 (UTC)
There are now three minor sevenths there. The 7/4 one is right out for all to see, and I tried to hide the 16/9 and 9/5 ones in a layer called foo. Here in Firefox I see them on this page, but not when I view the image file from its own page proper. Oh, well.
I'm pretty much done with this thing. I make no claim of ownership, of course; anyone with an SVG editor is free to go nuts with it. That said, I'll be keeping an eye out for suggestions for further elaboration or simplification or such. __Just plain Bill (talk) 02:13, 14 July 2008 (UTC)
I'm a little wary of referring to 7-limit ratios with unmodified diatonic names, since that naming scheme was developed only to reference (approximations of) 5-limit intervals. And there's no real standard for abbreviating terms like "septimal minor" or "supermajor". I think they should be unadorned. M2, on the other hand, could be placed above 9:8 and 10:9, because it can refer to either. Just as a quibble, I think 41-, 53-, and 72-tet have better claims to a place on the chart than 15- or 21-tet, due to usage (and enough notability to merit wikipedia articles), but they may be too fine-grained to show up well, and it's not a big deal either way. — Gwalla | Talk 03:06, 14 July 2008 (UTC)
You should label every line of that type or don't label them, you shouldn't just leave the one line out. See my concerns above regarding WP:NOTTEXTBOOK and WP:OR and Harmonic seventh. Hyacinth (talk) 03:48, 14 July 2008 (UTC)
It would be labelled, as "7:4". I supposed since it's called a septimal minor seventh that it can considered a type of minor seventh. Not sure what WP:NOTTEXTBOOK has to do with any of this though. — Gwalla | Talk 04:31, 14 July 2008 (UTC)
Now that I think about it, if 7:4 is a minor seventh, shouldn't 7:5 be labelled as a diminished fifth instead of an augmented fourth? It's the difference between a type of seventh and a type of third, which works out to a fifth. — Gwalla | Talk 17:01, 14 July 2008 (UTC)
As an aside, many of the images in Music and mathematics have the same problems as the old image here had: nonstandard abbreviations ("U1"), "perfect seventh", unusual units of measurement ("percent off of perfectly representing the harmonic identity"), etc. — Gwalla | Talk 04:31, 14 July 2008 (UTC)
No problem putting on 72-TET; it'll show up even in an image 800 pixels wide, which is what I see above. If you zoom in on the image, you'll see that each individual cent has its own tick mark, and that P4 and P5 are a little less than 2 cents off their 12-TET "equivalents." Not bad, considering that the lines were placed by eye in an attempt to get 3 or 4 decimal places of accuracy w.r.t. the log scale below the diagram. That's a beauty of SVG, although I'm not sure how a casual user would do that zooming.
and now we get to the nuts & bolts of it: what ratios ought to be shown, and how should they be labeled? Carry on... __Just plain Bill (talk) 10:46, 14 July 2008 (UTC)
I think my vote is for not too many intervals. I think the utility of the image is as a very quick way to gage how well various scales match important intervals, and with too many intervals, a quick impression becomes difficult. I would certainly eliminate P4 (4/3), because it is complementary with P5 (3/2): P4 is as far from one end of the octave as P5 is from the other, so both match any equal tempering of the octave equally well. Likewise, 8/5 and 6/5 can go because they are complementary with M3 (5/4) and 5/3 respectively.
I'm not advocating adding them, but some intervals not shown are 11/6, 9/7, and 9/5. Rracecarr (talk) 14:22, 14 July 2008 (UTC)
I don't think 4:3 should be dropped, because it's such a common and important interval. Also, because it is complementary with the perfect fifth, it shows the symmetry. I also think the 6:5 should stay, since it's the minor third. 8:5 can probably be dropped. The 11-limit and 13-limit intervals shown seem a bit arbitrary; I'd drop 11:7 and 13:7, and replace them with the octave equivalents of the harmonics, 11:8 and 13:8. The 11:9 neutral third is probably more important that 11:7 as well (and would illustrate a feature of 24-tet). Should we show the semitone ratios? — Gwalla | Talk 17:01, 14 July 2008 (UTC)
Sure, P4 is a common and important interval. But showing it does not add any information, since it matches identically well as P5. Similarly, I see nothing to be gained by showing both 6/5 and 5/3, since they also are complementary. The caption can point out these relationships, but drawing in extra lines that add no new information is not helpful, in my opinion. Rracecarr (talk) 18:23, 14 July 2008 (UTC)
Thanks Bill, that looks very refined with the additional labels and the title.
WRT, to the interval names: since they may be ambiguous without qualification or context, as Gwalla has noted, the names could be aligned with the 12-TET intervals (every 100 cents) and the caption could note the ambiguity.
WRT to what should be shown: As Rracecarr suggests, there shouldn't be too many (there is only so much room across the top of the graph). Each of the 12-TET intervals has an article that lists the nearby just intervals. That seems to imply 0, 1, or 2 just intervals for each interval in the 12-TET scale. And for consistency, the graph should show "m2".
I don't really understand Hyacinth's concern that this graph is OR, but I did try to find an example of this type of presentation at and did not, although the best sources were available as snippets or previews, so I could have missed something. Grove doesn't show a graph either. IMO, the ideal approach would be to cite an independent source that compares various ETs and JI in a table or graph. The second best would be to argue that the graph is simply presenting already documented facts in a new, though neutral, form. but here is a source that has a similar graph.
--Jtir (talk) 19:18, 14 July 2008 (UTC)
I don't understand the accusation of OR either. The OR policy is meant to prevent people from promoting pet theories that have not been accepted by anyone else. But this is, at most, simply presenting uncontroversial information in a new form (and I'd be kind of surprised if it's never been done before). It's the graphical equivalent of rephrasing (which is an accepted method of incorporating information while avoiding copyvio). As for Rracecarr's argument, I think we should all remember that these articles are meant as much for a lay audience as for specialists, especially general overview articles like this, and particularly explanatory graphics. While someone who is already familiar with just intonation may understand that 4:3 is the inversion of 3:2, that's not obvious to someone for whom this is all new, and neither is the fact that an equal temperament will match them both equally well (without doing the math). The 6:5 minor third is even more important to show, because it isn't matched just as well as the major third by equal divisions of the octave (equal divisions of the perfect fifth are a different story, but I don't think we should bring Wendy Carlos's scales into this). — Gwalla | Talk 21:28, 14 July 2008 (UTC)
OR? not really-- just a visual presentation of well-known facts IMO, derivable by anyone who did OK in high school math and has a shred of musical experience.
Count me with the ones who favor fewer intervals shown. If it was up to me, it would only show lines for P5, M3, and a couple of the 6:5 m3 ratios, with the need for P1 and P8 as endpoints being obvious. Maybe the 7:4 line, with or without a name, since it shows a poor fit to most ET scales, as mentioned before. I'd like to see a way to call attention to the fact that even though 12-TET looks like it comes pretty close to getting a P5 (and its inversion, P4) it is still off by enough to matter, once you start stacking fifths (e.g. just cello tuning gives a C that is flat to a piano with an identical A.)
Not much of a theoretician, me, I'd rather spend time learning creative voice-leading. Still, I'd rather not burden the chart with too much. Is it meant to be an introduction to how a boatload of just intervals mesh or don't mesh with ET, or is it meant to show how the most important ones do, or don't? __Just plain Bill (talk) 21:52, 14 July 2008 (UTC)
Well, the 9:8 does show the discrepancy with successive fifths, to some extent (9:8 is two 3:2s, transposed down an octave). I don't know if there's any better way of showing that, without extending the scale past a single octave. — Gwalla | Talk 23:00, 15 July 2008 (UTC)

One possible solution to the problem of excessive lines is to jettison the log2ratio scale (which is sort of redundant with cents anyway), and instead use the bottom line as a downward scale (cents below the high note). The left edge would be labeled P1 at the top and P8 at the bottom, and vice versa on the right. Each line would then do double duty as a ratio and its octave complement. Maybe this would be confusing, I don't know.

I do think we should include some of the more important consonant intervals from higher limits than 5 to give an impression of how well various equal temperaments accommodate these limits (at the very least, the 7th and 11th harmonics, and probably the 7:6 subminor third and one of the 11-limit neutral thirds). — Gwalla | Talk 07:29, 15 July 2008 (UTC)

Agreed, the log scale is a leftover of how the lines got placed in the drawing. I'd favor losing it entirely, without replacement, since that busy little cents scale accounts for a majority of the file size just now. If someone figures out how to trim the redundancy in its SVG description, it might be useful to have another cent scale at the bottom, in the regular forward direction; if someone is into it enough, the complementary relations will stand out anyway.
I like the idea of having a lot of intervals shown, and, with my skimpy theoretical expertise, will defer to almost anyone else's choice thereof. (I guess that weakens my earlier support of only showing a few. So be it.) I might approach that by making a difference in the font, weight, or color of the numeric ratio (depending on the limit used?) and adding some unobtrusive brackets tying groups of ratios and lines to the interval names. _Just plain Bill (talk) 12:40, 15 July 2008 (UTC)
Plus one for more rather than fewer intervals (current is about right, except "m2" needs to be added). Brackets are a good idea. I believe that the lines could be placed accurately with the cents scale alone. We also need to figure out a way to make it easy to display a higher resolution image — ATM some of the details (labels, cents) are barely legible. --Jtir (talk) 13:54, 15 July 2008 (UTC)
Color code by limit, perhaps? The cent numbering could probably be larger for readability without making the image much larger. — Gwalla | Talk 15:46, 15 July 2008 (UTC)
Coloring is a good idea. The interval name and the related interval ratios could all be one color. (Brackets or colored bars might provoke a prolonged discussion of where the "ends" should be.)
I agree that the font size of the cent numbering could be increased. Also, the SVG page size could be increased (not sure to what — 1000x450?) and the image link in the article could specify a reduced size. The syntax is the same as for raster images. Further, Firefox 3.0 can rescale SVG images. --Jtir (talk) 18:29, 15 July 2008 (UTC)
I really like Gwalla's idea of a reverse scale, so that each line can clearly represent two intervals. Rracecarr (talk) 18:40, 15 July 2008 (UTC)
I just realized Jtir had already suggested color coding by limit, right in the comment I was replying to. I know how to read, I swear! Anyway, I prefer coloring by limit rather than interval quality. The latter merely shows similarity in size, which is already visually indicated by distance. Color coding by limit would allow people to see at a glance roughly how well the various temperaments approximate each limit. — Gwalla | Talk 23:00, 15 July 2008 (UTC)
Actually, I was responding to your proposal for using color. :-) Could you do a mockup? (I'm not entirely sure I understand what you have in mind.) --Jtir (talk) 23:47, 15 July 2008 (UTC)
Now I'm really confused. O_o I swear, earlier today your suggestion preceded mine. Maybe I'm going insane. Anyway, sorry, but I can't do a mock-up. My SVG-fu is weak. — Gwalla | Talk
PNG would be fine for a mockup. WP renders SVG files as PNG, so by right-clicking on the image you can save it locally (default file name is 800px-Equaltemper.svg.png). --Jtir (talk) 12:13, 16 July 2008 (UTC)
I happen to think that color would be good for making a difference between various limits applied to the just interval ratios. I'd leave the lines themselves black, to avoid visual artifacts which might confuse or obscure the diagram. Colors, in this view, would only be applied to the numbers in the "ratio" row.
I don't favor coloring the variously limited takes on any given named interval, since they will be clustered together in their horizontal placement anyway. Hope that made sense.
Oh, and why not make it a bit bigger than nominally 1200px wide, say 1300? One cent per pixel might be neat. (It might be next week before I can fix up the image some more.) __Just plain Bill (talk) 15:10, 16 July 2008 (UTC)
1200 pixels for 1200 cents sounds like a deal. :-)
Confessing my ignorance ... having now realized that limit is a technical term, I agree that color coding by limit is a really good idea.
Since crowding is a problem in the ratio row, it could be split into two (or more) rows (based on limit?).
BTW, I added "m7" and some labels to the PNG version, since it will be in the article a while longer (and I wanted to practice my GIMP-fu :-)).--Jtir (talk) 21:41, 16 July 2008 (UTC)
work in progress -- comments invited

This time I let the machine handle the line placement, by cloning P1 and moving it by some number of cents. You can see there's room for more ratios...

__Just plain Bill (talk) 07:24, 18 July 2008 (UTC)

That's awesome. I see you have included the backward scale, so you can eliminate the 4 inversion pairs: [3:2 and 4:3], [6:5 and 5:3], [5:4 and 8:5], and [7:4 and 8:7]. I don't know the best way to do this. Maybe include another set of stripes at the bottom for interval and ratio labeling, with the labels on the top referring to the left side and those on the bottom referring to the right? Rracecarr (talk) 14:03, 18 July 2008 (UTC)

Thanks for the encouragement! I just uploaded a revision with pointers dropping from the limit bars. You can still see the way it was before, if you go to the image and look at the previous rev. As long as we're taking advantage of a huge image's resolution, I'd just as soon keep the inversion pairs. They show a certain symmetry available in music; usually the dominant and subdominant play different rôles, so any redundancy there is visually useful, IMO. __Just plain Bill (talk) 14:22, 18 July 2008 (UTC)
You are doing the work here, and I will leave it up to you. For your consideration, here are the reasons I'm in favor of getting rid of inversion pairs:
  1. The more lines you add, the harder it is get a visual impression of how well each ET matches the lines overall. Looking back at the original version of the figure, it is immediately apparent that 31-TET matches the three lines shown there better than any previous scale. With 16 lines, that kind of quick impression is not possible, at least for me. The 4 extras do not add any new information, since they are identical to 4 that are already there, but they do add to the visual clutter that makes a quick impression difficult.
  2. I do not think drawing both members of the inversion pairs shows the symmetry. Maybe it would if you showed the inversion of every interval (which would be silly) but as it stands there are 8 intervals whose inversions are not shown, and it is not at all obvious that the other 8 are symmetrically positioned. I think the symmetry would actually be emphasized much more clearly and explicitly by giving some of the lines two labels.
  3. If it seems like, say, 3:2 and 4:3 are both important intervals and should not be under-emphasized by allowing them only one line, that line can be made more prominent, for instance by drawing it thicker than 13:7 or other less important lines.
Just my 2 bits. Take it or leave it. Rracecarr (talk) 14:57, 18 July 2008 (UTC)
Wow! That's gorgeous, Bill. You have a real talent for graphic design. I am particularly impressed by the way you positioned the "INTERVAL" label and used a line to discreetly outline the interval abbreviations. Also, I like your choice of pastel colors and use of colored wedges as pointers. Positioning the lines semi-automatically is a very smart idea (I tried it (Transform dialogl), and it works great now that you have 1 pixel = 1 cent) (FYI, the lines are actually copies, not clones).
The reversed cents scale is going need to be fully explained in the caption or article — why it is in the graph is not at all intuitive. (It will need a label too — Rracecarr?). From Rracecarr's comments, it sounds like the graph does not consistently display inversion pairs. I like to see symmetry visually, and it seems like showing pairs would do this, if they could be be graphically tied together. Again, this feature of the graph will need further explanation — this description of inverted intervals might be a good start.
FYI, you could halve the size of the file, which is now at 1.11 MB, by cloning the reversed cents scale, instead of copying it, and then transforming it as desired. You could halve the size again by combining the cents scale into one path, as outlined here (I have tried both of these, and they seem to work, but I don't know if there are unwanted side-effects).
--Jtir (talk) 17:37, 18 July 2008 (UTC)
I would label the "cents" on the bottom scale "cents" just like the label on the upper scale. Intervals would also be labeled on the bottom: 2:1 at the left edge, 1:1 at the right edge, and whichever of the other intervals seem appropriate. The caption could explain that intervals in the top band are depicted relative to the left edge, and that those in the bottom band relative to the right edge. Rracecarr (talk) 19:04, 18 July 2008 (UTC)
OK, thanks. We would also need to explain somewhere, why this is being done (something brief in the caption, possibly linked to a fuller explanation). I'm not really qualified to do more than offer "suggestions". :-) --Jtir (talk) 19:15, 18 July 2008 (UTC)
The reason I'm in favor of the dual scale has nothing to do with music theory. I just think it's a more efficient way to present the information, and that a visually simpler graph is better. Also, I guess it makes it clear which pitches are inversions of each other, which may be a slight bonus. Unrelated question: is there a way to show that 3:2 and 4:3 are limit-3 as well as being perfect? Rracecarr (talk) 19:24, 18 July 2008 (UTC)

Only time for a few quick replies just now, on a strange machine here with something like a 640x480 pixel CRT...

I like it when the presentation gets out of the way and lets the content come through as easily as possible. That's something I care about, and enjoy doing. Your kind words are much appreciated. That business of labeling the intervals and ratios pretty much came from the crowding in that corner, plus trying to keep the overall image as compact as may be.

  • I haven't put all possibly meaningful intervals onto the chart yet. Is it aimed at folks still learning to understand what unisons and fifths are, or at an audience who's already into the subject a bit? Perhaps when I get back to the main machine I'll throw on a lot more, and we can have fun deciding which ones to prune away... no theoretical expert, me, but I knows what I knows. Could use a lot of input about which ones to keep.
  • If we have a reversed scales on top and bottom, I don't see the need for either of them to go past an equal-tempered tritone at 600 cents, cutting the width of the whole thing in half. I don't think I like that idea, because it calls on the viewer to do too much mental translation. I'd rather see it go from unison all the way to octave, and let thirds be thirds and sixths be sixths, and so on.
  • Somewhere in the caption or accompanying text there could be mention that limits are upper bounds, that a 5-limit system permits the use of 3-limit ratios, and so forth. I had that same thought about the "perfect" intervals (only 4 that I'm aware of, P1, P4, P5, & P8) also falling within the 3-limit scheme. I'm thinking of putting some white space between the 3-limit bar and the 5-limit bar to suggest that.
  • Actually,the lines started out as clones of the P1 line, but I de-linked them after moving them with the transform tool, so they'd stay put if I futzed with the "original."
  • I'll try making that cents scale into a single path, which might lose us a lot of the redundant formatting and resulting file bulk. I'll keep my mouth shut about how brilliant the idea of a text-based image generation language is for anything bigger than a tic-tac-toe grid... XML works just fine for a lot of things for a lot of folks, I guess, so no complaints here.
  • I didn't find that cloning things reduced the bulk of the description in my little experiments; as I understand it, it links the structures so you may modify them in groups... I need to to more exploring there.
  • Again, I'm looking for suggestions as to what other intervals are indispensible in your views, and which N-TET systems can be tossed, or which vital ones are still missing. We might even have two charts, an uncluttered one with only low-limit ratios, and a more elaborate one. (Note well, I didn't say a "complete" one ;-) ) __Just plain Bill (talk) 22:42, 18 July 2008 (UTC)
Just uploaded a rev with some extra white space below the limit 3 bar, and a boatload of reflected inversions. About the only one left without its inversion is 13:7 in the M7 neighborhood. At this point I'm winging it, not knowing what ratios are important to show, and which ones can be left out. Hyacinth, you want to weigh in? __Just plain Bill (talk) 05:12, 19 July 2008 (UTC)
One more thing before letting it percolate for a while: with careful choice of the lines we do show, it might be easier to spot things at a glance, the way Rracecarr is suggesting. Anybody got a sense of which inversions could be better shown by the member above 600 cents, and which ones below? For example, are there outstanding asymmetries there, kind of like one ratio being useful for a M7, while its inversion is not such a great m2? __Just plain Bill (talk) 05:35, 19 July 2008 (UTC)
  • Thanks for adding the additional intervals. Now I am starting to see the symmetry and believe that I could explain the utility of the reversed cents scale in the caption — it can be used to find the inverse interval by following an interval ratio line down to the reversed cents scale, reading the value, and then finding that value along the forward cents scale. Unfortunately, when I tried this, I almost immediately came across 13:7 whose inverse is not displayed. If there is a music theoretic reason for not including it, OK, but the inconsistency would need to be explained.
  • It seems to me Rracecarr's idea for displaying inversion pairs (with the folded axis that Bill inferred) could be adapted to a separate image whose only object would be to illustrate pairs (in a different article). A variant would be to lay out the ratios in a circle with the pairs at opposite ends of diameters, and the cents scale laid around the circumference. (Inkscape can do rotations very nicely.)
  • While zoomed in on the right side of the graph, I lose track of which ET scale is which. There may be enough room to add the scale names along the right side (coloring each scale bar might be an alternative). Also, the reversed cents scale needs a label.
  • "limits are upper bounds …" — Oh, now I see why they are called limits. Limit (music) could use a dose of WP:OBVIOUS. :-)
--Jtir (talk) 18:30, 19 July 2008 (UTC)
Yeah, I got lazy about putting on inversions for 13:7 and 11:7 sometime around 2:15 this AM, and wrapped it up for the night.
Let me give the "pairs at opposite ends of diameters" idea some thought. I see the symmetry here as a reflection about the TT line, not very circular. Might skate further out onto the thin ice of undecipherable original work that way, but such a presentation might reveal something. Stay tuned...
I've got the glimmerings of a plan to subtly color-code the N-TET bars. Probably saturate the color of 12-TET a little brighter than the others, to make it stand out in the crowd. That's the only one most of us ever have to deal with, anyway-- the others are pretty esoteric in my little world. Again the question: do we need to show 19-TET, 21-TET, and 22-TET? __Just plain Bill (talk) 20:47, 19 July 2008 (UTC)
New rev of image. File size cut in half by using clones for much of cents scale; thanks for suggesting that, Jtir. Attempts to color code the N-TET bars didn't come up with anything I liked; just applied some warmth to the 12-TET one. Let the pruning begin! __Just plain Bill (talk) 11:37, 20 July 2008 (UTC)
Thanks for your latest version with all pairs, second scale labels, and shaded 12-TET. Comments re size are here. It is ready to go live, IMO.
"not very circular … undecipherable original work". <g> OK, I tried to find a precedent that might justify my suggestion, and the closest was the circle of fifths, so I am withdrawing it.
--Jtir (talk) 19:04, 20 July 2008 (UTC)
Well, I was thinking that the bottom would have the reverse cents scale and the octave-complement ratios. That way, one line would suffice for e.g. 3:2 and 4:3. One way to do this would be to have only ratios with an odd numerator on the top. This would spread the lines out fairly well, I think. An advantage would be less crowding. A disadvantage would be that it'd be harder to compare relative matches to similar intervals (e.g. how well an ET matches 9:8 vs. 10:9). Without showing octave complements, I don't think the reverse cents line is that useful.
As for which temperaments should be included, I'd say: 5-tet (gamelan), 7-tet (gamelan, sort of; degenerate "meantone"), 12-tet, 19-tet (relatively popular among guitarists), 22-tet (related to ancient Indian theory, first good 7-limit-consistent ET), 24-tet (quarter tones, Arabic practice), 31-tet (Huygens/Fokker, optimal septimal meantone), 41-et (superior fifths to 12-tet), 53-tet (Chinese theory, Bosanquet's enharmonium, Turkish theory), and 72-tet (Byzantine theory, Joe Maneri, Boston Microtonal Society, Xenakis, Russian 72-tone school). As a general rule, I think we should only include temperaments that we have articles for, and I think 72-tet is a good upper bound.
The only problem I have with the color coding right now is that there isn't much contrast. The 11-limit and 13-limit lines are almost the same color, as is the unmarked line under 13-limit, and the 7-limit line isn't far off. The crossings become indistinct. The pastels look nice, but I think bold primaries might show up better.
I do, however, think that it is ready to "go live" in the article as-is. There's no reason that it can't be refined afterward. — Gwalla | Talk 23:22, 21 July 2008 (UTC)
I really appreciate your detailed particular suggestions. The latest image rev incorporates a lot of them.
I kept the limit colors pretty subdued because
  • I want them to fade behind the ratio numbers, so the viewer can see all those numbers in their own gestalt: as a bunch of ratios with various levels of simplicity.
  • Limited palette, low saturation, light value, is related to getting behind the numbers.
  • Limited palette helps the limits hang together as a single group of parameters in a moderately busy chart with some other things to say as well.
The reversed cents numbers got jettisoned, keeping the tick marks in case anyone wants to lay a ruler across a paper copy of this thing.
I'm not sure what you mean by octave complement ratios on the bottom. I definitely think the chart should span an octave, going from 0 to 1200 cents. As it is, I see a mirror symmetry pretty easily, about the 600 cent midline. Would you consider sketching something to show me? The scanned back of an envelope, or a simple sketch with a draw or paint tool would do, and I'd be more than happy to consider incorporating that idea into the chart.
One of the things I'd like to balance here is a clear simple exposition of a single notion, versus a more complex information-rich chart that may take some contemplation or digging to reveal stuff that doesn't pop out at first glance. Accepting, of course, that ideal as unreachable doesn't stop me from trying, and with my shallow knowledge of the subject matter, I can use your help in the attempt :-)
There are still a lot of ratios left off, ones with higher prime limits, ones that come "closer" to some desired ideal, and so on. I'd still like to massage the chart so it can show some of those (suggestions, please) and still be readable. __Just plain Bill (talk) 03:31, 22 July 2008 (UTC)
Here is an example of eliminating lines by allowing them to do double duty. I followed Gwalla's idea of putting odd numerators on top, even ones on the bottom. Obviously, this crude image is just to illustrate the idea, as you suggested. Rracecarr (talk) 15:40, 22 July 2008 (UTC)

Equal temper demo.png

Rracecarr has it. By octave complement ratios, I mean that there would be one line with the ratio 3:2 marked on the top, and 4:3 (an octave minus 3:2) marked on the bottom. One line would suffice for both. It turns out that the chart as it stands is not too busy, so there's no real reason to do it anyway. — Gwalla | Talk 16:04, 22 July 2008 (UTC)
I don't really see the point of having separate lines for "perfect" and 3-limit. 3:2 and 4:3 are, after all, the most basic 3-limit ratios. I don't think there's much to be gained by adding limits higher than 13, since they're so rarely invoked. I do think that, if we're going to have the 13-limit, then 13:8 and 16:13 should be labelled. The only other ratios I can think of that might be useful to show are the Pythagorean major third (81:64) and minor third (32:27). I don't think they're vital, though.
The colors are an improvement. They still make a weird sort of moire-like effect as the lines cross, though. Not sure how to fix that. Maybe outlines on the wedges? But that'd clash with the black text. — Gwalla | Talk 16:04, 22 July 2008 (UTC)
Cheers, you two! Funny, I don't get any moiré... you mean in the small patch where a wedge crosses a line, or the overall view?
In my view, a 9:8 is not a perfect interval, of which there are only 4, as mentioned. I may be misinformed about that. Could just as easily merge the two, losing the "perfect" label, I suppose. Open to persuasion there.
Tis quick response is all I've got time for atm, thanks again! later, __Just plain Bill (talk) 17:05, 22 July 2008 (UTC)
Thanks, Bill, for adding the graph to the article. With the image at 600px, the scale labels are almost illegible on my LCD. Maybe those labels could be enlarged slightly, perhaps to match the interval labels.
I also see an optical illusion in which the wedges seem to have varying brightness. Also the vertical ratio lines have slightly varying widths when rendered at low resolutions.
Following Gwalla's suggestion … Category:Equal temperaments includes 34 equal temperament and 88 equal temperament, neither of which are in the graph. The article on "34" is well sourced, and "88" is tagged with {{unreferenced}} (and a comment on the talk page questions its notability).
One reason for separating the ratios by limit is to reduce crowding along a single ratio row. Merging the "Perfect" row into the "Limit 3" row would not lead to crowding. However, doing so would obscure 4:3 and 3:2, which are, as I understand it, very important ratios.
--Jtir (talk) 19:24, 22 July 2008 (UTC)
That's a very nice sketch, Rracecarr. I believe that it could be used in the article, as is, to illustrate the complementary pairs. I don't favor removing lines from the current image, but would like to put more info in about how pairs of just intervals align with ET intervals. What I'm having trouble with is understanding the significance of this symmetry. I understand that each member of a pair is equally distant from its nearest ET interval, but what are the practical/musical/sonic implications? --Jtir (talk) 21:33, 22 July 2008 (UTC)
Ah, I missed 34. Good catch. 88 I'm not so sure about, but I may be biased. I meant replacing "perfect" and "3-limit" with just "3-limit". The term "perfect" is important to diatonic practice (it's the name for the intervals against the tonic that are identical in both major and minor), but not so much to tuning theory where it's really just a naming convention. Besides, that's just 6 intervals on one line, which is not very cluttered. That varying brightness is the effect I was trying to describe. I've only been looking on LCDs, so maybe it only shows up on that kind of display? — Gwalla | Talk 23:19, 22 July 2008 (UTC)
P+L3->L3 is what I thought you meant, and it sounds like a good idea. More on the optical illusion here. --Jtir (talk) 00:31, 23 July 2008 (UTC)

Now I see what you mean. I can live with it (not knowing a quick slick way to get rid of it.) I noticed those lines with extra weight too; I thought it was an artifact of the rendering decimation, but there were some lines hiding behind other lines -- now they're gone.

Cents number labels are now a bit bigger, without coarsening the image too much, or getting in the way of the more vital info. In my view, in the context of this particular chart, once you realize that they represent hundreds of cents, they can be small and fuzzy, but still useful. In the early stages I toyed with the idea of leaving them off entirely. The 600px width is really a thumbnail- serious viewing might better be done at a larger zoom.

For me, a key piece of what this graph shows is that 12-TET approximates fourths and fifths passably well, seconds not so well, and doesn't really even come close to just thirds (or sixths.) I knew that intellectually and aurally before, but this shows it graphically. The other equal divisions of an octave don't show up much in the practical part of my world. These days, about once a week I'm in a room with a piano. Someone said elsewhere that equal temperament, as a practical keyboard tuning matter, only showed up about a century ago, but gamelans have been around a bit longer. Sorry, rambling.

I've resisted the temptation to put on stuff like 64:81 alongside 8:5, since that's more of a circle-of-fifths than an equal temperament thing. More ratios might be added, if you think of some that would be useful to see here. __Just plain Bill (talk) 13:22, 23 July 2008 (UTC)

and for now, at least, I'm leaving all the inversions in place without changing the way they're labeled here. I grab them differently on a fingerboard, as I suppose one might grab them differently on a keyboard, but seeing the chart helps fill out my understanding of them, just as the grabbing helps fill out the understanding of the chart.

(Nice job, Rracecarr, that did help. I tried some similar stuff with this other chart, but it wasn't working for me, so it went on a back burner for now. I'm actually wondering whether I don't like your dots and lines better than my shaded rectangles.) __Just plain Bill (talk) 13:35, 23 July 2008 (UTC)

Thanks for the fixes, Bill. The line weights are very even now. Even with a magnifier, I couldn't quite read some of the text in the 600px thumbnail, so I increased it to 650px. I believe that the thumbnail only needs to show enough detail that a reader can see at a glance what the graph is about, and the axis labels are the best indicator of that, IMO. A possible way to reduce the the simultaneous contrast illusion would be to put a light grey background behind the ratio rows to reduce contrast slightly. Principles of color design: designing with electronic color has a chapter on the subject, but google books won't let me preview it. --Jtir (talk) 17:25, 23 July 2008 (UTC)

Two minor quibbles. One, having the limits stated and the limit color-codes start to the right of the 0¢/unison line obscures the scale a bit—it's hard to tell at first glance where things are measured from. On the other hand, putting them to the left would make the image even wider. Perhaps the limit labels should stay where they are, and the color lines should be extended to the unison line? Two, I think 13:8 (13th harmonic/tridecimal neutral sixth) and 16:13 (tridecimal neutral third, actually mentioned in that article) have a better claim to being shown than 13:7 and 14:13, which as far as I can tell do not have names in any listing of just intervals I can find. — Gwalla | Talk 23:52, 29 July 2008 (UTC)

Seems good to me; fixed it. __Just plain Bill (talk) 03:31, 30 July 2008 (UTC)
One more: It might help if the scale steps (in each temperament) were marked with a thin black line. As it is, there isn't really enough contrast between the lighter end of the gradient and the darker end to make them show up clearly. — Gwalla | Talk 03:59, 30 July 2008 (UTC)
The main reason for the steps being marked by a difference between greys is that can be truly pretty much zero-width at any zoom. It doesn't take much line width to obscure the 2-cent difference between P4, P5, and the nearest 12-TET steps. The JI lines are about as narrow as they can get and still show up nicely in the 800-wide presentation. If someone wants to make them dotted, go nuts... __Just plain Bill (talk) 12:47, 30 July 2008 (UTC)

A practical matter, FYI, and just for giggles: I tried to find the 13th harmonic on a cello's A string a few minutes ago. Up to about 10 or 11 they are pretty easy to find and keep separate, but after that it gets hairy, at least for a fat-fingered amateur like me. I'm pretty sure I found it, and it sounds like one of those notes that's oddly related to the rest of the series. Next time I'm in the same room with an accomplished trombone player I'll ask him about that. __Just plain Bill (talk) 15:21, 30 July 2008 (UTC)

After all the discussion and work you folks put into that illustration, how did it end up being inserted at the bottom of the article and tiny? It should be at the top instead of that Syntonic Tuning illustration. Anybody have a problem with moving it up and moving syntonic tuning down? I want to say though that the ET illustration is still incomplete. It only contains EDO ETs. The W. Carlos non-EDO ETs would be a good to add at least, as well as the Bohlen–Pierce_scale (ET version), which is 13 equal divisions per 3/1. Some other fun ones, though probably not "notable", include 1 ED per 16/15, 7 ED per 3/2, 14 ED per 9/4, 28 ED per 5/1, or various others that are near 12 EDO but divide some other just interval than 2/1. (talk) 01:55, 16 May 2015 (UTC)

Nevermind. I went ahead and moved it to the top. I also added "EDO" to the illustration description. Hope you like it there. It's a good illustration. Thanks for making it. (talk) 05:41, 16 May 2015 (UTC)

12-TET and Circle of fifths[edit]

A Question: Is the circle of fifths applied to 12-TET still called "circle of fifths"? I created the following graphic, but I do not know the "official name" of this concept, where the 6 flat and 6 sharp scales are really identical. Simply "equal temperament circle of fifths"?
Circle of fifths unrolled.svg
And how is this related to the Chromatic circle?
Greetings, mate2code 15:53, 18 July 2008 (UTC)

Well, maybe you already realize this, but the "circle of fifths" is only a "circle" in an equal temperament. In just intonation, for instance, there is a "line" or "chain" of fifths, which never gets back to exactly the same pitch. —Keenan Pepper 06:34, 19 July 2008 (UTC)

I wish it were so easy, but you mix up well temperament and the modern concept of equal temperament. In the German Wikipedia is a section about the difference.

Violinists do not necessarily treat F sharp and G flat as the same tone, but they also say "circle of fifths" - not "chain of fifths" or "Quintenspirale".

So probably the answer to my question is simply, that there is no special name, and all concepts of modulating between different keys are usually called "circle of fifths". Greetings, mate2code 13:06, 19 July 2008 (UTC)

Interpreting the graph[edit]

Pardon the question, but how should one interpret this graph? What exactly does it mean? What do all the shapes and colors mean? How did you construct it? What can be deduced from it? Thanks. Barak Sh (talk) 00:18, 22 July 2008 (UTC)

Sorry - the graphic above is just the raw material for an imagemap template like the one below. (Done now.)Indeed, without the (movable do) solfège names, the graphic is not easy to read. Moving your mouse over the graphic below will probably answer your question. Some additional hints if not:
Neighbour scales in the circle of fifths have 6 of 7 tones in common. This graphic shows which.
The backgruond colors mean nothing - they just mean, that one line is not the other. (Compare this.)
Red and blue stand for Major and minor.
Trigon pointing right, circle and trigon pointing left stand for the triads root, third and fifth:
Major triad:
Minor triad:

Circle of fifths unrolled, pythagorean comma.svg

(This table shows the true intervals without enharmonic change.) Greetings, mate2code 19:09, 22 July 2008 (UTC)
All clear now. Thanks. :-) Barak Sh (talk) 22:58, 22 July 2008 (UTC)
I'm still confused. Why are there diagonals alternating "fa ti fa ti fa ti"? Why those shapes? Why are some circles blank? What do the white and dark grey vertical lines signify, and why are some thick and others thin? You say the red is major and blue is minor, but they're applied to individual notes here. How can a single note, not an interval, be major or minor? Then you have notes with two overlapping shapes of different colors. Is that supposed to be both major and minor at the same time? Background colors mean nothing, they're just chartstuff, and just make the image more graphically busy and harder to decipher.— Gwalla | Talk 04:05, 30 July 2008 (UTC)

propose archiving[edit]

This talk page is getting unwieldy, so I propose archiving everything older than a year. --Jtir (talk) 21:43, 22 July 2008 (UTC)

Done. --Jtir (talk) 17:59, 29 July 2008 (UTC)

Comparison Table[edit]

In the comparison table, shown as the key of C major, the minor 2nd is given as C# when it is D-flat, Minor 3rd as D# when it should be E-flat - similar goes for the Dim 5th, Minor 6th and Minor 7th. I understand it's easier to have the sharp sign (#) as it is already on the computer keyboard, but the intervals are technically incorrect. Halfabeet (talk) 19:34, 13 September 2008 (UTC)

Inaccurate description of historical temperament[edit]

I believe that the "History" section requires a major revising, if not a complete rewriting.

The section on history has almost no discrimination between "equal temperament" and "well temperament." Although equal temperament have existed in theory since late Renaissance, it was 1)deemed impossible to achieve due to technical restriction, and 2)often deemed unusable and offensive, for having neither purity of consonant intervals nor tonal color. All of the historical "circular" temperament described in the section, meaning those temperaments that could play all 24 major/minor keys, were in fact "unequal" (which is "well temperament," a concept completely different from equal temperament with different aim--well temperament was for equal playability of intervals while equal temperament was for complete equality of intervals). Weickmeister was the pioneer of well temperament, or "wohl temperiert," as Weickmeister himself put forward and coined, and although he indeed discussed and popularized Mersenne's equal temperament theory in his posthumous work(which, in turn, is completely different from modern equal temperament of twelveth-root-of-two), the wording misleadingly represents him as THE advocate of equal temperament.

Until the twentieth century, there was absolutely no method of achieving equal temperament, even with the "a tuning fork tonometer in 1834" described in the text, since the beating between keys were to be counted manually--a method impossible to specify irrational number of beats, which is what exactly modern equal temperament is. Description of 18th century is thus completely inaccurate: at the time, the tuning system that dominated the scene was never "equal temperament" but "well temperament." Everything from Baroque to late Romantic period was either non-12TET meantone temperament(since 12tone equal temperament is meantone temperaments with 1/11 syntonic comma) or variations of well temperament.

Here are the sources.

Mondschatten (talk) 04:10, 15 February 2009 (UTC)

"Until the twentieth century, there was absolutely no method of achieving equal temperament" Is this really correct? If you look at a 12TET stringed instrument such as a Spanish guitar , the fretting positions up the neck to achieve 12TET are quite simple to calculate and the maths existed. AnotherPath (talk) 19:58, 25 October 2013 (UTC)

Equal temperament: History[edit]

I just read both this article and the one about Simon Stevin, and noticed quite the discrepancy.

Here it says: "It is possible that this idea was spread to Europe by way of trade, which intensified just at the moment when Zhu Zaiyu published his calculations. Within fifty-two years of Zhu's publication (1584), the same ideas had been published by Marin Mersenne and Simon Stevin."

According to the article about Simon Stevin: "Stevin was the first author in the West (1585, simultaneously with, and independently of, Zhu Zaiyu in China) to give a mathematically accurate specification for equal temperament. He appears to have been inspired by the writings of the Italian lutenist and musical theorist Vincenzo Galilei (father of Galileo Galilei), a onetime pupil of Gioseffo Zarlino."

I don't know which article is correct in this matter. It would be good if anyone could rectify this and find the correct facts about this. —Preceding unsigned comment added by (talk) 11:47, 20 February 2009 (UTC)

  • Simin Stevin never published his findings in his life time, hence had no influence on history of musics. Secondly, he used ratio of , further more his calculation contain numerous errors, hence he never did sucessfully resolve the equal temperament problem as Prince Zhu Zaiyu did. In other words, Simon Stevin never knew the number 1.059463, which appeared first time in the west in Marin Mersenne's paper in 1636 --Gisling (talk) 18:09, 26 May 2011 (UTC).
  • Simon Stevin never provided an accurate frequency ratio as Prince Zu Zaiyu did. Instead of the correct value of 1.059463094359295264561825

Simon Stevin provided the following frequency ratio in his unpublished manuscript.

  • semi tone 1.0595465
  • 1.5 tone 1.0600904
  • Wholetone 1.0593781
  • 1.5 tone 1.0600904
  • Ditone 1.0594758
  • Ditone and a half 1.0594046
  • Tritone 1.0593975
  • Tritone and a half 1.0594845
  • four-tone 1.0597014
  • four-tone-and-half 1.0595558
  • five-tone 1.0593477
  • five-tone-and-half 1.0594788
  • five-tone-and-half 1.0594788
  • full tone 1.0592000

(See Gene Cho, The discovery of musical equal temperament in China and Europe - Page 222). Cleary none of Simon Steven's number were accurate. Any one with rudimentary arithematic can see at once, that the frequence ratios were all different, that cannot be be true "equal temperament". The claim that he "give a mathematically accurate specification for equal temperament" is heresay only. The accurate answer 1.059463 never appeared in Simon Steven's writings. --Gisling (talk) 23:51, 4 August 2011 (UTC).

The claim that the calculation of was transmitted to the West from China seems to be pure speculation. Certainly, Mersenne would not have needed the help, and there were perfectly good reasons why the question arose which have nothing to do with China. I think perhaps the whole history thing should be moved to a separate article. Gene Ward Smith (talk) 12:59, 30 May 2011 (UTC)

    • Your counter claim has not basis, where is your reference ?? Show us please how Mersenne calculated his value, step by step, upload a

facimile ---Gisling (talk) 08:15, 2 June 2011 (UTC).

Gisling, you hãve inserted a large amount of material that is Original Research, and that is clearly against the rules here. Exceptional claims require exeptional sources, per WP:REDFLAG and WP:RS.--Galassi (talk) 08:41, 2 June 2011 (UTC)
    • Where is "Orignal research " ??, all properly sourced, not such thing as original research ??

A list of my sources in my personal library:

  • Zhu Zaiyu,The Complete Compendium of Music and Pitch, over 5000 pages.
  • Article by professor Gene of North Texas U The Significance of the Discovery of the Musical Equal Temperament In the Cultural History
  • German physicist, Herman Helmholz On the Sensations of Tone as a Physiological basis for the theory of music , p 258, 3rd edition, Longmans, Green and Co, London, 1895 Not reliable source ? You are probably too igorant about Helmholtz.
  • Robert Temple, The Genius of China, p209 not reliable source ??
  • The Shorter Science & Civilisation in China, An abridgement by Colin Ronan of Joseph Needham's orginal text, p385 not reliable source???

Professor Gene wrote a book The Discovery of Musical Equal Temperament in China and Europe in the Sixteenth Century Cho, Gene Jinsiong . It is out of print on, on used bookstore selling over $200, too steep for me. Fortunatelly there is a Chinese translation 东西方文化视野中的朱载堉及其学术成就availabe from, I order one copy, it will take two month to arrive.

--Gisling (talk) 09:38, 2 June 2011 (UTC).

Unequal Temperaments book and website[edit]

Dear friends,

The Unequal Temperaments book of 1978 was described-in writing-as the definitive reference on the matter by authorities such as John Barnes, Hubert Bédard, Kenneth Gilbert, Igor Kipnis, Rudolf Rasch and others.

In the 1990's I also developed the first professional-grade temperament spreadsheets.

Eventually I setup the "Unequal Temperaments" website, where I uploaded the spreadsheets which, kept permanently updated, are available for FREE. I also uploaded years ago a provisional "Update" to the book of 1978.

The website lately gives information on the recently released new version of Unequal Temperaments 2008, which includes a detailed chapter about the HISTORICAL EVOLUTION OF EQUAL TEMPERAMENT (The website does NOT sell the book)

I would find it useful to Wikipedia readers if my website was included among External Links:

Kind regards


Dr. Claudio Di Veroli (talk) 17:14, 26 February 2009 (UTC)

Fractal music ref[edit]

Is this a good reference, or not? How come?

Fractal Microtonal Music

__Just plain Bill (talk) 01:40, 25 April 2009 (UTC)

It appears to be some guy's personal explorations; self-published by a person with no evident claim to expertise or authority. I don't see why we would use it unless maybe as a secondary reference for a topic that discussed elsewhere. Dicklyon (talk) 04:07, 25 April 2009 (UTC)
Jim Kukula is a published computer scientist [1] The page in question itself cites 7 reliable resources. It also happens to present correct information on the subject. The claims that are apparently being supported are trivial and don't require a citation in the first place. The 160 characters in question need editing and possibly deletion, but not because of the citation or lack thereof, and hopefully by someone who knows something about equal temperament. beefman
As far as I can tell, whatever his qualifications in other fields, he is a musical hobbyist. His exposition is interesting, but it might be better to use some of his references directly. Does "lowest number of equal divisions", "second lowest number of equal divisions" and so on, have to do with the "claims being supported?" What puts those claims into the realm of triviality? (I hope you're not casting yourself as "someone who knows something about equal temperament" in any kind of unique sense here.) __Just plain Bill (talk) 13:03, 25 April 2009 (UTC)
Basic arithmetic is trivial and does not require citation. I do happen to be someone who knows something about equal temperament, but unfortunately, I'm spending all my time staving off damage to the article instead of improving it. beefman (talk) 21:39, 25 April 2009 (UTC)
He seemed to me to have a well-cited amateur essay on the subject, with the advantage that it was online vs. his sources which are largely not. If somebody wants to use Kukula's sources as references here, that's fine with me. Until then, I consider an inline ref with Kukula better than no inline ref at all. Binksternet (talk) 15:18, 25 April 2009 (UTC)
I don't really see it better than no ref at all; cite a real ref if you want to talk about this obscure stuff. Dicklyon (talk) 15:28, 25 April 2009 (UTC)
I have a suggestion for you too, Dicklyon. Concentrate on adding useful content to Wikipedia, in subject areas you understand, instead of wasting our time with nonsensical comments or deletions. Thanks in advance. beefman (talk) 21:39, 25 April 2009 (UTC)

Time to calm down a bit. I'll be happy to continue a reasoned discussion about this.

  • First, Beefman, your knowing the comfort of a Dvorak keyboard is a favorable point in my book. Bravo for that! I had to let it go years ago; for one thing, it made it difficult to share a keyboard.
  • OK, you know something about equal temperament. Don't feel like a lone voice crying in the wilderness about that. So do some of the rest of us. I admit I'm a lightweight compared to Dick Lyon, who studies and teaches the neurology, physiology, and digital modeling of acoustic phenomena for a living. For all I know, he may have dabbled in hybrid analog/digital models of it as well, and to a pretty fine-grained level.
  • In my book, trivial arithmetic is sums, differences, products and quotients. Raising numbers to fractional powers, as Kukula does in his piece, is not trivial.
  • Three edits to this article counts as "spending all my time staving off damage to the article" ?? Exaggeration is a poor choice of tool for this kind of discussion. In the interest of keeping things cool, I'll not address any other rhetorical gadgets that have been used here, and ask for some more substantial reasons to keep what amounts to a geocities blog as a solid encyclopedic reference.
  • Oh, wait: the claims made are so obvious that they don't need a reference. Which way will it be?

I'm ready to admit when I'm mistaken (it happens a lot) but I need some solid particular reasons to do so. For now, I'm going with "it's pretty close to being a blog, and not notable enough to show in an encyclopedia article."

Hi Bill. An inequality involving log(2^(29/17)) - log(3/2) doesn't require a citation. What would you cite, the S/N of your calculator? Not that this factoid is important or should be in the article (it shouldn't), but you won't improve the article by chopping this stuff out under the auspices of insufficient citation quality (give me a break). The article needs a major rewrite, and if you and other editors wish to attract people motivated and qualified to improve it (and most of the other articles in wikipedia), you have a strange way of going about it. --CKL
Carl, where in the Wikipedia article does it mention that inequality? __Just plain Bill (talk) 13:26, 26 April 2009 (UTC)

If someone wants to chase chapter and verse in the references Kukula cites, that would be a fine thing. __Just plain Bill (talk) 02:45, 26 April 2009 (UTC)

Hi again, Bill. It sounds like you're well on your way to producing the citation you claim is needed! Go get 'em! --CKL
Where did I ever claim this cite was needed? __Just plain Bill (talk) 13:26, 26 April 2009 (UTC)
"If someone wants to chase chapter and verse in the references Kukula cites, that would be a fine thing". beefman (talk) 06:19, 27 April 2009 (UTC)
That doesn't say we need this particular cite. Some of his references could be useful, and some seem to tend towards "music of the spheres" mysticism. I've got a copy of Danielou open in front of me now, looking at the place where he says: "Temperament, by disfiguring the major mode, has brought it onto an equal footing with many other modes equally disfigured by it. This explains why so many harmonic 'discoveries' followed the widespread use of the modern piano."
The book is more of an intellectual historical discourse on modes (e.g. in Chinese, Indian, and Greek music) and only mentions equal temperament in a disparaging coda towards the end. No mention at all of 41-TET nor 53-TET. __Just plain Bill (talk) 13:01, 27 April 2009 (UTC)
The math is still pretty simple, just geometric progressions, or arithmetic progressions on a log scale. What's "obscure" is the use of the fine (microtonal) equally tempered scales, picked to optimize fifths and/or fourths; by obscure I meant I haven't seen it before and it seems to only show up in some guy's GeoCities pages (which are about to disappear, I read on slashdot). I didn't look at his sources; if it turns out not to be obscure, and is represented in reliable sources, I have no problem with it. Dicklyon (talk) 04:26, 26 April 2009 (UTC)
There's nothing obscure about finding successive improvements to 3:2 approximations among equal temperaments. It has a long history, from Huygens, Newton, Helmholtz, Bosanquet, Wendy Carlos, and thousands of others. --CKL
Looking at the domain of a URL is a poor proxy for evaluating the content therein (of course I knew that's what you were doing from the start). Don't worry. When and if the geocities content goes down, I'll make sure to update the link. In the meantime, aren't there enough unreferenced equations posted by physicists who didn't notice their login had expired that you could be deleting? --CKL
Log scales can be simple and understandable, but I object to the use of "trivial" as a general-purpose rhetorical bludgeon. I'm asking for some substance here. What claims are supposedly being supported, and in what particular ways, by this reference? So far I haven't seen the link's defenders point out how it is relevant to the points presented in the Wikipedia article.
Are there notable musical traditions or genres actually use 41-tone or 53-tone systems? In my view, that's the kind of stuff that belongs in the article, the kind of obscurity that could stand some light shed on it. Where are notable musicians doing this, for an actual audience? __Just plain Bill (talk) 05:10, 26 April 2009 (UTC

I don't particularly care about refining approximations to 3:2. Fifths come pretty close even in 12_TET. The tallest head of the dragon here is major and minor thirds. Further divisions of equal temperament help a bit with those.

My problem with using the Kukula page as a reference is that it does not support the paragraphs it is attached to. The skimpy 41-TET paragraph says practically nothing, let alone anything needing this "cite." The 53-TET paragraph is a bit meatier, but the placement of the Kukula footnote seems to indicate it has something to say about 22-TET, which it doesn't. Keep it as an external link if you like, but as footnote(s), out it goes. __Just plain Bill (talk) 13:26, 26 April 2009 (UTC)

My problem is, instead of finding a better citation, or removing the citation, you and Dicklyon removed content from the article without attempting to edit it in any meaningful way -- you just clear cut, and used a complaint about a citation as justification. beefman (talk) 06:19, 27 April 2009 (UTC)
Diff __Just plain Bill (talk) 13:01, 27 April 2009 (UTC)

I doubt that I can add anything useful to the discussion here... but if folks decide they would like a link to a simple discussion of rationals and irrationals and 53-edo, I moved my soon-to-die geocities page to: Fractal Microtonal Music Kukulaj (talk) 12:23, 9 October 2009 (UTC) —Preceding unsigned comment added by Kukulaj (talkcontribs) 12:20, 9 October 2009 (UTC)

Hey, Jim, thanks for showing up here. I listened to your stuff, and found it interesting. It seems someone deleted that link as "spam" which it isn't, at least not in my opinion. Right now I intend to put it back; maybe it's time to rekindle this discussion in calmer tones. Be well, Just plain Bill (talk) 23:45, 9 October 2009 (UTC)

Corrected Reference[edit]

The broken link to the Huygens-Fokker foundation, at, can be replaced by the current address, I'm not sure how to do this on the actual page and preserve the text as it stands. —Preceding unsigned comment added by (talk) 08:34, 21 August 2009 (UTC)

Thank you. I have replaced the broken link. There were two others to the same site, but I simply removed them. I may look later to see if I can find an updated address for them at the current site. Rigaudon (talk) 09:03, 21 August 2009 (UTC)

Protest on Galassi's roll back[edit]

My contributions are all properly sourced, Galassi's roll back is against Wiki policy, this persion should be banned from wiki --Gisling (talk) 08:45, 2 June 2011 (UTC).

You must source your claims to realiable scholarly publication, not your own.--Galassi (talk) 08:50, 2 June 2011 (UTC)
    • Article by professor Gene of North Texas U, not reliable source ??
    • Herman Helmholz On the Sensations of Tone as a Physiological basis for the theory of music , p 258, 3rd edition, Longmans, Green and Co, London, 1895 Not reliable source ? You are probably too igorant about Helmholtz.
    • Robert Temple, The Genius of China, p209 not reliable source ??
    • The Shorter Science & Civilisation in China, An abridgement by Colin Ronan of Joseph Needham's orginal text, p385 not reliable source???

If you have any question about certain paragraph, the proper way is to use [citation needed] template, request for citation. You act are pure disruption, should be banned --09:02, 2 June 2011 (UTC)

Theory of transmission for China to the West needs more facts[edit]

I have some doubt about the validity of transmission of 12 TET from China to the west. Apparently there was already equal temperament of varous favour in Europe around the later half of 16th century. One point ins suspect is that Zhu Zaiyu's 12 TET was 100 CENTS, while Marin Mersenne's monochord was 99.8. If Mersenne got Zhu's data, it would be 100 also. Hence I move the paragraph about transmission from article to here.

Transmission to the West

It is possible that this idea was spread to Europe by way of trade, which intensified just at the moment when Zhu Zaiyu published his calculations. In 1580, the Cantonese government established biannual trade fair at Guangzhou lasting several weeks, Chinese and westerners exchanged goods and ideas. Two years earlier,[clarification needed] the great Jesuit Matteo Ricci arrived in Macao in 1582[1].

Matteo Ricci, who was in China at that time, was interested in calendar reform fashionable in the Ming dynasty court. In his private journal he wrote about Zhu Zaiyu's calendar reform; being a mathematician himself, Matteo plausibly knew Zhu Zaiyu's formula for his reform. Matteo Ricci was also a friend of the noted French scientist and musician Marin Mersenne; they shared common interest in astronomy and music, it is plausible that Matteo Ricci may have transmitted this key piece of information to Mersenne in correspondence.[citation needed] It is not surprising that the first published reference to 1.059463 was by Marin Mersenne's Harmonie Universelle in 1636.

    • I shall double check Dr Gene's book when I received it from mail, to see is there more concrete evidence. --Gisling (talk) 20:53, 5 June 2011 (UTC).
I agree with you, the paragraph seems to be conjecture.--Galassi (talk) 11:01, 6 June 2011 (UTC)
Interesting points Lastitem (talk) 13:55, 1 November 2011 (UTC)


  1. ^ Professor Gene J CHO(College of Music,University of North Texas) The Significance of the Discovery of the Musical Equal Temperament In the Cultural History

Galasi's roll back is disruptive[edit]

Galasi had twice rolled back my sourced edition without any just cause. you must provide you reason before rv other people;s contribution

Why you said nationalist POV ?

1. The first column figures of chord lengths are from Simon Stevin's own manuscript. These are FACTS, not opinion

2. The correct chord lengths were provided by Fokker, who was the editor of Simin Stevin's work. These are FACTS, not opinion

3. Professor Gene Cho is American citizen, taught in US university.

Where is your basis for "nationalist POV" The purpose of the following Stevin chord table is threefold;

semi-tone 9438 1.0595465 9438.7
Wholetone 8909 1.0593781
1.5 tone 8404 1.0600904 8409
Ditone 7936 1.0594758 7937
Ditone and a half 7491 1.0594046 7491.5
Tritone 7071 1.0593975 7071.1
Tritone and a half 6674 1.0594845 6674.2
four-tone 6298 1.0597014 6299
four-tone-and-half 5944 1.0595558 5946
five-tone 5611 1.0593477 5612.3
five-tone-and-half 5296 1.0594788 5297.2
full tone 1.0592000

a) To see what numbers were provided by Stevin, and to prove that Simon Stevin never obtained the correct frequency ratio for equal temperatment 1.059463094359295264561825.

b) To substantial The Cambridge history of western music theory statement that due to insufficient accuracy of his calculation, many of the chord length numbers he obtained were off by one or two units from the correct values. Actually, the chord length data of 1.5 tone provided by Stevin 8404 is off by five units from the correct value 8409(provided by Fokker).

c)The frequency ratio from tone to tone by Stevin is unequal,(FACTs) which is clearly not correct for "equal temperament".(Professor Gene's opinion based on facts)

- Why are you so afraid of people knowing the facts, even deleted Simon Stevin's own data ?? you are a nationalist, desperate to hide the facts !

I challagne Galasi to provide counter proof that a) Stevin was the first person in history to discover the correct number for equal temperament 1.059463094359295264561825

b)Stevin used a single frequency ratio for all tones

c)Stevin's numbers were correct, without errors.

Standup, Galasi, don't be a coward.

--Gisling (talk)

I don't know about the nationalist POV, but I don't think there's any question that equal-temperament figures were developed in China. (Whether the Western idea came from China or was developed independently is a separate question, and needs its own sources.) In any case, it's not necessary to quote all these sources verbatim (some of which are over a century old). I apologize to Gisling for being unclear in my first edit -- the quotations didn't have "quotation marks" so it was not clear that the text came from another source. /ninly(talk) 21:00, 15 August 2011 (UTC)

There are some odd things about this table. I can understand that there are supposed to be original inaccurate computations, and corrections to these given also. But even with this explanation, can anyone explain how the number 1.0594758 comes to be in the table? Is it itself a misprint, or does it point to one of the other numbers being a misprint? To that level of precision, I cannot see how 1.0594758 derives from the other numbers in either of the adjacent columns, whether before or after the correction. I think I can make sense of all the other numbers in the column headed "RATIO" to their full precision, except perhaps the last digit in a couple. In fact it is true that 1.0594758 = 8408/7936 but I there is no 8408 in the table. In fact I suspect that the entry 1.0594758 is probably an indication of a misprint at 8404 which probably should be 8408. (But oddly the 1.0600904 does match the current 8404.) I think whoever originated the RATIO column could make a few corrections. — Preceding unsigned comment added by (talk) 19:21, 10 February 2013 (UTC)

Definition of an octave[edit]

Shouldn't we mention something about an octave being a doubling (or halving) of the frequency? BobbyBoykin (talk) 20:00, 3 May 2012 (UTC)

Removed sentences without citations and dubious sources[edit]

I have removed the uncited parts of the article, and a few dubious sources used. I have also written a detailed account of Kenneth Robinson's theory on the history of equal temperament and Fritz A. Kuttner's criticism of it. If anyone wishes to access the Fritz A. Kuttner journal article, it's available on JSTOR.--Ninthabout (talk)

Middle C[edit]

A Guitar will sound WAY better when accompanied by a keyboard if the starting note is MIDDLE C, and the tuning is equally tempered in reference to this note, as opposed to A 440...

-Perk — Preceding unsigned comment added by (talk) 03:24, 5 April 2013 (UTC)

I doubt that. If both of them were really tuned to the same equal temperament, they would agree on all pitches, no matter what the "starting note" were. I suspect the guitar had some intonation problems, which caused the fretted pitches to be out of tune when the open strings were in tune, and your intentionally detuning it gave you a slightly better overall fit for the frets you were using most often. (talk) 02:26, 8 May 2015 (UTC)

Inconsistently abbreviated[edit]

The article describes the phrase as being inconsistently abbreviated as 12-TET, 12TET, 12tET, 12tet, 12-ET, 12ET, or 12et. No kidding! I've seen lots of variation on WP. It would be nice to have some consistency, varying with context only when there's some reason to do so. Maybe some are more common in American English and others in British English, for example, or sources writing on certain subjects tend to use some abbreviations while those discussing other subjects use others. I have no idea, but it's a topic of narrow enough interest that I wouldn't expect too much of that sort of thing. --Dan Wylie-Sears 2 (talk) 12:25, 15 December 2013 (UTC)

12TET probably makes the most sense editorially, since initializations should be capitalized, and I think hyphens are dropped during formation like spaces are (12-tone equal temperament). However, 12EDO is even better than 12TET informationally, because it also tells you what's being divided (12 equal divisions of the octave) (preposition "of" and article "the" are dropped when forming the initialization). I would vote for replacing 12-TET, 12TET, 12tET, 12tet, 12-ET, 12ET, or 12et with 12EDO where practical. (talk) 00:11, 8 May 2015 (UTC)

Syntonic tuning doesn't belong here[edit]

The syntonic tuning system and continuum should not be given as an example in an equal temperament article, especially so high in the article and with an illustration that makes it stand out and seem important to the topic. The syntonic continuum is primarily a tool for generating meantone tunings, because it always truncates the series at the octave, or wraps at the octave, or is modulus 2:1, however you want to say it. It's only at those particular accidents of exact alignment with the octave that the syntonic tuning system generates equal temperaments, and therefore only EDO temperaments. The number of equal temperaments it can generate is theoretically infinite, but the number of mean tone temperaments it can generate is infinitely larger than that. In other words it's like the infinity of real numbers being infinitely larger than and including the infinity of integers. A much better illustration for the article would be a graph of a sliding smallest generator interval creating a sequence of identical intervals indefinitely in both directions. That would produce all possible equal temperaments including non-EDO ones and would produce no other kinds of tuning, such as mean tone. (talk) 06:04, 5 May 2015 (UTC)

Moved to its own section lower in the article. (talk) 18:49, 17 May 2015 (UTC)

Rational semitone[edit]

Rational semitone

For any semitone that is a proper fraction of a whole tone, exactly one equal division of the octave lets the circle of fifths generate all the notes of the equal division while preserving the order of the notes. (That is, C is lower than D, D is lower than E, etc., and F is indeed sharper than F.) The number of divisions needed for the octave is seven times the number of divisions of a whole tone minus twice the number of divisions of the semitone. The corresponding fifth spans a number of divisions equal to four whole tones minus one semitone. Hence, for a semitone of one-half of a whole tone, the corresponding equal temperament scheme is 12-EDO with a fifth of seven divisions. A semitone of one-third of a whole tone corresponds to 19-EDO with a fifth of eleven divisions.

12-EDO is the equal temperament with the smallest number of divisions that allows for a rational semitone to preserve the desired properties concerning note order and the circle of fifths. It also has the desirable property of making the semitone exactly one-half of a whole tone. These are additional reasons why 12-EDO became the predominant form of equal temperament.

While each rational semitone corresponds to only one equal temperament, the reverse is not the case. For example, both a semitone of one-seventh, and a semitone of eight-ninths both use 47-EDO, which is the smallest number of divisions that has two different semitones. However, they have different values for the fifth, as a semitone of one-seventh uses a fifth of twenty-seven divisions while a semitone of eight ninths uses a fifth of twenty-eight divisions.

The above text is the full text of the "Rational semitone" section. I'm removing it because it's so poorly written that it only harms the article. It misuses terms (a "semitone" is half or roughly half of a tone, not a third, a quarter, a fifth, etc.), it stumbles over itself ("one equal division of an octave" would mean 1-EDO! "twice the number of divisions of the semitone" but the semitone is not divided, the whole tone is, according to its own definition), it leaves out critical statements (we can guess that it means a whole tone of 200 cents, but it never says it), it makes conclusions with no arguments, and it even gets its own math wrong (6*3 is 18, not 19). However, I'll store it here in case anybody wants to discuss how to make it sensible enough to return to the article. (talk) 07:20, 16 May 2015 (UTC)
Nope! A semitone is only exactly half a tone in 12-EDO, and only in 12-EDO is a semitone 100 cents and a tone 200 cents. This section is not defining whole tone as 200 cents. In quarter-comma meantone (about 31-EDO), which was very common in the 16th and 17th centuries, the chromatic semitone is about two-fifths of a whole tone, and the diatonic semitone is about three-fifths. In third-comma meantone (about 19-EDO), semitones are actually roughly one third of a tone.
The phrase "exactly one equal division of the octave" does not mean 1-EDO. It means that if you define the semitone as any fraction you like of a whole tone (e.g. half in 12-EDO, one-third in 19-EDO, etc.), you have just defined a unique n-EDO (where n has just been fixed by your initial definition), so that starting a circle of fifths from any note will generate all the others, and the notes are in the right order (i.e. you don't get silliness like F-sharp being lower than F-natural).
"Divisions of a whole tone" clearly means how many of the smallest intervals make up a whole tone. In 12-EDO, this is two; in 19-EDO, this is three; in 31-EDO, this is five. The semitone is also not necessarily the smallest interval: there are smaller ones, like the diesis, the difference between C and D, for exmaple. Are you overlooking this because it happens to be mapped to zero steps in 12-EDO? That's a special case, and it usually isn't.
So in 12-EDO, you set a semitone to exactly half a whole tone, and you find that n must be 12 for this to hold. You get a semitone of 1 step and a whole tone of 2 steps, so an octave takes 7(2) − 2(1) = 12 steps, and a fifth takes 4(2) − 1(1) = 7 steps. For 19-EDO, you get a 1-step chromatic semitone and a 3-step whole tone, so an octave takes 7(3) − 2(1) = 19, and a fifth 4(3) − 1(1) = 11 steps. (One thing that should be made clear: you need to put in a fraction that is 1/2 or lower, and calculate using the smaller semitone.)
Your statement that the article gets its maths wrong seems founded on an assumption that a tone always has to be 200 cents. It's not. The ideal tone would be the just 9:8 (major tone) or 10:9 (minor tone), but it can be tempered in many different ways if you want various different properties. 19-EDO has a tone of 189.47 cents (tempered 14.44 cents down from 9:8, and +7:0, and a semitone that is one-third of that at 63.16 cents.
Hence I have restored this section, as this critique seems to be mostly founded on assuming every system is like 12-EDO, which is designed to have a semitone that is exactly half a tone and a tone that is exactly one-sixth of an octave. This does not hold in general. The section probably does need more clarification (and I'm thinking about how to do this and make it absolutely clear that we are talking in general and not just about 12-EDO), but it is actually completely correct AFAICS. Double sharp (talk) 10:35, 16 May 2015 (UTC)
It's unfortunate that you did that instead of helping me make the text sensible, which is why I took the time to move it here and list some criticisms in the first place. You even said, "probably does need more clarification" but nothing you wrote above helped, because you didn't add it to the text. I think you read my comments too quickly too. For example, I wrote "a semitone is half or roughly half of a tone", but you must have thought I wrote "exactly", because your reaction was "A semitone is only exactly half a tone in 12-EDO" (which BTW isn't true, an obvious example being 24-EDO, which has a semitone that divides the tone exactly in two, and quarter tones that divide the semitone exactly in two). There are other places you misread me too, but I don't want to get into that. We are not important. The final text is what's important. Anyway, here's an idea: Why don't you bring the text back here so we can go over it sentence by sentence and fill in the missing parts? You obviously understand theory. I do too. Let's get some work done. Feel like it? It really is too poorly written to stay in the published area as is. It doesn't matter that there may be a valid point in the mind of the original contributor, it only matters what the text actually succeeds in communicating. One thing though, we need to stick to ETs if we do work on it. We can't employ examples of meantone (as you did above), or just (as I could offer that a pair of unequal semitones 16/15 and 15/14 can together split an 8/7 septimal whole tone). (talk) 01:22, 17 May 2015 (UTC)
Right, when I reference a particular EDO I really meant it and its multiples: so I should have written 12k or something. But the multiples will not let the circle of fifths generate every note.
I've tried to rewrite the section above, sticking to equal temperaments.

In this section, semitone and whole tone may not have their usual 12-EDO meanings, as it discusses how they may be tempered to produce desired relationships. Let the number of steps in a semitone be s, and the number of steps in a tone be t.

There is exactly one family of equal temperaments that fixes the semitone to any proper fraction of a whole tone, while keeping the notes in the right order (meaning that, for example, C, D, E, F, and F♯ are in ascending order if they preserve their usual relationships to C). That is, fixing q to a proper fraction in the relationship qs = t also defines a unique family of one equal temperament and its multiples that fulfill this relationship.

For example, where k is an integer, 12k-EDO will set q = 1/2, and 19k-EDO will set q = 1/3. The smallest multiples in these families (e.g. 12 and 19 above) will have the additional property of having no notes outside the circle of fifths. (This is not true in general; in 24-EDO, the half-sharps and half-flats are not in the circle of fifths generated starting from C.) The extreme cases are 5k-EDO, where q = 0 and the semitone becomes a unison, and 7k-EDO, where q = 1 and the semitone and tone are the same interval.

Once one knows how many steps a semitone and a tone are in this equal temperament, one can find the number of steps it has in the octave. An equal temperament fulfilling the above properties (including having no notes outside the circle of fifths) divides the octave into 7t − 2s steps, and the perfect fifth into 4t − s steps. If there are notes outside the circle of fifths, one must then multiply these results by n, which is the number of nonoverlapping circles of fifths required to generate all the notes (e.g. two in 24-EDO, six in 72-EDO). (One must take the small semitone for this purpose: 19-EDO has two semitones, one being 1/3 a tone and the other being 2/3.)

The smallest of these families is 12k-EDO, and in particular 12-EDO is the smallest equal temperament that has the above properties. Additionally, it also makes the semitone exactly half a whole tone, the simplest possible relationship. These are some of the reasons why 12-EDO has become the most commonly used equal temperament.

Each choice of fraction q for the relationship results in exactly one equal temperament family, but the converse is not true: 47-EDO has two different semitones, where one is 1/7 of a tone and the other is 8/9, which are not complements of each other like in 19-EDO (1/3 and 2/3). Taking each semitone results in a different choice of perfect fifth.

P.S. Come to think of it, this article has a problem: it can't seem to decide whether it wants to talk about 12-EDO, or EDOs in general. I think it really ought to be the latter, with 12-EDO having a natural article title as 12 equal temperament. Double sharp (talk) 03:10, 17 May 2015 (UTC)
I don't think the ET article has a problem re: 12-EDO; it's definitely about all Equal Temperaments. The opening defines all ETs, says 12-EDO is the most common one in Western music, and gives examples of others, which is pretty straightforward. I think I opened a can of worms with my move and invitation though. It's getting more arcane, not less. Sorry about that. Maybe we should drop it. I think your proposed text for this section is still trying to say everything all at once. We need to break the old text down sentence by sentence, state things in the simplest form, define each term as it is introduced, and give examples at every step, so that it progresses clearly with no jumps. Also, if this is coming from a source, it should be cited. Otherwise stuff like your new "That is, fixing q to a proper fraction in the relationship qs = t" starts to sound like original research, however useful it might be to your own work. Give me some time to digest. There's got to be a simpler exposition. Can you point me to the source so I can collect the original terminology? That's probably the best approach at this point. Whose term is "Rational semitone" in this context anyway? Out of this context, "rational semitone" would intuitively imply just semitones. Need to define the title (or change it if it's not sourced), and define "proper fraction of a whole tone" so that "rational" and "fraction" can't be confused with rational intervals (just intervals), because we're already on the log scale when we're talking interval names like "whole tone", so a "proper fraction of a whole tone" in this context would have an irrational frequency relationship (not, though, for example, in the case of any non-EDO ET whose generator is a just interval, such as a stack of 16/15, but those are not in this section if all of these must be EDO). Some other tweaks: Simplify the verbage, like "exactly one family" to "a family" and if it's EDO temperaments, say "family of EDO temperaments" not "family of temperaments" so the context of the article is narrowed clearly and immediately instead of making the reader figure that out later on their own. Also, not every proper fraction of a whole tone should be labelled a semitone, only the ones near the middle. For example, 2/3 or 3/2 of a whole tone yes, but 1/5 and 4/5 of a whole tone no. This section doesn't need to immediately open with a contrast against, and apology to, 12-EDO, especially since 12-EDO is in the group of ETs this section is about. It doesn't make sense to describe other ETs as tempered versions of 12-EDO, in "..this section...discusses how they [12-EDO intervals] may be tempered to produce desired relationships." I mean, all ETs are tempered, but they are not tempered versions of each other. Tempered means altered from just by spreading little errors across many pitches for the practical purpose of having a limited set of pitches to make acoustic instrument construction reasonable and not having it sound too terrible when you pseudo modulate around within the limited pitch set. Also whole tone and semitone are not defined by 12-EDO. They already had a meaning long before 12-EDO and they still have that larger meaning as anything near certain size ranges, which the ear naturally picks up on and organizes into a kind of scale grammar regardless of the tuning. So there's no need to apologize for them not being 200 and 100 cents. Especially since they are exactly that width in one of the ETs in the group. Thinking. Be back later... Can you get me that source? (talk) 20:25, 17 May 2015 (UTC)
I do not know what the original source for this was, as its original author did not cite it (and no, I didn't write it). Although if it is citable, I'm not sure how the equation is original research, as it does nothing but rephrase exactly what has previously been said. (If the source for this was looking at it primarily mathematically, I find it very likely that it would use "semitone" very generally – even including very edge cases like 1/5 and 4/5 of a tone – because it makes the maths simpler without needing to use different terms for special cases.)
I've changed the title to "proportions between semitone and whole tone". And I did mention the restrictions to ET right off the bat, with "There is exactly one family of equal temperaments that fixes the semitone to any proper fraction of a whole tone...".
I didn't want to talk about them as tempered versions of 12-EDO, but given that your first impression of what "whole tone" meant was 200 cents, as opposed to 9:8 or 10:9 ("(we can guess that it means a whole tone of 200 cents, but it never says it)"), I thought it was necessary to warn against that. (Perhaps it would help if it didn't come right after a section on 12-EDO, so I've moved it further down along with all the stuff talking about EDOs in general.) Double sharp (talk) 11:42, 19 May 2015 (UTC)
I think you might be confusing ET with EDO. All EDOs are ETs, but not all ETs are EDOs. If this section is restricted to EDOs, the opening sentence should state it. There's got to be a better (short) title. What is the central concept of the section? It's describing a subset of ETs, that are EDOs, and that can be completely traversed by a circle of whatever a fifth would be in each EDO (no notes are skipped, as can be the case in double-helix or triple-helix type ETs that have multiple interlocking cycles of fifths that never touch), and where the "semitone" is some rational division (n/m) of the width of a whole tone (which actually puts it on the log scale if you're talking frequency rations, which means it's actually likely to be an irrational frequency ratio except in rare cases), and where the note names that are ascribed when generating the circle of fifths don't get out of order after you wrap them all into the same octave. There's got to be a name for that subset of tunings. The section should use that name. I've been searching around but I can't find it yet. That's what's making it look like original research. It's unfortunate that neither you nor I wrote the original text, and the original contributor didn't cite anything. It an interesting topic, and it makes sense (assuming you fill in the missing details mentally), but writing-wise, it's a mess, and it's uncited. When I say "mess", the worst offense is that it's ambiguous. In other words, if you already know what it's trying to describe, then what it says is not untrue, but the text itself doesn't contain enough information to distinguish the one correct interpretation from multiple incorrect interpretations of the same text, making it useless to anyone new to the topic. That's ambiguity in a nutshell. About semitone being defined as the smallest interval in an ET: I disagree, and it's not valid to change the terminology to make the math easier. "Semi" means "roughly half of", not "smaller than". A semitone is not "any interval smaller than a tone". If it meant that, then quartertones would be called semitones, but they aren't. 1/5 of a tone is a fifth tone, not a semitone. The "tone" and "semitone" interval names represent various sizes of actual intervals in different systems, but they can't be stretched indefinitely, and they still keep that rough relationship to each other. If you need a term to make the math easier, then call it the smallest interval or call it the generator interval, which means the same thing, is super clear mathematically, and carries no interval-name connotations. Agreed, it's definitely important to state that the whole tone has different sizes in the different temperaments being described. I just mean that saying that non-12-EDOs are tempered versions of 12-EDO is not the way to explain that. Every EDO, indeed every ET, is valid on its own, and if they are "tempered" versions of anything, its of just, not of each other. I've a feeling this is going to take a while. I'll keep searching. Still working on it... Thanks for sticking. (talk) 20:13, 19 May 2015 (UTC)
BTW, this sort of misuse of "semi-" is rather old: in the old Latin interval names, "semi-" meant "one semitone smaller", not "half". For example, a perfect fourth was a "diatessaron", and a diminished fourth was a "semidiatessaron". This only really works as an explanation for the smaller semitones, though. Intervals as small as the quarter-tone may be able to work as semitones: similarly large inflections are usually performed by symphony musicians in live concerts, and nobody feels that anything is wrong! (See "Some Aspects of Perception, I: Sizes of Harmonic Intervals in Performance," Shackford, Charles, Journal of Music Theory, Vol. 5, No. 1, 1961, 162–202 and "Some Aspects of Perception, II: Interval Sizes and Tonal Dynamics in Performance," Shackford, Charles, Journal of Music Theory, Volume 6, No. 1, 1962, pp. 66–90.) Of course, this is all just after-the-fact rationalization, unless the original source (whatever it was) misused "semi-" this way, in which case we can mention the terminology, but use something else. "Generator interval", as you suggest, should work: I guess we ought to mention that in special cases it acts like a semitone? Double sharp (talk) 10:10, 26 May 2015 (UTC)

This looks somewhat similar, in how it relates the semitone:tone ratio to Western note names (and notes that things work best for tonal music when that ratio is close to 1:2). It gives some low examples (and please interpret "12" as meaning "any multiple of 12", etc.): 12 gives 1/2, 17 gives 1/3, 19 gives 2/3, 22 gives 1/4, 26 gives 3/4, 27 gives 1/5, 29 gives 2/5, 31 gives 3/5. Double sharp (talk) 12:59, 1 June 2015 (UTC)

Historical comparison[edit]

Sorry, I know verly little about the subject, I was interested in the historycal aspects. The comparison table of the values of the equal semitone obtained by different authors looks very suspicious: that different authors in different countries at different ages had computed it with the very same number of digits is for sure the least probable event ever devised... Esagherardo (talk) —Preceding undated comment added 07:26, 14 December 2015 (UTC)

Example notes for the table "Comparison to just intonation"[edit]

The table "Comparison to just intonation" often shows two examples notes. For example "Minor third" got "(D♯/E♭)". But from C, the example starting note, only E♭ is a minor third. D♯ is an augmented second. — Preceding unsigned comment added by Aths (talkcontribs) 19:45, 1 July 2016 (UTC)

General properties: General formulas for the equal-tempered interval[edit]

Mr Just plain Bill,

Please let me express my many thanks for correcting my grammar in General formula for calculating equal temperament, General properties section.

With regards and friendship, Georges Theodosiou, (talk) 11:17, 17 September 2016 (UTC)

Please, it is only collaborative editing. (You're welcome.) Best regards, Just plain Bill (talk) 17:13, 18 September 2016 (UTC)
Mr Just plain Bill, (I suppose you are the Author of this article), please let me express my sincere gratitude for the "welcome" and tell you the little story of "General formula for calculating equal temperament".
For first time I have posted it at Psaltologion by 30-01-09, 11:26, (2009 January 30), as an answer to Mr Zinoviev, a Byzantine Music lover, as well I am.
However I have to acknowledge that from when I posted it there, until I published at wiki, nobody responded, but from when I published at wiki (by 2015 December 2), I have received many thanks from music lovers.
I consider you music theoretician. Then please let me the question: do you consider this formula of some value for music theoreticians?
With regards and friendship, Georges Theodosiou,, Georges T. (talk) 07:50, 20 September 2016 (UTC)
Georges, I see you have registered an account. Congratulations!
I am by no means the "author" of this article; like every other wikipedia article, it is meant to be a collaboration among editors. I merely adjust format and orthography as I see a need. I would consider myself an interested amateur, not so much of a music theoretician. I did check the formula by substituting values for a perfect fifth in 12TET just intonation with a ratio of 3/2, and it gave t=701.955 or so, which comes as no surprise. I can imagine cases where the the formula may be useful. Best regards, Just plain Bill (talk) 17:00, 20 September 2016 (UTC)

Mr Just plain Bill, good morning (I live in France),

Please let me express my thanks for you replied, and for you can imagine cases where the formula may be useful.

In your computation unit is cent = octave. However you can use another unit, for example, unit = of the tone . Then, t = 344.247... units (hundredths of the tone). In this example, r = , b = , and d = 100.

With regards and friendship, Georges Theodosiou, Georges T. (talk) 10:15, 21 September 2016 (UTC)

The paragraph needs further work to improve the English and to make clearer what was intended. Perhaps someone else can contribute? Dbfirs 12:27, 24 September 2016 (UTC)
I've done a bit more to the paragraph, but I'm not a musician, so please check that I haven't made any errors. Dbfirs 11:48, 25 September 2016 (UTC)

Mr Dbfirs, let me express my thanks for your comments. I'm Greek living in France and my english are poor. Could you, or any other with well english, please, improve grammar, syntax, and phrasing? Merci d'avance! Now let me make some remarks:

1. Anyone familiar with unit cent, could check results in Examples. For example, ratio interval is 701.955... cents. Interval is 203.91 cents. Its hundredth is 2.0391 cents. Now, 701.955 cents divided by 2.0391 cents yields 344.247... hundredths of , that is result in 2nd Example. Then,

2. Formula only directly calculates tempered interval, from ratio interval, with any unit.

3. "Pitch depends to a lesser degree on the sound pressure level (loudness, volume) of the tone, especially at frequencies below 1,000 Hz and above 2,000 Hz. The pitch of lower tones gets lower as sound pressure increases. For instance, a tone of 200 Hz that is very loud seems one semitone lower in pitch than if it is just barely audible. Above 2,000 Hz, the pitch gets higher as the sound gets louder." (Pitch and frequency: last paragraph).

With regards and friendship — Preceding unsigned comment added by Georges T. (talkcontribs) 09:41, 27 September 2016 (UTC) Georges T. (talk) 09:49, 27 September 2016 (UTC)

I appreciate that it is difficult to write in encyclopaedic style in a third language. I couldn't even begin to do that in either French or Greek, so I admire your efforts. The logarithmic law applies only approximately to sound intensity, and you make a good point about the sound pressure level changing our perception of pitch. Personally, I think that the logarithmic treatment of pitch intervals can be derived without making an analogy with Frechner's formulation of the Weber-Frechner principle (there's really only one principle involved, since each law leads to the other). I'm not a music theorist, so perhaps someone else can comment on this? Best wishes from across the channel. Dbfirs 09:55, 27 September 2016 (UTC)
Mr Dbfirs please accept my many thanks for appreciate that it is difficult to write in encyclopaedic style in a third language, and for the best wishes across the channel.
I really don't see how a ratio of frequencies can be described as a stimulus intensity. This is unscientific use of language. I suggest that we remove the analogy with the Weber-Frechner observations, or only just mention it in passing as an analogy, since they made no claim about sound frequency. Dbfirs 11:07, 27 September 2016 (UTC)
Many thanks for your comment. For example, ffreq = 200 Hz is a stimulus, ffreq = 300 Hz is another, 400 Hz another. Consider ratios and . Former is 1.5 and latter 2. Clear, intensity 2 is greater than 1.5, like pressure (say intensity) 2 Pascals is greater than 1.5 Pascal. So, ratios represent frequencies intensity although in logarithmic scale. Regards. Georges T. (talk) 12:37, 27 September 2016 (UTC)
No, that is a very confusing analogy because the intensity of a sound is related to its volume, not its pitch. Dbfirs 12:45, 27 September 2016 (UTC)

Mr Dbfirs, please let me express my many thanks for your comment. Clear I mean pitch intensity. You can explain it, in the article, for english is native language for you. Regards. Georges T. (talk) 13:12, 27 September 2016 (UTC)

I don't know what you mean by "pitch intensity" because I would interpret that as volume, and you don't mean volume. Why not just pitch or frequency? Dbfirs 16:59, 27 September 2016 (UTC)

Mr. Dbfirs, please let me express my thanks for your reply. I just use Fechner law terminology. When law applies to pitch, intensity means pitch. I'm going to edit article. Regards. Georges T. (talk) 08:56, 28 September 2016 (UTC)

You are certainly persistent in your opinions. Can you link to anywhere that Fechner mentions pitch? Dbfirs 11:49, 30 September 2016 (UTC)
I have been wondering about that as well. My quick reading of the Weber–Fechner law article does not show that the Fechner law applies to pitch. Just plain Bill (talk) 15:04, 30 September 2016 (UTC)

Gentlemen Dbfirs and Just plain Bill. It just is my inspiration. Regards. Georges T. (talk) 14:10, 4 October 2016 (UTC)

Messrs Dbfirs and Just plain Bill. Please let me explain my understanding (inspiration) about.
It was known to Pythagoras, and perhaps to ancient Chinese philosophers, that our subjective sensation of sound pitch is proportional to the logarithm (exponent) of the ratio of two sounds fundamental frequencies. For example, ratios 9÷8 and (9÷8)^2. Our subjective sensation of latter is double of that of former.
To go further. Our subjective sensation needs two sounds, for compare their relative pitch. As with every quantity. For example, for understand magnitude of any length we need another length called unit. Comparing the two, we understand magnitude of other. Similar, for understand magnitude of any pitch, we need another pith, called unit pitch, e.g. cent. Indeed, difference is that, length is proportional to unit length, while pitch is logarithmically proportional to unit pitch, that is, logarithm of any ratio interval is proportional to logarithm of unit ratio interval.
Fechner just expressed this law more general including sound volume. Then, citation is not needed. Understanding of relative theory is needed. Regards. Georges T. (talk) 12:56, 5 October 2016 (UTC) Georges T. (talk) 13:17, 6 October 2016 (UTC)
Since Fechner never considered pitch, any reference to him is superfluous and should be removed as WP:original research. I wonder if the whole section should be removed? Dbfirs 19:04, 5 October 2016 (UTC)
Framing pitch perception in terms of Fechner's law seems like a stretch, or original research. In a context of imaginative brainstorming (tempête d'idées, or ιδεοκαταιγισμός?) it might be interesting to consider the metaphors of such a framing, but that would not make suitable encyclopedic text. I am tempted to put the section back to how it was in November 2015. Just plain Bill (talk) 02:24, 6 October 2016 (UTC)

Messrs Dbfirs and Just plain Bill. Please let me explain you that Fechner's law is general, including sound pitch and volume. Regards. Georges T. (talk) 12:48, 6 October 2016 (UTC)

No, that is just your interpretation and original research. Fechner never mentioned pitch. I tend to agree with Just plain Bill about reverting, especially because your additions appear to me to be too verbose and confusing, but I'm reluctant to upset you as a sincere editor who persists in his belief that he is right. If you can find any reference for your additions, then I might be persuaded to change my mind. Dbfirs 07:53, 7 October 2016 (UTC)

Gentlemen, you can read background knowledge (relative theory), at my talk page. Regards. Georges T. (talk) 08:30, 17 October 2016 (UTC)

So are you confirming that this is all WP:original research? Dbfirs 09:47, 17 October 2016 (UTC)
Mr Dbfris, please let me say that reliable, published sources (Sensation of tone, and Stanford Physics Dept. course online), exist in relative (background) theory. Only presentation and explanation are mine. Georges T. (talk) 05:13, 18 October 2016 (UTC)
If you wish to avoid having your additions deleted, then you need to provide in-line citations to the reference works that you are using. Your "Further Reading" is in the wrong place (it should go at the end of the article), and I cannot find your additions in that document, though I might have missed something. Also, please put comments about the article on this page, not in the article itself. Dbfirs 15:37, 19 October 2016 (UTC)

Mr Dbfris, Please let me ask: By 11:26, 25 September 2016‎, in Edit Summary you state: "... this section needs more work." Then, do you think work I have done so far is enough? Regarding threats for deleting, you could do that at any time. Regards. Georges T. (talk) 13:35, 21 October 2016 (UTC)

I wouldn't want to discourage you by deleting your work without a consensus here, but the problem is in "your work". If it is original research, then it doesn't belong in Wikipedia (by policy -- see Wikipedia:No original research), so you need to provide some in-line citations if you found the details in publications. Dbfirs 16:38, 21 October 2016 (UTC)
Mr Dbfris, please accept my many thanks for you respond immediately to my messages. I have copied background theory from my talk page and pasted to article. It includes many citations. I have 4 for Pythagoras research. Then, you should be satisfied, at last! In recent times Helmholtz undergone experimental research and proved psycho-acoustical law known to Pythagoras. Even formulas are widely known, but only for base interval 2 (octave). Only, background theory, general formulas (for any base interval), and their explanation are mine. Is it "new research"? To my view it is a small step ahead of an old and long research. That's all my contribution here. Regards. Georges T. (talk) 09:12, 22 October 2016 (UTC)

Mr Dbfris, please understand that we do not take the logarithm of any frequencies but the logarithm of the ratio of fundamental frequencies. As long our "subjective sensation is proportional to the logarithm of stimulus intensity", follows that, in our case, stimulus intensity, or intensity of stimulus is the ratio of fundamental frequencies. Regards. Georges T. (talk) 12:58, 22 October 2016 (UTC)

Please don't use other Wikipedia articles as in-line citations. That's not the way we do it. Please see WP:inline citations and WP:reliable sources. You may link to relevant articles in Wikipedia by using double brackets just to clarify terms used and to provide further reading from Wikipedia. Also, please don't use the word intensity for pitch, or for a ratio of pitches. In the context of sound it invariably refers to volume. You might also like to learn about the use of articles in English grammar. Dbfirs 15:48, 24 October 2016 (UTC)
Mr Moderator Dbfris, please let me say, I use citations from Wikipedia articles as I do from other websites. Regarding term intensity I deleted reference to Fechner's law, for it is not necessary in understanding formula, rather is encyclopedic information. Indeed, when it it applied to sound pitch, stimulus intensity or intensity of stimulus is the deference (interval) of two sounds in pitch, expressed mathematically by the ratio (not the difference) of corresponding fundamental frequencies, due to psycho-acoustical law. Georges T. (talk) 08:21, 25 October 2016 (UTC)
Dear Mr Georges T, I'm not a moderator, just an ordinary editor. I agree that Fechner's law is not necessary for the understanding of pitch intervals, and thank you for understanding about "intensity" of sound. Wikipedia internal links, where appropriate, use double brackets like this. Citations to external on-line references use a single bracket weblink between ref tags (<ref>[ details]</ref>, as you have been using), or you can use cite web (Template:Cite web) if you prefer. Best wishes. Dbfirs 09:17, 25 October 2016 (UTC)
Dear Mr Dbfirs, please accept my thanks for your instructions, and my sorrow for delay in respond. I'm homeless in France, proscribed (every where in Europe), "sans papier, marquée et compliquée" according to french hidden state, and always persecuted by it. So I'm forced to travel from town to town. I get internet access for free, usually in municipal libraries (Médiathèques), like this I'm just now, in town Niort, west France. Under these circumstances I try my best. I think have finished my contribution about equal temperament, and donate it - indeed through Wikipedia to which I'm very indebted - to music theoreticians, both professionals and amateurs. Last but not least, I have to inform you that as I have already said to Mr Just plain Bill "for first time I have posted it at Psaltologion by 30-01-09, 11:26, (2009 January 30), as an answer to Mr Zinoviev, a Byzantine Music lover, as well I am", although only the main formula, without background theory and explanation, and with only one example. So publishing in Wikipedia is the second time. I mean that Mr Coumparoulis owner of is justified to claim copyright as the first publisher. With regards and friendship. Georges T. (talk) 12:07, 26 October 2016 (UTC)

My dear mate, please let me complain for you do not comment my recent edits. They are interesting, especially negative tempered intervals. Did you ever hear about? My greetings to your Queen. Regards. Georges T. (talk) 08:35, 3 November 2016 (UTC)

I'm not sure to whom your last message is addressed, but I'm not in a position to pass on your message to our own dear Queen. Are you trolling? I've given up on trying to improve your additions to the article. Dbfirs 09:25, 3 November 2016 (UTC)
My dear mate please do not worry. Word "mate" is chiefly British according to Merriam-Webster dictionary, then as long you are British, it is clear whom my message is addressed to. Really I troll your contribution at least in grammar.
Please let me answer your question "General formulas for the equal-tempered interval: ????"
Euler's formula is only. Regards. Georges T. (talk) 10:25, 3 November 2016 (UTC)

My dear mate, please accept my many thanks for correcting my grammar. Yes i was meaning "appendix" and "Especially".

Also let me correct that I wrote in my message by 09:12, 22 October 2016 (UTC) that "Even formulas are widely known, but only for base interval 2 (octave).". I see in the article's subsection "Equal temperaments of non-octave intervals" base intervals 3 and . However I do not see general formula published before I publish it at psaltologion. So I stand on that "publishing in Wikipedia is the second time. I mean that Mr Coumparoulis owner of is justified to claim copyright as the first publisher.". Regards. Georges T. (talk) 11:33, 20 December 2016 (UTC)

Messrs Dbfirs and Just plain Bill. Please accept my many thanks for your collaboration, also my wishes for the new year. Regards. Georges T. (talk) 13:08, 25 December 2016 (UTC)

Ladies, Gentlemen, please let me express my sorrow for I can not make the example more easy. One should be familiar with the notion of temperament for easy understand it. Georges T. (talk) 09:12, 21 January 2017 (UTC) With regards and friendship. Georges Theodosiou. The Straw Man. Georges T. (talk) 14:28, 18 January 2017 (UTC)

Ladies Gentlemen, please let me say that section "General formulas for the equal-tempered interval" as it is, is confusing. I suggest either, whole section be deleted or be restored.

With regards and friendship, Georges Theodosiou, The Straw Man. Georges T. (talk) 13:28, 5 July 2017 (UTC)

Tutorials on psycho-acoustical law[edit]

Ladies, Gentlemen, please let me say, regarding template that example is confusing, that, Wikipedia does not permit tutorial in article, nor link to talk page. So I post following tutorials here, with the hope that beginners in equal temperament will read it.

Informal tutorial[edit]

We consider 4 lengths of a string: 90 cm, 80 cm, 81 cm, and 64 cm.
Let P90, P80, P81, P64, be sound pitches produced by these lengths when string is plucked.
Now we consider two intervals: P90 - P80, and, P81 - P64. Our subjective sensation is that, latter is double the former.
We want an arithmetic relationship between string's lengths, agreeing with our subjective sensation.
We try differences in lengths: 90 cm - 80 cm = 10 cm, and, 81 cm - 64 cm = 17 cm. Latter is not double the former.
Now we try ratios of lengths: 90/80 = 1.125, and, 81/64 = 1.265625. Latter only little greater than former.
Eventually we try logarithms of these ratios:
log90/80 = 0.05..., and, log81/64 = 0.1...
Marvelous! Latter is double the former.
At last, we found arithmetic relationship between string's lengths agreeing with our subjective sensation. We say that "our subjective sensation of intervals, P90 - P80 and P81 - P64, is proportional to the logarithms, 0.05... and 0.1..., of the ratios, 90/80 and 81/64, of the lengths".
By trying many lengths ratios, we have found that
"Our subjective sensation of the interval between two sound pitches produced by two different string's lengths, is proportional to the logarithm of the ratio of the lengths".
It is said: "psycho-acoustical law".
Same law is applied regarding fundamental frequencies as follows:
"Our subjective sensation of the interval between two sound pitches is proportional to the logarithm of the ratio of corresponding fundamental frequencies".
Remark: Fundamental frequency and string's length are inverse proportional.
Present version is mathematically correct. The before version was not, for I used term "inverse" in a liberal way. Mr. Wgrommel is right on it.
But present version provokes the question: "How is that ratio of fundamental frequencies and ratio of string's lengths are both logarithmically proportional to interval between pitches, while fundamental frequency and string's length are inverse proportional? I can not answer this question. Then I suggest my readers beginners in equal temperament, first understand psycho-acoustical law regarding fundamental frequencies, and then, having in mind they are inverse proportional to string's lengths, understand law regarding string's lengths. It is the easiest way for understand it, although indirect.

Formal tutorial[edit]

For math lovers I post following formal tutorial.

We consider 4 sound fundamental frequencies: F1, F2, F3, F4, and corresponding pitches P1, P2, P3, P4.
Now we compare intervals (differences): P1 - P2, and P3 - P4. Psycho-acoustical law states that: P1 - P2/P3 - P4 = logF1/F2/logF3/F4.

Any comment on these tutorials is well appreciated.

Note: I regard that above tutorials are best explaining psycho-acoustical law. However, if Mr. Dbfirs threatens me by offending, I may  explain it better.

With regards and friendship. Georges Theodosiou. The Straw Man. Georges T. (talk) 13:51, 26 January 2017 (UTC)

I shall not comment on the twists and turns of this discussion. IMHO, "our subjective sensation of the difference (interval) in pitch (tone) is inversely proportional to the exponent of the ratio of string (chord) lengths" does not make sense. Two quantities can be inversely proportional; e.g., the gravitational attraction between two masses is inversely proportional to the distance between them. But our subjective sensation is not a quantity. It seems that something like this is meant: "If three pitches are in continued proportion c:d:e, i.e. the ratio between c and d is the same as the ratio between d and e, we hear these subjectively as two equal intervals. Since then the ratio e/c is the square of the ratio d/c, we hear the former as an interval that can be described as "double" the latter." One could then invoke logarithms. Not being well-versed in this field, I have not changed the text, but I think a clarification of this sort is called for. Wgrommel (talk) 21:57, 18 February 2017 (UTC)

I agree that your explanation is clearer. I've been wondering how to simplify all this without offending Georges (to whom I was offering advice on avoiding deletion, per policy, not issuing any threat!). Dbfirs 22:07, 18 February 2017 (UTC)
My dear mate Mr. Dbfirs. Please let me say my understanding of simplify a tutorial. It means rather to add than to delete. For example "informal" tutorial is much longer than "rigorous" for it is formal simplified. Also please continue correct my grammar and phrasing as before. Regards. Georges T. (talk) 12:29, 4 March 2017 (UTC)
Mr Wgrommel, please accept my many thanks for you are the first one commented my tutorial. My old mate in this discussion Mr Dbfirs is now the second. Regarding "inverse proportional", I explain: "by sense, as greater the length as lower the pitch".
Our subjective sensation is psycho-acoustical quantity as gravitation is physical quantity. So psycho-acoustical law is similar to physical law.
As long I understand english (I'm Greek living in France) "difference" and "interval" are synonyms. Also chord is a string. Logarithm is exponent, "the logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number.".
You confuse 22 with 2 * 2. Although are equal, in general 2a is not equal with 2 * a. Consider 3 fundamental frequencies: 100 Hz, 200 Hz and 800 Hz. Our subjective sensation is that, interval 800 Hz/100 Hz is (not 4, but) 3 times interval 200 Hz/100 Hz. Latter is equal 2, former is equal 8, and 23 = 8. So we conclude that we comprehend pitch intervals with proportion (3) equal to exponent relating ratios of fundamental frequencies.
My dear old mate Mr. Dbfirs, please accept my many thanks for you are interested in not "offending Georges". Georges T. (talk) 09:45, 21 February 2017 (UTC)
Γεωργος, never fear -- I shall not confuse 2n with 2*n. If ratio b is the square of ratio a, the logarithm of b is twice that of a (log 9:4 = 2 * log 3:2); thus, as you say, the interval between C and E might be described as twice the interval between C and D. But I must disagree with the statement "Our subjective sensation is psycho-acoustical quantity." Subjective experience is generally qualitative, not quantitative. Yes, those of us with mathematical and musical training hear certain intervals as being equal to other intervals, and we can count intervals, but could the naive listener say that an octave is twelve times as big as a semitone? I respectfully doubt it. Wgrommel (talk) 21:44, 25 February 2017 (UTC)

Mr Wgrommel, many thanks for above message. I can add many things. For example that we call "ratio interval" or just "interval" is incorrect by math view. Ratio is not interval, difference, result of subtraction. Indeed we call it interval by the sense it corresponds to interval between pitches, according to our subjective sensation.

Psycho-acoustical law was known to Pythagoras and probably to ancient Egyptians and Chinese. Obviously they concluded it from many experiments. By the way let me do english humor. Mr. Dbfirs claims background theory is original work of mine! Rather it is original work of some Pharaoh. Only presentation is mine. Regards.

My guess would be that the basic facts were known to the Sumerians, long before the Pharaohs and Pythagoras, but I'd be surprised if they knew about logarithms. Dbfirs 12:01, 28 February 2017 (UTC)
My dear old and present mate, please let me be not surprised if Sumerians knew about logarithms, given that logarithm is just exponent. I'd be surprised if they knew about psycho-acoustical law. Regards. Georges T. (talk) 13:17, 28 February 2017 (UTC)
They probably knew about it in music, but didn't express it in such a complicated way. Dbfirs 09:59, 4 March 2017 (UTC)

Absolute unit[edit]

1. Main formula

We can calculate tempered interval of ratio one, by following formula:

t(r) = log(r) atu = ma atu (1)

"t" stands for "tempered", "r" for "ratio interval", "ma" for "absolute magnitude", "log" for "logarithm with base 10", and "atu" for "absolute tempered unit". It means that "log(r)" is just the magnitude of tempered interval. Unit "atu" is defined by this formula.

It is direct application of psycho-acoustical law. Indeed we can use a factor say f, as the number 1000 in 0.003 m * 1000 = 1 mm.

t(r) = log(r) * f satu = ms satu (2)

Where "satu" stands for "submultiple absolute tempered unit", and "ms" for "log(r) * f " that is magnitude with submultiple unit satu. For example f = 1000. Then, octave (ratio interval 2) is 301 satu, fourth (4/3) is 125 satu, fifth (3/2) is 176 satu, and major tone (9/8) is 51 satu.

Formal expression for the case of octave is (t = tempered): t(2) = 301 satu. Number 301 is magnitude for submultiple unit satu.

2. Derived Formula

From (2) we get: log(r) = ms/f and eventually:

For f = 1000, and ms = 301, we get r = 2, that is octave.

3. Remark

In general formulas, if we put b = 10 (base of log), d = f and m = ms we get above formulas. Indeed formula , is easier than . Georges T. (talk) 18:22, 22 April 2017 (UTC)

Any comment is well appreciated.

With regards and friendship Georges T. (talk) 14:03, 21 April 2017 (UTC)

Rewrote Syntonic Tuning to Regular Diatonic Tunings[edit]

I've rewritten this section along the lines of the new Regular Diatonic Tunings page for the reasons discussed in the talk page of that page. Robert Walker (talk) 23:02, 23 September 2016 (UTC)

External links modified[edit]

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"Examples" deleted[edit]

Any user who is vaguely numerate will not need these examples, and any user who is not numerate will not understand them.----Ehrenkater (talk) 14:21, 27 June 2017 (UTC)