Talk:Pythagorean tuning

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Added to the text about collapsing the series into a compact scale[edit]

I edited the text about collapsing the series of perfect fifths into a one octave scale (by shifting low or high notes up or down by one or more octaves). My intent was to provide the musical motivation for this part of the method. In the as-is text, it was stated that this compression was done by convention, but no rationale was given.

I hope this is an improvement but comments and corrections would be welcome (talk) 04:35, 28 December 2010 (UTC)


I'll expand this further in time, but this will do for now. I'd appreciate it if people could let me know if it makes sense - it makes sense to me, but I've studied all this stuff, and a list of unannotated numbers and cent values would make sense to me, so that doesn't mean much. Oh, and if anybody wants to prettify the table, please do so - my html skills are not up to the task. --Camembert

As I've thought about this since I wrote it, I've come to think that I'm really trying to write two or three articles in one here; one on Pythagorean tuning, one on what should be at Mathematics of the Western music scale, and one on tuning ratios generally (what a 9:4 is, and why it is equivalent to a 9:8). So I regard this as something of a work in progress, and one that might get broken up into bits in the future. --Camembert

I've added a note on number theory. I think modular arithmetic comes in to this too, because in tuning up in fifths we yank notes down to stay within the range of an octave. I am planning to make a PNG of the piano keyboard at some point. Maybe a long version of that would be good here, to show graphically the circle of 5ths stretched out, with the clash in the middle. -- Tarquin

Something like that would also be very useful for the articles I'm writing on small intervals like the syntonic comma. --Camembert

Okay, I'm talking rubbish: removed "Put in terms of number theory, no multiple of 1.5 is a power of 2." because it's completely wrong. 1.5 ^ 12 = 129.746337890625 , and 2^7 = 128. I think it's that. (note: "think"). Will ponder some more. -- Tarquin 11:26 Oct 30, 2002 (UTC)


Camembert, I had a look like you asked, and I don't think you're spending too much time on the basics here at all. There's only one slight sticking point for me where there's talk of all twelve fifths being tuned and the D's not being a perfect octave. I know what you mean, but I had to read it a couple of times. I'd have fixed it, but I'm not sure how else to say what you're trying to say there. Good job though, this tuning stuff is still a tricky subject and I'm impressed with the work you folks have done.JFQ

Thanks - I think I'll leave this article as is for now, and maybe come back and try to polish it a bit when I've done the other tuning systems articles (gulp). --Camembert
Yes, I'm VERY interested to see what you do for Mean Tone Temperament. I'm not sure that's ever been explained clearly by anyone. J.F.Quackenbush

Broken link[edit]



This article has been deemed too confusing by (apparently) a U.S. junior high school student (see Wikipedia:Reference_desk/Miscellaneous#PYTHAGORAS). Make of that what you will, but the discussion contains content and links that might help to make this article more accessible. Sandstein 17:47, 3 June 2006 (UTC)

Notes and frequencies[edit]

In the "Method" section, why were the letternames chosen? D and A are so nowhere near 200Hz and 300Hz that it's idiotic. The frequiencies given are an awful lot closer to G (196Hz) and D (294Hz). It's a handy illustration of the ratios being discussed, and such approximation of the frequencies (within 2%) is not unreasonable, but you can't pull note names out of thin air. You have to make an effort to sound like you know what you're talking about. 03:08, 12 May 2007 (UTC)

French Wikipedia[edit]

The article there is very detailed and contains many images and tables. Anyone here knows French and can translate the entire article to English? This will be very helpful! I'm willing to help with translating the text in the images (just tell me what to write). Barak Sh (talk) 03:32, 27 May 2008 (UTC)

Tuning up or down?[edit]

The table in the article says that in order to simplify the ratios involved, intervals are tuned down from unison as well as above. Is this the preferred method, or is only tuning upward also common? SharkD (talk) 02:37, 12 August 2008 (UTC)

It does not make any difference to the resulting frequencies. Only the numbers shown are smaller. The ratio of two consecutive notes in the circle of fifths is always the same 3:2. It is just where you choose to start with 1. If you put one at the start, you get all factors 3 in the enumerator. If you put 1 in the middle they are in the denominator in one half of the circle. −Woodstone (talk) 07:47, 12 August 2008 (UTC)
There is a difference. For instance, after re-modulating the interval to fit within the same octave according to the rules outlined in the article, the eighth fifth after unison has a ratio of 6561:4096 (815.64 cents, a minor sixth) when tuning upward. When tuning downward the ratio is 128:81 (792.18 cents). The difference between the two is the Pythagorean comma--the same as the difference between Eb and D# in the table within the article. The next fifth is 19683:16384 when tuning upward and 32:27 when tuning downward--also a difference equal to the Pythagorean comma. And so on. I haven't checked to see where and when wolf intervals occur. SharkD (talk) 09:11, 12 August 2008 (UTC)
The only difference is created by the choice where the "break" in the circle is located. The pythagorean comma is why the "circle" is actually not a circle but a spiral. To bend it into a circle, a piece the length of the comma needs to be cut. This does not influence what happens to tuning up or down. If the circle starts at 1 going up, the last one is 312/219=531441/524288=~1.014. If you start in the middle with 1, the highest is 36/29=729/512=~1.424 and the lowest is 210/36=1024/729=~1.405. The difference between the extremes is still the same (729/512)/(1024/729)=(729*729)/(512*1024)=531441/524288 and verified by 1.424/1.405=~1.014. −Woodstone (talk) 12:36, 12 August 2008 (UTC)
Still, the frequencies themselves are affected. For instance, the table and image in the article "break" at different points, leading to discrepancies. SharkD (talk) 16:25, 12 August 2008 (UTC)

Horribly confusing[edit]

Does this article need some serious looking at or am I missreading it?

Under the "Method" heading it states "The table below (starting at E flat) illustrates this". That has me confused. The table doesn't start on E flat.

It goes on to say "showing the note name, the ratio above D...." Why would it show the ratio above D if it started on E flat? The table shows the ratio above C, not D or E flat.

The article continues with "Pythagorean tuning uses the above 12 notes from E flat to D sharp shown above" thus compounding the confusion. —Preceding unsigned comment added by (talk) 01:57, 2 January 2009 (UTC)

You are right. It arose because people have changed the table without adapting the surrounding text. I may have a look to straighten it out. −Woodstone (talk) 12:29, 2 January 2009 (UTC)
Had a go at it. Should be in better shape now. Clearly a Pythagorean tuning is key-specific. In a piece in C, there will usually be harmonies including both flats an sharps. The center of D is chosen for the table to show equally many flats and sharps. I have not been able to find literature on where it is usual to place the cut in the circle. −Woodstone (talk) 14:35, 2 January 2009 (UTC)
That was a bit better but there were still a few notes wrong. You'd still got a few "Gs" in there that should have been "Ds". I've put it right (I think) —Preceding unsigned comment added by (talk) 15:49, 2 January 2009 (UTC)

This article makes no sense[edit]

I am very much interested in this article, but I simply cannot understand it. It does not seem to explain itself in plain English, using a number of technical terms that I do not understand. The article begins with three confusingly-named music files that all sound the same (what does “diatonic scale in 12 et” mean?).

The article then begins to exaplin that Pythagorean tuning is “based on a stack of perfect fifths, each tuned in the ratio 3:2”. Sorry, what does this mean? I didn’t come to this article because I’m a professor of music and already know these terms, I came here to try and learn.

The article goes on like this using unfamiliar terms with no explanation of the basics, assuming the reader knows absolutely everything about music apart from the article in question.

Please can somebody re-write this whole article to make it make some sense. I am actually a musician and I know a lot more about music than most people, yet even I find this article baffling.

Grand Dizzy (talk) 20:41, 21 February 2009 (UTC)


As someone who has never read anything about this sort of theory, I find the article easy to follow to a degree, and understand what is being said, and the 3 sound bites were a nice touch (even if I can't tell the difference), but can someone include somewhere why this tuning system is used? It says it has been used for thousands of years, but why? Why would you choose this method rather than the 'normal' tuning? Without this included I'm left thinking that it has no point to it. ArdClose (talk) 15:17, 7 April 2009 (UTC)

I added this sentence giving one of the motivations for this system:
"This interval is chosen because it is one of the most pleasing to the ear."
Note on the diversity of tuning systems. There have always been many tuning systems and always will be. Even setting aside different tastes in different times and places, harmony and melody make conflicting demands. Also, tuning of fixed pitch instruments require either impractical mechanical complexity, or a tuning system (called "even temperament", in which every chord except an octave is slightly out of tune) that singers and non-fixed pitch instruments could not tolerate. (talk) 22:14, 1 January 2011 (UTC)

Looping around[edit]

Below the tables it says, To get around this problem, Pythagorean tuning uses the above 12 notes from A flat to G sharp shown above [sic], and then places above the G sharp another A flat, starting the sequence again. Shouldn't this be "from A flat to C sharp"? I.e., instead of the last step in the table, return to the first line? —Preceding unsigned comment added by (talk) 21:30, 21 May 2009 (UTC)

A stack of "seven factors 2"? What's that?[edit]

In his recent edit to the "Method" section, Woodstone suggested to "avoid unnecessarily mentioning jargon like fifth, octave, which may be confusing to the unitiated", and substituted most occurrences of interval names such as "perfect fifth, and "octave", with expressions such as "3:2 ratio", "factor 2", "factor 3:2".

In my opinion, this terminology is not compatible with terms like "stack", "span", "move up", or "move down", which are extremely helpful when you describe tuning. These concepts refer to linear motion of the hand by a given distance or by a given number of keys along a keyboard or along a string. They are spatial ideas, you can perceive them. Human hears transform pitch multiplications into additions. The "distance" perceived between C and C' is the same as between C' and C", as suggested by the constant length of the octaves on a piano keyboard. On the contrary, you cannot perceive a ratio. You can "increase" a frequency "by a ratio", or by N times that ratio, but you cannot "add" multiplications. The concept of stack is additive. It is compatible with the concept of interval width in cents, or semitones, or scale steps, or staff positions. Not with the multiplicative concept of frequency ratio.

Thus, it is difficult to grasp the concept of spanning "a stack of 7 factors 2" (i.e. 7 octaves). Also "bring all the notes one span of a factor 2" appears to me more confusing to the layman than "bring all the notes in the basic octave (starting from the original D)". I believe that a reader should be forced, as soon as possible, to learn this basic terminology, which is widely used and based on very common visual an tactile perceptions strongly associated with the idea of note and music, such as the sequence of "ebony and ivory" on a piano keyboard, or the frets on the neck of a guitar, or the fingertips of a violinist moving up or down the strings of a violin. Can we really let a reader believe that a specific article describing a tuning system can be understood without knowing basic terminology associated with the concept of interval? That terminology is used in most articles about music on Wikipedia. It's like "real number" in mathematics. By the way, to explain this tuning system in the "Method" section is only necessary to use the words "fifth", and "octave". Is it really possible to understand the concept of musical interval without learning the names of these intervals?

Is it really of service to the reader to write the most specific section about this tuning system without ever mentioning the standard terms "fifth", "octave", "stack of fifths"?

By the way, the introduction does not refer to octaves or fiths and that's perfectly OK. I do accept the idea that Wikipedia should try to reduce unneeded complexity and facilitate access to laymen.

Paolo.dL (talk) 20:27, 2 July 2010 (UTC)

Using terms like a "perfect fifth" in the construction of a tuning misses the point. It makes the definition somewhat circular. It presupposes a particular scale or a keyboard. Pythagorean tuning does not. It starts out by dividing a string or air column in a well defined fixed length ratio and only uses the extremely stable interval of a factor two to normalise notes. At this point the "octave" (8 notes) has no meaning yet. It is then discovered that after 12 such created notes, the series bites almost in its tail, suggesting a 12-tone scale to be used. No white or black notes need to be defined. The notion "fifth" (5 white notes) needs much more sophisticated musical background to obtain meaning.
The scale is not based on "fifths" as can be seen from the different size fifths occurring), but on a ratio. If the word "stack" is a good choice is a different debate. See the picture of the logarithmic stack in just tuning for example. Stacking annual or monthly interest is not an unusual expression using a factor instead of an addition. Other suggestions are welcome.
We had a lot of complaints about the jargon in comparable music articles. After learning the terms like "fifth" or "octave", one tends to forget how illogical they really are. Five (or eight) what? The edit was an attempt to stay within the bounds of the method applied for tuning, without using terms that are part of a much more refined and sophisticated nomenclature, based on a specific culture.
Woodstone (talk) 01:11, 3 July 2010 (UTC)

I agree that the terms are illogical and based on an arbitrary convention. I sympathize with you fullheartedly. Moreover, the very first time I participated in a discussion (on Talk:Exterior algebra, section Exterior to what?), it was about illogical terminology due to a questionable translation from the German term "Ausdehnungslehre" ("Extension theory"; or, if you like, "Extended algebra", not "Exterior"!). My question was "Exterior to what?", quite similar to your very effective "Five (or eight) what?" :-).

Your intention was noble, but it led to horrible expressions such as "a stack of 7 factors 2" (i.e. 7 octaves), or "bring all the notes one span of a factor 2".

Your explanation about the ratio between different string lengths is also an excellent point, and you convinced me that this is the start point for the method section. I also suggest to refer to the length of the pipes of a panpipes or organ. However, after explaining that a 2/3 string length produces a sound which has a frequency 3/2 times higher, it is important to just explain that this interval was later called (arbitrarily, questionably, but consistently for centuries in Western music) a "fifth". And the interval 2:1 was called an octave. After that, you can (and must, unfortunately) use these interval "labels" to write a much clearer text. In other words, you need a term to refer to the concept "interval between two notes with 3:2 pitch ratio", and you cannot arbitrarily decide that the terminology you prefer ("3:2 factor") should be used, just because we (me included) hate standard terminology. That is what I was told on Talk:Exterior algebra.

Paolo.dL (talk) 10:07, 3 July 2010 (UTC)

Hi Paolo, please be aware that I personally have no problem understanding the standard musical terms. However, repeated complaints on this and other music theory pages state that the jargon made it difficult to gain entry into the field. Therefore I made a serious effort in reducing the amount of unnecessary jargon. If you can smooth the text, please feel welcome to do so. For string or pipe (I used the more technical air column), there is no need to explain about frequencies first. The point of Pythagorean tuning is that only factors 2 and 3 up and down are used. If they are interpreted as lengths, it just reverses what one would call up or down. The term fifth is largely irrelevant in the construction context and can be deferred to where a diatonic scale is defined. −Woodstone (talk) 13:29, 3 July 2010 (UTC)

Don't worry, I don't have doubts about your knowledge. You don't need to delete standard terminology (which you keep calling jargon) to make the text clear. You do need simple terms to label these two intervals, you can't keep caling them "3:2 factors". Even the notes have a name, they are not "note 1", "note 2". And it's not only impossible, but also not advisable to avoid using standard and frequently used terminology. The reader must be forced to learn it as soon as possible. Gradually, but necessarily. It would be nice to call these intervals musical "flights" or "ramps" or "twelfths" (12 "steps" = 2:1, rather than octaves), or "halves" or "sixths" (6 "steps", rather than fifths), "quarters" or "seconds" (= 2 "steps"), etc. or something like that, wouldn't it? But it's impossible. If a reader asks for non-standard terminology, he should use Simple Wikipedia. Your intention was good, your edit was too drastic.

I agree with the comment by User:Grand Dizzy above, who wrote: "The article then begins to explain that Pythagorean tuning is based on a stack of perfect fifths, each tuned in the ratio 3:2. Sorry, what does this mean?". I already agreed with you about the need to clean the text and to insert a few introductory satements in the method section. Yet, removing completely standard terminology (which is, by the way, linked to other pages) is not of service to the reader. For instance, on Simple Wikipedia, the page about real numbers starts with this sentence: "A real number is a rational or irrational number." Here, technical terminology - that you would probably call jargon - is used even before explaining its meaning! Can you see how important is to use appropriate terminology in an article? (Provided you give sufficient introductory explanation). --Paolo.dL (talk) 14:24, 3 July 2010 (UTC)

I would love to know the opinion of some other editor. In my opinion, although most of the advices given by Woodstone are precious and his concerns need to be addressed ASAP, his solution is too drastic, and his implementation in this specific case is a worse starting point than the previous version of the article (that was not written by me, by the way).

--Paolo.dL (talk) 14:58, 3 July 2010 (UTC)

It seems to me that fifth and octave are so ingrained in basic music theory that they would be understandable to most anyone who's learned just a little about music theory. Yes, when you are looking at anything other than a diatonic scale, the roots of their names are illogical. If you're using a tuning with a different number of notes than 12 per octave (or one which is built without regard to octaves), the standard note names are illogical as well. However, Pythagorean has 12 notes and is perfectly functional from a composition standpoint as diatonic. Further, to give an example of how meaningless the roots of words can be as they are used differently over time, just think of those months September, October, November, & December. They are clearly meant to number the months 7,8,9, & 10, but Julius and Augustus Caesar just squeezed themselves right in there after the months named after Gods and Goddesses. Illogical? Yes, but you're not likely to get people to change the last 4 months of the year to November, December, Undecember, and Duodecember now. — Preceding unsigned comment added by (talk) 15:14, 2 June 2012 (UTC)

Sign of the Pythagorean diminished second[edit]

We are discussing the sign of the Pythagorean diminished second in Talk:Pythagorean comma. If we apply the definition of diminished second given on Wikipedia, it should be negative (see also Pythagorean interval):

Name Short Ratio Cents ET
diminished second d2 524288/531441 -23.460 0
Pythagorean comma 531441/524288 23.460 0

Please share you opinion with us on Talk:Pythagorean comma. Do not answer here, because it would duplicate the discussion. This messagge will point the editors of this article to the other talk page. − Paolo.dL (talk) 22:10, 15 August 2010 (UTC)

We agreed that d2 should be negative in Pythagorean tuning. Paolo.dL (talk) 12:21, 17 August 2010 (UTC)

Sign of ε[edit]

In Talk:Pythagorean comma, we are discussing the sign of ε for Pythagorean tuning. ε is defined at the end of this section, as the positive difference between the size of the fifth and 700:

Definition of ε in Pythagorean tuning
By definition, in Pythagorean tuning 11 perfect fifths (P5 in the table) have a size of approximately 701.955 cents (700+ε cents, where ε ≈ 1.955 cents).

Please share you opinion with us on Talk:Pythagorean comma. Do not answer here, because it would duplicate the discussion. This messagge will point the editors of this article to the other talk page. − Paolo.dL (talk) 22:10, 15 August 2010 (UTC)

The Underlying Mathematics[edit]

The Pythagorean scale is made possible by the mathematical coincidence, discovered by the Pythagoreans, that is approximately equal to . In fact, their ratio is 1.01364..., within 1.37 %. That the third part of the octave is close to the pure major third is due to another, independent coincidence, that is approximately . In this case, 128/125 = 1.024, for an error of 2.4%.[1]
  1. ^ Manfred Schroeder. Fractals, Chaos, Power Laws, Minutes from an Infinite Paradise, W.H. Freeman, 1991. Reprinted by Dover, 2009.

I removed this text. It might be interesting, but there's no explanation about the use, in Pythagorean tuning, of the mentioned numbers:

  • 3^{12}
  • 2^{19}
  • 2^{7}
  • 5^{3}

Also, this note does not deserve to be the first section of the article. Paolo.dL (talk) 20:08, 27 February 2012 (UTC)

Pythagoras and unnecessary, and uncorrectly formatted quotations[edit]

The dedication to Pythagoras is very popular in modern handbooks (and probably since the "Latin middle ages"), but I changed the beginning, because there is no evidence (a general problem since Walter Burkert's Pythagoras study by the 1960s), that it was used to an earlier time than by Eratosthenes, if we would like believe Ptolemy and Boethius who copied Ptolemy's divisions and his ascriptions in his music treatise. Please check the quotation before note 1 and 2, because readers cannot be sure, who was really quoted here. I "ascribed" the quotation now to Leo Gunther, but quite frankly, what is said here, is so common, that it should be better said by someone's own words (what should a reader imagine to be a "major chord"?). Hence, I bet it was taken from the handbook of Benward and Saker. Platonykiss (talk) 14:04, 3 September 2013 (UTC)