# Talk:Just intonation/Archive 1

## Update external link to JIN

effective immediately, the URL for the Just Intonation Network has moved

Old URL: www.dnai.com/~jinetwk

New URL: www.justintonation.net

thank you.

--DBD (David B. Doty--Just Intonation Network)

Done - incidentally David, if you're still around here - you could have edited the page yourself. Wikipedia can be edited by anyone. See Wikipedia:Welcome, newcomers if you're interested. --Camembert (24 November 2004)

On the contrary - this link has nothing to do with Just Intonation and appears to be spam for personal injury lawyers. Removing. — Preceding unsigned comment added by 70.196.192.193 (talk) 14:59, 2 August 2012 (UTC)

## Continued Fractions and the Just Intonation Major Scale

I'm putting this section up front because it can be used to illuminate most discussions about harmonic ratios. It also clearly shows the short comings of the original article in extremely objective terms. I suggest that a re-write may be in order.

What are continued fractions and why are they illuminating? Continued Fractions is a well-known mathematical method that can find the simplest and most accurate possible fractional approximation to any arbitrary ratio. In some sense the continued fraction approximations are the best fractional approximations to any arbitrary ratio. Any fractional approximation to a ratio which is NOT a continued fraction approximation will either have larger integers in the numerator and/or denominator of the approximating fraction, or the difference of the fractional approximation from the approximated value will be greater (than a continued fraction approximation).

In other words, continued fractions can expose how harmonic an arbitrary ratio is by finding its best fractional approximations and examining how simple and accurate they are. For example, the first continued fraction approximation of a tempered perfect fifth that is closer than a semi-tone is 3/2 and differs from the tempered perfect fifth by only 1.95 cents.

Rather than give you the whole continued fraction thing I will give a simple method to use it.
0) First key the ratio to be approximated into a hand calculator, preferably one with a reciprocal function.
1) If the number is greater than one, subtract out the integer part and write the integer part down.
2) Take the reciprocal of residue that results from the subtraction.
3) repeat 1) & 2) a few times using the last result of 2) as the starting number for the next 1). This will give you a series of integers you have written down and perhaps a residue on the calculator.
If you exactly reverse the just described steps by adding the last integer to the current residue and taking the reciprocal, and backup through the list of integers in a similar fashion, you can reconstruction the original ratio you started with. Even more interesting, if you approximate reversing the original steps by zeroing the residue and starting with any intermediate integer and working backwards as just described, you will get a APPROXIMATION of the original arbitrary ratio. On the calculator you will get the answer in the form of a decimal fraction, but by doing integer calculation with paper and pencil you can get the answer as a "best" fractional approximation of the original ratio (there are a series of such approximations and which fraction in the series you get depends on which integer you decided to start with).
The more integers that you take, the more complex the fractional approximation and the closer it will be to the original number. If you use too many integers you will use up the precision of your calculator and the integers will be random and meaningless. If the original ratio was an exact fraction you will get just so many integers before all that is left as a residue is an integer with zero or almost zero value to the right of the decimal. (Often the result will not be exactly zero because of rounding errors due to the finite limit of precision of the calculator). Of course, the larger the numerator and denominator, the less harmonic significance a result has because it is more arbitrary than harmonic. Remember that the simpler the resulting fraction is, the more harmonic it "sounds".
In the fractional form it can be fun because you could start with a messy decimal fraction like 7/17 (e.g. 0.4117647...) which would give you the integers 2, 2, 3. Reconstructing would be 2 + 1/3, and then 2 + 3/7, and then 7/17. Or we could drop the last 3 and reconstruct just using 2, 2. This would give us 2 + 1/2 or 2/5. Notice that 2/5 is a good approximation to 7/17 with only an error of about 0.012...

Note that this process will work on any arbitrary ratio such as an irrational number that has no exact fractional representation because it would take an infinitely large integer numerator and denominator to "exactly" represent the irrational ratio.

Again, the fractional approximations you build in this way are in some sense the simplest and most accurate fractional approximations conceivable.
NOW COMES THE INTERESTING PART. If you take each note in a ET major scale, and find the first fraction of the corresponding continued fraction series (that is what you were finding with the hand calculator), that is closer than 1/2 the ratio between 2 adjacent half steps, THAT FRACTION WILL BE EXACTLY THE CORRESPONDING NOTE OF THE JI SCALE!!!!
What does this tell us? It tells us that several hundred years ago, musicians found the PUREST note (e.g. in some sense, the simplest harmonic ratio) corresponding to each note of the scale and put it together in what we now call the Just Intonation major scale. Now what is really going on some of the following comments make sense. The human ear could care less whether a not is one series or another, it only knows and recognizes harmonic relationships according to how simple the fractional ratio of a pair of tones is.
This shows that JI in the context of a continuing fraction analysis, is demonstratively the most harmonic scale possible based on 12 more or less equal intervals within an octave. This suggests that human ear has the capability of recognizing these harmonic relationships, if they are simple or "pure" enough. That makes these observations interesting. --HonestGent (talk) 03:25, 7 November 2009 (UTC)

## JI/ horn deletion

Hello, you noted that natural horns play far from just intonation, but the lead from the article says

In music, Just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by whole number ratios; that is, by positive rational numbers. Any interval tuned in this way is called a just interval; in other words, the two notes are members of the same harmonic series.

If natural horn players always modify pitches other than the key note, the deletion makes sense (my small exposure to them suggests they use the natural notes) but if it is because the 7th and 11th harmonics don't fit in the diatonic pattern it doesn't because these are rational intervals and members of the same harmonic series, and the same notes played from the trumpet marine. --Mireut 23:37, 4 February 2006 (UTC)

Actually, I think the whole section (as it was) is rather silly (especially given that there are only two instruments, one of which is rather bizarre). There are thousands of instruments capable of just intonation not mentioned here, many quite conventional. The question is more whether or not they are ever asked to.
However, on the Natural Horn specifically, I think I'd have to argue with you. It produces a harmonic series quite easily, and quite naturally. To produce other notes, yes, there is a stopping technique that is used to flatten pitches. The seventh harmonic is actually used a fair bit, though you're right that the 11th is hardly ever used (Benjamin Britten's Serenade is a fun exception). However, in its standard technique, the harmonic tones which are used (1,2,3,4,5,6,(7),8,9,10,12,(14),15,16) are indeed just.
You might argue that if this is true, the natural horn can only be played just in one key then. This is also true. A quick study of natural-horn

## Just tuning

I was going to merge the content below from Just tuning, but which "one possible scheme of implementing just intonation frequencies" does the table show? Hyacinth 10:29, 1 Apr 2005 (UTC)
It shows the normally used just intonation scale - I don't think that it has a special name. It can be constructed by 3 triads of 4:5:6 ratio that link to each other, e.g. F-A-C, C-E-G, G-B-D will make the scale of C. Yes, this should definitely be included. (3 April 2005)
Please Wikipedia:Sign your posts on talk pages. Thanks. Hyacinth 22:03, 3 Apr 2005 (UTC)

I am no expert, and I've not done any Wikipedia changes either, so forgive me if I'm wrong in what I'm doing (content) or how I'm doing it (method), but the main text gives 6/5 as a minor third, and I currently disagree.

Scholes' Oxford Companion to Music, eighth edition, in the section on intervals, says a minor third is a semitone below a major third, ie 15/16 * 5/4 = 75/64 and not 6/5 as stated in the main text. The Oxford Companion to Music also states that by going up a semitone an interval becomes an augmented interval and so a major tone (a second) would become an augmented second as follows: 9/8 * 16/15 = 6/5. Thus 6/5 is an augmented second, and 75/64 is a minor third. Ivan Urwin

There are many semitones. A minor third is a chromatic semitone (25/24) smaller than a major third.
I can also give you a reductio ad absurdum for your reasoning. If 6/5 is an augmented second, then 5/4 * 6/5 = 3/2 is not a perfect fifth, but a doubly augmented fourth. Then if 4/3 is a perfect fourth, 4/3 * 3/2 = 2/1 is not an octave, but an augmented seventh. —Keenan Pepper 00:32, 14 April 2006 (UTC)
Why not split the difference, and call a semitone the twelfth root of 2, or 1.059463...? right between 16/15 at 1.066667 and 25/24 at 1.041667. Even better, one could call a minor third 300 cents, or the fourth root of 2, again smack between those silly over-simplified integer ratios. Of course I'm kidding; thanks, KP! Ivan, I'm not familiar with your Scholes reference; how completely does it treat the differences between just tunings and various temperaments? I'm guessing that's where the oversimplification may lie. Just plain Bill 01:49, 14 April 2006 (UTC)
Okay, I think I missed a key word in the Scholes Oxford Guide to Music text, approximately like this ...
If an inverval be chromatically increased a semitone, it becomes augmented.
I am looking at this as a mathematician and so the musical terminology throws me somewhat: dividing by 5 being called thirds and dividing by 3 being called fifths, etc. If I were to rewrite the Scholes text as shown below and use 25/24 as a definitiion for a 'chromatic semitone' as per Keenan Pepper's remarks, then I'd agree.
If an inverval be increased by a chromatic semitone, it becomes augmented.
The way this arose was me looking at the ratios with my mathematical background. Prime factorisation of integers is unique. The only primes less than 10 are 2,3,5, and 7. I gather that 7 is used for the 'blue' note in blues, and that most western music just uses or approximates ratios based on 2, 3 and 5. With 2 being used to determine octaves and with notes an octave apart being named similarly, that brings practical ratios down to just determining the power of 3 and the power of 5. I was making a 2 dimensional table of the intervals, and putting names to the numbers with the help of a borrowed book, but it appears the complex terminology for simple mathematics got the better of me.
I am happy to drop my remarks and delete all this in a few days time (including Keenan's and Bill's remarks), but I'll give you chance to read it and object before I do. Maybe some moderator will do that. Maybe you two are the moderators. Whatever. Anyway, thanks guys.
Ivan Urwin
You're mostly right about the primes. See Limit (music).
The labeling of intervals as "seconds", "thirds", etc. corresponds to the number of steps they map to in 7-per-octave equal temperament. 3/2 maps to four steps, so it's a "fifth", 5/4 maps to two steps, so it's a "third", 16/15 maps to one step, so it's a "second", and 25/24 maps to zero steps, so it's a kind of "unison" or "prime". Intervals separated by 25/24 have the same name, for example the 6/5 "minor third" and the 5/4 "major third".
Conversations on talk pages are usually never deleted, only archived when they become too long, so don't worry about that. —Keenan Pepper 05:04, 15 April 2006 (UTC)
I'm lost here (not a moderator, by the way, just another netizen) when you speak of "determining the power of 3 and the power of 5." The interval of a fifth is just the musical pitch space between the first and fifth notes of a scale. Because I happen to be used to vibrating strings, a perfect fifth being a 3/2 frequency ratio now seems as obvious to me as the fact that x^2+y^2=1 makes a unit circle, just a matter of familiarity. I'm equally happy to talk about it or to send it to oblivion as you suggest.
We are lucky to have folks around like Keenan Pepper who can quickly point out the discrepancy in types of semitone, for example. Just plain Bill 03:56, 15 April 2006 (UTC)
This is the sort of table I had. You can see that going up a major tone consists of going right couple of cells, so looking at the entry 10/9 (minor tone), I could quickly see that a mojor tone higher than that would be a third, and similarly a major tone higher than a semitone would be a minor third.

 Power of 3 (fifths) -3 -2 -1 0 1 2 3 Power of 5 (thirds) -3 128/125 -2 256/225 128/75 32/25 48/25 -1 64/45 16/15 (semitone) 8/5 6/5 (minor third) 9/5 0 32/27 16/9 4/3 (fourth) 1/1 3/2 (fifth) 9/8 (major tone) 27/16 1 40/27 10/9 (minor tone) 5/3 (sixth) 5/4 (third) 15/8 (seventh) 45/32 135/128 2 25/18 25/24 25/16 75/64 225/128 3 125/64

The powers of two in the table just bring the ratios to within an octave. Clearly one could add more ratios to the table. I have just included it for illustration.

Ivan Urwin

That helps me make more sense of it. Thanks, and also to Keenan for pointing out the tonality diamond of Harry Partch. Until I sit with this some more, I have nothing really useful to add... cheers, Just plain Bill 14:46, 16 April 2006 (UTC)

#### content for merge:

Just intonation is any musical tuning in which the frequencies of notes are related by whole number ratios. This table shows one possible scheme of implementing just intonation frequencies.

Just tuning frequencies of all notes in each key based on A = 440 Hz when in the key of C. The just intonation scale ratios of 24:27:30:32:36:40:45 are used and each key note has the same frequency in the scales with +/- 1 sharp or flat.

Note that the 6th note in a key changes frequency by a ratio of 81/80 when it becomes the 2nd of the key with one more sharp or one less flat. All other notes retain the same frequency. In C all frequencies are an exact number of Hertz.

In just intonation incidentals tuning must be worked out on a case by case basis. Often the minor third and minor seventh take the ratios 28 and 42 when the tonic is taken as 24, so that in C the tuning for Eb and Bb would be 308 Hz and 462 Hz. These frequencies allow dominant seventh chords with frequency ratios of 4:5:6:7.

For frequencies in other octaves repeatedly double or halve the tabulated figures.

There is a difference between Gb and F# which amounts to a ratio of ${\displaystyle {3^{12}}/{2^{19}}}$ = 1.0136433 as discovered by Pythagoras.

 Key \ Note C Db D Eb E F Gb G Ab A Bb B Gb (6b) 278.123 309.026 347.654 370.831 417.185 463.539 494.442 Db (5b) 260.741 278.123 312.889 347.654 370.831 417.185 463.539 Ab (4b) 260.741 278.123 312.889 347.654 391.111 417.185 469.333 Eb (3b) 260.741 293.333 312.889 352 391.111 417.185 469.333 Bb (2b) 264 293.333 312.889 352 391.111 440 469.333 F (1b) 264 293.333 330 352 396 440 469.333 C (0) 264 297 330 352 396 440 495 G (1#) 264 297 330 371.25 396 445.5 495 D (2#) 278.438 297 334.125 371.25 396 445.5 495 A (3#) 278.438 297 334.125 371.25 417.656 445.5 501.188 E (4#) 278.438 313.242 334.125 375.891 417.656 445.5 501.188 B (5#) 281.918 313.242 334.125 375.891 417.656 469.863 501.188 F# (6#) 281.918 313.242 352.397 375.891 422.877 469.863 501.188 Key / Note C C# D D# E F F# G G# A A# B Equitempered 261.626 277.183 293.665 311.127 329.628 349.228 369.994 391.995 415.305 440.000 466.164 493.883

## Meantone

Shouldn't the section on meantone be deleted? Meantone is a temperament. How does it belong here any more than a section on 12-TET? Caviare (talk) 04:14, 3 August 2015 (UTC)

I can see a certain logic from the context in which it is presented, since meantone compromises the perfect fifths in order to obtain just major thirds. The juxtaposition with Pythagorean tuning is instructive concerning the conflict created by combining just thirds and just fifths in any tuning system, though this aspect really needs to be explained in order to justify (no pun intended) the inclusion of meantone at all.—Jerome Kohl (talk) 17:50, 3 August 2015 (UTC)

## Indian Scales

I will freely admit I am not an expert on Indian Classical music, but I study and play sitar and the Pythagorean Tuning is much closer to how my instrument is tuned than the Just Diatonic mentioned in the first sentence of the Indian Scales section. My instrument 'sounds correct' when Ga is ~404.3 cents versus 386.3 cents defined in the Just Diatonic tuning. A difference of 18 cents, while 404.3 is -3.5 cents relative to the Pythagorean tuning. Ni is also substantially 'off', 1104.6 cents on my sitar versus 1088.2 for Just Diatonic, a difference of 16.45 cents. And my Ni s 5 cents lower than the 1109.7 cents defined in Pythagorean. I am measuring the frequency of my sitar using the Peterson iStrobeSoft tuner. I tune it to have 0 beats relative to Sa for each note, and my sitar is in tune with my teacher's sitar, a master of the instrument. When I moved the Ga and Ni frets to match the Just Diatonic tuning, it sounds pretty awful so I don't think it is correct to say that the Indian scale is Just Diatonic.Chuckpwhite (talk) 04:01, 2 February 2016 (UTC)

The most commonly acknowledged system of shruti placement, keeping in mind that shrutis are not fixed pitches strictly speaking, posits the duality of ten possible pitch classes relative to a fixed tonic / Sa, and fifth / Pa. These may be considered as the Pythagorean twelve tone system together with the 5-limit alternatives of the ten degrees that are not tonic nor fifth. The 5-limit alternatives are a syntonic comma removed from their pythagorean relatives. Thus the twenty two shrutis may be represented by their relation to a theoretical twelve tone scale defined thusly; tonic / Sa / ratio 1:1 ; a 'semitone' / re / pythagorean ratio 256:243, and/or 5-limit ratio 16:15 ; 'whole tone' / Re / pythagorean ratio 9:8, and/or 5-limit ratio 10:9 ; 'semiditone' / Ga / pythagorean ratio 32:27, and/or 5-limit ratio 6:5 ; etcetera. This structure gives dual values for the intervals of re, Re, ga, Ga, Ma, ma, dha, Dha, ni, and Ni. These are useful to various contexts of interval movement in order to 'correct' intervals in terms of their origination and destination, and also in terms of placement and movement regarding the various deflections, undulations, glissandos and other grace ornamentation / gamak used in Indian classical music.

To add to these complications, one must also consider that variations of these intervals, such as the use of the septimal intervals in place of the Pythagorean, may be utilised in practice. This seems to be in keeping with the complex systems of 55 commas to the octave used by baroque western classical theory, including the various dieses amd commas, as well as the system of 66 shrutis to the octave. This last may indicate, not only an incorporatiom of higher n-limit interval ratios but also the possibility of superimposition of twenty two shruti systems. The mathematical legacies of Indian classical music are interesting to consider in conjunction with those of the Greek classical systems based upon tetrachord usage and mathematical string division. — Preceding unsigned comment added by Daniel Z. Franks (talkcontribs) 08:22, 29 February 2016 (UTC)

## Staff notations

The current 'Staff Notations' section concentrates on the extended Helmholtz-Ellis notation, however in my opinion this section is incomplete since it does not mention several other important systems.

I've been researching notation systems for free Just Intonation over the last couple of years. I have developed my own system (Rational Comma Notation, or RCN), and I am also aware of at least two other systems which should be included: Sagittal (which can notate both just and tempered tunings) and Kite's color notation.

One reason to include these three, instead of just extended-HE notation, is because extended HE cannot notate the whole of free Just Intonation, but only up to a low prime limit, at the moment the prime 61. However, all of RCN, Sagittal and Kite's color notations can notate the whole of free-JI. This is due to defining 'prime commas' by algorithm by every higher prime number. Conversely, in extended-HE, there is no algorithm, no higher prime commas, and no general system for notation higher prime alterations.

Here are primary sources for each of the three notation systems: RCN: [1] Sagittal: [2] Kite's color notation: [3]

Maybe someone else can provide secondary sources?

Thanks! Davidryan168 (talk) 13:34, 2 February 2017 (UTC)

I've researched into secondary sources for Sagittal, which is a system I've had a long term interest in as a music software developer as a potential way to show musical intervals to users. I think the citations I found make it sufficiently notable to justify a separate article on it, with several academic citations, and it's also included in the SMuF music font originally developed by Steinberg and now overseen by the W3D Music Notation Group. I propose making it as a separate article Sagittal (Music Notation System). The draft is in my user space here: User:Robertinventor/Sagittal. Corrections and citations welcome. I haven't been able to find much by way of secondary sources yet for the other two systems you mention, so they will depend on finding citations to support them. Robert Walker (talk) 14:30, 9 February 2017 (UTC)

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## Mistake in audio example description?

This doesn't make sense:

"In the equal temperament chords a roughness or beating can be heard at about 4 Hz and about 0.8 Hz"

Humans can't hear that range! It should be kHz, right? — Preceding unsigned comment added by 71.230.123.65 (talk) 18:53, 16 January 2018 (UTC)

Hz is correct. Beat (acoustics) may be heard as pulsing, or volume variation, at those rates. Just plain Bill (talk) 20:03, 16 January 2018 (UTC)