# Talk:Just intonation

## Key of examples

Not that there's anything wrong with it, but is there any reason for the examples being changed from C major to F major? Just curious. --Camembert (22 August 2003)

## Outline

My proposed outline:

1. introduction: Just intonation is any musical tuning in which the frequencies of notes are related by whole number ratios. Another way of considering just intonation is as being based on lower members of the harmonic series. Any interval tuned in this way is called a just interval. Intervals used are then capable of greater consonance and greater dissonance, however ratios of extrodinarily large numbers, such as 1024:927, are rarely purposefully included just tunings.
2. Why JI, Why ET
1. JI is good
1. "A fifth isn't a fifth unless its just"-Lou Harrison
2. Why isn't just intonation used much?
1. Circle of fifths: Loking at the Circle of fifths, it appears that if one where to stack enough perfect fifths, one would eventually (after twelve fifths) reach an octave of the original pitch, and this is true of equal tempered fifths. However, no matter how just perfect fifths are stacked, one never repeats a pitch, and modulation through the circle of fifths is impossible. The distance between the seventh octave and the twelfth fifth is called a pythagorean comma.
2. Wolf tone: When one composes music, of course, one rarely uses an infinite set of pitches, in what Lou Harrison calls the Free Style or extended just intonation. Rather one selects a finite set of pitches or a scale with a finite number, such as the diatonic scale below. Even if one creates a just "chromatic" scale with all the usual twelve tones, one is not able to modulate because of wolf intervals. The diatonic scale below allows a minor tone to occur next to a semitone which produces the awkward ratio 32/27 for Bb/G.
3. Just tunings
1. Limit: Composers often impose a limit on how complex the ratios used are: for example, a composer may write in "7-limit JI", meaning that no prime number larger than 7 features in the ratios they use. Under this scheme, the ratio 10:7, for example, would be permitted, but 11:7 would not be, as all non-prime numbers are octaves of, or mathematically and tonally related to, lower primes (example: 12 is an octave of 6, while 9 is a multiple of 3).
2. Diatonic Scale: It is possible to tune the familiar diatonic scale or chromatic scale in just intonation but many other justly tuned scales have also been used.
4. JI Composers: include Glenn Branca, Arnold Dreyblatt, Kyle Gann, Lou Harrison, Ben Johnston, Harry Partch, Terry Riley, LaMonte Young, James Tenney, Pauline Oliveros, Stuart Dempster, and Elodie Lauten.
5. conclusion

Hyacinth (30 January 2004)

## 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

## 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