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
   2. 1. "A fifth isn't a fifth unless its just"-Lou Harrison
   3. 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

http://www.musicmavericks.org/features/essay_justintonation.html

Hyacinth (30 January 2004)

Scope? Accuracy?

There seems to be a lot here that isn't to do with Just Intonation itself. Why is Phythagorian tuning here? Why is there 20th Century stuff? This might be helpful *context* for Just Intonation, but this isn't explained and is frankly confusing.

I don't see any mention of Mersenne, Zarlino and so on who discussed these things quite a long time ago and should presumably be definitive sources...

A look in Groves' (at Just Intonation, and Temperament) might help quite a lot here. Sorry I don't have time to take this on myself. — Preceding unsigned comment added by 94.1.250.30 (talk) 14:39, 7 March 2020 (UTC) Why is Pythagorean intonation in here? It's arguably the very basis for the rest and *is* just intonation. The theorists mentioned there are standing on it, and came long after. I would say feel free to edit and add that, there are numerous authors of this, but I'm not sure you're the person for that job.J Civil 17:49, 9 November 2021 (UTC)

Hello. I've just made some improvements and one or two corrections here. Harry Partch was a modern champion of just tuning, worth a link and mention I would have thought. I'll be back. Thelisteninghand (talk) 14:39, 4 June 2022 (UTC)

Something I feel like is worth noting is that, although Pythagorean tuning is technically equivalent to 3-limit JI, it has different goals behind it, at least in almost every presentation of it I've ever encountered. (I have multiple academic sources on hand that actually contrast what they call "Pythagorean tuning" and "just intonation," not treating one as part of the other at all—one even treats them as kind of opposite.) My understanding is that the point of Pythagorean tuning is so that you have maximally in-tune fifths (wolf aside), despite what it does to the major thirds. This comes from a historical perspective that puts the fifth significantly higher than the major third in consonance, which constrasts with modern sensibilities. JI, as I understand it, is more about having your tuning closely approximate the harmonic series "overall," at least for the lower harmonics (with whatever value of "lower" the composer prefers). Those who are partisan to JI often argue for its superiority based on its stronger consonances compared to equal temperaments, and it would be kind of strange to prefer Pythagorean tuning on that basis, since its thirds are more out-of-tune than in 12-TET. Because of this, it doesn't surprise me that I haven't usually seen Pythagorean tuning referred to as a kind of JI outside of this article and the Xenharmonic wiki. It's not wrong to do so per se, at least for a certain formulation of JI, but it seems strange to me to simply say that and not give more historical context beyond noting that Pythagorean tuning is very old; it kind of gives the impression that JI is "timeless," which I feel is pouring old wine into new bottles a bit. (The Online Etymology Dictionary gives the earliest usage of "just" in a musical sense as 1850, which is also around the time that the modern interest in and controversy over JI began.) I'd really like it if the "History" section here made all of this more clear, and in some ways the whole presentation of JI's development and goals throughout the article. Mesocarp (talk) 18:37, 26 November 2022 (UTC)

Just to be sure, reading over this now I worry it comes off as a bit anti-JI. I'm not at all (I love Harry Partch's music for example) and don't see any reason why anyone should even stick to a single tuning system or anything; I just wish there was more context given for the ideas, which I think would benefit everyone. Mesocarp (talk) 20:04, 26 November 2022 (UTC)

Problems of this article

There are many problems in this article, as appears from the warnings on top. Some of these problems already are obvious in the lead, which says:

In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals (and chords created by combining them) consist of tones from a single harmonic series of an implied fundamental. For example, in the diagram, if the notes G3 and C4 (labelled 3 and 4) are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just fourth

• Not all intervals can be said "just" or "pure", I think that the expression only concerns the octave, pure fifths and fourths, and pure major and minor thirds. (This should be checked in Sauveur's writings where the expression "pure" originates, but I have been unable to do so just now.)
• It is not true that "just intonation [...] is the tuning of musical intervals as whole numbers ratios" – the Pythagorean comma, for instance, 531441⁄524288, also is an interval formed of a ratio of whole numbers.
• Also, it is not true that "just intervals consist of tones from a single harmonic series of an implied fundamental": the just minor third in the perfect minor chord cannot be found in the harmonic series of the fundamental of that chord.

There is a somewhat better article on Intonation juste in the French Wikipedia, which could inspire the revision of the article in English. — Hucbald.SaintAmand (talk) 15:26, 11 January 2023 (UTC)

Sauveur, in Methode Générale Pour former les Systêmes temperés de Musique (available here), describes the "just diatonic system" as formed of degrees corresponding to the numbers 24 27 30 32 36 40 45 48 for C D E F G A B C. He adds that in this system all octaves are equal [48:24 = 2], as are the minor seconds [48:45 = 32:30 = 16:15]; the major seconds are inequal [being either 27:24 = 9:8 or 30:27 = 10:9]. He then says that some minor thirds are "just", but not all, and that the major thirds are all the same, and continues saying that most fourths and fifths are "just", but not all. As one sees, which intervals are called "just" is not entirely clear. This, so far as I can tell, is the first usage of the term "just" to describe intervals. — Hucbald.SaintAmand (talk) 15:51, 11 January 2023 (UTC)

Taking your comments in order:

1. Musicians (and music theorists) do sometimes refer to these other intervals as just or pure, even if Sauveur doesn't. It seems pedantic to insist that all vocabulary derive from a single source in 1707. All music terminology is applied inconsistently.

2. How about we just say "ratios of small whole numbers"? That would eliminate the Pythagorean comma problem, although it would leave it unexplained why some small ratios, such as 7:5 or 13:11, aren't considered part of this meaning. I'm comfortable with the vagueness of "small," since there are different opinions about some intervals. The French article breaks it down to primes, so that what I would consider a just-tuned second, 9:8, is broken into its prime factors, but that makes the explanation more complicated.

3. The minor third E-G can be derived from the fundamental C, even if you're using it in an E minor triad. The sentence you cite doesn't say the implied fundamental is the root of the chord. Even the Pythagorean comma could theoretically be derived from a single, impossibly-low fundamental frequency.

I agree, in general, though, that there's room for improvement, and the French article seems like it might be helpful. —Wahoofive (talk) 18:21, 11 January 2023 (UTC)

The French article says "Until the 20th century, just intonation aimed particularly at just consonances, fifths and fourths, thirds and sixths". No source is given, but this appears to make sense. It continues discussing Euler's extension to harmonic 7, which suggests that just intonation was limited before to numbers 1 to 6 (the Renaissance senario) until the development of the 7-limit diatonic scale.

Note that saying this does not imply that these numbers correspond to sounds being harmonic partials of a single fundamental. There is no reason to claim that "the minor third E–G can be derived from the fundamental C", because nobody would do that. It makes more sense to say that it can be derived from numbers 5 and 6. Describing the 5-limit C major scale as derived from the fundamental F, or the 5-limit A minor scale as derived from the fundamental B♭, as done here really makes little sense. Removing these mentions already would improve the article. — Hucbald.SaintAmand (talk) 09:32, 13 January 2023 (UTC)

I don't understand how the ratios are derived. You say "For example, in the diagram, if the notes G3 and C4 (labelled 3 and 4) are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just fourth." From this, I would assume I could also say that "the interval ratio between G3 and C4 is 3:4, a just fourth." Is this true? This example is confusing. How about saying what all the ratios are, starting at the low C and working your way up? It then goes on to discuss "the numbers 2 and 3 and their powers, such as 3:2, a perfect fifth, and 9:4, a major ninth." - I don't see from the first example how these ratios are derived. Could you please explain this in a more clear and thorough way? For example, what does the first number in the ratio refer to and what does the second number refer to? Some clarification of how these numbers are derived would be helpful. — Preceding unsigned comment added by 24.103.122.30 (talk) 12:52, 25 April 2023 (UTC)

There is indeed something confusing in this description of the interval ratios, particularly in the mention of the harmonic series. It should be corrected, but let's leave that for later. The intervals considered in Just Intonation are consonances, which were shown by the Pythagoreans in Greece about 27 centuries ago, on the consideration of string lengths, to correspond to simple ratios of whole numbers: 2:1 for the octave, 3:2 for the pure fifth, 4:3 for the pure fourth. Modern explanations often are based on concordances between the partials of the sounds considered (see Consonance and dissonance), which may explain why what you quote refers to harmonic partials. But once again, let's leave that for now.

Whether you describe the intervals as 2:1, 3:2 and 4:3, or 1:2, 2:3 and 3:4 depends (a) on whether you consider the intervals in ascending or in descending order and (b) whether you consider them to concern string lengths or frequencies. A pure fourth, for instance, is pure in either direction. The ratios are merely that, ratios. The numbers themselves describe a proportionality. 4:3, for instance, may refer to string lengths of 40 and 30 cm, or of 60 and 45, or whatever; or to frequencies of 400 and 300 Hz, or of 500 and 375 Hz, etc., as you want. The ratio is 4:3 in all cases. If 4 and 3 correspond to string lenths, then 4 denotes the lower sound (the longer string); if they correspond to frequencies, the reverse. Is that clearer? I hope so, but tell me if something remains unclear. — Hucbald.SaintAmand (talk) 19:38, 25 April 2023 (UTC)