Talk:Just intonation

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)

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)

As the tag at the top of the article states, there are multiple issues with this article. There are vast stretches like the "Terminology" section that have no sources. I removed a similar one called "Singing and scale-free instruments" which mused on ways to perform in just intonation.

The opening sentence used to jump right into frequencies and ratios. It now more clearly describes the natural phenomenon of just intonation as described in the Harvard Dictionary of Music. The lead is shorter and plainer now. There is a Definition section now. Most importantly, statements are being matched to citations.

Is anyone able to pitch in?Trumpetrep (talk) 18:29, 1 March 2026 (UTC)

There are several sentences in the article that I don't understand, but I am not sure enough of my English to correct them myself. Here follow (to begin with) those in the "Definition" section:

• "The frequency of each note in a pure interval will correspond to the whole number ratios in the harmonic series." How could a frequency correspond to a ratio – or, even more, to ratios (plural!)?
• "In the harmonic series on C, the 1st and 2nd notes form an octave in a 2:1 ratio. The fifth between the G and C is in a 3:2 ratio. The fourth is a 4:3 ratio." The octave is "just" (or "pure") in any intonation, and fourths merely are inversions of fifths. I thought that Just intonation was about combining just fifths (or fourths) with just major thirds, so that the ratios that should be mentioned are 3:2 and 5:4.
• "Just intonation also describes a tuning system that contains five or more pure intervals in an octave." This is a general definition (from the New Harvard Dictionary), but it would be better to name these intervals. One might think of the minor and major thirds, fourth and fifth, minor and major sixths, of which several are mere inversions of the main ones, major third and fifths. Would it not be better to say that Just intonation is a combination of just fifths and major thirds, and of their inversions and combinations?
• "Examples include Ptolemy's intense diatonic scale and 5-limit tuning, in addition to systems that use whole number intervals derived from the harmonic series." Doesn't this contradict the first sentence in this section, when it mentions systems "in addition" to those using whole number intervals (and what are "whole number intervals," why not "intervals corresponing to whole number ratios," as above?).

— Hucbald.SaintAmand (talk) 19:03, 1 March 2026 (UTC)

Thanks for the response. The language from the first 3 bullets reflect the cited Harvard and Grove entries. The 4th bullet is the vestigial stuff that we need to clean up.

I didn't realize language was a barrier. I'll see if I can find examples that are clearer. Trumpetrep (talk) 19:21, 1 March 2026 (UTC)

Just because that Harvard definition is easy to understand doesn't make it a good or useful definition. When people say "just intonation," they are almost certainly not referring to some abstract "five intervals within an octave"; what does that even mean? Which intervals? Why five? Also, "beatless" is imprecise; an interval doesn't simply have beating or not. Like, the 16:16 just minor second will certainly have beating audible, especially if played in a lower register.

In an attempt to give some information I know to help out: In the most general sense, just intonation is an approach/philosophy of tuning. It may be used in the sense of "small whole number ratios", or in a more general sense of "whole number ratios" in general. The Xenharmonic community has adopted the latter definition, and deemed the former as low-complexity just intonation (LCJI). This definition also seems more useful because accommodates terminology like the aforementioned "just minor second", which gets its name because it is found between more consonant intervals in a just intonation tuning system. Epixix (talk) 19:20, 1 March 2026 (UTC)

Thanks for your response, User:Epixix. As you can see from this talk page as well as the tags on the article, this page has a lot of issues. First, it needs to be much clearer for the general reader, which is our audience. Second, it needs to be supported by sources.

An easy to understand definition is both good and useful. Most importantly, this one can be referenced by page number with a link for everyone to see. The lead should lay the table for the article, and previously it did not.

What the previous lead sentence did was ignore an entire history of meaning for just intonation, which is a tuning of an interval that has no beats. The tuning system derived from an attempt to emulate that. Without first defining for the general reader what just originally meant, the search for an entire tuning system makes little sense. This article leapt two feet first into the admirable concerns of the Xenharmonic community, which befuddle even professional theorists like Hucbald.SaintAmand.

The goal of overhauling the article is to streamline information that is true, verified, and has encyclopedic value. That process will enable a cogent discussion of the more advanced concepts this article was in a hurry to discuss. Trumpetrep (talk) 19:32, 1 March 2026 (UTC)

The tough thing here is that you are going to find multiple definitions of the term "just intonation", all from verifiable sources. I can see how you want to introduce with the idea that "just" is related to pleasant/beatless (where I am using "beatless" loosely as I have mentioned in my previous reply. Here is an idea to accommodate that, by deferring the "tuning system" senses altogether (although it probably needs some copyediting):

In music theory, an interval is said to be justly tuned if it contains minimal beating. Specifically, this happens when the frequency ratio between two pitches is a small whole number ratio, as this will maximize the overlap of the notes' harmonics, and subsequently maximize their consonance when sounded together.

What do you think of this as an introduction paragraph? We can then follow it by something along the lines of: "just intonation refers to several related notions regarding justly tuned intervals..." Epixix (talk) 19:53, 1 March 2026 (UTC)

Thanks for your response.

For now, I think the lead is clear and based on the historical understanding of the term. The version that you have written is similarly in a hurry to introduce ratios when there is no express need to.

It also is indirect in ways that an encyclopedia should not be. There is no need to say "In music theory" to define it. We don't say "In warfare" to define a cannon. Also, the first occurrence of the term to be defined is the variation "justly tuned". The simpler the better for the general reader, which is our audience.

The Grove definition also sets the table by talking about beating "the term 'just intonation' refers to the consistent use of harmonic intervals tuned so pure that they do not beat". All the work that everyone did over millennia in pursuit of just intonation systems starts with that natural phenomenon. So should we, in my view.

Do you think you could find quality sources for the article? That would be an enormous help. For instance, the "Terminology" section doesn't have a single source.Trumpetrep (talk) 20:06, 1 March 2026 (UTC)

I'm focusing on the lead for now because I heard that a large percentage of readers don't read past that. To be honest, I don't see the issue with mentioning frequency ratios; it's the only way to meaningfully explain anything past the original "minimizing beating" goal.

When I was first learning about just intonation, I heard phrases like "resonates within the harmonic series" and had no idea what that meant. Even still, I don't like this phrasing; it is loosely using vocabulary and it doesn't explain anything if you don't already know what it's talking about. With my knowledge now, I know that it's talking about something like this: When an interval is tuned in a frequency ratio of whole numbers, the two tones can be interpreted as harmonics of an implied fundamental.

I do not think these things are too hard for a general reader to understand, as long as we don't obfuscate it in jargon. A good thing to help with this would be a diagram or piece of sheet music showing how an interval like 5:4 can be interpreted as the 4th and 5th harmonics of an implied fundamental. Epixix (talk) 20:20, 1 March 2026 (UTC)

I see what you are saying. You are right that there's nothing wrong with mentioning ratios. It's just a matter of doing it in a way that is clear and concise in the style of a lead section.

As you say, there are stumbling blocks all over the place for the general reader. Some might not know the harmonic series. Others may tune out when they see math involved. Hearing your experience of learning about the issue is illuminating. Thank you for sharing that.

I've adapted your language as the third sentence in the lead section. It hews closely to the Harvard & Grove definitions as well. That's the key. We have to be able to back up whatever we say with a reliable source.

Speaking of which, if you have any sources that you can recommend beyond what is already cited, it would be very helpful. Trumpetrep (talk) 20:38, 1 March 2026 (UTC)

What do you mean by "the" harmonic series? That's unclear. What harmonic series is it, i.e., what is its fundamental pitch?

There *is* an implication. Take the interval of a just major third, 5/4. This also means that these pitches are in the form of 4f and 5f, where f is the implied fundamental. f is the period of the resulting waveform when these two tones are played together. Try this with a tuner, which picks up on periodic waveforms; play a just major third, and it will report a note two octaves below your lower note, even though that fundamental isn't present as a frequency. See Missing fundamental.

https://masa.plainsound.org/pdfs/JI.pdf

https://en.xen.wiki/w/Just_intonation

https://www.youtube.com/watch?v=DzUhjxNEyOs Epixix (talk) 21:17, 1 March 2026 (UTC)

Cart is again before the horse here. You are far down the line with the example of a just major third.

The first sentence is referring to the extremely simple concept of harmonics. That is the first definition of the article. An octave is pure when there are no beats. That happens in every tuning system the world over because of the harmonic series.

There's nothing complicated in that, and there is no need to complicate it further in the second sentence of the article. Again, we are setting the table for the rest of the article. By establishing that the original phenomena are pure intervals, we can talk about the extremely complicated means by which so many people have tried to sustain it throughout entire tuning systems.

Thanks for the links. The Xenharmonic Wiki can't be used as a source, but it should be in the External Links section. Many of the links on Xenharmonic's Just intonation page are already listed here. Trumpetrep (talk) 21:54, 1 March 2026 (UTC)

Is User:MrOllie implying that User:Epixix is Marc Sabat? When MrOllie removed all of the Extended Helmholtz-Ellis JI Pitch Notation material, I didn't object because the edit was well within Wikipedia's guidelines. When he removed the External Link to Plain Sound, I undid it because the bar is much lower for External Links. The Plain Sound website is a far better resource than some of the other links I removed during the overhaul of this article.

When Epixix restored some material about Sabat and Schweinitz, MrOllie undid the edit and commented, "This was added by the author, citing his own website. That's clearly selfpromo." It's unclear what MrOllie means. So, I am asking about it here.

The article's focus on Extended Helmholtz-Ellis JI Pitch Notation was far too extensive when I started overhauling it. It's hard to argue that the system shouldn't be mentioned, however. If a conflict of interest is the only impediment to its inclusion, we can easily handle that problem. Trumpetrep (talk) 03:46, 14 April 2026 (UTC)

No, I was referring to the single purpose account who originally added the material (and the link to plainsound.org). The content is entirely based on sources written by Sabat as well. If a new version could be rewritten based on secondary sources, I wouldn't have any objection. MrOllie (talk) 11:03, 14 April 2026 (UTC)

Ah, thanks for explaining. Trumpetrep (talk) 14:20, 14 April 2026 (UTC)