Talk:Limit (music)

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Prime vs whole[edit]

Hmm, I'm not sure about this - isn't it the highest prime number (or at least the highest odd number), rather than the highest whole number? So in 3-limit, for example, a 9:8 is fine (or at the very least, a 4:3 is fine). --Camembert

You're absolutely right, but now I have to figure out why.--Hyacinth Found out!

I think highest prime divisor doesn't make sense. With high enough powers you can do almost anything. For example, 36:29 and 210:36 are both pretty close to a tritone. To get completely ridiculous, 2243:3153 is even closer. Phr 23:56, 15 February 2006 (UTC)
Yes, 729/512 and 1024/729 are both 3-limit (aka Pythagorean) tritones. What's your point? —Keenan Pepper 04:33, 16 February 2006 (UTC)
The point is that there is no such thing as an interval that can't be very closely approximated with high enough powers of 2 and 3. So describing an interval as 3-limit is meaningless unless the exponent is bounded somehow, preferably to low numbers. Am I missing something? Phr 23:42, 17 February 2006 (UTC)
Exactly. A rational number has a definite prime limit, but the ear cannot discern intervals if they are too complex. But the important thing is that any 3-limit interval can be precisely tuned by using a succesion of perfect fifths and octaves (for example 729/512 occurs in the Pythagorean diatonic scale), whereas an interval of any higher limit cannot. —Keenan Pepper 00:12, 18 February 2006 (UTC)
I'm taking the view that just intonation is only musically relevant because the ear can distinguish between, say, 1.5:1 (perfect fifth) and 1.4983:1 (equal tempered fifth). But if the ratios are sufficiently close, say within .0001 of each other, the ear cannot distinguish them and therefore they are for musical purposes exactly the same. So since every interval is has a 3-limit ratio within .0001 of it, for musical purposes there is no such thing as a non-Pythagorean interval and the mathematical distinction between 3-limit and non-3-limit intervals is meaningful for theoretical purposes only. Is that a fair description? (If that's the intended meaning, maybe the article should say so). Phr 03:06, 18 February 2006 (UTC)
From a mathematical point of view, it is fair to say that any tone can be accurately expressed as an infinite series of pythagorean tunings. However, it is not possible in tuning practice to go through such a series. To tune a pythagorean (3-limit) major third, for instance, you need to tune first 3:2, then 9:8 based on that 3:2, now 27:16, now 81:64. Thus it takes 5 tunings to get to that major third, each of which takes time and care. You cannot simply jump to the ratio 81:64, because it is not -audibly- tuneable. Can you understand how impractical it would be to have to tune everything from series' of fifths? - Rainwarrior 23:10, 7 May 2006 (UTC)

Rewrite by Namrevlis[edit]

I see some improvements, but I don't see any mention of odd limits or the difference between odd and prime limits, which to me is a crucial omission. —Keenan Pepper 13:40, 15 May 2006 (UTC)

Please, feel free to add information about odd limits. I rewrote the article in a broad attempt to replace much of the arithmetic with descriptive musical information.Namrevlis 02:37, 16 May 2006 (UTC)

Set notation[edit]

The following was originally written as a reply to the above. Headline inserted because the previous is one year old; the article has been substantially rewritten since. — Sebastian 18:28, 9 May 2007 (UTC)

Hope you'll be patient to my English... It is possible rewrite so:
In just intonation, any given interval can be expressed as the ratio between two frequencies, type 4/3 for the perfect fourth or 10/9 for the minor tone.
If these ratios to factorize, the limits for such intervals are defined as follows:
The odd limit selects a set of the notes of one name in all octaves and is the biggest multiplication of the all odd prime members of the factorization with same exponent sign and exponent modulus taken.
If to suppose frequency of the note C as 1, then C:1 = {C2:2-2, C1:2-1, C:20, c:21, c1:22, …, c5:26} in fact is a set of the notes С in all octaves, and each note of this set has the odd limit 1.
The notes F:4/3 = C:1×3-1×22 = {F2:3-1×20, F1:3-1×21, F:3-1×22, f:3-1×:23, f1:3-1×24, …, f5:3-1×28} have the odd limit 3 = 1×3|-1|;
the notes D-:10/9 = C:1×51×3-2×21 = {D2-:51×3-2×2-1, D1-:51×3-2×20, D-:51×3-2×21, d-:51×3-2×22, d1-:51×3-2×23, …, d5-:51×3-2×27} have the odd limit 9 = 1×3|-2|, because 1×3|-2| > 1×5|1|.
The prime limit is a largest prime in the factorization. The prime limit of the perfect fourth is 3 (the same as the odd limit), because in the factorization 3-1×22 the largest prime is 3, but the minor tone has a prime limit of 5, because in the factorization 51×3-2×21 the largest prime is 5. Commator 18:18, 7 May 2007 (UTC)

In principle, it is possible to write a set notation. However, I don't see how it would improve the article. If you want to be mathematically rigorous, it seems that modular arithmetic would be a more promising and elegant approach. The sets require a lot of terms for relatively little added value. Moreover, to be rigorous you would have to use infinite sets. — Sebastian 18:28, 9 May 2007 (UTC)


How is odd-limit tuning supported by modern psychoacoustics? I checked the reference, and the graph seems to support odd-limit tuning. But how did they arrive at the graph? Is it a theoretical model, or some kind of experiment? tedneeman (talk) 04:28, 10 October 2009 (UTC)

Failed verification: Partch's Genesis[edit]

While I don't think it's necessary for the exact word "complexity" to be found, a similar word should do, I have also failed to verify the reference by finding a description of "odd limit" or any description of "limit". Hyacinth (talk) 07:16, 29 May 2010 (UTC)

Yes, I agree with all of this. Partch uses the word "limit" frequently throughout Genesis without ever actually defining the term. It leaves the impression that he is using a concept he got from somewhere else, and assumes widespread familiarity on the part of his readers. However, it is still possible that Partch originated the term in one of his earlier writings, and is simply assuming that, by 1949 (the year of the original edition of Genesis), anyone reading his new book would be familiar with his previous work. It is possible to infer a great deal about this from Genesis (for example, all of the index entries for "limit" have an odd number appended: "limit of 3", "limit of 5", etc., and on page 88, Partch takes care to explain in a footnote that he means odd numbers rather than prime ones, because "the multiple 9 also assists in the delineation of tonality"). However, it is still true that he never plainly sets forth this principal in Genesis, so we must look elsewhere for it—whether in Partch's own writings, or in those of other authors. As to "complexity", the problem may be compounded by the associated references to Bart Hopkin and Paul Erlich, one or both of whom may equate Partch's scale of consonance/dissonance with simple/complex. Erlich should be easy to check, since his essay appears to be available online; Hopkin may take a little longer.—Jerome Kohl (talk) 19:06, 29 May 2010 (UTC)

While not necessarily intended in as precise a technical sense, there seems to be some precedent from William Smythe Babcock Mathews 1881: "The ninth is the genetic limit of harmony". —David Eppstein (talk) 19:39, 29 May 2010 (UTC)

Thanks for the pointer, which at least indicates caution needs to be exercised with respect to the origins of the term "limit" in this sense. In the meantime, I have, I think, straightened out the reference to "complex" and its equating to higher-limit systems. Unfortunately, in the process I also uncovered another failed claim verification.—Jerome Kohl (talk) 20:00, 29 May 2010 (UTC)

"Just major tenth" is linked to a wrong file[edit]

I cannot supply a this file myslef, so I leave it to the original uploader. —Preceding unsigned comment added by (talk) 17:39, 20 February 2011 (UTC)

Etymology of odd limit and prime limit, clarification[edit]

I searched through Genesis of a Music trying to find the official definition of "odd limit" and I can't find the term used anywhere. Was this invented by someone other than Partch? The references would seem to need work. Can we find a reference for the first usage of these terms? Is anything found in Genesis of a Music then a reference to odd limit and not prime limit?

If it's modulo octaves, then the interval 10:3 would be in the 5 odd-limit and 5 prime-limit? If not for the modulo octaves, this would be in the 3 odd-limit?

11:7 is 11 odd-limit and 11 prime-limit? 15:7 is 15 odd-limit and 7 prime-limit? 12:1 is in the 3 odd-limit and 3 prime-limit? (talk) 18:30, 4 September 2014 (UTC)

Odd limit is the greatest odd factor, so octaves do not make a difference in either case. The odd limit of 10:3 is 5, because it is 2·5:3. Basically, if either side of the ratio is even, divide it by 2 until you find an odd number, and then compare it to the other.
I could have sworn that Partch mentioned both but only really used one. It's been a while since I've read GoaM. — Gwalla | Talk

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