Talk:Limit (music)

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This article is part of the WikiProject Tunings, Temperaments, and Scales to improve Wikipedia's articles related to musical tunings, temperaments, and scales.

[edit] Prime vs whole

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)

[edit] Rewrite by Namrevlis

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)

[edit] Set notation

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)