Talk:Regular temperament

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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] Diophantine approximation

Hmmm - don't know how interesting it is for musicians, but the explanation with abelian groups could do with some diophantine approximation development.

Charles Matthews 13:35, 2 Mar 2004 (UTC)

[edit] Some questions and comments:

"Regular temperament is a system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios."

The first clause seems to be implying that an interval can only be rational, I'm assuming that is unintended? Are you allowing for irrational generators?

"If the generators are all of the prime numbers up to a given prime p, we have what is called p-limit just intonation. However, normally we aspire to adjust the tuning of one or more of these primes, to produce an actual "tempering" of the justly tuned ratios, as in meantone."

The first sentence of this paragraph - read in the context of the first paragraph - implies that a just intonation is a regular temperament. This seems backwards. The second sentence implies that a temperament is a tuning adjustment of a p-limit just intonation, which seems more correct.

"In mathematical terminology, the products of these generators defines a free abelian group. The number of independent generators is the rank of an abelian group, but one less than this number is sometimes called the dimension of the temperament. A "0-dimensional" tuning system with a single generator is normally regarded as having approximated an octave after a certain number of steps, and therefore of constituting an equal temperament. A linear or one-dimensional temperament has two generators, one of which is usually taken to be an octave or some equal subdivision of an octave; in the strict sense of linear temperament, a term due to Erv Wilson, one of the generators is taken to be an octave. A temperament with two generators can, however, always be called a "rank two" temperament."

It seems to me, that the dimension being equivalent to one less than the rank makes semantic sense *only* if the interval of equivalence - whether octave, tempered octave, tritave etc (or a subdivision of said interval of equivalence) is one of the generators? This paragraph seems to be implying that this needn't necessarily be so, maybe it's just the way it's written, or is there a good reason to allow for non-interval-of-equivalence generators not to be counted as a dimension?

"In studying regular temperaments, it is generally advantageous to regard the temperament as having both a map from p-limit just intonation for some prime p, and a tuning map to particular values for the generators."

Can we have an explanation of what a "tuning map" is? Some examples of both mappings would be really hepful for clarification here.

"From that point of view, it is no longer required that the generator of the temperament be independent, which is demoted to a mere matter of tuning; we are only interested in the rank of the group which is the image under the first map."

What is "first map" referring to? What does "image under" mean? Overall, this whole paragraph is unintelligible without further explanation and the example given doesn't help to clarify it for me.

Babygrow 17:34, 13 February 2006 (UTC)