Talk:Regular temperament

message: 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)
replies: text: Heaven forbid! This article is already overloaded with an excess of mathematical jargon such that it is incomprehensible for us normal folk. What this article REALLY needs is a worked example showing what you're talking about using prime generators to produce real frequency numbers. Abelian groups are overkill, the article is choking to death on them! Dlw20070716 (talk) 12:18, 28 July 2011 (UTC)
thread_title: Diophantine approximation

message: :"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)
replies: text: Perhaps the author of this article could try to imagine a quartet of rock musicians in from of him as he/she rewrites this article to be understandable. Anyone tempted to use terms named after mathematicians need not apply. I'm pretty sure that no mathematical concepts more complicated than logarithms are involved. Dlw20070716 (talk) 12:27, 28 July 2011 (UTC)
thread_title:  Some questions and comments: 

message: There appear to me to be two serious problems with this article. The second, and in my opinion less serious, is the issue of comprehensibility raised in several of the above comments.  The first, and more serious, in my opinion, is that "regular temperament", as traditionally used in most of the literature on the theory and history of temperaments means something significantly different from the concept described in the article.  I have now finished writing an article, Regular tuning and temperament, in which I have attempted to remedy these defects (and for whose title I have purloined the suggestion of Robert Walker in his comment above).  In my comments below I have elaborated a little further on what I see as the major problems with the current version of the article.
First, there is a large body of literature, dating from at least as early as the mid to late 19th century, and  continuing right up to the present day (12 instances of which are cited here), in which a regular tuning is defined to be one in which all fifths have the same size (with the possible exception of one per octave, if diminished sixths are taken to qualify as "fifths").  While it's true that the intervals of such a tuning will generate an abelian group that constitutes a "regular tuning" as defined by the article, only those of ranks 1 and 2 can be obtained in this way.  Those of rank 3 or more cannot.  Moreover, the intervals of tunings not classifed as "regular" according to traditional usage will generate tunings which are classed as "regular tunings" as described in this article.  In Ptolemy's syntonic diatonic scale, for instance, the perfect fifths between the first and fifth, third and seventh, fourth and eighth,  fifth and ninth, and sixth and tenth degrees of the scale are all perfectly just, whereas that between the second and sixth degrees is flatter than just by a syntonic comma. Its tuning is therefore not regular according to the traditional definition.  All the intervals of this scale, however, can be constructed from combinations of octaves and perfectly just major and minor thirds, and together generate the rank 3 tuning comprising all 5-limit intervals, which is a "regular tuning" according to the definition adopted by the article.
The concept of "regular temperament" described in this article appears to have been developed  mostly in the first decade of this century in internet forums and mailing lists, and on enthusiasts' websites.  While much of this material seems to have remained accessible, a fair proportion of it would appear to me to be quite difficult for those who were in on the original development to comprehend.  If it weren't for  two papers published recently in respectable academic journals by Milne, Sethares and Plamondon I would find it difficult to see how anyone could write an article on the concept that would satisfy Wikipedia's policy on verifiabily.  But even though I'm willing to grant that this can probably be done, the fact that the article is devoted entirely to this recently developed concept, and completely ignores the more traditional and much better documented one, means that it fails badly to satisfy Wikipedia's policy on neutral point of view.
Secondly, I strongly agree with the comments of editors Dlw20070716 and Babygrow above that the article's reliance on unexplained jargon makes it unnecessarily incomprehensible to all but a minuscule proportion of its potential readership.  I was a professional mathematician for 40 years (I'm now retired) and am reasonably well acquainted with all the mathematical concepts mentioned in the article.  I took up the study of music about 15 years ago, and have completed the AMEB's grade-5 music theory examination (with distinction).  When I looked up this article I had long been interested in the history and theory of tuning and temperament, and had already consulted quite a few good sources in the mainstream literature on those topics.  With that background, I expected to have few problems understanding the article, but instead found it almost totally incomprehensible, at least initially.  Probably the main cause of my confusion was my assumption that the article was simply giving a more general mathematical treatment of the traditional concept of "regular temperament" that I was already reasonably familiar with.  It wasn't until I realised that the concept being described by the article was a significantly different concept from the traditional one that I was able to winkle out a reasonably good understanding of it, mainly from the papers of Milne et al. cited above.
The article I've written is divided into two main sections.   The first deals with the traditional concept of regular tuning and temperament, while the second deals with the one described in this article.  I think everything mentioned in this article is covered more comprehensively, and I hope more understandably, in the second section of the new article.  So I would propose that after an appropriate time has elapsed for interested editors to compare the two, the text of this article be replaced with a redirect to the new one.
David Wilson (talk · cont) 12:57, 27 October 2018 (UTC)
thread_title:  New article