Talk:Regular temperament

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)

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)

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)

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)

Errors in the first paragraph

First, as another noted, temperament generators are generally NOT ratios. The term "frequency ratio" should be replaced by "interval". "Product of powers" should be replaced by "sum".

Also, 12-TET is rank-1, therefore it has only one generator, which is the 100¢ semitone. Musically, it seems like the fifth should be the generator. But mathematically, it isn't. Because you can't get a major 2nd or a minor 3rd by stacking fifths, without octave-reducing. But octave reduction means using the octave as an additional generator. But it's rank-1, so there can be only one generator. That's the semitone, because every 12-TET interval is obtainable by adding up semitones.

I would suggest moving the 12-TET example to later in the article, since it's so counter-intuitive, and using meantone as an example in the first paragraph.

I would also suggest explaining why temperaments are useful musically -- they approximate just intonation better than equal temperaments do, but have fewer wolf intervals than just intonation does. In just intonation, some chords sound great, some sound awful (because of the wolves). In a rank-2 temperament, most chords sound quite good, a few sound awful. In an equal temperament, all chords sound pretty good.SeventhHarmonic (talk) 22:46, 14 January 2017 (UTC)

Good points. On the first point, yes indeed. No need to say any more. Why not just "be bold" and fix it? But looking a bit further ahead, I think the whole lede needs to be rewritten. I have a draft in my user space which I meant to share but forgot. On those particular points, yes that makes sense that you can treat 12 equal as having a single generator which is just a single step of the equal temperament. But that's an unusual way of thinking about it though mathematically valid. I think another way of looking at it, which doesn't involve the octave as a generator, is to think of the octave as an equivalence relation. So you treat all notes that are a multiple of octaves apart as "equivalent". I think that's how musicians intuitively think about them, and also how it is presented in the literature. That gives us a way to think of it as "1-D" while still using the more musically motivated interval of the fifth as a generator.

Also agree that it's good to explain why the temperaments are useful. Anyway here is my draft for the lede, which explains the whole idea of a temperament. To explain it properly I found it was necessary to explain many other topics. The main prerequisites for understanding the idea are:

• Generators of a scale
• Just intonation
• Musical equivalence
• Tempering a musical interval
• Vanishing commas
• Graphing a tuning in 1D, 2D and higher dimensions.

Lede here: User:Robertinventor/Regular tunings and temperaments

It's rather long for a lede. But hard to see how it can be shortened significantly without impairing understanding. Splitting it into two articles could help perhaps. But a newbie reader will need to go through all those concepts and understand them, so if it is split into two articles, they will need clear guidance as to which article they need to read first before they read this one. I think this is a bit of a tricky one, to know how to do it, and welcome any suggestions. Do feel free to fix any obvious uncontroversial errors in my draft and to discuss anything that needs to be fixed that may need more thought Thanks! Robert Walker (talk) 12:55, 2 February 2017 (UTC)

New article

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)

Proposed "merger"

As foreshadowed in my comments immediately above, I propose that this article's page be turned into a redirect to the article Regular tuning and temperament. This will complete what is effectively a merger of this article into that one.

Since a consensus for this merge seems unlikely, I hereby withdraw it.

David Wilson (talk · cont) 01:19, 21 December 2018 (UTC)

The chief rationale for doing this is that the latter article explains the meaning of the term "regular temperament" as it is currently used in mainstream literature on the theory of music, whereas this one ignores that meaning completely. It concentrates instead on a meaning given to the term more recently by a relatively small number of musical theorists who appear to have been working chiefly in the area of microtonal music.

The redirect will not result in the subject matter covered in this article's being hidden from view, because that material is covered at least as comprehensively by the article the redirect will point to.
David Wilson (talk · cont) 12:16, 7 November 2018 (UTC)

David, as someone who is active in the microtonal community, there are some issues with this current page that ought to be corrected and are wrong even from the perspective of microtonal theory. I was just about to edit the errors on this page when I saw you had made another one and are proposing to merge the two.

Regarding your new page, it seems to me like it has valid material; both pages cite theorists that are either active in the microtonal community (such as Secor, Erlich, sometimes Milne) or who are often cited in the microtonal community (Bosanquet, Blackwood, Barbour, Ellis, etc). So now I'm in limbo, as I'm not sure what to edit. However, I would much rather see that page merged into this one than the other way around; the two are roughly equivalent but "regular temperament" is a much more common term than "regular tuning."

Finally, your ideas about the use of the Smith normal form are very interesting! Are you using it as a canonical representation for a temperament matrix? We tend to use Hermite form for that, but perhaps Smith form has its merits as well. If you are interested, there is a Facebook group where we discuss tuning-math, you are certainly welcome to join and we would love to have you. See here: https://www.facebook.com/groups/497105067092502 - Best, Battaglia01 (talk) 22:21, 12 November 2018 (UTC)

(edit conflict) I would have no objection to the merger proceeding in the opposite direction. The effect of either merger is essentially to replace the current text of the article Regular temperament with that of Regular tuning and temperament as Wikipedia's article on the subjects treated. I would be just as happy for that article to bear the former title as the latter.

I wouldn't lay any claim to the ideas in the section of the latter article on finding minimal sets of generators as being "mine". Finding such a minimal set of generators is a stock standard problem in the theory of finitely generated abelian groups, and I played no role in developing the now well-established techniques for solving it—including the one outlined in the article.

I've never come across the term "temperament matrix" before, so I can only make an educated guess as to what it might mean. I presume it refers to a matrix of a temperament mapping with respect to some bases of its domain and codomain (and presumably with the basis of the domain being the logarithms of the primes, or the primes themselves, if a multiplicative representation is being used). If so, then the Smith normal form mentioned in the article isn't one for any such matrix. It is instead the Smith normal form of the matrix of the relations satisfied by the original set of generators. I don't attach any special significance to it beyond the fact that the rank of its kernel is immediately recognisable by inspection, and tells you the rank of the group generated by the original set of generators. It's also true that you don't have to proceed all the way to obtaining the Smith normal form to solve the original problem. It's sufficient, in this case, to reduce the original matrix to lower triangular form—i.e. column-style Hermite normal form. Once the original matrix has been reduced to such a form, the unimodular matrix you've postmultiplied it by will have the properties needed to solve the original problem.

David Wilson (talk · cont) 23:30, 14 November 2018 (UTC)

Hello David - apologies for missing this, I hadn't "watched" the page so I didn't see your reply. My main point is that most of us in the microtonal theory community are using the same exact group-theoretic formalism that you are. For some prime p, the set of p-limit (or "p-smooth" to mathematicians) rationals forms a free abelian group under multiplication, representing a finite-rank subgroup of just intonation. In the group-theoretic sense, a temperament is, then, a torsionfree quotient group of this group, where the kernel is the vanishing commas.

An equivalent way to define a temperament is to first look at surjective homomorphisms from this group to the free abelian group Z^n, for some n of rank less than the original group. These can be represented by integer mapping matrices similar to yours. A "temperament," then, is an equivalence class of homomorphisms which have the same kernel (or, equivalently, which have row-equivalent matrices, each of which can be considered as representing a different basis for the temperament). This is equivalent to the first definition, but is sometimes a useful perspective as the row vectors form a sort of "dual group" representing rank-1 mappings (assuming your intervals are column vectors), which you can combine to form higher-rank temperaments.

There is some ambiguity with the term "temperament" in the literature. Sometimes the term is used to denote individual tunings, such as "quarter-comma meantone" or "third-comma meantone," as though they were distinct "temperaments." Sometimes, the entire equivalence class of tunings is considered a temperament in the abstract sense, such as the generic "meantone temperament" being the 5-limit temperament in which 81/80 vanishes, regardless of the particular tuning. Likewise, Secor's "miracle temperament" is usually considered as an abstract 11-limit entity in which certain commas vanish, and for which a range of tunings is acceptable. We have sometimes called these equivalence classes of tunings that share a common kernel "temperament classes," to avoid ambiguity with the other definition in which we consider each tuning a distinct "temperament."

I don't mind what convention we use. The one where a temperament involves a particular tuning seems to be most common. It would probably be best to have a section which addresses these different usages.

Beyond that, I could give some more examples of the group theory stuff, but you already seem to be more than familiar with all these concepts, and with important papers such as Erlich's and Secor's. So all I'll say is, most of the "modern" stuff not included in those papers is really just variations on this theme. For example, these days it is very common to look at temperaments derived from subgroups of the rationals that are not full p-limits - for example, the subgroup generated by primes <2,3,7,11,13>, skipping 5. But of course this doesn't really change much about the group-theoretic formalism.

My view is that this current page needs to be changed to reflect the above. I think that much of the material on the new page you've written is an excellent starting point.

The new page has a few minor quibbles I would change - for example, you say a regular temperament has every 3/2 tuned the same way. Typically, however, it is defined so that this property holds for each rational number, so that every instance of that rational is tuned the same way. Or more simply, we want the "tuning map" from the group of tempered intervals to the real numbers (representing a tuning in cents) to be a group homomorphism. So the criterion that the tuning map is "regular" just means that it is a linear map.

Likewise, you seem to be defining a temperament as a tuning of the chromatic scale (which I assume means 12 notes). This is another sense in which the term "temperament" has also been used historically - not just as a particular abstract "temperament class," and not even as a particular tuning of said "temperament class," but as a particular, finite set of pitches derived from that. However this is not the universal definition in the literature, and for example some of the other material cited on your page (such as Secor's miracle temperament) doesn't even fit this definition. Miracle temperament is not usually considered to be a 12-note tuning - it is typically considered as an abstract group of pitches, of which a 21-note chromatic scale is commonly used (along with 31 and 41 notes). So I would want to change the definition to reflect this variance in the usage.

That being said the general focus seems to be in the ballpark, so if what stuff I've written above seems sensible to you, I would vote in favor of the merger. Battaglia01 (talk) 03:18, 28 November 2018 (UTC)

Apologies for the delay in getting back to this. Unfortunately, the Wikipedia editor completely lost an earlier very long reply I was in the process of preparing.

I'm concerned that the third last and second last paragraph of your immediately preceding comment seem to imply that the article should treat the microtonal community's redefintions of the terms "regular tuning" and "regular temperament" as if they are the only ones of current significance. This is simply not true, and the current version of the article Regular temperament's implict assumption that it is was the subject of the first of the two objections I raised against it.

There's a large body of what I would call "mainstream literature" on the theory of music, a small selection of which is cited in footnote 4 of the Regular tuning and temperament article, where "regular temperaments" are defined simply to be tunings whose fifths are all the same size. Even when it's not stated explicitly that the chromatic scale is the object of the tuning, it's nevertheless always obvious that it's what's being assumed. As I noted in the second of my objections, and as you appear to recognise in your preceding remarks, the concept traditionally referred to by this term in this literature is significantly different from what is now being referred to by the microtonal community's recent use of the term. I'm very strongly of the opinion that any Wikipedia article on this subject must treat the former as a now very well-established concept, and clearly differentiate it from the latter. I would very strongly object to its being treated as no longer current, or having been superseded by, or being merely a special case of, the microtonal community's concept. None of those things is true.

The traditional definition of "regular temperament" is not simply a historical curiosity, now superseded by the microtonal community's version. It's the one currently given in Mark Lindley's entry Temperaments in New Grove and Jeremy Montagu's entry, Temperament, in the Oxford Companion to Music. It's also the only one I've been able to find in any treatises devoted to the subject of scales and temperament—including Barbour's Tuning and Temperament: A Historical Survey (1951), Lloyd & Boyle's Intervals, scales and temperaments (1978), Jorgensen's Tuning: containing the perfection of eighteenth-century temperament, the lost art of nineteenth-century temperament, and the science of equal-temperament, complete with instructions for aural and electronic tuning (1991), Lindley & Turner-Smith's Mathematical Models of Musical Scales: A New Approach (1993), Donahue's A Guide to Musical Temperament (2005), Di Veroli's Unequal Temperaments: Theory, History and Practice (3rd ed., 2013), and Dolata's Meantone Temperaments on Lutes and Viols (2016). There's also a large body of relatively recent musical literature which, while not exactly being treatises on temperament itself, do nevertheless deal with it to some extent, and adopt exactly the same definition of "regular temperament" as the aforementioned treatises—Duffin's How Equal Temperament Ruined Harmony And Why You Should Care (2007) and the Dolmetsch music dictionary, for example. In none of this literature—with the exception of those noted below—have I ever come across an instance of any version of the microtonal community's definition of "regular temperament" being used.

The exceptions are Milne, Sethares and Plamondon's journal articles, Isomorphic Controllers and Dynamic Tuning: Invariant Fingering Over a Tuning Continuum (2007), Tuning Continua and Keyboard Layouts (2008), their slightly earlier Open University research report, The X_System (2006), and miscellaneous online resources such as the Facebook group you mention, the Xenharmonic wiki, the Tuning-Math mailing list, Graham Breed's website, Gene Ward Smith's Xenharmony web pages, Joe Monzo's Tonalsoft Encyclopedia, and the like. But although these latter online resources might be very useful for understanding the material they cover, I don't see how—with the possible exception of the Tonalsoft encyclopedia—they could be regarded as satisfying Wikipedia's criteria for reliable sources, which explicitly exclude personal websites, open wikis, internet forum postings, social media postings and the like from normally being considered reliable.

In the Regular tuning and temperament article, I've treated the the traditional concepts of regular tuning and regular temperament and the ones recently developed by the microtonal community in completelty separate sections—the first in the Description and examples section, and the second in the More recent alternative concept section—, and tried to make it clear that they're quite different pairs of concepts. The presumption that the chromatic scale (or some extension of it) is the object of a regular tuning, for instance, is limited to the first four paragraphs of the lead, where the traditional concepts are introduced, and the Description and examples section, where they're described in more detail. In the More recent alternative concept section, that condition is explicitly dropped, as is the requirement that a regular tuning must contain an octave or a fifth of any sort, whether tempered or just. If the present version of the article doesn't make this clear, then it certainly needs to be amended so that it does.

I'm certain that the Description and examples section of the article contains a reasonably accurate account of the concepts of regular tuning and regular temperament as they're presented in the traditional music theory literature I've cited above. Nevertheless, like all material in any Wikipedia article, it is of course subject to future improvements and corrections, provided only that these can be verified by consulting reliable sources.

The material in the More recent alternative concept section is mostly based on my reading of the journal articles of Milne et al., again cited above. While I've tried to be conscientious in not misrepresenting anything I've read, I'm somewhat less certain of how successful I've been.

David Wilson (talk · cont) 05:53, 20 December 2018 (UTC)

David, thanks for the reply. I tried to express, in my last reply, that the present definition of "regular temperament" on this current page is *not* the one used in microtonal community -- the one that talks about "syntonic" temperament (Milne's term) and so on. So, when you talk about the microtonal community's "redefinition" of the term, I don't know which particular subset of the community you are talking about. For example, the definitions presented in Milne's work are very different from those in the other communities you mention, and I would agree that some of what Milne's done (such as the "syntonic temperament") is a redefinition in the sense that other work is not.

However, this is somewhat irrelevant since I agree, of course, that online xenharmonic discussion forums are not a "reliable" source. Rather, my point is that papers you are already citing as reliable use the term in the way I describe.

Two important references you didn't mention in your reply above, which use the definition of "regular temperament" I mentioned, are Erlich's "A Middle Path" paper, another is Secor's papers on Miracle temperament. You have already cited these papers as reliable in your current version of the page, so my point is that the term is already being used the way I describe (or something equivalent) in papers you've already considered reliable enough to cite. Certainly these papers are cited, routinely, in the other discussion forums you mention (the Yahoo! tuning list, and so on).

My point isn't so much that the modern definition has "superceded" the old one, it's that historically, authors simply did not study tuning systems generated by anything other than a perfect fifth before a certain point in time. If you only consider such tuning systems, then saying "each instance of the perfect fifth must be tuned identically" and "each instance of the generator(s) must be tuned identically" is the same thing.

When authors started considering temperaments generated by things other than a perfect fifth, they typically used the definition where the generators are tuned identically, not just the "perfect fifths." And again, this isn't just a modern "microtonal internet community" thing; Secor's paper on miracle temperament is 43 years old.

I would have no problem putting *all* of these definitions in a section about the history of regular temperament, what scales were considered at which points in time, and how different authors have used the term. Furthermore, I would consider the page to be inadequate if it *didn't* have that.

But my main objection with the way the page is now is that you seem to be claiming that there is some context in which people used the term "regular temperament" to include tuning systems *not* generated by fifth, but simultaneously demanded that *only* the fifths be tuned identically.

I don't think this particular combination of criteria has *ever* appeared in the literature. If it has, I would be interested in seeing a citation.

But my understanding is, either people only cared about fifths and focused on that, or they extended their focus to other generating intervals and changed the identical-tuning criterion to all generators, not just fifths.

The correct thing to do in this case would be to add a note that earlier authors did not consider regular temperaments generated by intervals other than that of a perfect fifth, hence the criterion that all instances of the "perfect fifth" are the same size is equivalent to the criterion that all instances of each "generator" are the same size. Battaglia01 (talk) 18:45, 20 December 2018 (UTC)

Despite my best efforts, it's obvious that I've completely failed to articulate my position clearly. I hasten to reassure you that I accept full responsibility for this and don't intend my recognition of this fact to be in any way an adverse reflection on your comments. Nevertheless, I'm unwilling to engage in further discussion along these lines. I've therefore withdrawn my proposal to merge the two articles. To avoid having two overlapping articles with conflicting material, and since I'm the sole author of the Regular tuning and temperament article (apart from a single edit which wikilinked the word "secor"), I've moved it back to my user space .

I suggest you now simply proceed with your original plan of amending the Regular temperament article. If you think there's anything in the Regular tuning and temperament article worth making use of, please feel free to do so.

David Wilson (talk · cont) 03:24, 21 December 2018 (UTC)

@Battaglia01:

Until very recently I had been completely baffled by the following statement of yours, because I could not think of anything I had written that even remotely resembled making the claim you say I appear to have made:

" … you seem to be claiming that there is some context in which people used the term "regular temperament" to include tuning systems *not* generated by fifth, but simultaneously demanded that *only* the fifths be tuned identically."

However, just recently, I've realised that you're probably referring to the following statement of mine:

"There are, however, some more extended scales for which the construction of regular tunings requires two or more chains of fifths offset from each other by intervals which cannot be formed from combinations of whole numbers of fifths and octaves."

There are two classes of historically discussed systems I was referring to in that statement:

• Implementations of just intonation in which several interleaved chains of just perfect fifths are offset from one another by intervals of a whole number of just major thirds. Several such systems are described on pages 108–13 of J. Murray Barbour's Tuning and Temperament: A Historical Survey.
• Equal divisions of the octave in which the number, n, of unit intervals making up the octave, and the number m, making up the fifth are not coprime. In this case, the "generator", the unit interval of size 1⁄n of an octave, cannot be expressed as a combination of a whole number of fifths and a whole number of octaves. If  g def= gcd( m, n )  , then the system will comprise g interleaved chains of fifths, offset from one another by a whole number of unit intervals, and each containing n⁄g notes per octave. Of the systems Barbour discusses on pages 113–32 of Tuning and Temperament, 24-, 25-, 28-, 34-, 36-, 58-, 84-, 87-, 112-, and 612-EDO are of this type, and of these, all except 25-, 28- and 84-EDO are said by Barbour to have been discussed by others before him.

I have not claimed (and would certainly never do so) that anyone has ever "demanded that only the fifths be tuned identically" in systems they regard as regular. I merely contend that in both traditional and current mainstream literature on the theory of music, "regular temperament" is in fact defined to mean one whose fifths are all the same size. In theory it would be possible for someone to take a system satisfying this definition, declare various of its intervals of different sizes to be its major thirds (for instance), while maintaining, consistently with the definition, that it was a regular sytem. In practice I'm not aware of anyone's having ever done such a thing.

Barbour, for instance, defines "regular temperament" to mean one in which "all fifths save one are of the same size", but he takes all the major thirds of any given temperament to be the same size (usually, but not always, the one which is closest to its just value). The interval Barbour is referring to with his "save one" is in fact a diminished sixth rather than a fifth, and presumably that's the reason why many theorists don't feel it necessary to mention it as an exception to the specification that all fifths be of the same size.

David Wilson (talk · cont) 09:35, 28 December 2018 (UTC)

David - I see what you are saying now. My point was just that the way this was originally written, you were using the definition of "regular temperament" where all the fifths are the same size, but if you are going to go into something like miracle temperament -- which I think is certainly a good thing -- this criterion is necessary but not sufficient (and does not match the definition used by those two authors). So I thought that was worth mentioning.

It looks like you changed this beginning on the version of the article on your userspace, so this may not be relevant anymore. The bigger point I wanted to make is that, yes, sure, I agree there is sort of an "online microtonal community" that uses terms loosely, but I wasn't trying to cite that in my comments.

Beyond that, I thought what you had was good, I just wanted to change a few things for wording. Perhaps the best way is for me to make a revision to your page, and see what you think? Perhaps we can move forward that way. Battaglia01 (talk) 20:01, 5 January 2019 (UTC)

Hello David, I agree with much of what you and the other commenters say here and would like to help. I am also more of a mathematician than a musician. Here are my brief thoughts: (1) can you point to mainstream literature using the term regular temperament? Not doubting you, but I couldn't find any (with only a very quick look). It isn't covered in Grade 5 Theory, that's for sure :-) (2) I believe that mainstream literature may use tuning and temperament interchangeably; I think it's important to note that the newer theory you refer to makes a clear distinction between them. (3) I think there's a case for 2 pages; the first part of your new page would be one of them: focussing on the mainstream use, which I think is mainly about tunings; the second would focus on the newer theory and would make use of the second part of your page and the old page, and is mostly about temperaments. (4) Your presentation of the newer theory is quite different from how I would do it, so would like to talk about that. Limmata (talk) 21:52, 14 November 2018 (UTC)

• Concerning (1): Of the 13 sources listed in footnote 4 in the Regular tuning and temperament article, 10 use the term "regular temperament" precisely as described in the article's lead. The three others, (Lindley, 1984; Jorgensen, 1991; and Lindley, 2001), use the expression "regular meantone temperament" to refer to the same concept. A Google Books or Google Scholar search on the expression "regular temperament" will return a plethora of other sources (along with lots of false positives). Contrary to Norman Carey's apparent attribution—in his 1998 Ph.D. thesis, Distribution Modulo 1 and Musical Scales—of coinage of the term to J. Murray Barbour in the middle of the 20th century, it had already been used in exactly the same sense as early as 1895 by Ernest Bergholt in an article spanning several issues of Musical opinion and music trade review.
• Concerning (2): While it's true that some writers of mainstream music literature use the terms "tuning" and "temperament" as interchangeable synonyms, it's certainly not true, in general, of most. Of the sources cited in footnote 4, Barbour (1951), Lindley (1984) and (2001), Lindley & Turner-Smith (1993), Jorgensen (1991), Duffin (2007), Blood (2013), Di Veroli (2013) and Dolata (2016) quite clearly use the terms "temperament" and "tuning" to mean different things, the former term referring to a special instance of the latter in which at least some consonances are tempered rather than pure. Of all the sources cited in the article, the only ones I'm aware of that unequivocally do use the terms "tuning" and "temperament" as synonyms are Bosanquet (1876) and (1889), Ellis (1885), and Lloyd & Boyle (1978). Of course, it's also true that in the vast majority of modern mainstream western music literature it's implicitly assumed that the notes of the chromatic scale have been tuned to equal temperament. Since this is both a temperament and a tuning, according to the standard terminology, it may well be referred to indiscriminately as either, thus giving rise to an erroneous impression that the terms are completely interchangeable.
• Concerning (3): In my opinion, the concepts referred to by traditional terminology and that adopted by modern microtonal theorists are sufficiently closely related that it's preferable for them both to be treated in the same article. Although I'd be willing to entertain the idea of having separate articles, I'm very strongly of the opinion that if either of the articles is to carry the title "regular temperament", it must be the one on the traditional use of that term, and not the one on modern microtonal theorists' redifinition of it. On this point, it's simply not true that the modern mainstream use of "regular temperament" is "mainly about tunings".
• Concerning (4): The material on the modern microtonal concepts, like everything in Wikipedia's article space, is open to improvement by anyone sufficiently familiar with whatever's verifible from reliable sources. I'd be very interested in seeing an outline of how you would have treated the topic.
  

David Wilson (talk · cont) 05:35, 23 November 2018 (UTC)