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Help me out here. Is this article supposed to be about one tuning called "Syntonic temperament", or is it supposed to be about a system for generating a continuum of different tunings? It seems to be blurring those concepts. The title and opening statement suggest one particular tuning, but the illustration and the rest of the body text suggest multiple tunings. 108.60.216.202 ( talk) 06:16, 25 May 2015 (UTC)
This page seems to be describing what is normally called a " Meantone temperament" To quote from the website [1]
"
But the chart shown includes tunings with the fifth wider than the pure fifth. So it's hard to be sure what it is about. The more general term here is a Linear temperament - in other words any rank 2 regular temperament with one of the generators the octave. In its most general case the other generator in a Linear temperament doesn't have to be a fifth, but can be of any size. So, I think there are two options here, really. We could just merge this with meantone, or we could merge it with Linear temperament which currently redirects to Regular temperament. As it stands, syntonic temperament seems to be a term coined by a single author and only used on the MusicaFacta website. Robert Walker ( talk) 17:05, 12 July 2016 (UTC)
#redirect [[Linear temperament]]
#redirect [[Meantone temperament]]
Let me answer you here for the time being, before I make my objections on your Wikipedia:WikiProject Council/Proposals/Microtonal Music, Tuning, Temperaments and Scales page, about which I have to think further. I am a historian of music theory, which means that I not only considered the history of temperaments, but also that of non European tunings e.g. in Oriental (maqam) music. My feeling is that microtonal music forms a very little part of that ensemble (it would be a gross mistake to consider maqam music as microtonal, even although the mistake is at times done by Arabic musicians). I believe that the confusion made here, calling "syntonic temperament" something that boils down to being (extended) meantone temperament is a case in point. Microtonal websites (such as those concerned with xenharmonic music) often appear to me as playgrounds for arithmetic games (which may be interesting if only because they turn back to the Antique Pythagorean conception of music as another name for arithmetics), but which have little to do with real music. As you can see, I am an old-fashioned theorist; but I keep up to it.
To turn back to the matter of merging or deleting this article, I had thought we should leave a chance to the original author(s) to react to the proposition. I suppose that after now about sixth months, they had their chance and ditn't take it. Then remains the problem that several other articles may need corrections as a result of the move. Some of my ideas (and a few of my mistakes) about all this can be read on the Talk:Schismatic_temperament page.
Hucbald.SaintAmand ( talk) 09:31, 13 July 2016 (UTC)
Robert, you write:
Well, we obviously do not share the same idea of what microtonal might be. To say that anything different from ET temperament is microtonal makes no sense to me because ET is at best an utopy. It has so to say never been used in real music, unless with electronic instruments. Even pianos have octaves that are slightly too large, because of the inharmonicity of stiff strings; guitars the same. Wind instruments hardly could play in true ET and stringed instruments usually adapt their intonation to the context.
One might consider that microtonal music is that which uses more than 12 tones in the octave. But that, of course, would exclude the musics that you quote, maqam, medieval or Gamelan. If more different notes are used in their heptatonc (or possibly pentatonic) scales, it merely results from the notes being mobile; and to consider a movable note as consisting in several different notes is a very dangerous stretch of the mind. To consider Pythagorean tuning a microtonal tuning seems to me counterproductive, because it merges to saying (with Julen Torma, "Euphorismes", in Écrits définitivement incomplets) that "Everything is the same thing. Everything therefore is very sufficiently good". To consider that the various tunings in which Bach's Well tempered keyboard has been played (or could be played) as variously microtonal is utterly problematic.
Any definition of "tuning" should take account of the fact that many musics make use of mobile (or movable) notes – that is, notes that are not tuned stricto sensu. We would easily agree, I trust, about what a temperament is: in addition to make use of tempered intervals, it is also meant for instruments of fixed tones, producing a limited number of tones. From the present discussion, it appears to me that a tuning too must concern limited numbers of tones. Violin strings are tuned (usually in pure or just fifths); but can one say that the notes played also are tuned (especially if the open strings are not actually played)? In what kind of tuning does a violin play? (Or a maqam player, or a singer, etc.?)
You may consider that this is a wrong way to ask the question, but I don't think so. I think that these questions are not enough considered, that they are rejected as apparently futile or unduly complex. But in the end, they cannot be avoided, and this is one of my main objections against the whole subject of "microtonal music" as it is too often presented. There is a lot to add, and I'd gladly continue the discussion if you feel like it; but I will not bother you further if you don't. — Hucbald.SaintAmand ( talk) 13:00, 13 July 2016 (UTC)
Yes, Robert, 12-ET is an interesting theoretical case. It has been around, conceptually, since Antiquity, since when one decided that the tone could be divided into two semitones, implying that the two halves could be equal. But I don't follow you when you write that
Medieval lutes at times were tuned to a good approximation of ET, but their tuning was not a temperament. One made use of the 18:17 ratio, which amounts to 98.95 cents, but is an integer ratio nevertheless. There was no way to conceive a true tempered semitone, nor to geometrically represent it, before the late 16th century and Simon Stevin's continuous fractions (and later, of course, logarithms). Also, fretted instruments at times had double frets for the two enharmonic notes (i.e. they made a distinction between the diatonic and chromatic semitones). This was the case especially with larger instruments (viola da gamba); for shorter ones, strings were slightly pulled sideways while playing to obtain different intervals. ET was approximated on the lute because the frets, crossing all strings in one straight line, could not make the distinction between the two forms of the semitone (diatonic and chromatic), but certainly not because it was "the simplest way to fret": it certainly was not. And players often used expedients in order not to have to play in ET. This remained true even in Bach's time, when theorists (and Bach himself) advocated against ET because it would have reduced the difference between the various keys. To say that lutes were tuned in ET isn't really wrong, but it certainly is anachronic. [Margo Schulter has an expertise on medieval tunings, indeed, but not every scholar on this topic, me included, always agrees with her.]
Well, if we agree that 12-ET is more conceptual than real, the same can be said of any other ET. To say that a singer sings in 7-ET, or 17-ET, or 19-ET, really is a matter of conception. 19-ET, in particular, can so to say not be differentiated from 1/3-comma meantone (taking two units for the diatonic semitone): one hardly could say whether anyone sings in 1/3-comma meantone or in 19-ET, unless one knows her or his conception. There is a tendency, among, say, Xenharmonicians, to reduce everything to xx-ET and to precisely quantify microintervals. This is not wrong in itself, but it gives a rather distorted image or the realities of former times.
This again may be true conceptually, but it certainly was not thought so in former times because an infinite number of notes was unthinkable. (Infinite mathematical series again belong with continuous fractions and logarithms.) Meantone tunings were conceived at most with 17 notes, that is the diatonic scale + 5 sharps and 5 flats. The idea of double sharps and double flats is a recent one, and Xenharmonicians are way beyond that. Take a look at List_of_pitch_intervals#List: one could argue that no singable or playable interval is absent of the list; but to claim that sung or played intervals in real music do reflect those of this list is, to me, an aberration – and, for the ordinary Wikipedia readers, a misleading. Using integer frequency ratios implicitly claims that these are ratios between harmonic overtones; but what sense is there to imagine harmonic numbers in three figures and more, when hearing is concerned?
This all to say that I have problems, as a historian of music theory, to accept the Wikipedia articles about which we are speaking as vulgarizing concepts of earlier times. I don't mean that these articles are wrong, nor do I have excessive difficulties understanding what they do. I merely mean that I don't think they provide the simple explanations that the average Wikipedia reader is hoping to find. — Hucbald.SaintAmand ( talk) 21:17, 13 July 2016 (UTC)
And there have been many other experiments over the years. Notable recent one of course, Adriaan Fokker' organ - he was composing for 31 equal back in the 1940s?
Robert Walker ( talk) 10:10, 14 July 2016 (UTC)
...18:17 , which happens to make a very good prescription for placing the frets down the neck of a lute for equal temperament. It puts the octave shy of the string's midpoint by some 1/3 of 1% of the total length (comparable on a tenor lute to the width of the fret itself). This might be considered a defect from a certain theoretical point of view, but in reality 18:17 works better than twelfth root of 2 as the latter makes no allowance for the string's greater tension when it is pressed down to the fret. On a good instrument (that is, with a low action) the 18:17 rule renders the string just about long enough to compensate
To Robert Walker. Robert, I'll organize my answer around many quotations of yours. First:
Whether medieval musicians aimed for the Pythagorean tuning is not clear. It is true that the theorists mentioned only this tuning, but one may wonder whether singers really sang Pythagorean thirds, when music began to use thirds. As early as the 14th century, keyboardists had found what has been called (by Helmholtz or Ellis?) the "schismatic change" (Schismatische Verwechslung; it really is a commatic change) to avoid Pythagorean thirds, tuning the black keys as Pythagorean flats and using them as sharps in major thirds of 388 cents. This is documented, among others, in several papers by Mark Lindley. The problem with lute tunings may not have arisen long before, so that it is not at all sure that lutenists ever aimed for Pythagorean tuning.
Indeed, they could have calculated or constructed the square root of the square root of the cubic root. But I think it is quite known that they didn't. It certainly is known that Simon Stevin was the first to approximate the 12th root of 2 by continuous fractions, and Napier's logarithms were soon recognized, in the 17th century, to at last allow calculating it. This all produced a major change in "music" itself: it had been up to then another name for the science of ratios of integers and, as such, had belonged to the quadrivium; it was a science of discontinuous quantities. But the new mathematics changed that, music lost its place in the quadrivium (which disappeared altogether), it had to seek its models in the sciences of language (mainly rhetorics), and eventually was rejected from university curricula. There were other reasons, of course, but the developments of mathematics were part of it.
I do believe that the situation in 1661 was drastically different from that in 1555. Vicentino apparently was not aware that 31-note meantone approximated 31-ET. So far as I am aware, Salinas knew of an instrument giving 31-tone meantone, but did not know its maker; there is no "report on Vicentino's Archicembalo" by Salinas, nor any mention of it in Huygens who, in addition, apparently did not know Vicentino's book. (Be aware of this before you modify the Archicembalo article!) Huygens does refer to Salinas, who does refer to an Italian instrument, but neither mentions Vicentino. Vicentino himself did not think that his instrument could serve for extended meantone (i.e. an extended range of tonalities in meantone), but conceived it for playing microtones (1/2 or 1/3 semitones) for which he devised a special notation. Other similar instruments of the time more often aim at extended just intonation, which is quite a different matter. (Extended meantone would have made it possible to play "tonalities" that were very close in pitch and identical in all other respects. Vicentino aimed at playing what he would have considered "enharmonic tonalities", i.e. modes or keys returning to what he believed was the enharmonic system of the Greek.)
This is not by Helmholtz, it is found in the additions by Ellis in his English translation. But Helmholtz does mention the 53 division (e.g. p. 328 of the translation). This is Holder's (18th c.) or even Mercator's (late 16th c.) approximation of the Pythagorean tuning, with 9 "commas" (53th root of 2) in the tone, 5 in the chromatic semitone, 4 in the diatonic one; it had been known, conceptually, in Greek Antiquity. But this was theoretical speculation, which left few traces in actual music.
I don't think this goes back any farther than the 18th century (I could find the source, but not just now). The Cairo congres of 1932 advocated 24 equal quarter tones for Arabic maqam music, and I have had Arabic students who were convinced that this went back to the early Middle Ages...
The schisma was described long ago as the difference between the Pythagorean and the syntonic commas (this is not a definition, because schisma really means any very little interval), which can be calculated in a variety of ways. I am not convinced that 32805:32768 (or 3^8*5 : 2^15) makes this so much clearer.
I am not sure of what you mean here – as a matter of fact, I don't understand what "tempering-out" means. I suppose that what you mean is that 5-et is a temperament where the 5ths are tempered so that they reach unison in the end, instead of a diatonic semitone; that is, 5-et is a "positive" temperament where the 5ths are augmented by 1/5 diatonic semitone. I feel far-fetched to call this "tempering-out".
This author, Mark Lindley, is the one I mentioned above re the "schismatic change": you'll find his papers on this in his (long) list on Academia.edu. He is one of the best scholars on the history of tunings and may usually be trusted.
My point, il all this, is that the mathematical discourse about microtones and tunings appears to afford all kinds of exact quantifications, applied to a historical reality that never had any use of this kind of exactitude. One may get the illusion that mathematical definitions are more rigorous and therefore better than the earlier fuzzy definitions, but music, like any other language, cannot be dealt with in such terms. Tuning appears to be of paramount importance because it would allow an exact definition of the various notes, of their exact frequency, etc. But notes are first of all semiotic categories – while even these are today mistaken for classes (e.g. in pitch-class set theory). A semiotic category fully escapes mathematical definition. — Hucbald.SaintAmand ( talk) 11:39, 15 July 2016 (UTC)
@ Hucbald.SaintAmand: I've just had an idea. Wikipedia doesn't yet have an article on linear temperaments, it just redirects to regular temperament. So - as an interim measure, why not make this into a stub article on linear temperaments? I think I can do that quite easily, just rewrite it, explain what a linear temperament is. Then can say that for tempered fifths, you get twelve tone tunings but if you temper the fifth flat enough you will temper out the diatonic semitone and the result is five equal, and if you temper it sharp enough, you temper out the chromatic semitone, resulting in seven equal. Then say that Andrew Milne has coined the word "Syntonic temperament" to covert the continuum of tunings in this range. Whether that is notable enough to be mentioned can be a matter for further discussion but as an interim measure can do it like that. And I can also say that tunings with fifths sharper than pure are called schizmatic temperaments and flatter than pure called meantone temperaments - and add a "citations needed" for the schizmatic temperaments so hopefully someone can come up with the citations you've been looking for there.
Then can add a "please expand". As is, it is only mentioning a tiny subset of the linear temperaments, and experts can come in later on with temperaments based on generators other than a fifth, I know there are lots of those but am not expert on them :).
~The advantage is as an interim measure it involves hardly any work and we can do it a bit at a time. Even if the other articles still say "syntonic temperament" well after clicking through at least they will see syntonic temperament mentioned further down the page. And they won't puzzlingly find themselves on the meantone page which I think would be very baffling, if you are there already, or if for instance you are on some other schizmatic temperament type tuning, click on syntonic temperament and you end up thinking it is a form of meantone which might be very wrong. Then after doing that can go through and replace occurrences of "syntonic temperament" by "linear temperament" and at every point in the editing process then wikipedia's articles are linked together in a reasonably sensible way. What do you think? Robert Walker ( talk) 14:22, 13 July 2016 (UTC)
Margo Schulter has just suggested this off wiki. This is both
It would cover the range from 5-et to 7-et, but not including the two end points. The word "Regular" here is understood in the sense of a homomorphism map from the pythagorean diatpnic such that all intervals of the same type are tuned the same, wherever they occur in the tuning. As in Dynamic Intonation for Synthesizer Performance by Benjamin Frederick Denckla (1995). In other words any scale consisting of "tones" and "semitones" arranged in the sequence TTSTTTS and adding up to the octave with all the Ts tuned the same way and all the Ss tuned the same way counts as a regular diatonic tuning.
This also includes to all linear temperaments within Easely Blackwood's "Range of Recognizabilty" in his "The Structure of Recognizable Diatonic Tunings" for diatonic tunings with the fifth tempered to between 4/7 and 3/5 of an octave, with major and minor seconds both positive and the major second larger than the minor second, though his "range of recognizability" is more restrictive than "regular diatonic tuning", for instance requiring the diatonic semitone to be at least 25 cents in size. See Carlo Serafini's review for a summary.
This has the advantage that we need minimal rewrites of wikipedia, just replace "syntonic temperament" by "regular diatonic tuning" everywhere and rename the article, and is clearly noteworthy enough to deserve an article of its own I think. @ Hucbald.SaintAmand: - any thoughts on this? If there are no objections or other suggestions, I think I'll follow WP:BOLD and just do it, and rewrite the article also to make it all clearer. Robert Walker ( talk) 12:04, 19 July 2016 (UTC)
There is only one historical or logical definition of a "meantone", unless I am mistaken:
I see no historical or logical reason to limit the definition of (historical) meantone tunings to "negative" temperaments, i.e. those tempering the 5ths to flatter that 700 cents. Equal temperament is a meantone tuning, as is Pythagorean tuning and also the "positive" temperaments, producing 5ths sharper than 702 cents.
I very much dislike the idea of "tempering out" (or "distributing") the syntonic comma, which to me is meaningless. No historical temperament ever tempered anything else that the fifth. And a "meantone tuning" with 5ths larger than 700 cents widens the syntonic comma... A circulating 17-note "Pythagorean" tuning needs 5ths of 706 cents; it is "Pythagorean" in that the diatonic semitone is smaller than the chromatic one; but the major third still consists of two tones. Etc.
Hucbald.SaintAmand ( talk) 20:30, 24 July 2016 (UTC)
Robert, you assume that when I define the mean tone as "the mean of the major third", I mean "the mean of the pure third", but that is not the case. I mean that the mean is the mean of the particular major third of that particular temperament. 12-ET is a meantone because the tone (200 cents) is the mean of the tempered third (400 cents). [This means quite a lot of means, but I hope you can understand what I mean ;–))]
J. Murray Barbour writes (Tuning and temperament, p. 31): "Strictly, there is only one meantone temperament. But theorists have been inclined to lump together under that head all sorts of systems". And he later comes to equate meantone temperaments with what Bosanquet called "regular" temperaments in 1876, namely the temperaments with only one size of fifth. This also is what I understand by "meantone temperament". Barbour further describes 2/7-comma and 1/3-comma, then discusses the possibility that the temperament was progressively reduced from 1/4-comma to 1/11-[syntonic] comma, which is equal (he says) to 1/12-Pythagorean comma.
[By the way, the name of Bosanquet should probably be given in the Regular Diatonic Tunings article, as the originator of the term "regular".]
I always considered that the range of historical meantone temperaments extended from 1/3 to 0, i.e. 5ths from 695 to 702 cents. The possibility of negative temperament (fifths wider than pure) is a bit farfetched, yet a closed 17-ET, which strictly speaking may be considered a meantone and in any case a "regular" temperament, has 5ths of 706 cents. It is described by Sauveur as the "Arabian scale" and indeed was almost exactly al-Farabi's system in the 14th century. The 22-note division, which has been equated with the "Hindoo scale", is a regular temperament with 5ths of 709 cents. Note that not all multiple divisions produce "negative" 5ths, as Vicentino's 31 division evidences: it is a 1/4-comma meantone. But if 31-ET can be described as a meantone, I don't see why 17-ET or 22-ET (or any ET, for that matter) could not.
As to my mention of the semitone as a possible generator of the 12-note scale, it results from a reflection on the idea that the generators are the numbers that are prime to 12: 1, 5, 7 and 11 – the semitone and the fourth, and their inversions. Indeed, 12 has the property to have that many dividers, which makes it an excellent choice for combinatorial possibilities. But I admit that it is not extremely helpful here (it is helpful for a reflection on modes of limited transposition in 12-ET, because these obviously are based on the divisions of 12 in equal parts).
Hucbald.SaintAmand ( talk) 06:40, 25 July 2016 (UTC)
For an image of my conception of meantone temperaments, see Meantone.jpg, which also appears in Meantone temperament. Hucbald.SaintAmand ( talk) 08:09, 25 July 2016 (UTC)
(@ Hucbald.SaintAmand: - just to say finished editing this post. Robert Walker ( talk) 14:38, 26 July 2016 (UTC))
Hi Hucbald, been talking about this to Margo so far, anyway she agrees with you about the meantone being a logical name for the entire gamut of regular diatonic tunings. She prefers to use the word "eventone", I think to avoid confusion with the historical usage. Anyway she suggests, though I'd say we can probably use this informally on the talk pages only, not for the wikipedia articles unless we can establish that it is used in WP:RS for wikipedia - that one could generalize the idea by using commas that require fifths to be tempered wide instead of narrow. E.g.
etc.
In the xenharmonic wiki they also have a distinction between positive and negative temperaments, which puzzled me at first, as positive there means the fifth is wider than 700 cents, not 3/2. Margo says that she uses those words also though and gave an explanation that made sense to me. She tends to say tunings with fifths larger than 700 cents are "positive," in the sense that twelve fifths exceed 7 octaves (version of the pythagorean comma using the tempered fifths is positive), and corresponding sharps are higher than flats (e.g. C-Db-C#-D). That is true throughout the regular diatonic range of 700.0-720.0 cents
On the meantone / shizmatic then she says that the 700 cents is a logical transition point between the two as it is the point where a regular or meantone mapping of 5/4 (+4 fifths) and a schismatic mapping (-8 fifths) yield an identical 400 cents.
Then on the schismatic temperaments, for her, they occupy a narrow region around the 3/2.
It means that for example a fifth at 702.227 cents (0.272 cents or 1/14 of this septimal schisma wide) produces pure 7/4 minor sevenths from -14 fifths and large 8/7 tones.
BTW, hope you don't mind a minor correction. I'm saying this because I know if I was in your position I would want it and I think most academics are the same.
On the 17 tone tuning, she thinks you are probably thinking not of the ninth century theorist Al-Farabi (c. 870-950), but of Safi al-Din al-Urmawi (one possible transliteration), c. 1216-1294.
If so, it was a 17-note Pythagorean tuning , or alternatively, another system by the same theorist setting the Wusta Zalzal at 72/59 or 344.7 cents which is similar to Ibn Sina (c. 980-1037), based on a Wusta Zalzal at 39/32. None of these systems is close to 17-ed2, and they involve chains of just 3/2 fifths -- a single chain with Safi al-Din's Pythagorean scheme, or more than one chain in his alternative scheme with the Wusta Zalzal or middle third at 72/59.
I'm sure she'd be delighted to give more detail if you want to email her about it :).
BTW I know it's a little awkward that I'm kind of relaying what Margo says to you, but I think she is wise to not get too involved in editing wikipedia at this stage at least, and if commenting on the talk page it is rather close to editing the encyclopedia. The reason being that she is so close to the material and it makes it rather hard to be objective or to know where to stop and how much to put into wikipedia or to make judgements about what is OR.
You must have some of the same issues yourself. I have that with some other topics in wikipedia, that it takes care because I am so close to it myself. But not so much here as I'm not at all involved in tuning theory development myself. I find the maths interesting, but it has moved way beyond any possibility of me contributing to the development of the maths itself unless I dedicated a fair bit of time to it, which I prefer to spend on other things like programming, writing my articles etc, given how many capable tuning theorists there are. And interested in the history in a general way also but that's not for me either to immerse myself in the historical part of the subject either.
So I feel I can be reasonably objective / neutral in wikipedia in this topic area, as much as anyone could be anyway. Robert Walker ( talk) 01:22, 26 July 2016 (UTC)
Hucbald, okay a few remarks again.
"Negative System — A regular system whose fifth has a ratio smaller than 3:2. Positive System — A regular system whose fifth has a ratio larger than 3:2"
Since this page is summarized elsewhere in wikipedia using the idea of a syntonic temperament, we need a short summary we can use in its place. I've tried this out in the equal temperament page here to get us started: Equal temperament#Regular Diatonic Tunings.
I've also done a shorter version in the page on Musical Tunings here: Musical tuning#Systems for the twelve-note chromatic scale and have updated the Template:Musical tuning.
Eventually we should change all the pages that link here so that they refer to these tunings as regular diatonic tunings instead of syntonic temperament as per the discussion above. I've done a minor edit along those lines here 31 equal temperament. It's easiest to find those pages with a google site search for "syntonic temperament" as a "what links here" check lists all the pages which use the Template:Musical tuning Robert Walker ( talk) 02:06, 24 September 2016 (UTC)
"A chain of three fifths generates a minor third (A, D, G, C)" This is a chain of three fourths, isn't it? 79.79.171.35 ( talk) 13:07, 3 January 2017 (UTC)
"A chain of four equal sized fifths (E.g C, G, D, A, E) generates a major third consisting of two whole tones
79.79.170.213 ( talk) 13:24, 7 January 2017 (UTC)A chain of three fourths generates a minor third (A, D, G, C)"
"This paper describes Dynamic Tonality, a system of realtime alterations to tuning and timbre that extends the framework of tonality to include new structural resources such as polyphonic tuning bends, tuning progressions, and temperament modulations. These new resources could prepare art music for the 21st Century"
Oh, just re-read what you said about "Valid Tuning Range", I get what you are saying there. Though the valid tuning range is a different thing from the range over which the syntonic comma is customarily used to define tunings. As after all it requires use of negative amounts of the comma apart from anything else. As for the scope of this page I lead out with a definition that the reader would find easy to understand which also makes it easy to understand why it is called a "regular diatonic tuning":
"any musical scale consisting of "tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's."
That's a well defined continuum of tunings, and I'm not sure therefore how one can be unsure what it is about? I then go on to give the more technical definition from the thesis in which "Regular Diatonic Tuning" is defined, as
"Regular" here is understood in the sense of a mapping from Pythagorean diatonic such that all the interval relationships are preserved".
They could be defined in many other ways too. But it's the same continuum of tunings, no matter how you define them. Just different definitions which mark out the same entities. I'm trained as a mathematician, so it might be that I am coming to this from a slightly different perspective, so that things that seem identical to me look different to others? For me the continuum of tunings is what this page is about, and how you define them doesn't really matter so long as you mark out the same tunings, and it just seemed that in an article addressed to the ordinary non mathematical reader, it's best to start off with the simplest easiest to understand definition. And of all the definitions, the TTS TTTS definition seemed by far the easiest to both state and to understand. That's the motivation for doing it that way. Robert Walker ( talk) 03:56, 31 March 2017 (UTC)
I just added this which may help with exposition and to tie the article together:
"In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size (TTTS or a permutation of that) which makes it a Linear temperament with the tempered fifth as a generator."
It's one of those things that seems self evident, to anyone with the musical background to understand the construction of a diatonic scale, like "2 + 2 = 4" with just a bit of reflection so haven't bothered with either citing it or giving a formal proof. Robert Walker ( talk) 01:02, 2 April 2017 (UTC)
Gentlepersons,
Let me again thank you all for your learned attention to the matter of the "syntonic temperament." Again, let me emphasize that the lead authors of the seminal papers -- Prof. William Sethares and Dr. Andrew Milne -- both know infinitely more than I do about both music and mathematics. I am a mere marketing slime, who happened to read about the Wicki-Hayden keyboard and realized that its having "the same fingering in all keys" HAD TO BE the result of some correspondence between the geometric pattern of the notes and the deep structure of music. Reading Sethares' book "Tuning, Timbre, Spectrum, Scale" introduced me to the notion that consonance arose solely from the alignment of a tuning's notes with a timbre's partials. Everything else arose from those two observations, with nearly all of the heavy lifting having been done by Milne and Sethares (et al.) while I kibbitzed. So, whenever either of them corrects me, they are right, and I am wrong.
That said, please allow me to propose that Wikipedia's stand-alone article on the syntonic temperament should be restored, with only a brief summary of it included in Wikipedia's article on Regular diatonic tunings.
Here's why: The inclusion of the syntonic temperament in the "regular diatonic tunings" article is problematic, because it assumes that all tunings of the syntonic temperament are regular and diatonic, when, in fact, both of these assumptions are false.
First, "diatonic:" The valid tuning range of the syntonic temperament includes an endpoint at which the minor second has a width of zero (5-tet) and another endpoint at which it has a width equal to the major second (7-tet), neither of which are diatonic.
Second, "regular:" Consider the discussion of "Related Just Intonations" on the pages numbered as 79-80 in Spectral tools for Dynamic Tonality and audio morphing, which describes the Tone Diamond used in Dynamic Tonality synthesizers. Moving the control-point within the Tone Diamond systematically shifts the tuning away from regularity. To quote a paragraph on the page labeled as 80:
The syntonic temperament is therefore not, strictly speaking, a regular tuning, nor a family of regular tunings, nor even a regular tuning system. It embraces closely-related irregular tunings, too.
In summary: because the syntonic temperament embraces tunings that are not regular and not diatonic, having Wikipedia's only discussion of the syntonic temperament be subsumed into Wikipedia's article on "Regular diatonic tunings" is illogical and inappropriate. It attempts to squeeze a hypercubic discussion into a cubic hole.
That's why I initially put my description of the Syntonic temperament into its own Wikipedia article, and then added references from other articles to it (and vice versa), rather than attempting to shoehorn it into any existing article. This "where to put this discussion?" problem is just one small piece of evidence that the ideas embodied Dynamic Tonality, and hence in Musica Facta (the research project into Dynamic Tonality), the syntonic temperament, and Guido 2.0 (the aspect of Musica Facta focused on music education) may be incommensurable with those of music-making's current paradigm.
I apologize for not presenting these arguments when the idea first arose of merging the discussion of the syntonic temperament article into some other pre-existing article.
I hereby propose reverting the article on the Syntonic temperament to being stand-alone, and adding a brief summary of it, with a reference to the full article, in the Regular diatonic tunings article.
Respectfully,
Jim Plamondon -- JimPlamondon ( talk) 06:44, 26 April 2020 (UTC)
Gentlepersons,
Thank you again for your continued attention to this matter. :-)
In the Talk above, people have repeatedly questioned the relationship between the syntonic temperament and timbre. Temperament, they say, has nothing to do with timbre; the linking of the two, which runs throughout Sethares, Milne, Plamondon, Prechtl, Tiedje, et al., they see as aberrant.
I think I now understand where y'all are coming from. I suspect that the microtonal community has assumed the use of Just Intonation and Harmonic Timbres for so long that they have forgotten that they are even making an assumption. They often write about ratios of small whole numbers, as Pythagoras did, as if the ratios themselves were the source of tonal meaning -- when it has been abundantly clear since Joseph Sauveur, or, for the doubtful, since Helmholtz, and inescapably since Plomp & Levelt that these intervals arise from the specific pattern of vibrations defined by the Harmonic Series, not from some magic of numerology.
Alternatively put: Sethares, Milne, Plamondon, et al. are explicitly relating tempered pseudo-Just tunings and tempered pseudo-Harmonic timbres, whereas microtonalists and previous music theoreticians have implicitly related [un-]tempered Just tunings and untempered Harmonic timbres.
Examples of the above implicit assumptions in the Talk section of this article:
Sethares, Milne, Plamondon, Prechtl, Tiedje, et al. have offered something that is truly novel, and y'all seem to be working very hard to deprecate its novelty and force it into the lower-dimensional framework of what has historically been done. Our work COMBINES:
All of these innovations have been documented in the peer-reviewed literature -- and some non-peer-reviewed technical reports -- for over a decade. Our Computer Music Journal article was still, the last time I checked, the most-often downloaded article in its history. Our "sight reading music theory" article has been downloaded from ResearchGate over 3500 times. To claim that these innovations, or the ideas that they document, or the terms that they use, are not "established," begs the question of what definition of "established" you are using.
When combined, our system's components enable a systematic expansion of the frontiers of tonality and novel tonal effects. However, each component by itself is of little particular interest. How useful is an automotive transmission without an engine? A computer without electricity? A browser without the Internet? So, our explanation of each component explicitly refers to the other components.
Y'all, I suspect, are not making such explicit connections among your uses of "temperament," "timbre," etc., because youimplicitly assume:
In summary, we are explicitly stating relationships that you are assuming implicitly.
The solution is to update all of the other articles on microtonality and music theory to make their implicit assumptions explicit.
Y'all are resisting this. Your resistance is manifesting itself in an attempt to shoehorn our innovations into a box defined by your lower-dimensional implicit assumptions, in which the interconnections and synergies must be stripped away to fit in those lower-dimension boxes. This looks to me to be a clear-cut case of the Semmelweis Reflex. That reflex is a natural human and institutional reaction to novel ideas, so I don't blame y'all in any way for exhibiting it. I have every confidence that I am displaying similar human reflexes and cognitive biases. :-)
I am happy to see that the discussion surrounding the original Syntonic temperament article led to the creation of the Regular diatonic tunings article, which is a boon to Wikipedia.
As a first step towards resolving the "problem of assumptions" described above (and the fact that the Syntonic temperament includes non-regular and non-diatonic tunings), I will de-merge the Syntonic temperament article out of the Regular diatonic tunings article sometime in early June 2020.
In the meantime, I will add to the article on Dynamic Tonality a section that discusses (a) the implicitly-assumed relationships described above, and (b) the explicitly-defined relationships made by Dynamic Tonality. I can them update the discussion of the Syntonic Temperament to refer to this section in the Dynamic Tonality article. I can then add a reference to this same "implicit assumptions" section to all other articles on traditional music theory. This will clarify the distinction between Dynamic Tonality and everything that came before.
That should resolve all of the issues raised in this Talk page, and improve Wikipedia's discussion of music theory considerably.
I look forward to your comments on this suggestion.
Respectfully, -- JimPlamondon ( talk) 04:49, 2 May 2020 (UTC)
Maybe about 80% of this article seems to correspond to a fringe theory by user JimPlamondon. This is against wikipedia's philosophy, as you can see here. In the fringe theories page. This has nothing to do with whether I agree or disagree with this theory, it is an objective fact that this is not broadly supported and as such should be deleted. Wikipedia is not a place for promoting your own theories, it should reflect what is broadly acceped in a field, there are many other websites that can be used to promote or support this research.
In a nutshell, as my reference states: To maintain a neutral point of view, an idea that is not broadly supported by scholarship in its field must not be given undue weight. I don't think this deserves extra discussion, and JimPlamondon's own comments on the matter make this very clear. I will procede to delete everything associated with Dynamic Tonality IgnacioPickering ( talk) 23:52, 17 July 2021 (UTC)
Ignacio, et al.: Wikipedia states that "fringe theories in science depart significantly from mainstream science and have little or no scientific support." The content that you deleted is, by that definition, not a fringe theory, in that it is well-supported by numerous publications in the peer-reviewed scientific literature. Indeed, the content that you deleted contains more references (7) that the content that you did not delete (4).
This leads to the second criteria for identifying fringe theories: Parity of sources. Four of the seven sources cited in the content you deleted are to respected peer-reviewed publications, such as the Computer Music Journal (H-Index: 41), the Journal of Mathematics and Music (H-Index: 14), and the Proceedings of the College Music Society. Only three of the seven sources are not similarly peer-reviewed. By comparison, of the four sources cited in the non-deleted content, the first refers to a master's thesis; the second to a non-peer-reviewed book; the third to a non-peer-reviewed review of said book; the fourth to a non-peer-reviewed wiki post. None but the first of these four sources can credibly be claimed to be peer-reviewed. The other three can reasonably be categorized as "self-published texts," which Wikipedia's description of fringe theories cites as a sign of fringe theories. In short: The deleted content is demonstrably less fringy than the non-deleted content.
Please note also that publications on the syntonic temperament and Dynamic Tonality are cited frequently, according Google Scholar:
Paper | Citations |
---|---|
Spectral tools for dynamic tonality and audio morphing | 52 |
Isomorphic controllers and dynamic tuning: Invariant fingering over a tuning continuum | 51 |
Tuning continua and keyboard layouts | 39 |
Dynamic tonality: Extending the framework of tonality into the 21st century | 7 |
This is an impressive level of citation for papers in the humanities (such as music theory), in which 65% of papers are never cited. These citations prove the "broad acceptance" of the content that you deleted.
I would go a step further, and point out that many of the papers supporting the syntonic temperament and Dynamic Tonality include mathematical proofs of their claims. Such proofs are the gold standard for claims made about models of phenomena in the natural world, such as musical tunings. The citations supporting the non-deleted content contain no such mathematical proofs. Therefore, the theory of the non-deleted content is less strongly supported by its citations than is the content that you deleted.
Finally, as Wikipedia's discussion of fringe theories states, "One important barometer for determining the notability and level of acceptance of fringe ideas related to science, history or other academic pursuits is the presence or absence of peer-reviewed research on the subject." As shown above, the content that you deleted meets this standard much better than the content that you did not delete.
Therefore, the deleted content should have been retained, and the non-deleted content should have been deleted as a fringe theory.
However, Ignacio, I am not unreasonable. I will not press for the deletion of the obviously-fringy theory represented by the content that you did not delete. I will only insist that you follow Wikipedia's rules fairly, and therefore not delete the material on the syntonic temperament and Dynamic Tonality. -- JimPlamondon ( talk) 03:27, 18 July 2021 (UTC)
The article claims: "The small difference in pitch between [a chromatic and diatonic semitone] is called a comma, usually prefixed by the name of the tuning system that generates it, such as a syntonic comma (21.5 ¢)..."
This is clearly incorrect. While the Pythagorean comma does work out such a difference, as seen on the corresponding page, the syntonic comma is not the difference between the chromatic (25:24) and diatonic (16:15) semitones of just intonation, equivalently the minor second and augmented unison - that yields 128:125, i.e. the (enharmonic) diesis.
The syntonic comma (81:80) - the one referred to when one speaks of "fractional-comma meantone" - is a ratio between the major tone (9:8) and minor tone (10:9) of just intonation - which is not a regular diatonic tuning, as it uses two different tones. Equivalently, it's the difference between a just-intonation ditone and major third.
216.209.47.135 ( talk) 07:57, 31 January 2024 (UTC)
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![]() | This article was nominated for merging with Meantone temperament on October 2010. The result of the discussion was oppose. |
Help me out here. Is this article supposed to be about one tuning called "Syntonic temperament", or is it supposed to be about a system for generating a continuum of different tunings? It seems to be blurring those concepts. The title and opening statement suggest one particular tuning, but the illustration and the rest of the body text suggest multiple tunings. 108.60.216.202 ( talk) 06:16, 25 May 2015 (UTC)
This page seems to be describing what is normally called a " Meantone temperament" To quote from the website [1]
"
But the chart shown includes tunings with the fifth wider than the pure fifth. So it's hard to be sure what it is about. The more general term here is a Linear temperament - in other words any rank 2 regular temperament with one of the generators the octave. In its most general case the other generator in a Linear temperament doesn't have to be a fifth, but can be of any size. So, I think there are two options here, really. We could just merge this with meantone, or we could merge it with Linear temperament which currently redirects to Regular temperament. As it stands, syntonic temperament seems to be a term coined by a single author and only used on the MusicaFacta website. Robert Walker ( talk) 17:05, 12 July 2016 (UTC)
#redirect [[Linear temperament]]
#redirect [[Meantone temperament]]
Let me answer you here for the time being, before I make my objections on your Wikipedia:WikiProject Council/Proposals/Microtonal Music, Tuning, Temperaments and Scales page, about which I have to think further. I am a historian of music theory, which means that I not only considered the history of temperaments, but also that of non European tunings e.g. in Oriental (maqam) music. My feeling is that microtonal music forms a very little part of that ensemble (it would be a gross mistake to consider maqam music as microtonal, even although the mistake is at times done by Arabic musicians). I believe that the confusion made here, calling "syntonic temperament" something that boils down to being (extended) meantone temperament is a case in point. Microtonal websites (such as those concerned with xenharmonic music) often appear to me as playgrounds for arithmetic games (which may be interesting if only because they turn back to the Antique Pythagorean conception of music as another name for arithmetics), but which have little to do with real music. As you can see, I am an old-fashioned theorist; but I keep up to it.
To turn back to the matter of merging or deleting this article, I had thought we should leave a chance to the original author(s) to react to the proposition. I suppose that after now about sixth months, they had their chance and ditn't take it. Then remains the problem that several other articles may need corrections as a result of the move. Some of my ideas (and a few of my mistakes) about all this can be read on the Talk:Schismatic_temperament page.
Hucbald.SaintAmand ( talk) 09:31, 13 July 2016 (UTC)
Robert, you write:
Well, we obviously do not share the same idea of what microtonal might be. To say that anything different from ET temperament is microtonal makes no sense to me because ET is at best an utopy. It has so to say never been used in real music, unless with electronic instruments. Even pianos have octaves that are slightly too large, because of the inharmonicity of stiff strings; guitars the same. Wind instruments hardly could play in true ET and stringed instruments usually adapt their intonation to the context.
One might consider that microtonal music is that which uses more than 12 tones in the octave. But that, of course, would exclude the musics that you quote, maqam, medieval or Gamelan. If more different notes are used in their heptatonc (or possibly pentatonic) scales, it merely results from the notes being mobile; and to consider a movable note as consisting in several different notes is a very dangerous stretch of the mind. To consider Pythagorean tuning a microtonal tuning seems to me counterproductive, because it merges to saying (with Julen Torma, "Euphorismes", in Écrits définitivement incomplets) that "Everything is the same thing. Everything therefore is very sufficiently good". To consider that the various tunings in which Bach's Well tempered keyboard has been played (or could be played) as variously microtonal is utterly problematic.
Any definition of "tuning" should take account of the fact that many musics make use of mobile (or movable) notes – that is, notes that are not tuned stricto sensu. We would easily agree, I trust, about what a temperament is: in addition to make use of tempered intervals, it is also meant for instruments of fixed tones, producing a limited number of tones. From the present discussion, it appears to me that a tuning too must concern limited numbers of tones. Violin strings are tuned (usually in pure or just fifths); but can one say that the notes played also are tuned (especially if the open strings are not actually played)? In what kind of tuning does a violin play? (Or a maqam player, or a singer, etc.?)
You may consider that this is a wrong way to ask the question, but I don't think so. I think that these questions are not enough considered, that they are rejected as apparently futile or unduly complex. But in the end, they cannot be avoided, and this is one of my main objections against the whole subject of "microtonal music" as it is too often presented. There is a lot to add, and I'd gladly continue the discussion if you feel like it; but I will not bother you further if you don't. — Hucbald.SaintAmand ( talk) 13:00, 13 July 2016 (UTC)
Yes, Robert, 12-ET is an interesting theoretical case. It has been around, conceptually, since Antiquity, since when one decided that the tone could be divided into two semitones, implying that the two halves could be equal. But I don't follow you when you write that
Medieval lutes at times were tuned to a good approximation of ET, but their tuning was not a temperament. One made use of the 18:17 ratio, which amounts to 98.95 cents, but is an integer ratio nevertheless. There was no way to conceive a true tempered semitone, nor to geometrically represent it, before the late 16th century and Simon Stevin's continuous fractions (and later, of course, logarithms). Also, fretted instruments at times had double frets for the two enharmonic notes (i.e. they made a distinction between the diatonic and chromatic semitones). This was the case especially with larger instruments (viola da gamba); for shorter ones, strings were slightly pulled sideways while playing to obtain different intervals. ET was approximated on the lute because the frets, crossing all strings in one straight line, could not make the distinction between the two forms of the semitone (diatonic and chromatic), but certainly not because it was "the simplest way to fret": it certainly was not. And players often used expedients in order not to have to play in ET. This remained true even in Bach's time, when theorists (and Bach himself) advocated against ET because it would have reduced the difference between the various keys. To say that lutes were tuned in ET isn't really wrong, but it certainly is anachronic. [Margo Schulter has an expertise on medieval tunings, indeed, but not every scholar on this topic, me included, always agrees with her.]
Well, if we agree that 12-ET is more conceptual than real, the same can be said of any other ET. To say that a singer sings in 7-ET, or 17-ET, or 19-ET, really is a matter of conception. 19-ET, in particular, can so to say not be differentiated from 1/3-comma meantone (taking two units for the diatonic semitone): one hardly could say whether anyone sings in 1/3-comma meantone or in 19-ET, unless one knows her or his conception. There is a tendency, among, say, Xenharmonicians, to reduce everything to xx-ET and to precisely quantify microintervals. This is not wrong in itself, but it gives a rather distorted image or the realities of former times.
This again may be true conceptually, but it certainly was not thought so in former times because an infinite number of notes was unthinkable. (Infinite mathematical series again belong with continuous fractions and logarithms.) Meantone tunings were conceived at most with 17 notes, that is the diatonic scale + 5 sharps and 5 flats. The idea of double sharps and double flats is a recent one, and Xenharmonicians are way beyond that. Take a look at List_of_pitch_intervals#List: one could argue that no singable or playable interval is absent of the list; but to claim that sung or played intervals in real music do reflect those of this list is, to me, an aberration – and, for the ordinary Wikipedia readers, a misleading. Using integer frequency ratios implicitly claims that these are ratios between harmonic overtones; but what sense is there to imagine harmonic numbers in three figures and more, when hearing is concerned?
This all to say that I have problems, as a historian of music theory, to accept the Wikipedia articles about which we are speaking as vulgarizing concepts of earlier times. I don't mean that these articles are wrong, nor do I have excessive difficulties understanding what they do. I merely mean that I don't think they provide the simple explanations that the average Wikipedia reader is hoping to find. — Hucbald.SaintAmand ( talk) 21:17, 13 July 2016 (UTC)
And there have been many other experiments over the years. Notable recent one of course, Adriaan Fokker' organ - he was composing for 31 equal back in the 1940s?
Robert Walker ( talk) 10:10, 14 July 2016 (UTC)
...18:17 , which happens to make a very good prescription for placing the frets down the neck of a lute for equal temperament. It puts the octave shy of the string's midpoint by some 1/3 of 1% of the total length (comparable on a tenor lute to the width of the fret itself). This might be considered a defect from a certain theoretical point of view, but in reality 18:17 works better than twelfth root of 2 as the latter makes no allowance for the string's greater tension when it is pressed down to the fret. On a good instrument (that is, with a low action) the 18:17 rule renders the string just about long enough to compensate
To Robert Walker. Robert, I'll organize my answer around many quotations of yours. First:
Whether medieval musicians aimed for the Pythagorean tuning is not clear. It is true that the theorists mentioned only this tuning, but one may wonder whether singers really sang Pythagorean thirds, when music began to use thirds. As early as the 14th century, keyboardists had found what has been called (by Helmholtz or Ellis?) the "schismatic change" (Schismatische Verwechslung; it really is a commatic change) to avoid Pythagorean thirds, tuning the black keys as Pythagorean flats and using them as sharps in major thirds of 388 cents. This is documented, among others, in several papers by Mark Lindley. The problem with lute tunings may not have arisen long before, so that it is not at all sure that lutenists ever aimed for Pythagorean tuning.
Indeed, they could have calculated or constructed the square root of the square root of the cubic root. But I think it is quite known that they didn't. It certainly is known that Simon Stevin was the first to approximate the 12th root of 2 by continuous fractions, and Napier's logarithms were soon recognized, in the 17th century, to at last allow calculating it. This all produced a major change in "music" itself: it had been up to then another name for the science of ratios of integers and, as such, had belonged to the quadrivium; it was a science of discontinuous quantities. But the new mathematics changed that, music lost its place in the quadrivium (which disappeared altogether), it had to seek its models in the sciences of language (mainly rhetorics), and eventually was rejected from university curricula. There were other reasons, of course, but the developments of mathematics were part of it.
I do believe that the situation in 1661 was drastically different from that in 1555. Vicentino apparently was not aware that 31-note meantone approximated 31-ET. So far as I am aware, Salinas knew of an instrument giving 31-tone meantone, but did not know its maker; there is no "report on Vicentino's Archicembalo" by Salinas, nor any mention of it in Huygens who, in addition, apparently did not know Vicentino's book. (Be aware of this before you modify the Archicembalo article!) Huygens does refer to Salinas, who does refer to an Italian instrument, but neither mentions Vicentino. Vicentino himself did not think that his instrument could serve for extended meantone (i.e. an extended range of tonalities in meantone), but conceived it for playing microtones (1/2 or 1/3 semitones) for which he devised a special notation. Other similar instruments of the time more often aim at extended just intonation, which is quite a different matter. (Extended meantone would have made it possible to play "tonalities" that were very close in pitch and identical in all other respects. Vicentino aimed at playing what he would have considered "enharmonic tonalities", i.e. modes or keys returning to what he believed was the enharmonic system of the Greek.)
This is not by Helmholtz, it is found in the additions by Ellis in his English translation. But Helmholtz does mention the 53 division (e.g. p. 328 of the translation). This is Holder's (18th c.) or even Mercator's (late 16th c.) approximation of the Pythagorean tuning, with 9 "commas" (53th root of 2) in the tone, 5 in the chromatic semitone, 4 in the diatonic one; it had been known, conceptually, in Greek Antiquity. But this was theoretical speculation, which left few traces in actual music.
I don't think this goes back any farther than the 18th century (I could find the source, but not just now). The Cairo congres of 1932 advocated 24 equal quarter tones for Arabic maqam music, and I have had Arabic students who were convinced that this went back to the early Middle Ages...
The schisma was described long ago as the difference between the Pythagorean and the syntonic commas (this is not a definition, because schisma really means any very little interval), which can be calculated in a variety of ways. I am not convinced that 32805:32768 (or 3^8*5 : 2^15) makes this so much clearer.
I am not sure of what you mean here – as a matter of fact, I don't understand what "tempering-out" means. I suppose that what you mean is that 5-et is a temperament where the 5ths are tempered so that they reach unison in the end, instead of a diatonic semitone; that is, 5-et is a "positive" temperament where the 5ths are augmented by 1/5 diatonic semitone. I feel far-fetched to call this "tempering-out".
This author, Mark Lindley, is the one I mentioned above re the "schismatic change": you'll find his papers on this in his (long) list on Academia.edu. He is one of the best scholars on the history of tunings and may usually be trusted.
My point, il all this, is that the mathematical discourse about microtones and tunings appears to afford all kinds of exact quantifications, applied to a historical reality that never had any use of this kind of exactitude. One may get the illusion that mathematical definitions are more rigorous and therefore better than the earlier fuzzy definitions, but music, like any other language, cannot be dealt with in such terms. Tuning appears to be of paramount importance because it would allow an exact definition of the various notes, of their exact frequency, etc. But notes are first of all semiotic categories – while even these are today mistaken for classes (e.g. in pitch-class set theory). A semiotic category fully escapes mathematical definition. — Hucbald.SaintAmand ( talk) 11:39, 15 July 2016 (UTC)
@ Hucbald.SaintAmand: I've just had an idea. Wikipedia doesn't yet have an article on linear temperaments, it just redirects to regular temperament. So - as an interim measure, why not make this into a stub article on linear temperaments? I think I can do that quite easily, just rewrite it, explain what a linear temperament is. Then can say that for tempered fifths, you get twelve tone tunings but if you temper the fifth flat enough you will temper out the diatonic semitone and the result is five equal, and if you temper it sharp enough, you temper out the chromatic semitone, resulting in seven equal. Then say that Andrew Milne has coined the word "Syntonic temperament" to covert the continuum of tunings in this range. Whether that is notable enough to be mentioned can be a matter for further discussion but as an interim measure can do it like that. And I can also say that tunings with fifths sharper than pure are called schizmatic temperaments and flatter than pure called meantone temperaments - and add a "citations needed" for the schizmatic temperaments so hopefully someone can come up with the citations you've been looking for there.
Then can add a "please expand". As is, it is only mentioning a tiny subset of the linear temperaments, and experts can come in later on with temperaments based on generators other than a fifth, I know there are lots of those but am not expert on them :).
~The advantage is as an interim measure it involves hardly any work and we can do it a bit at a time. Even if the other articles still say "syntonic temperament" well after clicking through at least they will see syntonic temperament mentioned further down the page. And they won't puzzlingly find themselves on the meantone page which I think would be very baffling, if you are there already, or if for instance you are on some other schizmatic temperament type tuning, click on syntonic temperament and you end up thinking it is a form of meantone which might be very wrong. Then after doing that can go through and replace occurrences of "syntonic temperament" by "linear temperament" and at every point in the editing process then wikipedia's articles are linked together in a reasonably sensible way. What do you think? Robert Walker ( talk) 14:22, 13 July 2016 (UTC)
Margo Schulter has just suggested this off wiki. This is both
It would cover the range from 5-et to 7-et, but not including the two end points. The word "Regular" here is understood in the sense of a homomorphism map from the pythagorean diatpnic such that all intervals of the same type are tuned the same, wherever they occur in the tuning. As in Dynamic Intonation for Synthesizer Performance by Benjamin Frederick Denckla (1995). In other words any scale consisting of "tones" and "semitones" arranged in the sequence TTSTTTS and adding up to the octave with all the Ts tuned the same way and all the Ss tuned the same way counts as a regular diatonic tuning.
This also includes to all linear temperaments within Easely Blackwood's "Range of Recognizabilty" in his "The Structure of Recognizable Diatonic Tunings" for diatonic tunings with the fifth tempered to between 4/7 and 3/5 of an octave, with major and minor seconds both positive and the major second larger than the minor second, though his "range of recognizability" is more restrictive than "regular diatonic tuning", for instance requiring the diatonic semitone to be at least 25 cents in size. See Carlo Serafini's review for a summary.
This has the advantage that we need minimal rewrites of wikipedia, just replace "syntonic temperament" by "regular diatonic tuning" everywhere and rename the article, and is clearly noteworthy enough to deserve an article of its own I think. @ Hucbald.SaintAmand: - any thoughts on this? If there are no objections or other suggestions, I think I'll follow WP:BOLD and just do it, and rewrite the article also to make it all clearer. Robert Walker ( talk) 12:04, 19 July 2016 (UTC)
There is only one historical or logical definition of a "meantone", unless I am mistaken:
I see no historical or logical reason to limit the definition of (historical) meantone tunings to "negative" temperaments, i.e. those tempering the 5ths to flatter that 700 cents. Equal temperament is a meantone tuning, as is Pythagorean tuning and also the "positive" temperaments, producing 5ths sharper than 702 cents.
I very much dislike the idea of "tempering out" (or "distributing") the syntonic comma, which to me is meaningless. No historical temperament ever tempered anything else that the fifth. And a "meantone tuning" with 5ths larger than 700 cents widens the syntonic comma... A circulating 17-note "Pythagorean" tuning needs 5ths of 706 cents; it is "Pythagorean" in that the diatonic semitone is smaller than the chromatic one; but the major third still consists of two tones. Etc.
Hucbald.SaintAmand ( talk) 20:30, 24 July 2016 (UTC)
Robert, you assume that when I define the mean tone as "the mean of the major third", I mean "the mean of the pure third", but that is not the case. I mean that the mean is the mean of the particular major third of that particular temperament. 12-ET is a meantone because the tone (200 cents) is the mean of the tempered third (400 cents). [This means quite a lot of means, but I hope you can understand what I mean ;–))]
J. Murray Barbour writes (Tuning and temperament, p. 31): "Strictly, there is only one meantone temperament. But theorists have been inclined to lump together under that head all sorts of systems". And he later comes to equate meantone temperaments with what Bosanquet called "regular" temperaments in 1876, namely the temperaments with only one size of fifth. This also is what I understand by "meantone temperament". Barbour further describes 2/7-comma and 1/3-comma, then discusses the possibility that the temperament was progressively reduced from 1/4-comma to 1/11-[syntonic] comma, which is equal (he says) to 1/12-Pythagorean comma.
[By the way, the name of Bosanquet should probably be given in the Regular Diatonic Tunings article, as the originator of the term "regular".]
I always considered that the range of historical meantone temperaments extended from 1/3 to 0, i.e. 5ths from 695 to 702 cents. The possibility of negative temperament (fifths wider than pure) is a bit farfetched, yet a closed 17-ET, which strictly speaking may be considered a meantone and in any case a "regular" temperament, has 5ths of 706 cents. It is described by Sauveur as the "Arabian scale" and indeed was almost exactly al-Farabi's system in the 14th century. The 22-note division, which has been equated with the "Hindoo scale", is a regular temperament with 5ths of 709 cents. Note that not all multiple divisions produce "negative" 5ths, as Vicentino's 31 division evidences: it is a 1/4-comma meantone. But if 31-ET can be described as a meantone, I don't see why 17-ET or 22-ET (or any ET, for that matter) could not.
As to my mention of the semitone as a possible generator of the 12-note scale, it results from a reflection on the idea that the generators are the numbers that are prime to 12: 1, 5, 7 and 11 – the semitone and the fourth, and their inversions. Indeed, 12 has the property to have that many dividers, which makes it an excellent choice for combinatorial possibilities. But I admit that it is not extremely helpful here (it is helpful for a reflection on modes of limited transposition in 12-ET, because these obviously are based on the divisions of 12 in equal parts).
Hucbald.SaintAmand ( talk) 06:40, 25 July 2016 (UTC)
For an image of my conception of meantone temperaments, see Meantone.jpg, which also appears in Meantone temperament. Hucbald.SaintAmand ( talk) 08:09, 25 July 2016 (UTC)
(@ Hucbald.SaintAmand: - just to say finished editing this post. Robert Walker ( talk) 14:38, 26 July 2016 (UTC))
Hi Hucbald, been talking about this to Margo so far, anyway she agrees with you about the meantone being a logical name for the entire gamut of regular diatonic tunings. She prefers to use the word "eventone", I think to avoid confusion with the historical usage. Anyway she suggests, though I'd say we can probably use this informally on the talk pages only, not for the wikipedia articles unless we can establish that it is used in WP:RS for wikipedia - that one could generalize the idea by using commas that require fifths to be tempered wide instead of narrow. E.g.
etc.
In the xenharmonic wiki they also have a distinction between positive and negative temperaments, which puzzled me at first, as positive there means the fifth is wider than 700 cents, not 3/2. Margo says that she uses those words also though and gave an explanation that made sense to me. She tends to say tunings with fifths larger than 700 cents are "positive," in the sense that twelve fifths exceed 7 octaves (version of the pythagorean comma using the tempered fifths is positive), and corresponding sharps are higher than flats (e.g. C-Db-C#-D). That is true throughout the regular diatonic range of 700.0-720.0 cents
On the meantone / shizmatic then she says that the 700 cents is a logical transition point between the two as it is the point where a regular or meantone mapping of 5/4 (+4 fifths) and a schismatic mapping (-8 fifths) yield an identical 400 cents.
Then on the schismatic temperaments, for her, they occupy a narrow region around the 3/2.
It means that for example a fifth at 702.227 cents (0.272 cents or 1/14 of this septimal schisma wide) produces pure 7/4 minor sevenths from -14 fifths and large 8/7 tones.
BTW, hope you don't mind a minor correction. I'm saying this because I know if I was in your position I would want it and I think most academics are the same.
On the 17 tone tuning, she thinks you are probably thinking not of the ninth century theorist Al-Farabi (c. 870-950), but of Safi al-Din al-Urmawi (one possible transliteration), c. 1216-1294.
If so, it was a 17-note Pythagorean tuning , or alternatively, another system by the same theorist setting the Wusta Zalzal at 72/59 or 344.7 cents which is similar to Ibn Sina (c. 980-1037), based on a Wusta Zalzal at 39/32. None of these systems is close to 17-ed2, and they involve chains of just 3/2 fifths -- a single chain with Safi al-Din's Pythagorean scheme, or more than one chain in his alternative scheme with the Wusta Zalzal or middle third at 72/59.
I'm sure she'd be delighted to give more detail if you want to email her about it :).
BTW I know it's a little awkward that I'm kind of relaying what Margo says to you, but I think she is wise to not get too involved in editing wikipedia at this stage at least, and if commenting on the talk page it is rather close to editing the encyclopedia. The reason being that she is so close to the material and it makes it rather hard to be objective or to know where to stop and how much to put into wikipedia or to make judgements about what is OR.
You must have some of the same issues yourself. I have that with some other topics in wikipedia, that it takes care because I am so close to it myself. But not so much here as I'm not at all involved in tuning theory development myself. I find the maths interesting, but it has moved way beyond any possibility of me contributing to the development of the maths itself unless I dedicated a fair bit of time to it, which I prefer to spend on other things like programming, writing my articles etc, given how many capable tuning theorists there are. And interested in the history in a general way also but that's not for me either to immerse myself in the historical part of the subject either.
So I feel I can be reasonably objective / neutral in wikipedia in this topic area, as much as anyone could be anyway. Robert Walker ( talk) 01:22, 26 July 2016 (UTC)
Hucbald, okay a few remarks again.
"Negative System — A regular system whose fifth has a ratio smaller than 3:2. Positive System — A regular system whose fifth has a ratio larger than 3:2"
Since this page is summarized elsewhere in wikipedia using the idea of a syntonic temperament, we need a short summary we can use in its place. I've tried this out in the equal temperament page here to get us started: Equal temperament#Regular Diatonic Tunings.
I've also done a shorter version in the page on Musical Tunings here: Musical tuning#Systems for the twelve-note chromatic scale and have updated the Template:Musical tuning.
Eventually we should change all the pages that link here so that they refer to these tunings as regular diatonic tunings instead of syntonic temperament as per the discussion above. I've done a minor edit along those lines here 31 equal temperament. It's easiest to find those pages with a google site search for "syntonic temperament" as a "what links here" check lists all the pages which use the Template:Musical tuning Robert Walker ( talk) 02:06, 24 September 2016 (UTC)
"A chain of three fifths generates a minor third (A, D, G, C)" This is a chain of three fourths, isn't it? 79.79.171.35 ( talk) 13:07, 3 January 2017 (UTC)
"A chain of four equal sized fifths (E.g C, G, D, A, E) generates a major third consisting of two whole tones
79.79.170.213 ( talk) 13:24, 7 January 2017 (UTC)A chain of three fourths generates a minor third (A, D, G, C)"
"This paper describes Dynamic Tonality, a system of realtime alterations to tuning and timbre that extends the framework of tonality to include new structural resources such as polyphonic tuning bends, tuning progressions, and temperament modulations. These new resources could prepare art music for the 21st Century"
Oh, just re-read what you said about "Valid Tuning Range", I get what you are saying there. Though the valid tuning range is a different thing from the range over which the syntonic comma is customarily used to define tunings. As after all it requires use of negative amounts of the comma apart from anything else. As for the scope of this page I lead out with a definition that the reader would find easy to understand which also makes it easy to understand why it is called a "regular diatonic tuning":
"any musical scale consisting of "tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's."
That's a well defined continuum of tunings, and I'm not sure therefore how one can be unsure what it is about? I then go on to give the more technical definition from the thesis in which "Regular Diatonic Tuning" is defined, as
"Regular" here is understood in the sense of a mapping from Pythagorean diatonic such that all the interval relationships are preserved".
They could be defined in many other ways too. But it's the same continuum of tunings, no matter how you define them. Just different definitions which mark out the same entities. I'm trained as a mathematician, so it might be that I am coming to this from a slightly different perspective, so that things that seem identical to me look different to others? For me the continuum of tunings is what this page is about, and how you define them doesn't really matter so long as you mark out the same tunings, and it just seemed that in an article addressed to the ordinary non mathematical reader, it's best to start off with the simplest easiest to understand definition. And of all the definitions, the TTS TTTS definition seemed by far the easiest to both state and to understand. That's the motivation for doing it that way. Robert Walker ( talk) 03:56, 31 March 2017 (UTC)
I just added this which may help with exposition and to tie the article together:
"In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size (TTTS or a permutation of that) which makes it a Linear temperament with the tempered fifth as a generator."
It's one of those things that seems self evident, to anyone with the musical background to understand the construction of a diatonic scale, like "2 + 2 = 4" with just a bit of reflection so haven't bothered with either citing it or giving a formal proof. Robert Walker ( talk) 01:02, 2 April 2017 (UTC)
Gentlepersons,
Let me again thank you all for your learned attention to the matter of the "syntonic temperament." Again, let me emphasize that the lead authors of the seminal papers -- Prof. William Sethares and Dr. Andrew Milne -- both know infinitely more than I do about both music and mathematics. I am a mere marketing slime, who happened to read about the Wicki-Hayden keyboard and realized that its having "the same fingering in all keys" HAD TO BE the result of some correspondence between the geometric pattern of the notes and the deep structure of music. Reading Sethares' book "Tuning, Timbre, Spectrum, Scale" introduced me to the notion that consonance arose solely from the alignment of a tuning's notes with a timbre's partials. Everything else arose from those two observations, with nearly all of the heavy lifting having been done by Milne and Sethares (et al.) while I kibbitzed. So, whenever either of them corrects me, they are right, and I am wrong.
That said, please allow me to propose that Wikipedia's stand-alone article on the syntonic temperament should be restored, with only a brief summary of it included in Wikipedia's article on Regular diatonic tunings.
Here's why: The inclusion of the syntonic temperament in the "regular diatonic tunings" article is problematic, because it assumes that all tunings of the syntonic temperament are regular and diatonic, when, in fact, both of these assumptions are false.
First, "diatonic:" The valid tuning range of the syntonic temperament includes an endpoint at which the minor second has a width of zero (5-tet) and another endpoint at which it has a width equal to the major second (7-tet), neither of which are diatonic.
Second, "regular:" Consider the discussion of "Related Just Intonations" on the pages numbered as 79-80 in Spectral tools for Dynamic Tonality and audio morphing, which describes the Tone Diamond used in Dynamic Tonality synthesizers. Moving the control-point within the Tone Diamond systematically shifts the tuning away from regularity. To quote a paragraph on the page labeled as 80:
The syntonic temperament is therefore not, strictly speaking, a regular tuning, nor a family of regular tunings, nor even a regular tuning system. It embraces closely-related irregular tunings, too.
In summary: because the syntonic temperament embraces tunings that are not regular and not diatonic, having Wikipedia's only discussion of the syntonic temperament be subsumed into Wikipedia's article on "Regular diatonic tunings" is illogical and inappropriate. It attempts to squeeze a hypercubic discussion into a cubic hole.
That's why I initially put my description of the Syntonic temperament into its own Wikipedia article, and then added references from other articles to it (and vice versa), rather than attempting to shoehorn it into any existing article. This "where to put this discussion?" problem is just one small piece of evidence that the ideas embodied Dynamic Tonality, and hence in Musica Facta (the research project into Dynamic Tonality), the syntonic temperament, and Guido 2.0 (the aspect of Musica Facta focused on music education) may be incommensurable with those of music-making's current paradigm.
I apologize for not presenting these arguments when the idea first arose of merging the discussion of the syntonic temperament article into some other pre-existing article.
I hereby propose reverting the article on the Syntonic temperament to being stand-alone, and adding a brief summary of it, with a reference to the full article, in the Regular diatonic tunings article.
Respectfully,
Jim Plamondon -- JimPlamondon ( talk) 06:44, 26 April 2020 (UTC)
Gentlepersons,
Thank you again for your continued attention to this matter. :-)
In the Talk above, people have repeatedly questioned the relationship between the syntonic temperament and timbre. Temperament, they say, has nothing to do with timbre; the linking of the two, which runs throughout Sethares, Milne, Plamondon, Prechtl, Tiedje, et al., they see as aberrant.
I think I now understand where y'all are coming from. I suspect that the microtonal community has assumed the use of Just Intonation and Harmonic Timbres for so long that they have forgotten that they are even making an assumption. They often write about ratios of small whole numbers, as Pythagoras did, as if the ratios themselves were the source of tonal meaning -- when it has been abundantly clear since Joseph Sauveur, or, for the doubtful, since Helmholtz, and inescapably since Plomp & Levelt that these intervals arise from the specific pattern of vibrations defined by the Harmonic Series, not from some magic of numerology.
Alternatively put: Sethares, Milne, Plamondon, et al. are explicitly relating tempered pseudo-Just tunings and tempered pseudo-Harmonic timbres, whereas microtonalists and previous music theoreticians have implicitly related [un-]tempered Just tunings and untempered Harmonic timbres.
Examples of the above implicit assumptions in the Talk section of this article:
Sethares, Milne, Plamondon, Prechtl, Tiedje, et al. have offered something that is truly novel, and y'all seem to be working very hard to deprecate its novelty and force it into the lower-dimensional framework of what has historically been done. Our work COMBINES:
All of these innovations have been documented in the peer-reviewed literature -- and some non-peer-reviewed technical reports -- for over a decade. Our Computer Music Journal article was still, the last time I checked, the most-often downloaded article in its history. Our "sight reading music theory" article has been downloaded from ResearchGate over 3500 times. To claim that these innovations, or the ideas that they document, or the terms that they use, are not "established," begs the question of what definition of "established" you are using.
When combined, our system's components enable a systematic expansion of the frontiers of tonality and novel tonal effects. However, each component by itself is of little particular interest. How useful is an automotive transmission without an engine? A computer without electricity? A browser without the Internet? So, our explanation of each component explicitly refers to the other components.
Y'all, I suspect, are not making such explicit connections among your uses of "temperament," "timbre," etc., because youimplicitly assume:
In summary, we are explicitly stating relationships that you are assuming implicitly.
The solution is to update all of the other articles on microtonality and music theory to make their implicit assumptions explicit.
Y'all are resisting this. Your resistance is manifesting itself in an attempt to shoehorn our innovations into a box defined by your lower-dimensional implicit assumptions, in which the interconnections and synergies must be stripped away to fit in those lower-dimension boxes. This looks to me to be a clear-cut case of the Semmelweis Reflex. That reflex is a natural human and institutional reaction to novel ideas, so I don't blame y'all in any way for exhibiting it. I have every confidence that I am displaying similar human reflexes and cognitive biases. :-)
I am happy to see that the discussion surrounding the original Syntonic temperament article led to the creation of the Regular diatonic tunings article, which is a boon to Wikipedia.
As a first step towards resolving the "problem of assumptions" described above (and the fact that the Syntonic temperament includes non-regular and non-diatonic tunings), I will de-merge the Syntonic temperament article out of the Regular diatonic tunings article sometime in early June 2020.
In the meantime, I will add to the article on Dynamic Tonality a section that discusses (a) the implicitly-assumed relationships described above, and (b) the explicitly-defined relationships made by Dynamic Tonality. I can them update the discussion of the Syntonic Temperament to refer to this section in the Dynamic Tonality article. I can then add a reference to this same "implicit assumptions" section to all other articles on traditional music theory. This will clarify the distinction between Dynamic Tonality and everything that came before.
That should resolve all of the issues raised in this Talk page, and improve Wikipedia's discussion of music theory considerably.
I look forward to your comments on this suggestion.
Respectfully, -- JimPlamondon ( talk) 04:49, 2 May 2020 (UTC)
Maybe about 80% of this article seems to correspond to a fringe theory by user JimPlamondon. This is against wikipedia's philosophy, as you can see here. In the fringe theories page. This has nothing to do with whether I agree or disagree with this theory, it is an objective fact that this is not broadly supported and as such should be deleted. Wikipedia is not a place for promoting your own theories, it should reflect what is broadly acceped in a field, there are many other websites that can be used to promote or support this research.
In a nutshell, as my reference states: To maintain a neutral point of view, an idea that is not broadly supported by scholarship in its field must not be given undue weight. I don't think this deserves extra discussion, and JimPlamondon's own comments on the matter make this very clear. I will procede to delete everything associated with Dynamic Tonality IgnacioPickering ( talk) 23:52, 17 July 2021 (UTC)
Ignacio, et al.: Wikipedia states that "fringe theories in science depart significantly from mainstream science and have little or no scientific support." The content that you deleted is, by that definition, not a fringe theory, in that it is well-supported by numerous publications in the peer-reviewed scientific literature. Indeed, the content that you deleted contains more references (7) that the content that you did not delete (4).
This leads to the second criteria for identifying fringe theories: Parity of sources. Four of the seven sources cited in the content you deleted are to respected peer-reviewed publications, such as the Computer Music Journal (H-Index: 41), the Journal of Mathematics and Music (H-Index: 14), and the Proceedings of the College Music Society. Only three of the seven sources are not similarly peer-reviewed. By comparison, of the four sources cited in the non-deleted content, the first refers to a master's thesis; the second to a non-peer-reviewed book; the third to a non-peer-reviewed review of said book; the fourth to a non-peer-reviewed wiki post. None but the first of these four sources can credibly be claimed to be peer-reviewed. The other three can reasonably be categorized as "self-published texts," which Wikipedia's description of fringe theories cites as a sign of fringe theories. In short: The deleted content is demonstrably less fringy than the non-deleted content.
Please note also that publications on the syntonic temperament and Dynamic Tonality are cited frequently, according Google Scholar:
Paper | Citations |
---|---|
Spectral tools for dynamic tonality and audio morphing | 52 |
Isomorphic controllers and dynamic tuning: Invariant fingering over a tuning continuum | 51 |
Tuning continua and keyboard layouts | 39 |
Dynamic tonality: Extending the framework of tonality into the 21st century | 7 |
This is an impressive level of citation for papers in the humanities (such as music theory), in which 65% of papers are never cited. These citations prove the "broad acceptance" of the content that you deleted.
I would go a step further, and point out that many of the papers supporting the syntonic temperament and Dynamic Tonality include mathematical proofs of their claims. Such proofs are the gold standard for claims made about models of phenomena in the natural world, such as musical tunings. The citations supporting the non-deleted content contain no such mathematical proofs. Therefore, the theory of the non-deleted content is less strongly supported by its citations than is the content that you deleted.
Finally, as Wikipedia's discussion of fringe theories states, "One important barometer for determining the notability and level of acceptance of fringe ideas related to science, history or other academic pursuits is the presence or absence of peer-reviewed research on the subject." As shown above, the content that you deleted meets this standard much better than the content that you did not delete.
Therefore, the deleted content should have been retained, and the non-deleted content should have been deleted as a fringe theory.
However, Ignacio, I am not unreasonable. I will not press for the deletion of the obviously-fringy theory represented by the content that you did not delete. I will only insist that you follow Wikipedia's rules fairly, and therefore not delete the material on the syntonic temperament and Dynamic Tonality. -- JimPlamondon ( talk) 03:27, 18 July 2021 (UTC)
The article claims: "The small difference in pitch between [a chromatic and diatonic semitone] is called a comma, usually prefixed by the name of the tuning system that generates it, such as a syntonic comma (21.5 ¢)..."
This is clearly incorrect. While the Pythagorean comma does work out such a difference, as seen on the corresponding page, the syntonic comma is not the difference between the chromatic (25:24) and diatonic (16:15) semitones of just intonation, equivalently the minor second and augmented unison - that yields 128:125, i.e. the (enharmonic) diesis.
The syntonic comma (81:80) - the one referred to when one speaks of "fractional-comma meantone" - is a ratio between the major tone (9:8) and minor tone (10:9) of just intonation - which is not a regular diatonic tuning, as it uses two different tones. Equivalently, it's the difference between a just-intonation ditone and major third.
216.209.47.135 ( talk) 07:57, 31 January 2024 (UTC)