Talk:List of pitch intervals/Archive 2

Archive 1

Archive 2

List or errors in the article

I must apologize for what follows, but I HATE this article, which I consider one of the most carelessly written I ever saw on WP (or elsewhere). Having devoted much of my life to the study of historical tunings and temperaments, I cannot easily admit that this history be treated so lightly. I do understand that computers have made it possible to tune anything and to produce any type of temperament. Some may even think that it is possible to make music with that: it is up to them to prove so (Nicola Vicentino had tried some 500 years ago, without much success). Anyway, I decided to produce a list of errors or inadequate formulations in the article. This will take me some time, but here follows a first instalment, concerning the introductory text of the article, its "lead" so to say. I will soon add a list of inadequacies in the list itself. This list of errors may be considered a list of suggestions for improvement of the article, but I don't intend myself to realize any of these improvements.

• The introductory text does not give a single reference to published evidence.
• "Below is a list of musical intervals [why not "pitch intervals", as in the title?] encountered in tuning or temperament." Such wording appears to suggest that one may encounter these intervals while tuning an instrument. It might be more correct to write "in tunings or temperaments" (plural), referring to the results of the tuning. But one may wonder what type of instrument could be tuned to many of these intervals. Does one tune a computer? It might be more accurate to say that these intervals are encountered in (modern) theories of tuning or temperament, or something of the kind.
• "Intervals with lower prime limit are used more often." A properly unjustifiable statement. How does one determine, say, that 4ths (3-limit) are used more often than pure major 3ds (5-limit); are 'natural 7ths' (7-limit, about 969 cents) used more often than minor 7ths (about 1000 cents)?
• "Any n-limit entry is also m-limit for m>n." What this odd statement apparently means is that any n-limit interval can also be encountered in systems of higher limit, but not of lower limit. This could be better formulated by saying that the limit represents the system of lowest order in which the interval can be encountered: it is a low limit.
• "Similarly, septimal, undecimal, tridecimal, and septendecimal mean, respectively, 7, 11, 13, and 17-limit just intonation." The list itself also contains entries for "19-limit just intonation" and one can assume that any higher prime number could also form a limit (one has to stop somewhere in practice, but in theory there is no ... limit). But what is the point in calling all of them "just"? What does "just" mean if anything is just? Historically, "just intonation" has meant 5-limit, and there is no reason to change this historical meaning.
• "By definition every tone in a 3-limit unit can also be part of a 5-limit tuning and so on." This appears fully redundant with "any n-limit is also m-limit for m>n", see above. Any interval of a given limit can also be part of a system (or tuning) of higher limit.
• "In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in a meantone, each fifth is narrowed by the same small amount". A complex way to convey a simple idea. A meantone is a "regular" tuning, i.e. using only one size of fifth (but different from doubly augmented 4ths or diminished 6ths). In Pythagorean tuning, the fifths are perfect; in quarter-comma meantone the fifths are tempered (narrowed) by 1/4 (syntonic) comma; in 12-T equal temperament, the fifths are tempered by 1/12 (Pythagorean) comma; etc.
• "Equal-tempered refers to X-tone equal temperament with intervals corresponding to X multiples per octave." Apparently, "multiple" here means "division"… 12-T equal temperament corresponds to 12 divisions per octave.

2d instalment – Intervals from 0 to 100 cents (I refer to the intervals by their value in Cents, as given in the first column of the table).

• 0.00 This is a list of intervals. It makes no sense to call an interval the "tonic" (even if the fault exists in the reference cited, Kyle Gann's website) or a "fundamental". There probably is a confusion with another possible list, that of harmonic partials, in which partial 1 indeed would be the fundamental (and, by a common aberration, the "tonic"). Partch's definition of the interval as monophony also is odd. That the interval of unison in a list of intervals from C should be a C does not seem to need a reference to John Fonville's book.

• Other intervals are defined as "n^th harmonic", which strictly isn't an interval either; one may understand that what is meant is the interval between the harmonic and the highest power of 2 below it, but it would be better to say so.

• 0.01536 This interval is missing in the table. It is the Atom of Kirnberger, for a definition of which see Schisma.
• 3.99 An eptaméride is not the same thing as a savart and neither exactly corresponds to 10^1/1000. The eptaméride divides the octave by the decimal log of 2 counted with three decimals and multiplied by 1000, i.e. 301 parts of 3.9867 cents each; the savart does the same but using log(2) to the fifth decimal, i.e. 301,03 parts of 3.9863 cents each. They are therefore but approximations of 10^1/1000 (which counts an infinity of decimals), rounded to three or five decimals.
• 7.71 Marvel comma should refer to [Marvel temperaments]
• 13.79 Starling comma should refer to [Starling family]
• 21.82 This interval is missing in the table. It is the unit interval in 55-T ET, the pendant to 53-T ET, with diatonic semitones of 5 commas instead of 4. In this reasonably well documented temperament, all intervals corresponing to the chromatic scale are between 12-T ET and 1/4-comma meantone intervals.
• 27.91 This interval is missing in the table. It is Sauveur's méride, the 1/43 part of the octave. The heptaméride is the 1/7 part of the méride. Sauveur considered that the mérides were sufficient to describe "the tempered intervals of the diatonic system" (Principes d'acoustique, 1701, p. 12).
• 38.71 31-T ET deserves a reference. It is the tuning system advocated by Nicola Vicentino in the 16th century and discussed among others by Constantin Huygens. It provides an excellent approximation of 1/4-comma meantone.
• 63.16, missing in the table, should be mentioned. It is 19-T ET, with a chromatic semitone of 1/3 tone, a diatonic one of 2/3 tone. It provides an excellent approximation of 1/3-comma meantone.
• 90.22, reference [11]: considering that Helmholtz' treatise in Ellis' translation counts only 576 pages, the reference to page 644 must be a mistake. The expression "Low semitone" does not seem used in Ellis' translation, who speaks (p. 329) of the "small semitone or limma 256:243"; but Ellis describes a variety of intervals as "small semitones".
• 98.95 One wonders where the Huygens-Fokker Foundation found this "Arabic lute index finger". Al-Farabi (10^th century) describes 18:17 as one of the several positions that he names "neighbor to the index" (the index itself being at 9:8, 204 cents). 18:17 is exactly at the middle of the distance to 9:8 along the string (dividing a string distance in equal parts is a most interesting way to establish intervals). Other "neighbors to the index" include 256:243 (the Pythagorean limma), 162:149 and 54:49 (both missing in the table, also obtained by equidistance, but that will be for the next instalment).[To be continued]

There will be no third instalment, because I think I identified the general problems of this article, for which I will rather open a new section: see below, "Why this article does not achieve what it claims". – Hucbald.SaintAmand (talk) 19:01, 21 August 2014 (UTC)

While fully sympathizing with you on this, Hucbald, one small thing needs to be pointed out. Your complaint that "the introductory text does not give a single reference to published evidence" should not be a problem, supposing (as you do) that we are talking about the article's "lead" (or "lede"). The lead should never have references in it, for the simple reason that it should only summarize what is presented more amply in the main body of the article. However, only the first two paragraphs here are actually the article's lead. Beginning with "Some terminology used in the list:" we have got what should be a section with a separate header, and indeed you are correct that these terms all need references. If on the other hand this really is meant as part of the lead, then sources for the terms should be added at each appearance in the list. This would be cumbersome, however, so I suggest a "Terminology" header replace that phrase I just quoted, and references be demanded for any terms that are not in the "sky is blue" category.—Jerome Kohl (talk) 20:31, 21 August 2014 (UTC)

Critics should bear in mind that this is a "list" article, not a treatise on tuning and temperament. It aims to collect all possibly relevant intervals in a systematic categorised way, without theorising about them. Many of the remarks above are valid, so why don't you improve the header part accordingly? −Woodstone (talk) 05:59, 22 August 2014 (UTC)

You're "errors" are, at most, slight problems with the wording of the introduction. Hyacinth (talk) 09:58, 22 August 2014 (UTC)

If you want. But so many slight problems make a big problem. – Hucbald.SaintAmand (talk) 10:18, 22 August 2014 (UTC)

So, why don't you fix the errors you find? That's kind of the point of a wiki. The central premise, even. Be bold. — Gwalla | Talk 18:14, 22 August 2014 (UTC)

Because, as Hyacinth rigthly stressed, these are not "errors", merely "slight problems". I don't want in any way to endorse the general attitude of this article with respect to tunings and temperaments, because this attitude, even if may seem to lead only to "slight problems", denies all what I believe history of theory may be about.

I mentioned above the case of the "Arabic lute index finger" (98.95 cents). Anyone having ever played a lute (Arabic or other), or any other stringed instrument (a violin, for instance) knows that the normal position of the index finger is about a tone higher than the empty string. But some guy, probably having misread Arabic theory, declares otherwise, without leaving any possibility to check what he states. And somebody else, believing, as seems to be the case on WP, that anything published is true, reproduces it here. I am not myself an enthusiastic Wikipedian, in such conditions.

Al-Farabi, as I shortly explained, construed some of his intervals as equidistant divisions of the string. But this principle, equidistant division, even although it was conceived a thousand years ago (!), seems uninteresting to the self-appointed mathematicians building the kind of interval list we are dealing with here. I didn't check (I probably will), but I trust that many of the intervals so construed merely are absent of the list, their importance in Arabic theory notwithstanding. Max Weber, the early 20th-century sociologist, wrote in his book about musical sociology (I am merely quoting a vague reminiscense) that equidistant divisions have been an important technique in early times of music: he, at least, had some respect for these ancient theories.

I'd be willing to participate to the redaction of this article if I understood what it is all about. It implicitly claims to concern universal tunings, but in fact deals only with modern Western tunings.

Enough about this. In view of the above, I won't fix anything in this article. I repeat that anyone should feel free to make use of my comments (even if some, I'm ready to admit, are mere expressions of my irritation), and I might even answer questions, if any. But my comments ain't systematic, and it is not my intention to make any effort to make them more complete: I have other things to do.

Hucbald.SaintAmand (talk) 21:29, 22 August 2014 (UTC)

Dear Hucbald: While the point is well-taken, I must correct an error which you have now stated twice: As far as Wikipedia is concerned, if something is published it is verifiable—"truth" does not come into it. The famous formulation is, "The threshold for inclusion on Wikipedia is verifiability, not truth". This can be infuriating at times, especially so when the necessary refuting source cannot easily be found, but in the long run I think it is better to mention a well-documented error and then debunk it by citing the contrary evidence from an even more reputable source than it is either to disregard or attempt to conceal wrong-headed but widely disseminated notions that some bright child is only going to unearth later and try to insert as refutation of the stronger position we have worked so hard to establish in the first instance.—Jerome Kohl (talk) 21:52, 22 August 2014 (UTC)

Of course, Jerome, and we both know that truth may not exist. Change "true" in "verifiable": what I wrote becomes "believing ... that anything published is verifiable". I don't think that Wikipedia is that naive, but some Wikipedians are. If something is unverifiable in the source, it remains unverifiable once quoted. Woodstone wrote (above) that this article "is not a treatise on tuning and temperament" and does not want to theorize. Agreed, but it should at least base itself on verifiable theory... Hucbald.SaintAmand (talk) 17:32, 23 August 2014 (UTC)

Ah, but by the Wikipedia definition here, appearing in a reliable, published source is the way claims are verified (theories are another matter entirely). This is, amongst other things, a form of insurance against libel action. The tricky part of course is in demonstrating the reliability of a source. Even a source that is universally regarded as being reliable generally may contain the occasional error. In (Wikipedia) practice, any published book or article in a journal or magazine is more likely to be regarded as reliable than, for example, a self-published website or, worse, a blog. There are of course self-published books and magazines, too, so at a certain point the reputation of the publisher becomes involved. This in turn can lead to disagreement amongst editors, which is usually resolved through (sometimes tedious) discussion. The "open editing" of Wikipedia makes this inevitable, I think. As I said before, this can take some getting used to, and I have heard of many cases of superbly qualified editors giving up on Wikipedia in frustration. I do not want you to become one of them.—Jerome Kohl (talk) 19:54, 23 August 2014 (UTC)

Jerome, contrarily to those editors who gave up Wikipedia, I am not self-confident enough to give up. But I'll always feel free to express my opinion, even if it enters in conflict with WP's policies. After all, what I am doing here is just that, expressing my opinion, and this too is one reason why I don't want to work on the article itself. I note however that the Huygens-Fokker website cites none of its authors (I think to know some of them, but that changes nothing), and I remain totally puzzled at the idea of an "Arabic lute index finger": what to they mean by that? On the one hand, the index on the lute is moving as needed; on the other hand, most of the theorists who tried to define the position of the index on the 'Oud placed it a tone, not a semitone, higher than the open string. When I quote al-Farabi, I give an information that is verifiable (i.e. falsifiable – it could be proved to be false, if that were the case). The one given by the Huygens-Fokker foundation is neither, because they don't even bother to explain what it is supposed to mean. – Hucbald.SaintAmand (talk) 21:06, 23 August 2014 (UTC)

I am relieved to learn you will not be giving up easily! Opinions of course may be expressed freely on talk pages—in fact, one of their main purposes is to discuss what we think ought to be changed about an article, and this often involves exchanging opinions with other editors. As to the "Arabic lute index finger", I really have no idea what they can be talking about, but unsigned articles on websites are always of suspect reliability. This reminds me of a "reliable source" that was called to my attention thirty or more years ago now. This was when the mandolin method was first published in the Mel Bay series, and the person who indignantly told me about it was the owner of a music shop who had just received a copy as part of a consignment of new editions. Whoever had written this book was obviously a guitarist who may have picked up the mandolin occasionally, but plainly had never learned to play it properly. He instructed exactly what you describe as the "Arabic lute index finger", placing the index finger on the first fret, but then compounded the problem by adding fingers chromatically above it. My informant was in fact a professional mandolinist and violinist, and of course the mandolin is fingered very much like the violin: diatonically, starting with the index finger a whole step from the nut. He wailed that, because of the huge coverage in the market commanded by the Mel Bay name, whole generations of beginners were going to be brought up trying to play the mandolin the wrong way and would give up in frustration upon discovering they could play practically nothing in the mandolin repertoire using this method. I imagine this book is still in circulation, and I am afraid to look at what the Wikipedia article on the mandolin might have in it.—Jerome Kohl (talk) 22:37, 23 August 2014 (UTC)

Meantone

I corrected the definition of meantone temperaments in the "Terminology" section. I don't think that a question asked (without answer) on a forum can count as a verifiable source for 1/2-comma meantone. Pending the answer, I don't think that any meantone with fifths narrowed more than a 1/3-comma ever were in use. But I have no time to verify sources, so that my correction also needs a citation. Murray Barbour's book on Tuning and Temperament certainly is one, but I've no time to give page numbers just now. – Hucbald.SaintAmand (talk) 21:10, 27 August 2014 (UTC)

OK. So is usage in a program enough? [1] BartekChom (talk) 21:46, 27 August 2014 (UTC)

I suppose so; it would be even better if you could provide a reference, but I presume that it may not be easy to find. The [Logic Pro website] does not mention meantone at all, or I were unable to find it. But a sofware like Logic Pro can in fact produce any tuning or temperament, known or unknown: it allows inventing them...

What is certain on the other hand is that Fogliano (about whom the question was asked on the forum that you had mentioned) never described 1/2-comma meantone (Lodovico Fogliano, [Musica theorica], Venice, 1529, sectio tertia). – Hucbald.SaintAmand (talk) 10:55, 28 August 2014 (UTC)

The new picture in the lead is an excellent improvement on the doubtful former one. Actually it now could easily be enhanced by adding the most common 5-limit just tuning. That would give some information on the 4 most fundamental systems: ET, pythagorean, quarter-comma, just. If the picture becomes too cluttered one could leave out one or more of the other ones. −Woodstone (talk) 16:36, 28 August 2014 (UTC)

Glad you like it. I once drew it with just intonation added, but that does indeed make it somewhat cluttered (removing one of the meantone temperaments wouldn't really help).

Just intonation is made of segments of Pythagorean intonation separated by a syntonic comma. It could be represented in a Tonnetz, as this:

		Eb	Bb
F	C	G	D
A	E	B	F#
C#	G#

(This is a somewhat unusual presentation because the major thirds, in each column, must be read downwards; the advantage is that reading from left to right and from top to bottom, you get the full cycle of fifths). Each of the rows forms a segment of Pythagorean tuning (perfect fifths), parallel to the blue line in my figure. To pass from one segment to the next, you must go down in the figure by the distance of a syntonic comma, and get there a new segment of line parallel to the blue one. The line representing just intonation therefore would form a zigzag crossing the whole. It could be added, I'll think of it. But I trust that anyone with an idea of what just intonation really is easily would figure it out; and the others may not be much helped by an improved figure.

Note that I speak here of "just intonation", not of 5-limit. Neither ET nor the two meantone temperaments represented can be expressed in terms of a prime limit; Pythagorean tuning is 3-limit, and therefore also 5-limit. I don't see how one could justify all this without reaching in a mess... Comparing historical systems under their historical names, Pythagorean tuning, ET, meantone temperament and just intonation, is so much clearer. [This is not to say that I don't understand the logic behind prime limits; I merely think that they should be used with more caution.] – Hucbald.SaintAmand (talk) 18:22, 28 August 2014 (UTC)

Most of the intervals listed with a "M" for "Meantone" in the "M" column of the table actually do not belong to meantone temperament. Correcting that will be quite a work... – Hucbald.SaintAmand (talk) 17:54, 31 August 2014 (UTC)

Why this article does not achieve what it claims

It took me some time to identify in general terms the shortcomings of this article: my apologies for this. The article claims to be

a list of musical intervals encountered in tuning or temperament.

and one would suppose to find among others intervals found in historical temperaments. But one is soon disappointed.

The article mentions meantone, "the most common of which is quarter-comma meantone", and it refers to the article Meantone temperament where one would search in vain for some indication of the history of such tunings... Whatever it be, one may expect that "List of pitch intervals" would at least give the intervals of 1/4-comma meantone on a keyboard of 12 notes in the octave; but it gives (or identifies) only two, the minor third (310,36 cents) and the perfect fifth (696,58 cents). The just major third is there too, of course, and the list gives a "just augmented fifth" that turns out to also belong to 1/4 comma meantone, but neither is identified as being a 1/4 comma meantone interval. ALL other intervals of 1/4 comma meantone are missing (that is, more than half of its intervals). No mention is made of so many other varieties of meantone, 2/7 comma, 1/3 comma, 1/5 comma,, 2/9 comma, 3/10 comma, etc. etc., all meantone tunings described in any good history of tuning and temperament. The article does refer to the List of meantone intervals, it is true, but none of these is found there either.

The same is true of all kinds of tuning, the numerous historical varieties of just intonation (i.e. 5-limit!), or the irregular systems described by Grammateus, Ganassi, Artusi, Colonna, Mersenne, Rameau, Kirnberger, Werckmeister, Marpurg, etc. Nothing is said of the tuning of fretted lutes, the only mention of Arabic tuning is meaningless (or unverifiable), etc. etc.

In short, this article is about some intervals, but I fail to understand how the choice has been made; certainly, it is not about intervals "encountered in tuning or temperament".

Hucbald.SaintAmand (talk) 08:32, 24 August 2014 (UTC)

That's called being incomplete, isn't it? Hyacinth (talk) 12:47, 25 August 2014 (UTC)

Explicitly, what is your definition of "tuning" and what is your definition of "temperament"? Hyacinth (talk) 13:56, 25 August 2014 (UTC)

A tuning is a way to tune an instrument. Here, more specifically, a way to tune an instrument of fixed pitches (i.e., excluding the members of the violin family, which also are tuned, but usually with one single interval, or wind instruments with finger holes, the tuning of which involves other problems). Traditional instruments of fixed pitches include the keyboard instruments, the harp, the fretted instruments, etc., to which were added some electric and electronic instruments in the 20th century. As J. Murray Barbour wrote, "The tuning of musical instruments is as ancient as the musical scale. In fact, it is much older than the scale as we ordinarily understand it." (Tuning and Temperament, East Lansing, 1951, p. 1.) Mark Lindley, in the New Grove Online, gives three definitions of Tuning: (1) "The adjustment, generally made before a musical performance, of the intervals or the overall pitch level of an instrument"; (2) "the set of notes to which an instrument is tuned"; (3) "the 'tuning system' employed, referring to a model of the scale corresponding to some mathematical division of the octave." It is this third use that retains us here.

A temperament is a special type of tuning in which one or more interval(s) is (are) tuned false in order to resolve problems linked with the limited number of pitches available and, in many cases, the necessity to repeat them at the octave. Mark Lindley defines temperaments as "Tunings of the scale in which some or all of the concords are made slightly impure in order that few or none will be left distastefully so".

Murray Barbour gives other, more concise and in a way less explicit definitions, in which "tuning" and "temperament" are mutually exclusive, contrarily to the situation in Lindley's (or my) definitions:

• Temperament – A system, some or all of whose intervals cannot be be expressed in rational numbers.
• Tuning – A system all of whose intervals can be expressed in rational numbers.

(Murray Barbour, op. cit., p. xii)

The Pythagorean system is a tuning: it involves no tempered interval; Meantone temperament tempers the 5th, in order to tune the major or the minor 3rd closer to pure; Just intonation again is a tuning, involving no tempered interval.

Hyacinth, I am very sorry to be engaged in such a quarrel about this article. I merely think that if the authors want to change common and generally accepted definitions of what they claim to be speaking of, they should at least say so. Let me remind you that this discussion began when somebody suggested that this article should replace another one devoted to "musical intervals". I would never have bothered looking at "List of pitch intervals" without that. Of course, the article 'merely' is incomplete; I fear, however, that it could not be made complete, because there is something wrong in its conception itself.

The 12-degree meantone temperaments discussed by Murray Barbour, which range from 1/3 to 1/10 comma meantone (i.e. the fifth tempered by any amount between about 8 and 2 cents), yield the following values rounded to the nearest cent (that is that any figure here may stand for several, if more decimals were accepted).

C: 0

C#/Db: 64 72 70 76 79 83 85 89 92 95 97 99

D: 190 191 192 193 194 195 197 198 199 200

D#/Eb: 269 301 302 303 305 307 308 309 310 312 313 316

E: 379 383 384 386 389 390 394 395 396 397 398 399

F: 500 501 502 503 504 505

F#: 569 574 576 579 582 585 586 587 590 593 596 598 599

G: 695 696 697 698 699 700

G#/Ab: 758 768 773 777 781 787 791 794 797 798 808 812 814 817

A: 884 887 888 890 892 893 895 896 897 898 899

Bb: 1000 1001 1002 1003 1005 1006 1007 1008 1009 1010

B: 1074 1078 1080 1083 1085 1088 1090 1092 1095 1097 1098 1099

C: 1200

None of these intervals (except 0 and 1200) can be expressed as a ratio - i.e. none of them belongs to any n-limit! The article makes so much case of these n-limits, without realizing that temperaments, by definition, cannot fit within any of them!!!

Hucbald.SaintAmand (talk) 14:34, 25 August 2014 (UTC)

I need to take responsibility here, I believe, as (though I haven't actually gone through the history to be sure) I think that it was I who added that stuff about "encountered in tuning and temperaments". And I think I did so more or less unthinkingly because some of those intervals (such as major tone, Pythagorean comma) may be encountered when discussing tuning or temperament. I know for sure that most of them are not used in music-making in the ordinary way. That is why I made the now-suppressed List of musical intervals, so that ordinary people could come to Wikipedia and find out about the intervals that are commonly used or discussed in Western-style music discourse. And yes, I'm aware that there is an element of systematic bias in such a statement.

It was also I who gave this horrible mess its current title, for much the same reasons. If it's wrong, please go ahead and move it, to some appropriate title - List of intonational intervals? List of intervals you will never encounter as a working musician? List of totally fantastic made-up intervals with no real-world application whatsoever?

What I'd like to know now is this: if these intervals and this terminology (all this stuff about prime limits and the like) are not used in music-making (as they surely are not) and are not used in tuning (as Hucbald says more eloquently and cogently than I could - I can't fault him on one word), then where exactly are they used? Or is this all just the musical equivalent of "the length of the side of the great pyramid of Cheops measured in East Anglian barleycorns is exactly the same as the height of Rouen cathedral measured in lignes of the pied du Roi prior to the reform of 1622"? Justlettersandnumbers (talk) 18:50, 25 August 2014 (UTC)

I do think indeed that it would be important to know what this article is about. I belive that its list of references gives some cue. They are not so numerous:

– An article by John Fonville, "Ben Johnston's Extended Just Intonation: A Guide for Interpreters", Perspectives of New Music, 1991.

– A "List of intervals" anonymously published on the web by the Huygens-Fokker Foundation, which I take to be a Dutch gang assembling (a) experts in historical tuning and temperaments (some of whom I know); (b) fanatics of microtonal music. Even their own [website] does not give any name to make them less anonymous.

– A web site by Kyle Gann (I have no idea who he is), with somewhat odd and uncontrolled ideas about the question.

– A book by Ján Haluška, who I take to be one of those mathematicians fascinated by the odd mathematics of musical systems; I don't know him, but I know some of his kins quoted in his book.

– The website of Xenharmony, apparently a project linked to the Huygens-Foffer Foundation, or of similar aims.

– A few citations from older books (Ellis, Helmholtz), probably intended to somehow legitimate the whole.

In short, the article seems to stem form enthusiasts of modern microtonal music. Some of them do make music with that. I believe therefore that the article would be perfectly legitimate, if it only described the context in which it is proposed. But when it claimed to list "musical intervals" at large, I could not but react. The intervals listed here perhaps belong to some (modern) type of microtonal music. I suggest therefore that the article be renamed "List of intervals used in some modern microtonal music" – or anything better, provided the authors explain what...

Hucbald.SaintAmand (talk) 20:27, 25 August 2014 (UTC)

I think I can help in a very small way by directing you to the Wikipedia article on Kyle Gann, who is a respected American music critic long associated with the Village Voice, and also a composer and professor of music. He has also been known to drop in as an editor at Wikipedia from time to time.—Jerome Kohl (talk) 20:42, 25 August 2014 (UTC)

Thanks, Jerome. Following your message, I had a closer look at Kyle Gann's [Anatomy of an Octave]. I had found it odd that he named the interval of unison the "tonic". I see now however (1) that his list is not one of intervals, but one of pitches, which makes much more sense (but does not justify "tonic"), (2) that the origin of this "tonic" is in Daniélou's Tableau comparatif des intervalles musicaux (1958) to which Gann gives a link (more about this below). Gann clearly declares what his list is, and implicitly what it is not:

• Once again, it is a list of pitches, not of intervals; it gives "more than 700 pitches within an octave".
• Gann does not claim that these pitches are "musical"; at best, one is left to imagine that they may be used in music.
• He gives precise and systematic criteria for inclusion in the list:

– all pitches result from ratios, which means that none stems from a temperament

– the list includes (1) all harmonics up to 128 (reduced inside one octave, even if he doesn't say so, but it is obvious); (2) all ratios between whole numbers up to 32; (3) 5-limit ratios up to 1024; (4) 11-limit ratios up to 128; (5) 31-limit ratios up to 64; (6) a choice of historically important ratios.

No justification is given for these criteria, but no claim is made, either. The reader at least gets a clear idea of what (s)he is reading, and is left to decide for her(him)self whether it is interesting.

I didn't yet read Daniélou's Tableau comparatif, which counts iv+143 pages (and I am not sure I will do so). Daniélou is a controversed Indologist and the introduction to his Tableau includes somewhat puzzling statements, for instance (my translations):

• "All musicians have noted that, in the infinite scale of frequencies, if one chose an arbitrary point as tonic, or fundamental sound, some other sounds appeared to harmonise with this basic sound, appeared to form precise, definite, ineluctable intervals, and any deviation from these particular points appeared offensive to the ear and was immediately described as either too high or too low sound". (p. i)
• "It seemed to the ear that when the voice meandered on the scale of frequencies, one found, with respect to the tonic, a series of precise points corresponding to perfectly expressive intervals, kinds of melodic words." (p. i)
• "These elementary observations on the intervals led to the division of the octave in twenty-two of twenty-four regions within each of which was found a perfect interval of precise meaning. [...] This scale, known to the Greek, was called enharmonic (Enarmonikos) and was considered in the Hellenic world and its heirs as the basis of all musical scales. The Arabs took over the Greek tradition which they discussed and interpreted in a manner at times differing from that of the Europeans." (pp. i-ii)

Daniélou also states that his table does not include ratios higher than 7-limit, unless they are superparticular or they are "mentioned in the diverse writings about music theory". It must be stressed that his table is given as a list of intervals, while it should be read as a list of pitches (Daniélou gives frequencies with respect to the "scale of the physicists", i.e A4=426.6 Hz, because he believes that C should be a power of 2).

Reading this table reminded me that there exists another interesting list of pitches (or intervals), based on different premises, in Hugo Riemann's Musiklexicon, at least in the 3d edition of its French translation, that I must have somewhere in my library.

All in all, I think that Wikipedia might offer various lists of [relative] pitches (rather than intervals), e.g. pitches in this or that (family of) historical temperament(s), or in this or that prime limit, or whatever, each clearly stating what it concerns, possibly with a disambiguation article refering to these various lists. I might let me be convinced to participate in parts of such a project, which seems to me more reasonable than trying to make sense of a "List of pitch intervals".

– Hucbald.SaintAmand (talk) 09:45, 26 August 2014 (UTC)

I do not think it is helpful to refer to one of the sources as a "gang of fanatics". I also don't think it's all that important that Gann is a bit fuzzy with the term "pitch": his list is by relative distance (in ratios and cents), not by absolute frequencies. — Gwalla | Talk 18:52, 26 August 2014 (UTC)

Keep in mind that English is not my native language. My Webster defines a "gang" as "a group of persons having informal and usually close social relations", and it is in this sense that I understood it. And I maintain that some of them are fanatics ("marked by excessive enthusiasm and often intense uncritical devotion", Webster again) of microtonal music. As to Gann, did I ever said that he is "fuzzy" with the term "pitch"? I merely said that his choice of proposing a list of pitches "makes more sense" than a list of intervals. I am perfectly aware that pitches may denote relative distances. On the other hand, he is "fuzzy" with the term "tonic". – Hucbald.SaintAmand (talk) 19:58, 26 August 2014 (UTC)

"Gang" typically implies criminal activity, though it is also used for informal but close-knit groups of young friends. Similarly, "fanatic" has very negative connotations, of insanity or obsession. And you didn't say he was being fuzzy with the term "pitch", I did: it looked like you were saying that it is significant that he calls the members of his list "pitches", while I don't think it is. — Gwalla | Talk 22:13, 27 August 2014 (UTC)

You will be hard pressed to prove that an interval isn't used. Hyacinth (talk) 07:42, 5 September 2014 (UTC)

Given the definition of tuning as "a way to tune an instrument", how many ways are there to tune instruments? Hyacinth (talk) 07:48, 5 September 2014 (UTC)

Let’s first agree that what we are speaking of here is not the tuning as performed by instrument makers or professional tuners, but the building of systems, in a sense approaching that of the Greek systema teleion – the overall scalar arrangement underlying music. The tuning of instruments is involved mainly because some of them provide a concrete illustration of the system: particularly keyboards (many theory textbooks begin with an image of a keyboard, meant to represent the system), also harps, guitars, etc.; more generally, instruments 'of fixed pitches'.

Not all musical cultures know an underlying scalar system of this type; I won't discuss that aspect now. "Tuning", as discussed here, is mainly (but not exclusively) a matter of Western music. There are, I think, three main ways of 'tuning', of building musical systems: (1) tuning consonances, (2) temperaments, and (3) equidistant division, which I’ll consider in turn.

1. Tuning consonances. Consonances have been associated either with numeric ratios of whole numbers, or with harmonic partials; these two boil down to the same, because the harmonic series is the series of whole numbers. Three such systems are documented in Western history: Pythagorean tuning, based on tuning perfect fifths (3:2); "just" tuning (also called Zarlino’s system), tuning perfect fifths and major thirds (3:2 and 5:4); and unsuccessful 18th-century attempts (mainly by Euler) to add to these consonances the 'natural seventh' (7:4) – unsuccessful because the natural seventh does not easily fit in the diatonic/chromatic system of common practice Western music. Harry Partch rekindled the reflection on 'consonances' with his concept of prime limit, extending the number of harmonics considered … without limit.

2. Temperaments have been conceived, from the late 15th or early 16th century onwards, mainly for instruments of fixed pitches with 12 pitches in the octave. Temperaments consist in 'tempering' a consonant interval (usually the fifth), i.e. making it very slightly dissonant, in order to increase the compatibility of the degrees between themselves. There are mainly two types of temperaments:

– regular temperaments (also called meantone temperaments), tempering all fifths by the same amount;

– irregular temperaments, often meant to make some keys closer to just intonation than others.

Regular (especially equal) temperaments can of course be extended to produce as many subdivisions in the octave as one wants; several such extended systems are historically documented because they provide satisfactory approximations of systems calculated otherwise.

3. Equidistant divisions should not be confused with equal temperaments. They consist in dividing a geometrical space, e.g. between frets on a fingerboard or between fingerholes on a wind instrument, in equal distances. (Equal temperaments divide in equal intervals, resulting in distances in an exponential series.) I have seen a folk 'spinet' (a monochord) where the maker placed the frets using a stapler and a centimeter, putting a staple at each centimeter. More often, intervals obtained by 'consonance' (perfect fourths, in most cases, 4:3) are divided in several (e.g. three) 'equidistant' intervals: this is the case in several medieval descriptions of Arabic musical systems. Equidistant divisions coincide with consonant divisions, but are conceived on a different principle. The string of an Arabic lute, for instance, could be divided in 40 equidistant spaces, of which only the last 10 are used: 30 spaces produce the perfect fourth to the open string (30/40=3/4), 32 spaces produce the just major third (32/40=4/5); the other divisions produce less usual intervals. This corresponds to dividing the space of the 4th in 10 equidistant spaces. Although these 10 intervals (39/40, 38/40=19/20, 37/40, 36/40=9/10, 35/40=7/8, 34/40=17/20, 33/40, 32/40=4/5, 31/40 and 30/40=3/4) are relatively well documented in medieval Arabic theory, only 4 of them are listed here, which once again makes me wonder what this article is about.

Does this answer your question? – Hucbald.SaintAmand (talk) 18:15, 5 September 2014 (UTC)

If there are an infinite number of ways to tune instruments, and there are an infinite number of pitches, you may never be satisfied with this article's completeness. Hyacinth (talk) 23:34, 5 September 2014 (UTC)

Exactly, Hyacinth. This is why I wrote above: "All in all, I think that Wikipedia might offer various lists of [relative] pitches (rather than intervals), e.g. pitches in this or that (family of) historical temperament(s), or in this or that prime limit, or whatever, each clearly stating what it concerns, possibly with a disambiguation article refering to these various lists." Even a list of "prime-limit intervals" (or "relative pitches") could not be complete, unless it stated from the start which limits were retained. A list of Pythagorean intervals or relative pitches could be made reasonably complete. A list of meantone intervals or pitches could at least say which meantone temperaments are retained, and to how many degrees they are extended. A list of intervals or pitches in medieval Arabic theory could be more or less complete, and could easily say which theorists are considered. Etc.

My only complain about the present article is that it is unable to clearly state which intervals were retained, and why. I tried to clarify things by modifying the lead, but it remains that this article merely cannot in any way be 'complete'.

Hucbald.SaintAmand (talk) 07:46, 6 September 2014 (UTC)

See: Wikipedia:Wikipedia is a work in progress. Hyacinth (talk) 20:00, 8 September 2014 (UTC)

Newer image

Why not make the image in the introduction (File:Meantone.jpg) bigger? Hyacinth (talk) 20:01, 8 September 2014 (UTC)

It seems unclear what the syntonic comma is between. Maybe the line should go from symbol to symbol. Hyacinth (talk) 21:21, 8 September 2014 (UTC)

Instead of creating too much clutter, adding a 5-limit just chromatic scale (8/5 6/5 9/5 4/3 1/1 3/2 9/8 5/3 5/4 15/8) would help clarify. See: File:Meantone comparison.png. Hyacinth (talk) 21:25, 8 September 2014 (UTC)

[The answer below was written before your suggestions for changes in the image itself and concern only your suggestion to changing its size. It may however in a way answer tot the whole.]

Well, I don't know, Hyacinth, maybe merely because as its author I didn't want to push too far... Also, I thought that it nicely occupied the space left at the right of the table of contents. You have more experience in Wikipedian matters and I'll gladly give in to your opinion.

On a more general level, I think, Hyacinth, that we should stop arguing about this article, and that we should better seriously consider how to improve it or to replace it by something better. I think that the core of the problem has to do with this matter of prime limit, which is not treated as rigorously as it should, neither here nor in the other articles that I have read.

You induced me to reread Ellis' Additions to Helmholtz' treatise, and this made me realize that the concept of prime limit may somehow stem from there. Indeed, Ellis does speak of "septimal" chords, based on harmonic 7, and claims for instance that the "septimal minor triad G ⁷Bb D is by far superior to the Pythagorean minor triad D F A". The notation "⁷Bb" is striking. The passage raises several questions, for instance why he describes the Pythagorean triad as D F A, not G Bb D, etc. He also speaks of septendecimal chords, using harmonic 17 ("¹⁷Db").

These considerations do not belong to this particular article on "pitch intervals", but rather on the one on prime limit. But there exists an array of articles on such matters, also including those on the various types of temperaments, which do deserve some of our energy. My figure about which you raise the question above also does not really belong in this article, but more probably in one on meantone tuning and temperaments.

We won't solve any of this, and we may keep quarreling without true reason, unless we attack the problem from the start. This is a problem similar to that about diatonic/chromatic/enharmonic, as recently discussed on the Talk:Music_theory page, if I remember well, without that we reached any concrete conclusion.

Such a project begins with an inventory of the articles that need revision and coordination. But it requires many collaborations. I think that here too, an appeal should be made to the participants to the Music Theory Project. I would more easily participate to that sort of thing than to hopelessly try to improve the List of pitch intervals article as it is.

I am not an iconoclast, Hyacinth, even if at times I do not show enough patience. I do hope we can get out of the quarrel and begin something worthwhile.

Hucbald.SaintAmand (talk) 21:53, 8 September 2014 (UTC)

Let me add a few comments to your proposed image including just intonation. I shall first stress that what an image in the lead of this article really should show is 3-limit, 5-limit, etc. There is no other reason for the image to show meantone temperaments (which are not really present in the list itself) than that it is mentioned in the terminology section. But a representation of 3-limit and 5-limit intonations is an interesting challenge, worth a discussion. Needless to say, such an image would better belong to the Prime limit article. But imagining it here raises interesting questions.

Let's begin with 3-limit, i.e. extended Pythagorean tuning. The figure as it stands is limited to 12 degrees (plus one for the enharmony) because of the chromatic scale and the usual keyboard of 12 degrees in the octave. If one wants to show Pythagorean tuning extended to more than these, one merely has to extend the line representing it (blue in my figure, red in yours) on both sides, in order to reach as many degrees as desired. Or else, one may conceive the drawing as if drawn around a cylinder, with G# at the right on the same vertical axis as Ab on the left, and with a second turn after G#, continuing with D#, A#, E#, etc., and a third turn 'before' Ab, descending Db, Gb, Cb, etc.: the extended Pythagorean would then be represented as a spiral (this is a well-known representation). The unfolded cylinder would show several lines parallel to the main one, and the number of parallel lines would depend on the extension given to the Pythagorean system. This would form a convenient illustration of 3-limit, and a complete one if the limit of the extension were stated. [I didn't check the extension of the 3-limit in the list; I can only suspect (or hope) that it may be complete to some unstated extension.] I will not draw the figure, but I may represent it in tabular form as follows:

816

G#

D#

A#

E#

B#

Fx

Cx

Gx

Dx

Ax

Ex

Bx

F###

840

792

Ab

Eb

Bb

F

C

G

D

A

E

B

F#

C#

G#

816

768

Bbbb

Fbb

Cbb

Gbb

Dbb

Abb

Ebb

Bbb

Fb

Cb

Gb

Db

Ab

792

i.e. three lines, each with the same slope as the line labelled "Pythagorean", and each a Pythagorean comma higher than the preceeding one.

In your representation of just intonation, your brown zigzag line labelled "just", presented in the form of a Tonnetz (fifths as rows, major thirds as columns), would be like this:

		Ab	Eb	Bb
	F	C	G	D
	A	E	B
F#	C#	G#

As you can see, your presentation is somewhat irregular, with F# being the major third of no other note: F# should better stand at the right of B, i.e. on the same line as A E B in your drawing. [Note also that the four lines of this Tonnetz are distant from each other by a syntonic comma, as is usual in such representations.] It is inherent to just intonation itself that it can be (and has been) presented in many different ways: any representation that you would chose would represent only one version of just intonation. As a matter of fact, one would need each degree both as the fifth of another and as the major third of another still. This soon results in way more than 12 degrees, e.g. in instruments like Vicentino's arcicembalo with 36 keys in the octave, and in the Tonnetz itself being a torus: see File:TonnetzTorus.gif.

Never mind. Our problem now is to represent extended 5-limit intonation. That can be achieved by extending each of the fragmentary brown lines in your image at both sides, to reach both Ab and G# in pure fifths. One would again have parallel sloping lines as in the extended 3-limit described above, but now the lines would be a syntonic comma apart instead of a Pythagorean one! Combining 3-limit and 5-limit in a single figure would result in something like this:

816

Ab

Eb

Bb

F

C

G

D

A

E

B

F#

C#

G#

840

814

Ab

Eb

Bb

F

C

G

D

A

E

B

F#

C#

G#

838

792

Ab

Eb

Bb

F

C

G

D

A

E

B

F#

C#

G#

816

770

Ab

Eb

Bb

F

C

G

D

A

E

B

F#

C#

G#

794

768

Ab

Eb

Bb

F

C

G

D

A

E

B

F#

C#

G#

792

with the parallel lines 816-840 and 814-838, or 770-794 and 768=792 being a schisma (~2 cents) apart, the difference between a Pythagorean and a syntonic comma. Such a figure would indeed represent extended 3- and 5-limits, with a number of intervals to be measured from C in the center. In addition, the lines a syntonic comma apart could also be extended to spiral aroung the cylinder, but the whole would soon become utterly unreadable, reducing to a set of parallel lines 2 cents apart.

To sum up, if my figure were to be changed, (1) the two lines representing meantone temperaments should be removed, as they have no place here; (b) both the "Pythagorean" line and your zigzag line representing just intonation should be extended to represent 3- and 5-limits. I cannot figure how to represent 7-limit, unless by adding a third dimension – our computer screens cannot yet do that, but it will come.

Enough for today. – Hucbald.SaintAmand (talk) 08:16, 9 September 2014 (UTC)

Sorry, I believe I meant the display size, and not the image itself. But I was being a sarcastic jerk, for which I apologize. In the meantime it appears that I have created an image: File:Meantone comparison.png, as well as File:Meantone comparison just.png and File:Meantone comparison Pythagorean.png. Hyacinth (talk) 20:51, 27 May 2015 (UTC)

These indeed are interesting images, Hyacinth. Yet, I have a problem with the representation of just intonation. Such images work well with "regular" temperaments (i.e. temperaments with one size of 5th), as they are meant to show the size of the 5ths in comparison with some other case (ET, or Pythagorean, etc.). They hardly show the size of 3ds, which are considered resultant from three or four 5ths. They also show the size of the 5ths in the case of just intonation, particularly the fact that some 5ths are not the same size as others; but they hardly show why. It is only when one observes that the line representing just intonation sort of follows that representing 1/4 comma meantone that one might begin to suspect the reason. An additional problem is that just intonation can be built in many ways. (This already was discussed above, 9 September 2014, when I noted that in your description F# is the 3d of no other note.) In your comparison meantone-just, the increasing divergence between the lines at the right of the figure is caused by your particular construction of just intonation. If you had built a series of three 5ths A-E-B-F# (as is more usual), the figure would be less divergent. It would be tricky to explain this to the reader (I don't even know whether you can follow what I just tried to explain!). — Hucbald.SaintAmand (talk) 14:04, 28 May 2015 (UTC)

Hyacinth, although you removed your last changes before I saw them, they made me realize that I was wrong on my last point: your comparison meantone-just did include an A-B-E-F# span, which I overlooked. Sorry for that. (Or else you changed your image, but I don't think so.)

On the other hand, I have some doubts about the inclusion of the "schismatic temperament", which obviously belongs to another level of reflexion. Meantone temperaments, ET and Pythagorean tunings are all historical tunings, documented since centuries, while the schismatic temperament (1) is a recent construction (2007, I think); (2) has never been and couldn't be utilized to tune real instruments; (3) is only a mathematical construct aiming at explaining micro intervals. Not that this would be a wrong purpose, but that it distorts the image given, mixing historical facts with recent speculations. I may be too traditional a musicologist too easily accept this, but my opinion on this point might be worth recording here, if not in the article itself... Hucbald.SaintAmand (talk) 20:19, 1 June 2015 (UTC)

The image is also "distorted" (I assume you mean "distorting") in that it contains five other unlabeled syntonic commas. Hyacinth (talk) 04:28, 2 June 2015 (UTC)

I'm sure Helmholtz died before 2007, yet his On the Sensations of Tone mentions "skhismic temperament" on page 435. Hyacinth (talk) 05:02, 2 June 2015 (UTC)

;-)) You should never forget that English is not my native language. It is not your image that is distorted, it is the view that it gives of the situation (the image given by your schema, if you want). As to Helmholtz, several points:

1. The mention of the "skhismic temperament" on p. 435 is among the "Additions by the translator", i.e. by Ellis. What Helmholtz himself has to say about this can be found on p. 280 ff. of the translation, about the "Arabic and Persian musical system"; see especially Ellis' footnote * on p. 281.

2.1. I fail to see why Ellis calls this a "temperament". He says that the 5ths have to be "perfect", 701,955 cents [note that a temperament, as Lindley evidenced, always tempers the 5ths], i.e. Pythagorean fifths, and the Skhisma disregarded, which results in his "K" (the comma) amounting to 23.460 cents (i.e. the Pythagorean comma) and "T" (the major third) 384,360 cents, i.e. a Pythagorean 3d diminished by a Pythagorean (instead of a syntonic) comma.

2.2. This is not a temperament because no interval is tempered. The fifths are perfect and the only question is how one comes to such thirds. Ellis gives the answer when he says (p. 435) that c:fb is "such a major third", then quotes a:db:e and e:ab:b as "quite smooth" triads (see also p. 280). This is what Liberty Manik (Das arabische Tonsystem im Mittelalter, 1969) described as Safi al Din's Schismatische Verwechslung, and what was used by Arnaut de Zwolle and others in the 15th century as an early just intonation (as commented by Mark Lindley in several papers — see Schismatic temperament#History of schismatic temperaments).

3. It certainly is not what your figure describes as "schismatic temperament", in which (if I read correctly) the 5ths are tempered by 1/12 of a schisma and measure 701,8 cents — that is that the difference between Ab at the left and G# at the right of your line for the schismatic temperament is a syntonic comma of 21,506 cents instead of a Pythagorean one of 23,460 cents. This is a 1/12-schisma temperament and it belongs to the schismatic temperaments described by Milne, Sethares and Plamondon in Computer Music Journal 31/4 (2007) — but I agree that this is not the earliest description of such things.

Note that the Schismatic temperament article is a mess, probably another product of the gang of microtonalists [I know that my use "gang" here is improper English but, as said above, I am not a native speaker]. I fail to see what "tempering the schisma to a unison" might mean; the article at times speaks of "tempering out", at other times of "tempering to (some interval)", neither of which makes sense to me [and this is not because I don't understand English]; the name "Helmholtz temperament" is undocumented and probably comes from [2]; Mark Lindley never said (nor believed) that schismatic tuning as described in the article ever was in use during the late Middle Ages: there is a confusion here between Ellis' (or Safi al Din's) skhismic temperament (see above) and the "schismatic temperaments" to which the article is devoted. Helmholtz did describe something ressembling a schismatic temperament (p. 512 of the German version, 316 of the translation), but did not describe it in terms of fractions of a schisma: he used 1/8 of the amount by which a 5th is tempered in 12-T ET — it boils down to almost the same, but not exactly.

Hucbald.SaintAmand (talk) 09:29, 2 June 2015 (UTC)

You got the year 2007 from Syntonic temperament. Hyacinth (talk) 17:14, 2 June 2015 (UTC)

2007 is the date of the paper by Milne e.a. in Computer Music Journal. I just had a look to Syntonic temperament, which I had never seen before. It seems almost identical with Schismatic temperament. Both might better be reassembled under, say, "Regular temperament", which I don't think is a term coined by Erv Wilson; it is used, if my memory doesn't fail, by Murray Barbour and several others. Regular temperaments are also meantone ones... Wikipedia obviously does not provide (and does not have) a very clear idea of all this. — Hucbald.SaintAmand (talk) 19:23, 2 June 2015 (UTC)

See: Schismatic temperament#Construction. Hyacinth (talk) 19:04, 3 June 2015 (UTC)

Lets look at this together, Hyacinth.

In Pythagorean tuning all notes are tuned as a number of perfect fifths (701.96 cents). The major third above C, E, is considered five fifths above C.

We will easily agree that E really is four fifths above C, not five. Never mind.

This causes the Pythagorean major third, E+ (407.82 cents), to differ from the just major third, E♮ (386.31 cents): the Pythagorean third is sharper than the just third by 21.51 cents (a syntonic comma).

Helmholtz's "skhismic temperament" (Helmholtz/Ellis 1885 p. 435) instead uses the note eight fifths below C, F♭ (384.36 cents), the Pythagorean diminished fourth or schismatic major third. Though spelled "incorrectly" for a major third, this note is only 1.95 cents (a schisma) flat of E♮, and thus more in tune than the Pythagorean major third. As Helmholtz puts it, "the Fifths should be perfect and the Skhisma should be disregarded.

I think that it is Ellis who says that at this point, not Helmholtz, but never mind. This, if I may repeat myself, is Safi al Din's Schismatische Verwechslung as discussed by L. Manik, or Arnaut de Zwolle's (and others') tuning system as discussed by M. Lindley. It is NOT a temperament: the 5ths remain pure, as clearly stated later in the same article (Schismatic temperament#History of schismatic temperaments: "Mark Lindley and Ronald Turner-Smith (1993) argue that schismatic tuning was briefly in use during the late medieval period. This was not temperament but merely 12-tone Pythagorean tuning, though typically tuned from G♭ to B in ascending just fifths and descending just fourths, instead of the prevalent A♭ to C♯ or E♭ to G♯ schemes."

In his 1/8th-schisma "Helmholtzian temperament" (Helmholtz/Ellis 1885 p. 435) the note eight fifths below C is used as the major third above C, but it, rather than the perfect fifth, is assumed to be perfectly in tune.

No, by no means! In Helmholtz/Ellis/Safi al Din/Arnaut de Zwolle's tuning, the 5ths remain perfectly in tune, nothing is tempered by 1/8th-schisma, this is a confusion between two different systems! The 3d is a schisma flat, but this is assumed negligeable. The Wikipedia article, I repeat, confuses here two systems, both described by Helmholtz, one that Ellis names "skhismic temperament" (improperly so because it is not a temperament), and the other Helmholtz' tuning of his harmonium as described on p. 316 of Ellis' translation (and which truly is a temperament). Even this temperament, however, is not strictly speaking an 1/8th-schisma temperament, because Helmholtz clearly says that the 5ths are tempered by 1/8th of the difference between a perfect 5th and a 5th in equal temperament. This difference is almost equal to a schisma, but not exactly so. And, as a result, the major third in this temperament of Helmholtz is exactly in tune, not a schisma flat. Because the harmonium has 34 keys in the octave, it cannot really be said that the major third is C-F♭, even although in essence that is what it is.

As Helmholtz puts it, "the major Thirds are taken perfect, and the Skhisma is disregarded.

This is an inaccurate repeat of a quotation made two lines above and once again, it is not Helmholtz, but Ellis who puts this. What Ellis really writes (p. 435) is "The conditions is that the Fifths should be perfect and the Skhisma should be disregarded". That is to say, the fifths should remain perfect (as in Pythagorean tuning), and the resulting difference of a schisma in the thirds should be neglected. There is no mention whatsoever — never, neither in Helmholtz nor in Ellis — of a temperament by 1/8th-schisma. With all due indulgence for Wikipedia, this article is obviouly confused and mistaken. It renders unduly complex things that in essence are much simpler. — Hucbald.SaintAmand (talk) 21:35, 3 June 2015 (UTC)

Moved to: Talk:Schismatic temperament#Image
Hyacinth (talk) 15:18, 4 June 2015 (UTC)

SVG file wrong

Personally, I believe what the svg file shows is wrong. For example, perfect fourth in Pythagorean tuning is about 498 cents but 500 in 12-TET. However, on the svg file, it shows the perfect fourth in 12-TET is shorter than Pythagorean tuning.

Mscdancer (talk) 15:34, 4 May 2013 (UTC)

Most often people create a chromatic scale in Pythagorean tuning by creating fifths both above and below the tonic or first note. The chromatic scale used in this picture only uses fifths above the starting point. Thus there is no perfect fourth, but rather an augmented third (enharmonic). While the Pythagorean perfect fourth is below 500 cents, the Pythagorean augmented third is above 500 cents.

Fifth	Letter	Interval	Just cents	ET cents	Relation to ET
0	C	Unison	0	0	Equal
1	G	Pythagorean perfect fifth	701.96	700	Above
2	D	Pythagorean major second	203.92	200	Above
3	A	Pythagorean major sixth	905.87	900	Above
4	E+	Pythagorean major third	407.82	400	Above
5	B+	Pythagorean major seventh	1109.78	1100	Above
6	F#++	Pythagorean augmented fourth	611.73	600	Above
7	C#++	Pythagorean augmented unison	113.69	100	Above
8	G#++	Pythagorean augmented fifth	815.64	800	Above
9	D#++	Pythagorean augmented second	317.60	300	Above
10	A#+++	Pythagorean augmented sixth	1019.55	1000	Above
11	E#+++	Pythagorean augmented third	521.51	500	Above
12	B#+++	Pythagorean augmented seventh	1223.46	1200	Above

Hyacinth (talk) 00:19, 17 August 2013 (UTC)

Sorry, I made the above image. What can I do to fix it? SharkD  Talk  00:26, 13 November 2016 (UTC)