Talk:Schismatic temperament

Spelling
As I understand it, the story of the different spellings is as follows:

Helmholtz's original German book used the term "schismatische Verwechslung". I probably got that wrong, but it would naturally fit "schismatic temperament" in English.

Ellis translated it as "skhismatic" using a more modern Greek transliteration.

Other writers (dunno who) called it "schismatic" which is already an English word (a heretical movement that breaks off from the One True Church or somesuch) and consistent with "schism".

John Chalmers used the word "skhismic" to show its Greek roots, leave out the "at" which doesn't mean anything, and avoid unfortunate associations with heretical movements.

I then have to take the blame for introducing a third variant, because there wasn't an established standard, and I wanted a simple, familiar spelling. I don't think this action really deserves to be recorded in the first paragraph of the actual article. Whatever spelling you use here will become the standard, no need for apologies.

X31eq 12:56, 17 December 2005 (UTC)


 * I discover this article somewhat late, and my question/comment probably comes too late. However, here it is: I am surprized at the mention that "Helmholtz's original German book used the term schismatische Verwechslung". It is not very easy to check, but it seems that Helmholtz not only didn't say schismatische Verwechslung, but also never used the term schisma. This at least is what Liberty Manik writes in his Das Arabische Tonsystem. If it were not true, I'd be interested to have a more precise reference. Hucbald.SaintAmand (talk) 17:56, 5 June 2015 (UTC)

lack of examples with frequency-ratios
I find this article really hard to understand, and i think this happens because of the lack of examples. Does a typical schismatic temperament consist of 12 or 24 or 36 tones per octave? Is there a typical example in the first place? And if there is no typical example, what would just some example be like? i would appreciate the addition of one or two examples to this article by some knowing person very much. 62.134.176.86 23:33, 18 May 2006 (UTC)


 * There are no frequency ratios because schismatic temperament is a temperament, not just intonation, so the intervals don't correspond exactly to rational numbers. There is only one schismatic temperament, and it extends infinitely in both directions. Since there is only one schismatic temperament, it doesn't make sense to provide "examples". Nevertheless, I'll try to improve the article according to your suggestion. —Keenan Pepper 00:45, 19 May 2006 (UTC)


 * Thanks for the fast answer! With frequency ratios i mean in this case an actual rational number or something like 2^(1/12) as a factor between frequencies. But you are right, an irrational ratio is not really a ratio. So what i meant is something like a relative frequency table. So it's your point. Concerning the examples, i still think that a range of the infinitely extending temperament is still an example for a temperament. In my opinion a temperament is something that can be seen on real instruments. But one could argue that an infinitely extending temperament is more something like a temperament, because it's more abstract. 62.134.229.109 11:58, 19 May 2006 (UTC)

"If spelled according to their construction in the tuning" How? And why?

 * An advantage of meantone over schismatic tunings is that in meantone, the interval ratios of 5:4 and 6:5 are represented by the major third and minor third, respectively. In schismatic tunings, they're represented by the diminished fourth and augmented second (if spelled according to their construction in the tuning).

This needs explanation. Why would they be spelled differently? How do you spell "according to their construction" (it's not clear how the conventional ABCDEFG sequence is necessarily specific to Pythagorean/meantone tunings). Why couldn't people dodge the whole issue by spelling notes according to the meantone notes they are closest to? &mdash; Gwalla | Talk 01:50, 14 March 2008 (UTC)


 * See enharmonic. Hyacinth (talk) 16:23, 2 June 2015 (UTC)


 * A temperament always is a stack of fifths (Pythagorean tuning, meantone and schismatic temperaments are stacks of equal fifths). Spelling according to their construction is spelling according to the series of fifths involved. In meantone, this becomes, say, C-G-D-A-E, because the third is supposed reached after four 5ths; in schismatic tuning, it is C-F-G-E-A-D-G-C-F♭, eight 5ths.
 * The problem of spelling arises because tunings often are performed on keyboards (or other instruments of fixed tones) where each successive 5th is materialized on one key. In the 15th century, keyboards have been tuned in perfect fifths in the sequence, say, G♭-D♭-A♭-E♭-B♭-F-C-G-D-A-E-B, as follows:


 * || D♭ ||  || E♭ ||   ||   || G♭ ||   || A♭ ||   || B♭ ||    ||
 * C ||  || D ||   || E    || F ||   || G ||   ||  A ||   || B    || C
 * }
 * (This was discussed, in the 1980's I think, by Mark Lindley in several articles on the origin of meantone temperament and just intonation.)
 * In this case, black keys produced almost perfect thirds with the white key a major third below: B-E♭, D-G♭, E-A♭ and A-D♭, but these four flat keys of course were understood (and named) as sharps. The spelling according the construction (as described in several 15th-century treatises) differs from the spelling according to usage. (The reason why the schismatic enharmonies happen only on black keys has to do with the principles of musica ficta — this is another discussion.) In schismatic tunings, so far as I can tell, the spelling according to construction always is different from the one according to usage (i.e. there is always an enharmony involved), and this may well be one of their most characteristic aspects.
 * This case is quite simple because the tuning produces no more than 12 pitch classes. I don't know how one does if the tuning is to produce, say, 171 pitch classes in the octave (see below). The largest number of keys I ever knew to exist on a real instrument was 53 in the octave (this was an instrument tuned in Holderian commas and supposed to teach the students how to play "in tune", in the Conservatoire where I studied. It had been described in a booklet, see . When I saw it, the instrument was abandoned in an attic; but there are other examples.).
 * Hucbald.SaintAmand (talk) 06:41, 5 June 2015 (UTC)
 * This case is quite simple because the tuning produces no more than 12 pitch classes. I don't know how one does if the tuning is to produce, say, 171 pitch classes in the octave (see below). The largest number of keys I ever knew to exist on a real instrument was 53 in the octave (this was an instrument tuned in Holderian commas and supposed to teach the students how to play "in tune", in the Conservatoire where I studied. It had been described in a booklet, see . When I saw it, the instrument was abandoned in an attic; but there are other examples.).
 * Hucbald.SaintAmand (talk) 06:41, 5 June 2015 (UTC)


 * It seems the Musical temperament article needs to address the use of fifths in its introduction. Hyacinth (talk) 00:40, 7 April 2017 (UTC)
 * Meantone temperament says that it, "tempers the syntonic comma to unison." Hyacinth (talk) 01:07, 7 April 2017 (UTC)
 * This question is addressed in Talk:Meantone_temperament. I am at loss to make my point accepted, though, and therefore won't change anything for the time being. — Hucbald.SaintAmand (talk) 09:40, 7 April 2017 (UTC)

Image


There are two tunings are proposed by Helmholtz/Ellis, one of which uses tempered fifths. They are given two different names. What you took as an "inaccurate repeat" describing one tuning, was really a quotation about the second of two tunings. "Disregarded" is used in two different senses. Hyacinth (talk) 15:27, 4 June 2015 (UTC)
 * "the Fifths should be perfect and the Skhisma should be disregarded" describes "skhismic temperament" [sic]
 * "the major Thirds are taken perfect, and the Skhisma is disregarded" describes "Helmholtzian temperament" [no sic]


 * [The comment above by Hyacinth is in answer to my continued (and somewhat boring) criticism at the end of Talk:List_of_pitch_intervals: what follows would remain obscure for anyone who wouldn't have seen that exchange.]
 * Agreed, Hyacinth, I was too fast about that. As I have seen Schismatic temperament only in the midst of your recent corrections, I won't try to figure out how it was before and after. Let me only stress points that I think still need your attention:


 * To say that "one tempers the schisma to unison" makes little sense (as a matter of fact, I fail to see what sense it could make). As Lindley rightly stresses in his Mathematical models, the only interval that is tempered, in a temperament, is the 5th. The 5ths of equal temperament, for instance, are tempered by 1/2-Pythagorean comma, so that the Pythagorean comma vanishes — it is not, however, "tempered to unison". In 1/4-(syntonic) comma meantone, the fifths are tempered so as to make that particular syntonic comma vanish by which the perfect third is lower than the Pythagorean one — but the syntonic comma is neither "tempered to unison", nor "tempered out", nor "tempered" everywhere: it does not completely vanish for the minor third. It would be necessary to state which particular schisma is concerned by a schismatic temperament.
 * To say that the schismatic temperament "is also called the schismic temperament, Helmholtz temperament, or quasi-Pythagorean temperament" fails to make the point (which is made later in the article) that these terms migh refer to similar, yet different systems or temperaments.
 * To speak of "Helmholtz's 'skhismic temperament'" is misleading, because Ellis (not Helmholtz!) names a first tuning the "skhismic temperament" and a second, different tuning the "Helmholtzian temperament".
 * It might be worth explaining that Ellis' "skhismic temperament" had been first imagined in the late Middle Ages, at a time when Pythagorean intonation still was considered some divine truth to be respected at all times. Safi al Din apparently thought that the "schismatic change" could provide a Pythagorean justification of the Arabic "quarter tones", but he himself later recognized that this was a failure. Occidental musicians tried to produce "Pythagorean perfect thirds", until Ramos de Pareja (1486, I think) denounced this hypocrisy. It might be added that the medieval occidental tuning of this kind never extended to F♭ but only to G♭, because the enharmonic/schismatic major thirds could only involve black keys of the keyboard. Lindley listed several versions of this tuning, in several of his articles. (I didn't check whether they are mentioned in his Mathematical Models.)
 * "In his eighth-schisma "Helmholtzian temperament"... Once again, what is tempered here by 1/8-schisma is the 5th, and not the schisma itself. More important, this temperament is never described as an 1/8-schisma temperament in Helmholtz (and not clearly in Ellis). I admit that this is due to some kind of astonishing mistake from their part, yet it is a fact. Helmholtz says that the temperament (of the 5th) should amount to 1/8 of the difference between a perfect fifth and a tempered one in ET; Ellis does not clearly repeat this (see however note * on p. 316 of his translation).
 * [It might also be interesting to note that Ellis, in footnote + of p. 316, writes that "such a Fifth [...] is hopeless to tune exactly. Hence these Fifths can only be regarded as products of calculation which could not be realised". One cannot claim that this is no more true today, in the age of computers, because 1/8-schisma is certainly quite lower than the limit of human perception.]
 * "As Ellis puts it, 'the major Thirds are taken perfect, and the Skhisma is disregarded [tempered out]'." Equating "disregarded" with "tempered out" once again raises the question of the meaning of "tempered out". Ellis said "disregarded", but certainly not "tempered out".
 * "Whereas schismatic temperaments achieve a ratio with a number [of] perfect fifths"... Even if this is true, it took me quite some time to figure out what it meant. Schismatic temperaments apparently want to achieve one single ratio, 5:4, the perfect major third, as does meantone temperament. They do so not with perfect fifths (it would be mathematically impossible), but with tempered fifths. The main difference between meantone temperaments and schismatic temperaments, on first reflexion, appears to be that schismatic temperaments also involve an enharmony: they take C-F♭ instead of C-E to represent the perfect major third (i.e. starting from (5/4)^8 instead of (5/4)^4), or C-D♯ instead of C-E♭ for the minor third, which must somehow involve tempering by 1/9-schisma. I cannot figure out which other "ratio" (=interval expressed in form of a ratio) such a temperament could aim at. But "schismatic temperaments" in microtonal music apparently aim at arbitrary microintervals which, I suppose, could all be expressed as ratios, as complex as these may be...
 * "schismatic temperaments are often described by what fraction of the schisma is used to alter the perfect fifths". This is the only case when they can be described as "temperaments", for all the reasons discussed above. A temperament, by definition, consists in alterations of the perfect fifths. In order to reach an interval expressed as a ratio, the temperament by definition must also be expressed as a ratio (i.e. a fraction of some interval).
 * "Various equal temperaments lead to schismatic tunings". Any temperament dividing the octave in a large number of intervals easily becomes an easy approximation of any other similar temperament. Dividing the octave in 53 (or 55) is a approximate division in commas. 171-ET indeed is "hard to reach"... Ain't we speaking about music ?
 * "Helmholtz had a special Physharmonica" — a reference is indeed much needed. The term, so far as I have been able to check, is found neither in Helmholtz nor in Ellis. A Physharmonica is but an early-19th-century version of the more common harmonica; I doubt it still was in use in Helmholtz'time.
 * "schismatic is in some respects an easier way to introduce approach justly tuned thirds into a Pythagorean harmonic fabric than meantone"... Beside the fact that the phrase appears syntactically and semantically incorrect ("introduce", "approach", "into", "harmonic fabric"?), I'd say "in all respects". The introduction of meantone (and just intonation) at the turn of the 16th century represented a revolution, the abandonment of a millenial theoretic Pythagorean tradition, after somewhat futile attempts (including "skhismic temperament") to reconcile perfect thirds with Pythagorean tuning.
 * "schismatic substitutions just outlined": this is the first and only occurrence of the word "substitution" in the whole article.
 * Etc. This will be enough for tonight.
 * Hucbald.SaintAmand (talk) 20:59, 4 June 2015 (UTC)


 * How would you know proper English? Note that Ellis writes on p.435, "The Comma and the Fifth are therefore imperceptibly flattened."
 * There is no such thing as a "perfect third".
 * It doesn't matter if Ellis draws a square a doesn't label it a square, it still is one.
 * Where, in the wikipedia article, does the description of "Helmholtzian temperament" mention a schisma being tempered? Quote, if you can, the use of the number "700" or "1.955" in the book.
 * You should go see musical temperament before you start asserting definitions of "musical temperament".
 * Hyacinth (talk) 22:25, 5 June 2015 (UTC)

Hyacinth, I am perfectly aware of the shortcomings of my English; but if you take anything I write as personal attacks, even switching to another language would not help.
 * Ellis writes "The Comma and the Fifth are therefore imperceptibly flattened."

So what? I suppose you mention this to fault my statement that a temperament always is a temperament of the fifth. But there is no contradiction: tempering the fifths indeed usually results in other intervals being flattened.
 * There is no such thing as a "perfect third".

I will agree with this only if we agree that there also is no such thing as a "perfect fifth". We have a choice between such terms as "perfect", "pure", or "just". I personally prefer "just", because (a) it refers to just intonation and (b) it is a term used since the time of Sauveur, I think, in the early 18th century, by the French Académie des Sciences: it at least has the venerability of that institution. But never mind. What I mean is a 5:4 major third (or possibly a 6:5 minor third).
 * It doesn't matter if Ellis draws a square a doesn't label it a square, it still is one.

I'm sorry, I don't see what you aim at here. Let me say in passing, though, that it may matter that the word "schisma" apparently does not belong to the vocabulary of Helmholtz, up to the point that it never appears in his writings. As a matter of fact, it does not appear in Adelung's Wörterbuch of 1808 nor in J. and W. Grimm's of 1854, and in Brockhaus' Conversations-Lexicon of 1809-1811 or in Meyer's Konversationslexikon of 1888 only in reference with the Christian schism. Riemann, in his Musik-Lexikon (1882), rightly describes the schisma and stresses (perhaps in answer to Helmhlotz) that "the schisma expresses [...] almost exactly the difference between a pure (reine) fifth and the fifth of 12-note equal temperament, which is also named 'schisma'". None of this, of course, prevents a schisma for being a schisma, but it seems to me revealing about the history of the term and its (non) usage by Helmholtz and Ellis.
 * Where, in the wikipedia article, does the description of "Helmholtzian temperament" mention a schisma being tempered?

In the lead: "A schismatic temperament is a musical tuning system that results from tempering the schisma [...]. It is also called the schismic temperament, Helmholtz temperament, or quasi-Pythagorean temperament." Or do you make a difference between "Helmholtz temperament" and "Helmholtzian temperament"?
 * Quote, if you can, the use of the number "700" or "1.955" in the book.

In what book? If you mean Ellis'translation of Helmholtz, "700 cents" (I suppose that this is what you have in mind) appears on pp. 318, 432, 450, 465, 520, 522, 525, 526 and 556. 1.955 does not seem to appear as such, but there are several cases of 701.95.
 * You should go see musical temperament before you start asserting definitions of "musical temperament".

Hyacinth, I hope you don't take Wikipedia as an authority on temperament? I, for one, began researching the history of temperaments long before the beginnings of Wikipedia, and I consider that an article who dares quote this:
 * The word "Temperament" in Musical Temperament means the process of tempering and adjusting through tone calculation the desired tones of a high and low pitch; it denotes the use of a mathematical ratio of intervals to define the octave, the fifth, the fourth, the third and the semitones of a scale where the musical notes or tones have a logical mathematical relationship to each other".

probably does not understand what "temperament" means, and might even not understand English. I at least don't understand "tone calculation"; "the desired tones of a high and low pitch", and I think the author (Toni Morrow Wyatt? At www.lulu.com? I thought self-publication did not count as a reliable reference) of this quotation does not really understand what a "ratio" is.

Hucbald.SaintAmand (talk) 08:04, 6 June 2015 (UTC)


 * Sorry, I meant see the unusable state of the article about a concept whose definition you keep bringing up. Hyacinth (talk) 21:32, 6 June 2015 (UTC)
 * A website in no way guaruantees self-publication. However, the quote is not by Wyatt, and instead is by the publisher. I've removed it. Hyacinth (talk) 21:34, 6 June 2015 (UTC)

A revision of the article?
I wonder whether the time would not be ripe for a more thorough revision of the article. Schismatic temperament significantly improved these last days, despite (or, I hope, partly thanks to) our somewhat cahotic exchanges. Yet, I think it could become even clearer if it took better account of the history of what is called schismatic temperament today. It already is obvious that two different tuning systems are involved, which I'll discuss in turn below.


 * 1. The tuning by schismatic exchange, which Ellis unfortunately named skhismatic tuning. This is an obsolete tuning system, first documented in Arabic treatises of the 13th century (among which Safi al-Din al-Urmawi). The aim of al-Urmawi was to provide a Pythagorean explanation of the "quarter tones" (the "Zalzalian" intervals of music of the Near and Middle East). To achieve this, he extended the series of "just" fifths from G♭ to A♯, 17 degrees, in an attempt to equate the difference of a "half flat" or "half sharp" between degrees of the Oriental system with the difference between Pythagorean enharmonies (G♭/F♯, D♭/C♯, etc.). This was justified by the enormous faith in Pythagorean theory, even in Persia and as late as the 13th century. The attempt failed, however, because Zalzalian intervals differ from the diatonic ones by about two commas, while Pythagorean enharmonies differ by only one comma. This tuning system is documented among others in R. G. Kiesewetter, Die Musik der Araber, 1842, the principal source of Helmhotz, and later in L. Manik, Das Arabische Tonsystem im Mittelalter, 1969, and many others in more recent times. So far as I can tell, Manik is responsible for the expression Schismatische Verwechslung, that I translate here as Schismatic exchange; "commatic exchange" might be more appropriate, but it sounds somewhat funny.
 * A similar tuning has been in use in Occident in the late 14th and the 15th century, as described among others by M. Lindley in several papers available on his webpage on Academia.edu (this is a long list; one of the most directly relevant papers is "Pythagorean Intonation and the Rise of the Triad", 1980). This keyboard tuning was limited to 12 keys in the octave and knew several forms; in all cases, the keys tuned by enharmony were black keys. Once again, this tuning originated in a reluctance to abandon Pythagorean theory; that the theory eventually was abandoned is linked to the important changes by which the Western world passed from the Middle Ages to the Renaissance. The tuning by schismatic exchange did not survive these changes and was superseded by meantone tunings early in the 16th century (and, in theory at least, by just intonation at some later point in the century).


 * 2. The Helmholtzian temperament was proposed by Helmholtz as a revivification of the Arabic tuning; Helmholtz was not aware of the Western version. The idea was to slightly temper the fifths of Pythagorean tuning so as to end after 8 fifths, not on a slightly detuned major third by enharmony (say, C-F♭ for C-E), but on a "just" one (5:4). The difference to be gained is a schisma, so that each fifth had to be tempered by 1/8-schisma. Helmholtz however, oddly enough, apparently was not aware of anything named a "schisma", so that he said that the amount by which each fifth was to be tempered was 1/8 the difference between a "just" fifth and a tempered one in 12t-ET. Ellis, Helmholtz' English translator, gave a more exact description of this temperament, which he called "Helmholtzian".
 * Note that the present article states that "eighth-schisma has much much more accurate perfect fifths and minor thirds". I presume that this is a slip of the pen for "major third". It is not true for the minor one, unless this also is taken by enharmony, say C-D♯ for C-E♭, which means 9 fifths on the sharp side; a schismatic tuning (or Helmholtzian temperament) that would produce both good major and minor thirds above C would have to extend from F♭ to D♯, and if it wanted also A to be a good minor third, and A♭ a goor major third below C, it would have to extend from B♭♭ to G♯, etc. — this is the reason why schismatic tunings (like just intonation) are hardly more than utopies: they can produce very good approximations of many intervals, but very few diatonic scales coherent withing themselves. [This confines to the complexities that I wanted to avoid. One may get an intuition of this in the idea that a coherent diatonic scale in schismatic intonation should have all its degrees taken as enharmonies with respect to its "tonic", something like C E♭♭ F♭ E or G♭♭, F or A♭♭, G or B♭♭, C♭...]

Helmholtz' temperament does not seem to have been widely used, but recent webpages oddly describe it (not that it'd be odd to describe the tuning, but the descriptions are odd). A few comments about these:
 * — the unsigned page at x31eq.com oddly decides to call s and r (for the Greek letters sigma and rho — ???) what everybody else would call the major and minor semitones. It then gives a confused explanation of 7+5 and 5+7 scales, which boils down to stating whether it is the major or the minor semitone which is the diatonic one (and the other the chromatic one). It is well known that meantone temperaments have the diatonic semitone as the major one and that the Pythagorean tuning has it the other way around; but x31eq apparently prefers complex explanations (why make things simple if making them complex gives the impression that you are more intelligent?). To explain this in a manner that anyone with elementary musical formation will understand, the tone is said to be made up of 9 commas (let's assume that this is true); the semitone is either 4 commas or 5 commas. Now, a diatonic octave is made of five tones, each of 9 commas, 45 commas together, and two semitones which can be either 5 commas, with a total of 55 commas (5x9 + 2x5; this is the Holder scale properly speaking, with commas of 1/55-octage), or 4 commas, 53 in total (5x9 + 2x4; this gives a good approximation of 1/4-comma meantone). If one compares this with the image given by Hyacinth in the previous section of this talk page, the lines on this image that go up from left to right have a diatonic semitone that is the major semitone; the lines that go down produce the minor semitone as diatonic semitone. To call the first "schismatic"... well, why not?
 * — the "xenharmonic" page on what is called "[schismatic family]" uses a langage that I don't understand, or that I prefer not to understand.
 * — the tonalsoft pages make a clear difference between what they call schismic tuning (my "tuning by schismatic exchange") and schismic temperament, which it does not really describe. Then it turns to the same language as xenharmonic.

I offer the above as suggestions for a revision of the article (which, I add, may have an opportunity to disentangle the whole matter by making it simple ). But I won't rewrite the article myself for these reasons: (1) I do not have the time just now to check all needed references; (2) I cannot understand the "xenharmonic" language, which to me seems to come from another planet (or to have come after some schism &#x1F60A;); (3) my own idea would be that this whole affair ends with Helmholtz/Ellis and subsequently can only be treated as a historical phenomenon (as did L. Manik, M. Lindley, and some of the Arabic researchers that I know).

Hucbald.SaintAmand (talk) 10:58, 7 June 2015 (UTC)


 * This article could probably be trimmed and merged with Schisma. Hyacinth (talk) 16:57, 7 June 2015 (UTC)


 * Yes, why not? But then, if possible, without the phrase "Tempering out the schisma leads to schismatic temperament". I had not seen that article either, and I find most interesting to read that the musical sense of "schisma" was introduced by Ellis — this probably requires a reference, but it makes sense in view of my description above. I'd also like to see the mention of grad, as a unit defined by Werckmeister, checked in the treatises of this author — it is so much better to refer to the primary sources. Unfortunately, Werckmeister's treatises probably were published in Frakturschrift (Gothic font), which for the time being makes OCR difficult. I'll see what I can do. Hucbald.SaintAmand (talk) 17:30, 7 June 2015 (UTC)

Skhismic temperament: Helmholtz or Ellis? (or al-Urmawi?)
Hyacinth, I saw your recent change to the article. You are right that the name "skhismic temperament" is Ellis'; but this tuning nevertheless had been described by Helmholtz, p. 280 ff of the translation, and Helmholtz [rightly] described it as "the Arabic and Persian musical system". As said above, this is a 13th-century tuning system. All this will be resolved, of course, with the possible rewriting of the article and/or its possible merging with Schisma... Hucbald.SaintAmand (talk) 19:29, 7 June 2015 (UTC)

May I repeat that the statement in the article, "In both eighth-schisma tuning and quarter-comma meantone the octave and major third are just, but eighth-schisma has much much more accurate perfect fifths and minor thirds (less than a quarter of a cent off from just intonation)" needs more explanation. The minor thirds in eighth-schisma are problematic. This can be seen in Hyacinth's figure illustrating the "Image" section of this talk page ("Meantone comparison.png"). Pure minor thirds are obtained by 1/3-comma meantone: note particularly the position of E♭, marked by the vertical line "syntonic comma", and that of A, at the left of the same line marking E. Then figure out where the line for the schismatic temperament might reach values similar to those of 1/3-comma meantone: the position of E♭, at about +16 cents, would be reached when the "schismatic" line reaches D♯, outside the figure at the right, after nine fifths from C; A, at about -16 cents, would be reached by the "schismatic" line at B♭♭ at the left, again after 9 fifths. These would be highly correct approximations of "just" minor thirds, deviating by only 1/8 schisma, but they would require the tuning to extend on 18 fifths from B♭♭ to D♯, which would be difficult to say the least. Hucbald.SaintAmand (talk) 20:40, 12 June 2015 (UTC)