Talk:Holdrian comma

Challenging the numbers

These numbers make no sense. From the description it seems like it should be 2⁸⁴:3⁵³, which is about 3.62 cents, not 22.6415. But then "eight to nine" of them wouldn't make a neutral second. Also, 176777/177147 isn't really related to anything, it just happens to be close to 3.62 cents. I would edit this but I don't know which part is correct. —Keenan Pepper 08:32, 10 September 2005 (UTC)

See Touma (1996). Hyacinth 07:44, 11 September 2005 (UTC)

I second Keenan's comments. Furthermore this page says the Arabian comma is a 53rd equal division of an octave, which IS 22.64 cents. That's the only online reference I can find, other than WP mirrors and porn sites. I don't have the book you mention, but how relevant a concept is this? —Wahoofive (talk) 17:06, 14 September 2005 (UTC)

From The Music of the Arabs by Habib Hassan Touma, p.23:

"...so called Arabian comma, also know as the Holdrian comma, whose value is ${\sqrt[{53}]{2}}$, or 22.6415 cents...

"Indeed, as early as 45 B.C., the Chinese Ching-Fang had calculated the value of this comma. He discovered that the highest tone in a row of fifty-three natural fifths built one on top of the other is almost identical to the lowest tone of the row, if the fifty-third is tranposed down by thirty-one octaves. Thus the ratio of the lowest tone of the row to the transposed highest tone (that is, (3/2)⁵³ minus 2³¹) is 176777/177147, which corresponds to the value of the Arabian comma."

Hyacinth 23:57, 14 September 2005 (UTC)

Well, this just proves you can't trust everything you read in a book. Did he really write (3/2)⁵³ minus 2³¹? You can't subtract interval ratios. Surely he meant divided by. Try them on your calculator. But I reiterate Keenan's comment that if you do divide them, you get an interval of about 3 cents, not 22. So if that quote is accurate, that book belongs with the UFO abduction books. —Wahoofive (talk) 01:35, 15 September 2005 (UTC)

With hindsight, Touma obviously doesn't mean minus in the technical sense, but in the "take away" sense, in this case being the same as "down by". Hyacinth 10:36, 5 April 2006 (UTC)

Do you have a source which indicates you can't always trust what you read?

First, in Touma's defense (and I couldn't tell Touma's gender) the book was translated, and the error may have appeared there.

Second, the book is "A specialized work by an expert; for large music collections.?Bonnie Jo Dopp, Montgomery Cty. Dept. of Public Libs., Md."

Last, you may be able to read the page yourself at https://www.amazon.com/gp/reader/1574670816/103-7020248-0680614?checkSum=F4HSdoXlCoMxZA1muuxBSw%2few5aFmH3gE0lXwjdqtSk=&p=S01E&keywords=arabian%20comma&ref%5f=sib%5fvae%5fpg%5f23&twc=7

Hyacinth 07:52, 15 September 2005 (UTC)

I don't need to look inside the book (and besides, Amazon won't let me) — I believe you that it's a direct quote. But it's just wrong. This book says 2+2=5, and I'm baffled by your unwillingness to verify this yourself. I'm tempted to put this page on AFD as a hoax; I can't find any other references to "Arabian comma" or "Holdrian comma" anywhere else. Please don't put the table below anywhere in Wikipedia. The ratio shown doesn't agree with the number of cents (and 22 cents is about a 1/4 of a semitone, not an eighth) and it's just codifying an arithmetical mistake. —Wahoofive (talk) 15:17, 15 September 2005 (UTC)

"Books are not made to be believed, but to be subjected to inquiry." [William of Baskerville: original in Italian] -Umberto Eco, Il nome della rosa, 1980

"Books must follow sciences, and not sciences books." -Francis Bacon (1561-1626), Proposition touching Amendment of Laws.

"The multitude of books is making us ignorant." -Voltaire, French author, humanist, rationalist, & satirist (1694 - 1778)

"Truly, associating with bad books is often more dangerous than associating with bad people." -Wilhelm Hauff (1802-1827), Das Buch und die Leserwelt; ORIGINAL: "Wahrhaftig, der Umgang mit schlechten Büchern ist oft gefährlicher als mit schlechten Menschen."

[1]

How would I go about verifying this myself? Should I check it against the alternative information you all have yet to provide? Should I get degrees in math, Arab, and music and guess as to what Touma meant and which parts are typos?. Hyacinth 22:09, 15 September 2005 (UTC)

Verifying this yourself

Okay, get out your calculator. To "add" intervals, you multiply their interval ratios. So going up 53 perfect fifths leads to

$\left({\frac {3}{2}}\right)^{53}$

and going down 31 octaves is

$\left({\frac {1}{2}}\right)^{31}$

Multiplying these together gives you

${\frac {3^{53}}{2^{84}}}$

as Keenan mentions above. Applying the formula given on Cent (music) we get

$cents=3986\log _{10}\left({\frac {3^{53}}{2^{84}}}\right)$

which evaluates to about 3.62 cents. Therefore this can't equal the 22-cent interval mentioned. Now if you divide the octave into 53 equal parts, you do get this 22.6-cent interval, but that's not supported by either the source you quoted nor any other source I have yet found, so I can't regard it as legitimate. All of the Arab-music websites I've looked at describe their third as being somewhat smaller than a true major third, but none that I could find quantify it with the precision we're looking for.

As for the 176777/177147 ratio, this would only make sense if those numbers somehow related to powers of two or three. The denominator is 3¹¹, but the numerator is, as far as I can tell, prime, so that doesn't have much to do with interval ratios as we understand them.

Table

Holdrian comma
# semitones	# cents	just interval
about 1/4	22.6415	176777/177147

Excerpts from Harvard Dictionary of Music entry on "Arab music"

Most theorists discuss intervals and tetrachord species at great length, often presenting them through various frettings on [a kind of lute]. In the 8th and 9th centures, according to Ibn al-Munajjim, its fretting was:

open string	g(0)	c'	f'	bb
first finger	a (204)	d' (702)	g' (1200=0)	c'' (498)
second finger	bb' (294)	eb' (792)	ab' (90)	db'' (588)
third finger		e' (906)	a' (204)	d'' (702)
fourth finger	c' (498)	f' (996)	bb' (294)	eb'' (792)
				e'' (906)

The notes f'-e'' formed a normative series from which eight diatonic modes were derived, the second and third finger notes being mutually exclusive....

During the 9th and 10th centuries Persian influence again made itself felt....The most far-reaching innovation was the introduction ... of a neutral third. By the 11th century the notes found within the tetrachord (i.e. on any one string of the [lute]) were g a^b a ^x a b^b b ^x b c' (x is halfway between b and [natural]).... In the 13th century these were integrated into the Pythagorean system by Safi al-Din, being placed one comma below the diatonic intervals so that the tetrachord becomes in theory g(0) a^b (90) a^-c (180) a (204) b^b (294) b^-c (384) b (408) c' (498).

---

The word "comma" in the last sentence is unqualified, and the "-c" notes are about 24 cents below the major thirds, which is at least in the ballpark of what we're talking about here. But this stuff about the 53 perfect fifths is just bogus. —Wahoofive (talk) 16:03, 15 September 2005 (UTC)

On second thought, surely he means a Pythagorean comma. —Wahoofive (talk) 16:06, 15 September 2005 (UTC)

Excerpts from New Grove article on Arab music

"The treatises of Safi al-Din (d. 1294) supply the analytical framework which was used by nearly all the major writers of the following two centuries....The neutral intervals, difficult to reconcile with the traditional stress on the primacy of simple ratios, were now treated virtually as just intonation intervals. The octave was ... divided into two conjunct tetrachords and a whole tone, each tetrachord being made up of two whole tones and a limma, and each whole tone of two limmas and a comma...."

"At the beginning of the 19th century the Lebanese theorist Mikail Mashaqa, in his Risala al-shihabiyya fi al-sina'a al-mausiqiyya (Treatise on the art of music for the Emir Shihab), introduced a new system for analysing scales, which is now accepted in much of the Near East. In this system an octave is divided into 24 intervals of approximately a quarter-tone (each about 50 cents). This type of scale division makes it possible to transpose modes containing the neutral 3rd to any scale degree. The computation of the exact sizes of those quarter-tones as they occur in practice is rather complicated, and several alternatives were presented at the Cairo Congress on Arab Music in 1932. Some of the scale systems discussed at this meeting were obtained through mathematical computation and some were established experimentally....The differences between theorists and musicians, as well as modern research...indicate that none of the systems provide an accurate description of musical practice. They are merely convenient tools for prescriptive and didactic purposes. The quarter-tone system is, however, still used in describing the tonal material of the modes."

Excerpts from the New Grove article on China

"Three of the most important and related theoretical concepts [in early theoretical treatises dating back to the 4th century BC] are the establishment and calculation of the 12 fundamental pitches, the idea of scales and that of modes....The earliest account of the intervallic relationships of the 12 pitches is documented in Lu-shih ch'un-ch'iu (3rd century BC); the method of their calculation is the simple application of the Pythagorean (cycle of 5ths) method...."

"For centures theorists tried to solve the [problem of the Pythagorean comma] by devising a system of notes which could rotate their functions identically; various methods were attempted. In the 1st century BC Ching Fang tried calculating up the the 60th note, while Ch'ien Yueh-chih (5th century) went as far as the 360th note....The complex series of notes which resulted was not carried out in any known musical practice, nor did it have any influence on later theoretical developments...."

"The first attempt at creating an equal-tempered scale was made by Ho Ch'eng-t'ien (5th century), who lowered the frequency of each of the tones of the Pythagorean series by a simple factor so that the 13th note was exactly twice the frequency of [the fundamental]...Chu Tsai-yu (16th century) finally created an equal-tempered scale of 12 notes by successively dividing the fundamental number by the 12th root of 2."

And? Hyacinth 07:11, 17 September 2005 (UTC)

And this contradicts Touma's assertion that Ching-Fang's research had anything to do with the "Arabian comma"; rather, he was trying to temper the scale to avoid the Pythagorean comma. Between that and the meaningless fraction (not to mention the bad arithmetic shown above), I'd say Touma's credibility is shot. —Wahoofive (talk) 16:24, 17 September 2005 (UTC)

Deletion

I think this article should be deleted, and I intend to list it on AfD unless someone can give a good reason why it should stay. The article appears to be based on a single passage in a book, and this passage is deeply confused, mentioning three different intervals and talking about them as though they were the same thing. Two of these intervals are relevant to the 53-tone system (in this list of intervals they are referred to as the 53-tone comma and Mercator's comma). The third interval appears to have no relevance to anything, as others have already pointed out. --Zundark 11:16, 19 November 2005 (UTC)

I second that, unless someone can get some better sources and figure out what interval this actually is. —Keenan Pepper 16:29, 19 November 2005 (UTC)

OK. nobody has objected, so I've added it to AfD: Wikipedia:Articles_for_deletion/Arabian_comma. --Zundark 11:34, 22 November 2005 (UTC)

— JIP | Talk 06:54, 28 November 2005 (UTC)

What Touma says

I have the third, expanded edition of Touma's book in the original German (1989, but still seems to be the current edition). He mentions the "Arabian comma" on page 49, putting it in scare quotes (the Pythagorean and syntonic commas don't have these). It's $53root2$. A term like "Holdrian" is neither on this page nor in the index; this is the only difference between the book and the wikipedia article.

He goes on to describe the Mercator comma, attributed by him to Ching Fang. Although this is a real komma, he still calls it "Arabian comma" in the same scare quotes.

On page 50, he gives the maqamat Rast and Navahand in terms of steps of an unspecified comma. They do add up to 53 for the whole octave. As in the article, this is qualified: all diatonic steps except c-d are medium seconds; no mention of the flattened notes. I always took this to mean that theory and practice don't have much to do with each other, but it would be nice the know for sure.

Meanwhile, I have asked for Touma's address at the publishing house. — 84.160.147.123 (klaus) 10:44, 7 December 2005

The comment on the structure of Nihavent - citing Touma - does not make any sense, i.e. "has medium seconds between d–e♭, e–f, g–a♭, a♭–b♭, and b♭–c', a medium second being somewhere in between 8 and 9 commas". Is this what the reference is saying? kupirijo (talk) 17:53, 21 December 2019 (UTC)

The meaning of "comma"

I'm just wondering how the fifty-third root of two is considered a comma?

A comma is supposed to be the difference between two intervals ("difference" being divisive, of course). The pythagorean comma is the difference between twelve fifths and seven octaves, a syntonic comma the difference between four fifths and a major seventeenth, etc...

It doesn't seem to make sense to be calling this thing a comma if it doesn't describe a difference between two other practical intervals. The other definition as the difference between fifty-three fifths and a thirty-one octave actually describes something that would usually be called a comma.

It also makes no sense that a 45BC mathematician would have calculated the fifty-third root of anything. It does however, make quite a bit of sense for him to have calculated the difference (again, I mean quotient, just in case) between $(3/2)^{53}$ and $(2/1)^{31}$. A few hundred years later Boethius wrote similar calculations in his writings on music.

It was this reference to writings from 45BC that caused me to check this discussion page, where I was surprised by the level of confusion. I think either we've got this definition wrong at the moment, or there are two meanings in use (I've never seen the term before, personally) and our description fails to make this evident. Rainwarrior 04:08, 3 April 2006 (UTC)

Doesn't it help to explain what the comma is to describe it in other terms? Hyacinth 10:32, 3 April 2006 (UTC)

I'm sorry I dont understand what you are asking. My suggestion is that $(3/2)^{53}/(2/1)^{31}$ is a comma and ${\sqrt[{53}]{2}}$ is not, and furthermore the reference to Ching-Fang indicates (by way of the history of mathematics) that the interval in question is indeed a rational number and not the fifty-third root of two.

If you are asking me to again explain what a comma is, I will try: A comma is a measure of the compromise between two rational intervals available in a tuning system. (e.g. The pythagorean comma quantifies the difference between a cycle of twelve fifths and an octave.) An equal-tempered tuning system has no commas because it has no irregularity in its interval structure.

As it stands, I see this as a flaw in the article, and think there is an ambiguity about the definition which the current form of the page has failed to address. Does this answer your question? (I'm sorry I don't quite understand it.) Rainwarrior 04:34, 4 April 2006 (UTC)

Ahah! I understand now what is going on here. The comma is indeed $(3/2)^{53}/(2/1)^{31}$, and what we've got on the page is wrong. The problem is that this page is trying to describe two things, and only one of those things is a comma (which unfortunately isn't even described on the page!).

The other thing that is being described is the 53 tone equal tempered system, which is practically no different from a cycle of 53 natural fifths. 53-TET is technically unrelated to this comma, but it is very, very strongly similar to the 53-Pythagorean (what I'm going to call a cycle of 53 just fifths for now) tuning of which this comma is an integral part.

The difference between 53-TET and 53-Pythagorean is that a 53-TET fifth is tempered by one 53rd of a holdrian comma, which is already a tiny interval to begin with (someone said it was less than 4 cents above), which means the difference is practically insignificant. There is no acoustic instrument you could tune accurately enough to make the distinction.

That said, there is a theoretical difference between 53-TET and 53-Pythagorean, and this comma is one of those differences (commas essentially are a theoretical construct, and deserve to be acknowledges as such). The article should make this clear, and also point out the strong similarity. 53 equal temperament should also be updated (and much of the information currently on this page should be moved there. The turkish scales will also need some clarification, but should be moved as well.). I'll do this stuff later on in the week if noone else want's to. Rainwarrior 06:01, 4 April 2006 (UTC)

Okay, I read William Holder's book. A big part of the problem is that Holder shouldn't have called this thing a comma in the first place. Mercator used the logarithmic approximation of ${\sqrt[{55}]{2}}$ for the syntonic comma familiar in the meantone tuning used during his time. ${\sqrt[{53}]{2}}$ he called an "articificial" comma, which was used to consider a 53-TET scale, which Holder preferred because of how close to just intervals it was. It is rather unfortunate nomenclature to call this a comma simply because it was a small fudge away from something else that only approximated a particular type of comma.

I don't have access to Touma's book, but based on your quotation I would guess that either Ching-Fang miscalculated, or Touma did, or Touma was quoting from sources which he did not understand, or the translator lacked the technical knowledge of tuning to properly translate that passage. I think it's most probable that Touma believes Ching-Fang calculated something that he did not. I tried hard to find anything related to tuning that would generate the number 176777/177147, which I could not. If someone would care to enter a larger quotation from the book we could take a closer look at it. Rainwarrior 20:14, 4 April 2006 (UTC)

Okay, I've made the changes, and I think the article is more accurate now. I've removed the Ching-Fang reference, because whatever he did calculate is not very relevant to the Holdrian comma. The difference between 53 fifths and 31 octaves is relevant to 53-TET, but only very obscurely related to the Holdrian comma. Rainwarrior 21:38, 4 April 2006 (UTC)

Thanks, your "difference" as "division" comment clued me into the root of the above conflict. As such I reverted the article to a sourced version by me. Hyacinth 10:42, 5 April 2006 (UTC)

My source was Holder's definition of the comma. (I'll retype it out here if I need to.) It is, after all, called the "Holdrian Comma". It has already been explained on this talk page that 22 cents is neither the difference between 53 fifths and 31 octaves, nor is it 176777/177147. If a source has invalid math, you shouldn't put that math up on Wikipedia. You also removed my explanation of why this would have recieved the unusual name "comma" in the first place, which I think is important to its definition. The only thing thing that I can't cite a source for is the commentary on turkish music theory, which I didn't actually meaningfully change (except to make it clearer how a "medium second" relates to these commas). Rainwarrior 12:11, 5 April 2006 (UTC)

Just to be clear, I've struck out my previous comments which I later found incorrect. The comma was indeed the interval that the page had been describing, the problem was that it wasn't a "real" comma, and there was conflicting information on the page. The subtraction isn't actually the root of the problem, it is merely one of many indications of a lack of understanding of the underlying mathematics on Touma's part. The number 176777/177147 is not related to the calculation described using either divison or subtraction. The subtractive figure $(3^{53}-2^{22})/2^{5}3$ is irreduceable (the divisive figure is also trivially unreduceable, given that its numerator and denominator are relatively prime). If Ching-Fang did the calculation with subtraction (which I doubt, because he must have understood intervals to be multiplicative to even get that far) he would have come up with either of these large irreduceable numbers which Touma did not quote. If Touma is going to claim that Ching-Fang accurately calculated a particular number, he will at least have to get that number right before anyone should believe him. Rainwarrior 12:48, 5 April 2006 (UTC)

I found a better source on Ching Fang. What Ching Fang calculated was Mercator's Comma, and not the Holdrian Comma. As such, a reference to him here is not appropriate, but I amended 53-TET to explain in more detail his accomplishment. Rainwarrior 16:49, 5 April 2006 (UTC)

I used this information to fill in the red link for Ching Fang. Rainwarrior 17:45, 5 April 2006 (UTC)

The number 177147/176777 is not a very good approximation to Mercator's comma, but it is a good approimation if we insist that the numerator be a power of three. This no doubt has some relevance to the way Ching Fang carried out his computation. We can find Fang-like approximations to Mercator's comma by taking its reciprocal, multiplying by a power of three, and rounding to the nearest integer, and the Ching Fang approximation appears there. Why 3^11 in the numerator I don't know. Gene Ward Smith 05:01, 7 May 2006 (UTC)

You can find the full tables for Ching Fang's calculations in the journal article I used as source for his wiki article (I think it's online at JSTOR; if you don't have access to journals I could probably provide you with it). I briefly described his method of calculation on that page, if you haven't seen it. Let me check with a good calculator: Mercator's is about 1.00209031404..., Ching Fang's is 1.00209303246..., this is not very good? At any rate, I didn't think it was really relevant to Holder's comma, so I moved it over to 53 equal temperament. - Rainwarrior 17:23, 7 May 2006 (UTC)

The article is confusing in my opinion! Ιt confuses the Holdrian comma (i.e. $1200*\log _{2}({\sqrt[{53}]{2}})\approx 22.6415\,\mathrm {cents}$ with the Mercator's comma (https://en.xen.wiki/w/Comma)$1200*\log _{2}(19383245667680019896796723/19342813113834066795298816)\approx 3.6150\,\mathrm {cents}$. I have recently added the value of Mercator's comma but I still think it needs more clarification. In addition the following sentence is very ambiguous Thus Mercator's comma and the Holdrian comma are two distinct but related intervals. Basically, if I may say it compares apples with oranges --kupirijo (talk) 18:28, 19 November 2019 (UTC)

My own feeling is that the article as you corrected it is mistaken about Mercator's comma, in the statement added by you that it is the difference between 31 octaves and 53 just fifths, i.e. 3.6150 cents. You give no reference to this statement in the article, and you refer only here above to Mercator's comma, which itself does not further justify it.

The only thing that is not entirely clear in the article is whether Mercator's comma is the octave divided in 53 or 55. But there is no indication, ever, that it could be "the amount by which 53 fifths exceed 31 octaves", as the Xenharmonic page claims. This must result from a misunderstanding of how Mercator (or Holder) compared his division of the octave with a purely Pythagorean one.

I'll try to check Mercator's and Holder's own wrintings. (I am not a very good Wikipedian in that I usually don't trust secondary sources and prefer to refer to the primary ones.) But think of it in the meanwhile: the article, I think, was reasonably coherent in his definition of Holder's and Mercator's comma before your recent addition. — Hucbald.SaintAmand (talk) 20:53, 19 November 2019 (UTC)

Thank you for your reply. I see your point and I agree with you that we should resort to the primary sources. Let's revisit the topic soon. I will try and do some research as well. Perhaps send an e-mail to Xenharmonic. kupirijo (talk) 07:07, 20 November 2019 (UTC)

Joe Monzo seems to have written some books regarding this http://www.tonalsoft.com/enc/m/mercator-comma.aspx kupirijo (talk) 07:13, 20 November 2019 (UTC)

The problem with the Tonalsoft website is that it is once again one author alone, giving no reference. He writes "Because Mercator recognized this small anomaly (or comma, in its general meaning), it is known today as 'Mercator's comma' or more simply the 'mercator comma'", but doesn't say who "knows" it today under that name, nor who named it that way. At least, he does not say that Mercator gave it that name. To me, an interval of 3.6 cents would be a schisma, not a comma, but I will also have to check my Greek dictionary. Murray Barbour, in his Tuning and Temperament, says that Mercator proposed the division in 53 but never uses the terms "Mercator's comma" – and does not speak of Mercator recognizing the "anomaly". I'll see whether I can find the primary sources. — Hucbald.SaintAmand (talk) 08:50, 20 November 2019 (UTC)

Thank you! In the past when speaking to mathematicians/musicians, I have always described this ≈3.615 cent discrepancy as a "comma of commas", but I guess a σχίσμα sounds nicer. Actually, this interval is the difference between a "Pythagorean comma" and a "41-tone comma". In addtion, even in the Xenharmonic page a limma (λείμμα) is regarded as a big-type comma, but I must admit that I prefer having the distinction and therefore the following size relationships: limmas > commas > skhismas or limmata > commata > skhismata :) If you can find the primary source that would be great! kupirijo (talk) 12:29, 20 November 2019 (UTC)

It appears that Mercator never published on this matter and that all that is known about it comes from Holder's description of "a manuscript of his". Holder is absolutely clear about Mercator's comma: "[...] Supposing a Comma to be 1/53 part of Diapason; for better Accommodation rather than according to the true Partition 1/55; which 1/53 he calls an Artificial Comma." That is to say that Mercator approximated the "natural comma" (probably the Pythagorean comma) by dividing the octave in 53 "artificial" commas. The comma of Mercator is mentioned in several 19th-century texts as the 53th part of the octave. It is therefore not merely related to the Holdrian comma, it actually is the same.

The difference between 53 fifths and 31 octaves is described by Ellis, in his translation of Helmholtz, p. 432, as the Mercatorial. Ellis writes: "Fifty-three Fifths up and thirty-one Octaves down give what may be called a Mercatorial, because on it depends the advantage arising from the use of Mercator's 53 division of the Octave." There is no question of "Mercator's comma".

I have now reconstructed some of the history of the Ching Fang claim. This is first mentioned in Maurice Courant, "Chine et Corée", in Lavignac and La Laurencie's Encyclopédie de la musique et dictionnaire du Conservatoire, vol. I, 1913, p. 88. The discussion concerns the production of the Chinese lyǔ (pitch pipes) by a cycle of fifths that never reaches the octave. Ching Fang therefore described the production of a cycle of 60 ascending fifths and descending fourths, producing a scale of 60 lyǔ in the octave, that is a microtonal scale with 60 degrees, most of which separated by a Pythagorean comma and some by 2/3 of this comma. Murray Barbour (Tuning and Temperament, p. 124) refers to Courant's article, stressing that "King Fâng observed that the 54th note was almost identical with the first note," as Courant indeed mentioned, but less clearly. The next step is in Ernest G. McClain and Ming Shui Hung's "Chinese Cyclic Tunings in Late Antiquity", Ethnomusicology 23/2 (1979), pp. 208-214, who explain how Ching Fang performed his calculations, which make no use of fractional numbers and rest on clever approximations.

The important point in this probably is that Ching Fang noted that his 54th note was almost identical with the first one. One could indeed express that saying that 53 fifths are almost identical to 31 octaves, but Ching Fang never said it in this form and probably had only a confused notion of the octave. Ellis thought in these terms and gave the name mercatorial to the difference, about 3.6 cents, but did not name it a comma.

Herman Rechberger, in Scales and Modes around the World (2018), p. 258, writes that "Ching Fang (78-37 BC), a Chinese music theorist, observed that a series of 53 just fifths ((3/2)⁵³) is very nearly equal to 31 octaves ((2/1)³¹). He calculated this difference with six digit accuracy to be 177147/176776 (1,002099). Later the same observation was made by the mathematician and music theorist Nicholas Mercator (c.1620-1687) who calculated this value precisely as (3⁵³/2³¹), which is known as Mercator's comma (~3.615 cents)." Either this guy has sources that are unknown to the rest of the world, or all this is wishfull thinking. Neither Ching Fang nor Mercator had means to calculate any of this "precisely", even had they tried to do so – nor do we have the means to know "precisely" what they did.

I will soon rewrite the section of the article about Mercator's comma, but I must first let pass my irritation at reading all this nonsense. — Hucbald.SaintAmand (talk) 16:17, 20 November 2019 (UTC)

Many thanks User:Hucbald.SaintAmand for such a thorough "detective work"! kupirijo (talk) 19:50, 20 November 2019 (UTC)

Also shouldn't it say (3⁵³/2³¹⁺⁵³) not (3⁵³/2³¹)? kupirijo (talk) 20:05, 20 November 2019 (UTC)

I don't think so (although indeed I may have seen 2⁸⁴, or things like that, in my browsing through publications on microtonality). 2³¹ really means 31 octaves (power 31 of the ratio of the octave, 2/1). I fail to see what meaning 31+53 could have, but I would not be surprized that people use this without realizing its meaning and not really minding: their point, after all, mainly is to seem more intelligent than us all, not necessarily to understand what they write... — Hucbald.SaintAmand (talk) 20:20, 20 November 2019 (UTC)

31+53 does make sense here. The claim is that 53 fifths is approximately equal to 31 octaves — in other words, that $(3/2)^{53}/2^{31}$ is approximately 1. But $(3/2)^{53}/2^{31}=3^{53}/2^{31+53}$. --Zundark (talk) 08:42, 21 November 2019 (UTC)

Indeed, my mistake. I didn't notice Rechberger passing from $(3/2)^{53}$ to $3^{53}$. His definition of what he calls "Mercator's comma" as $3^{53}/2^{31}$ is definitely wrong. I suppose in addition that computing such high powers was no easy trick for 17th century mathematicians, who were not yet so fluent with logarithms. — Hucbald.SaintAmand (talk) 10:11, 21 November 2019 (UTC)

In addition, note that Rechberger, as I quote him above, says of Nicholas Mercator that he was a "music theorist", while Mercator did not publish a word about music theory, that the only thing we know about his relation to music comes from a few lines in Holder, concerning what ultimately was more a mathematical than a music theoretical problem. I cannot stand that ... — Hucbald.SaintAmand (talk) 20:30, 20 November 2019 (UTC)

I just wanted to point out that in 53 equal temperament#History and use, Mercator's comma is defined as $(3/2)^{53}/2^{31}$ with reference to the tonalsoft website. kupirijo (talk) 08:00, 22 November 2019 (UTC)

Indeed, this appears to be the common definition of Mercator's comma among microtonalists, in blatant contradiction to what formerly was considered Mercator's comma, namely ${\sqrt[{53}]{2}}$. Rainwarrior opened the present discussion 15 years ago stating that a comma cannot be an irrational number. This is true in Greek theory, but ceases to be true once one thinks in terms of equal divisions of the octave. In addition, if ${\sqrt[{53}]{2}}$ is not truly a comma, what are we to make of the "Holdrian" comma, the subject of this article? The reason to call ${\sqrt[{53}]{2}}$ a comma is that it is a good approximation of the Pythagorean comma (and a reasonable one of the syntonic comma).

Indeed a very good approximation to the Pythagorean and syntonic commas (which are rational commas). I understand Rainwarrior's concern about the Holdrian comma not being a just interval but it is probably the only small ET interval to be a good approximation to two of the most well-known commas (check out my recent edits on commas in the Template:Intervals). For example the smallest interval in 72-TET is not called a comma. In Byzantine music for example, with which I am familiar and which uses 72-TET, $2^{1/72}$-sized steps are called morio/moria (μόριο/μόρια) kupirijo (talk) 00:19, 24 November 2019 (UTC)

I think that the only solution for the article is to give both definitions of Mercator's comma in the article, first as ${\sqrt[{53}]{2}}$, giving sources (there are many 19th-century ones), then as $(3/2)^{53}/2^{31}$ (Ellis' "mercatorial"), a definition that seemingly appeared in the 2d half of the 20th century. But I'd like to find a serious source for this second definition, not merely a website (Tonalsoft) without reference. — Hucbald.SaintAmand (talk) 09:51, 22 November 2019 (UTC)

I don't really remember exactly what was in my mind 15 years ago, but I think I might have been a little bit idealistic about the definition of comma at the time. It should be some small difference between two tuning references, at least. I don't know that it needs to be rational? I'm not sure what the source of my assertion there was, and really if there are good sources that refer to the rooty figure as a common then they should be acknowledged. - Rainwarrior (talk) 07:08, 23 November 2019 (UTC)

We were all younger 15 years ago, Rainwarrior (or, in my case, I should better say 50 years ago ;–)). In any case, I find this whole discussion most entertaining, and I want to thank all participants for this: it is not that frequent on WP. I hope to soon be able to propose an alternative version of the section on Mercator's comma. I am still strungling with the "Ching-Fang claim". I spoke two days ago with one of the world's best knowers of early Chinese theory, who told me that, even although Maurice Courant's article in Lavignac's Encyclopédie de la musique is of very high quality (it appears to have been the source of all subsequent works, including Touma's, about Ching Fang), it is for him out of the question that any theorist of the 1st century BC should have computed such complex roots as those involved here. This makes sense and I suspect that what happened is that clearly approximative calculations were taken more and more for exact ones, forgetting that the means for the exact ones were developed only centuries later. — Hucbald.SaintAmand (talk) 21:16, 23 November 2019 (UTC)

Oh, that comment jogs my memory a bit. Ching Fang was absolutely not calculating ${\sqrt[{53}]{2}}$. I too think such a thing would have been incredibly anachronistic in the history of math, centuries before logarithms or complex analysis. He was calculating an approximation of $(3/2)^{53}/(2/1)^{31}$, and if you check his article you'll find a description of his method. The root observation being made here, is that a cycle of 53 pythagorean 5ths is incredibly close to being a perfect circle... but in modern terms that means "53-equal temperament", and for many practical purposes they might be considered the same thing (and deserve to be discussed together). However, we should avoid confusing the two so directly in the article, and certainly we shouldn't suggest Ching Fang was trying to calculate a 53rd root. I think the article right now is a bit loose and seems to imply equivalence by accident. Most directly he discovered how close 53 fifths are to being circular, and a little less directly that means he discovered the suitability of 53-equal temperament, I suppose, but it's probably worth a few words to make this clearer rather than just dropping his name in the middle of discussion of Mercator. - Rainwarrior (talk) 06:22, 24 November 2019 (UTC)

Thank you Rainwarrior. Perhaps Ching Fang's calculations are more relevant in the 53 equal temperament article. This article is about the Holdrian comma. May I also suggest having a separate article for Mercator's comma? kupirijo (talk) 09:09, 24 November 2019 (UTC)

Rainwarrior and kupirijo, I now reread the main sources concerning Ching Fang, Courant's article in the Encyclopédie de la musique et dictionnaire du Conservatoire, E.G. McClain and Ming Shui Hung's article in Ethnomusicology, and the mentions in J. M. Barbour's Tuning and Temperament. All three recognize the extreme precision of Ching Fang's calculation of 60 lü (pipe lengths) or chun (string lengths). All three note that as a result, the 54th lü or chun in this calculation comes very close to the first one, to within 3,6 cents. But Barbour is the only one to say that Ching Fang himself (or his Chinese commentators) noticed this close proximity, and he provides no source for this; it is doubtful whether Barbour had any other source than Courant. Courant says something about the regular alternance of fifths upwards and fourths downwards in the calculation of Ching Fang being somehow perturbed between the 53d and 54th lü, but I did not yet fully understand what he means. McClain and Ming stress that "the significance of Ching Fang's work lies mainly in the insight later theorists could gain from it", and I suspect that what he refers to is the recognition that the spiral of fifths reaches "near agreement with the reference tone after 53 consecutive operations – as Mersenne, Kircher, and Mercator were to learn in the 17th century", implying that this was not recognized before the 17th century.

Turning back to the Holdrian and Mercator's comma, the fact is that, in one definition at least, they are the same interval, ${\sqrt[{53}]{2}}$ – this is the reason why Mercator's comma must be mentioned in the present article. Holder is not entirely clear about this (I must reread and understand what he says in the Treatise) and seems to first state that Mercator divided the octave in 55, not 53. (This, as you know, results from the fact that the octave counts 5 tones of 9 commas and two diatonic semitones of either 4 or 5 commas, resulting in a total of either 53 or 55.) Of course, if a separate article is written about Mercator's comma, a mere mention and a link would suffice here. The problem remains, I think, that an article about Mercator's comma may become extremely polemic. The situation is as follows: after Holder and up to recently, Mercator's comma is described as ${\sqrt[{53}]{2}}$ – Google provides several sources for this. Ellis mentions $(3/2)^{53}/(2/1)^{31}$ and names it ("what may be called") a "mercatorial", a name that he probably coined in reverence (rather than reference) to Mercator. Barbour (and to a much lesser extent Courant and McClain & Ming) say that this interval, $(3/2)^{53}/(2/1)^{31}$, may have been conceived by Ching Fang. And at some point later (McClain & Ming's article is of 1979!) $(3/2)^{53}/(2/1)^{31}$ was described as Mercator's comma!!! An article about Mercator's comma may state all this, and probably could document it, but it obviously is "original research"... What do you (and what do others) think? — Hucbald.SaintAmand (talk) 10:37, 24 November 2019 (UTC)

Looking over the article, I'm not really sure why this comma needs its own page? The lead explains what it is and gives 3 names for it, then makes the claim that it was "widely used as a measurement of tuning in William Holder's time" (which seems dubious to me). The next paragraph jumps to talking about Boethius and the connection is already getting kind of loose. Next it describes "Mercator's comma", which is yet another digression without having established why the idea of this comma is supposed to be significant. Finally the whole "Arabian comma" section kind of hints at where someone actually had a use for Holder's comma, and might have both named it and applied it.

To me the organization of this information is backwards. There might be a worthy article about a branch of Turkish music theory, under which this comma might be incorporated as a subheading, rather than the other way around, but that'd be another article with its own justification. Much of the other information is either already part of, or could be migrated to 53 equal temperament, or maybe to William Holder. (The Arabian comma section also seems like information that would be fine at 53 equal temperament.) As-is, the article makes a bunch of vague connections, and drops a few references, leaving a lot of open questions behind, but without providing any compelling reason why someone would want to sort out the puzzle of a niche technical term used in very small circles. Personally, I suggest redirecting this article to 53 equal temperament, and migrating any useful connections and references there or to other relevant articles. - Rainwarrior (talk) 06:49, 25 November 2019 (UTC)

Hucbald.SaintAmand and Rainwarrior, I agree with all your comments. It seems that sentences of the article need to be incorporated either into William Holder or 53 equal temperament articles depending on relevance, in the same style as quarter tone is incorporated into 24 equal temperament. kupirijo (talk) 12:23, 25 November 2019 (UTC)

Requested audio

I have added an audio example to the article. Hyacinth (talk) 08:14, 9 August 2008 (UTC)

Arabian comma?

There is something wrong in the section describing the Arabian comma in this article. I never noticed it before, but now I am completely puzzled – all the more so that I cannot verify Toma's book just now.

The article says that the makam rast has the following set of Holdrian commas: 9 8* 5* 9 9 8 5 for the scale c d e f g a b c'. No explanation is given for the sign * (there should be one!), which probably corresponds to the statement that follows, that "in common Arabic and Turkish practice, the third note e in rast is lower than in this theory, almost exactly halfway between western major and minor thirds, i.e. closer to 6,5 commas above d and 6,5 below f". That is to say that 8* and 5* would stand for twice 6,5 "in common Arabic and Turkish practice". What is unclear for me (apart for the lack of explanation of the *) is to what other common practice the scale with 8 and 5 would correspond. Is there a common makam practice outside the Arabic and Turkish ones?

Makam nihavend is then described as "similar to the Western minor scale", 9 4 9 9 4 9 9, c d e♭ f g a♭ b♭ c'. Up to there, no problem. But the text continues saying that nihavend "has medium seconds between d–e♭, e–f, g–a♭, a♭–b♭, and b♭–c', a medium second being somewhere in between 8 and 9 commas." But 9 commas is a major second, and a minor second is 4 or 5 commas; a medium second should then be 6,5 or 7 commas, not "between 8 and 9". One may suppose that the mention of "medium seconds between d–e♭, e–f" really means "between d–e♭, e♭–f": the case would then be identical with that of rast in "common Arabic and Turkish practice".

Further, g in all these descriptions is 31 commas above c, and therefore there remains 22 commas between g and c', for a total of 53 commas in the octave. If however there are medium seconds of 6,5 commas between g–a♭, a♭–b♭, and b♭–c', they make a total of only 19,5 commas: there are 2,5 commas missing, probably indicating that one of the seconds should be major (9 commas) instead of medium.

Either nivahend is "similar to the Western minor scale", or it has medium seconds. And if it has medium seconds, these must come in pairs, dividing a minor third of 13 commas in two equal parts. In its present state, the section is unclear and while I can see the error, I have no idea of how to correct it. — Hucbald.SaintAmand (talk) 20:43, 25 October 2017 (UTC)

Holdrian comma and 53 equal temperament

I open this new section because I think that our discussion reached a point where it no more concerns the meaning of "comma" but rather whether an article about the Holdrian comma is justified. Rainwarrior's suggestion that this article could be integrated in 53 equal temperament seems to me rather questionable. The 53 equal temperament article tries to make us believe that 53-EDO was conceived in Chinese Antiquity, that it was precisely calculated by Nicholas Mercator, then by William Holder, etc.

The truth is that none of these authors conceived even only the idea of 53 equal temperament. Ching Feng in the first century BC described a theoretical system of 60 (not 53!) degrees in the octave, produced by a cycle of 59 just fifths. J. Murray Barbour claimed in 1951 that Ching Feng had noted how close the 53d degree of his system came to the octave, but provides no evidence to this effect. Mercator realized that Pythagorean tuning, with tones of 9 commas and diatonic semitones of 5 commas, counted 55 commas (which he apparently called "natural commas") in the octave, but thought it preferable to consider "artificial" commas of 1/53 octave, which allowed a better description of just intonation, the tuning system discussed by most music theorists in his time. This is what Holder reported soon after – and this is what the Holdrian comma article says.

Helmholtz discusses how just intonation can be approximated by a scale of 53 notes, mentions that this had already been suggested by Mercator, mentions Bosanquet's harmonium producing this scale, and describes it as counting 9 8 5 9 8 9 5 commas, where 9 is the major tone, 8 the minor tone and 5 the diatonic semitone. There is of course no question of using the 53 equal temperament as such. Ellis further discusses the difference between 53 just fifths and 31 octaves and calls it a Mercatorial "because on it depends the advantage arising from the use of Mercator's 53 division of the Octave." He also calls "Mercator's fifths" the fifths that would be tempered by 1/53 of the Mercatorial.

The 53 equal temperament article would have us believe that all these authors were thinking of a 53-note equal temperament, which would find its justification in their theories. That, to me, is backwards organization. All the theorists mentioned above were thinking of commas as a means to describe ordinary tunings. I don't know whether a Holdrian comma article is needed but, certainly, the discussion of the Holdrian comma could not be included in the 53 equal temperament article because that would give an entirely false idea of its historical purpose. The information available here might be moved to the Comma (music) article, but there too one finds strange ideas (such as that of "tempering out" the comma: see Talk:Meantone temperament#Tempering the syntonic comma to unison???). — Hucbald.SaintAmand (talk) 09:29, 28 November 2019 (UTC)

The 53 equal temperament article has problems of its own, but it's a large and important topic that deserves an article. This comma really does not, it deserves to be a subsection in William Holder and/or 53 equal temperament, or maybe just a footnote. It's an obscure technical term, and not much more. I don't want to dwell too much about the problems of the 53 equal temperament article, but Ching Fang's calculation definitely is relevant. He did not come up with 53-equal temperament, but discovering the closeness of the 53rd 5th is a very fundamentally important component of why 53-equal temperament exists. He did not propose an octave of 60 tones, he decide to calculate 60 perfect 5ths as an experiment to see what he would learn. The most interesting thing he learned was how close the 53rd was. If you want to dispute that he observed this, I dunno what to say in response, but I'm quite a bit incredulous that it was just made up by Barbour. What do you think Ching Fang was calculating these intervals for, if not for the purpose of comparing them? Just producing some pointless table of numbers? To be honest, this argument is not important enough to me to track down the original text and learn ancient Chinese to prove it to you, so you can do what you will from here, but I want to strongly suggest that it would have been absurd of him not to notice this.

So maybe the history section of 53 equal temperament is muddled. Apparently you think it's trying to assert that Ching Fang discovered 53-equal temperament. I don't think it says that, but since you want to read it that way, maybe it's worth trying to rewrite in some a clearer way. That article has to discuss that a cycle of 53 fifths comes so close to unity, because that is part of the meaning of 53-equal temperament. You can't just separate that critical insight from the topic. Similarly this "Holdrian comma" is the fundamental interval of 53-equal temperament, it has no better home than that article. Finally, in more practical terms, the difference between a cycle of 53 perfect fifts and 53-equal temperament is below normal thresholds of pitch perception, and it would be difficult to tune any real musical instrument to such precision as to make any difference between these two scales at all. They belong together as one topic, and the reason 53-equal temperament is a notable one hinges on their relationship. The history of one is part of the history of the other. The ideal/theoretical 53-equal temperament doesn't exist until you have the development of logarithmic math, and that does seem like something worth making clear in the article, but the pythagorean idea is still part of its history. - Rainwarrior (talk) 10:12, 28 November 2019 (UTC)

Also as a user of 53 equal temperament and a player of Ottoman classical music/Greek tradition music I would like to mention that perhaps we will have to find the original articles by Hüseyin Sadettin Arel (also see tr:Hüseyin Sadeddin Arel), the music theorist responsible for the modern notation of Ottoman Music (the so-called Arel-Ezgi-Uzdilek system) that might lead us to when this 53-TET idea got started. Also I suspect that some of his predecessors already had spoken about this (e.g. Rauf Yekta). Actually, I am seriously thinking of writing a review article about this in English and submit it to some musicology journal. kupirijo (talk) 13:05, 28 November 2019 (UTC)

Rainwarrior, I too don't read Chinese and I won't learn it either. But I did read E. McClain and Ming Shui Hung. "Chinese Cyclic Tunings in Late Antiquity", Ethnomusicology 23/2, 1979. pp. 205–224, which according to the 53 equal temperament article is the source of the claim that Ching Fang "observed that a series of 53 just fifths ([​3⁄2]⁵³) is very nearly equal to 31 octaves (2³¹. He calculated this difference with six-digit accuracy to be ​177147⁄176776." McClain and Ming nowhere say that and these two phrases are an exact quotation of another book, Herman Rechberger, Scales and Modes around the World (2018), p. 258 – or, more probably, this book is an exact quotation of Wikipedia. I am too busy with more important things to wonder what Ching Fang's intention was (McClain and Ming do mention the symbolism of 60, etc.). I have done a lot of work on the history of temperaments and I do believe that the Holdrian comma was of some importance in this history, among others as an approximation of Pythagorean tuning with a precision better that 1 cent for each of its degrees. I would certainly not call it an "obscure term". I think that to reduce it to a mere forerunner of 53-EDO falsifies the history. But I'll stop here and let you arrange the articles as you wish. — Hucbald.SaintAmand (talk) 18:49, 28 November 2019 (UTC)

The McClain and Hung article DOES say that. It's not a quote. See: Ethnomusicology May 1979 page 212 which contains the 177147/176776 figure in the middle of the page. Page 208 describes the computation and that it has 6 digit precision. The exact phrase I believe was written by me in 2007, attempting to clarify that Ching Fang did not come up with 53-equal-temperament (the article before this mentioned him in an unclear and uncited way), and provided that source for the nature of what he actually determined. The 2018 book I guess is quoting me? The exact phrasing seems to be mine, but the information is derived from the cited source.

Anyway, if you look at that edit (or even the discussion about it that's further up on this very page, from 2006), I think you can see that my intention was to clarify that Ching Fang did not invent 53-equal-temperament, but that his observation was still relevant to exist existence, and an important historical note. I think I'd still suggest this article be merged with 53 Equal Temperament, but I'm unclear what argument is being made. - Rainwarrior (talk) 23:38, 30 December 2022 (UTC)

Just to clarify my position is that this comma article does not do enough to justify being its own separate article from 53 EDO. They are so strongly related that both articles would benefit from being combined. Redundancies would be eliminated. The information would mutually benefit. I'm not making an argue for the relative importance of the comma. I just think it would be served better as a combined article. - Rainwarrior (talk) 00:25, 31 December 2022 (UTC)

I decided to perform the merge. The information looks better to me together that 53 equal temperament. I believe this will help further editing improve both sides of the article. - Rainwarrior (talk) 00:43, 31 December 2022 (UTC)