Talk:Holdrian comma

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Contents 1 Challenging the numbers 1.1 Verifying this yourself 2 Table 3 Excerpts from Harvard Dictionary of Music entry on "Arab music" 4 Excerpts from New Grove article on Arab music 5 Excerpts from the New Grove article on China 6 Deletion 7 What Touma says 8 The meaning of "comma"

[edit] 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 , 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)

[edit] Verifying this yourself

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

and going down 31 octaves is

Multiplying these together gives you

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

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.

[edit] Table

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

[edit] 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)

[edit] 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."

[edit] 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)

[edit] 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)

This article was nominated for deletion on November 22, 2005. The result of the discussion was no consensus.

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

[edit] 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. klaus

[edit] 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 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 for the syntonic comma familiar in the meantone tuning used during his time. 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 (3⁵³ − 2²²) / 2⁵3 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)