# Fivefold division of the whole tone - What does it mean?

I am recently reading this article: Jan W. Herlinger. Journal of the American Musicological Society. Vol. 34, No. 2 (Summer, 1981), pp. 193-216 (Title: Marchetto's Division of the Whole Tone). In the beginning, the author says the following:

He [Marchetto] became the first theorist to propose division of the whole tone into five equal parts, which he called dieses, and to present musical examples showing chromaticism and skips of an augmented fourth.[see picture below] []1

So I start wondering what does "division of the whole tone into five equal parts" mean. I am confused because a whole tone has only two semitones and can never be divided into more than two parts. So, where does the "five" come from. After reading the rest of the article, I get no clue.

Two or three pages later, the author presents Marchetto's argument of the fivefold division of the whole tone. Marchetto starts by saying that the continuum (which I understand as the set of all real numbers in mathematics) can be divided into infinitely many segments. However, this fact does not seem to be used later. Then, Marchetto suddenly claims the following:

Marchetto further states that the perfect division of the continuum and its parts is the ninefold division, and relates the ninefold division of a particular continuum-the string of a musical instrument - to the production of the whole tone, which he has earlier defined as the "regular" distance from one pitch to the next.

No explanation is given for why "nine" is the perfect way to divide the continuum.

Also, the whole tones is "demonstrated" to be indivisible into two equal parts - but I think we can - it can be divided into two semitones.

It is quite clear that I am heading in the wrong direction. Could anyone help me understand this?

## 5 Answers

A whole tone is a ratio of notes. In "just" intonation, there are two whole tones, 9/8 and 10/9. In equal temperament, there is a single whole tone with a ratio of 2^(1/6).

What the author is proposing is to divide one of these (or other) whole tones into 5 parts. One way (with equal temperament) would be to use 2^(1/60) (the 60th root of 2) for a halfstep and 2^(1/30) for a whole step; such a division has the usual equal temperament as a subset. Any such division is possible in theory. A few have been used in practice. (Vincenzo used a ratio of 18/17 for a semitone making a wholetone 19/17; this works well for many stringed instruments.)

The question is how the results sound. Personally, I like dividing a octave in to 53 equal parts of ratio 2^(1/53). It's not useful in practice.

This whole thing comes about because no number of musical fifths will equal any number of octaves (except for zero). Similarly for thirds, etc. It's another temperament.(There are a whole lot more than four now.)

• Wouldn't 2^1/60 make it ten (or eleven, since when people talk about dividing an interval into parts, AIUI they count both ends)? So if anything it should be 2^1/24 or 2^1/30. 1/24 would work well with the existing semitones of equal temperament. Nov 6 '19 at 15:55
• You're right; should be 2^(1/30) for even division.
– ttw
Nov 6 '19 at 16:14
• Would you mind editing the `2^(1/30)` into the answer itself? I can't do it for you as that would be a single character change, and only the author can do such small edits. Nov 6 '19 at 21:17

According to Benjamin Wardhaugh's Music: Experiment and Mathematics in England, 1653-1705, p.37, Marchetto's system was 31-equal temperament.

Here is one way that 31-equal can be justified: Let us divide the octave into some number N of equal parts, where N is yet to be determined. Marchetto made two further assumptions, namely that all whole tones are equal, and are to be divided so that

• an augmented unison (e.g. C-C#) is 2 parts
• a minor 2nd (e.g. C#-D) is 3 parts.

Then, assembling wider intervals by combining narrower ones, we conclude that:

• a whole tone (e.g. C-D, D-E or F-G) is 5 parts
• a major 3rd (e.g. C-E) is 5+5=10 parts
• a minor 3rd (e.g. C#-E or E-G) is 3+5=8 parts
• a perfect 4th (e.g. D-G or G-C) is 5+8=13 parts
• a perfect 5th (e.g. C-G) is 10+8=5+13=18 parts
• an octave (e.g. C-C) is 18+13=31 parts, so N=31.

Marchetto decided to treat all whole tones as equal; that is, to choose a mean-tone temperament. There is a range of mean-tone temperaments. Why would he choose to divide the whole tone in this way? An advantage of 31-equal is that it gives good major thirds: at about 387.1 cents, they are only about 0.8 cents wide, whereas 12-equal's major thirds are 400 cents, which is about 13.7 cents wide. Unfortunately, 31-equal's perfect fifths, at 696.8 cents, are about 5.2 cents narrow, whereas 12-equal's are 700 cents, which is only 2.0 cents narrow.

As for the ninefold division, this might be a reference to 53-equal, which among temperaments derived from dividing the octave into N equal parts, provides very accurate fifths and thirds, more accurate than any other N until we get to N=118. To get such accuracy, the temperament's approximations to the greater tone 9:8 and lesser tone 10:9 must be different, as ttw's answer mentioned. They are 9 and 8 parts, respectively.

• I don't see how the "decision to treat all whole tones as equal is tantamount to choosing a mean-tone temperament." It seems rather that a decision to treat whole tones as equal is by definition choosing a mean-tone temperament. Nov 8 '19 at 4:22
• OK, @phoog, I've edited my answer. Nov 8 '19 at 7:41

Already the ancient Greek have known smaller intervals than semitones. (Die absolute Harmonik der Griechen, J.L. Paul).

Marchetto assumes the standard 3-Limit or "Pythagorean" tuning of the music theories of his time in his description of the ratios of the basic intervals:*

"octave"2:1 "5th" 3:2 "4th" 4:3 "whole tone"9:8 unison 1:1

the "whole tone" specified as 9:8 ... it becomes clear that Marchetto's emphasis on 8 and 9:

This would probably have been more clearly expressed if the diagrams had looked like divisions of a monochord string, rather than an abstract multiplicative design.

For further information look up: http://www.tonalsoft.com/monzo/marchetto/marchetto.aspx

• So, why does he think that dividing into three equal parts is better than two equal parts? Nov 6 '19 at 12:49
• So, what does this mean "Now but it is so, that the furthest division into which the continuum may be divided, if we wish to embrace every division so that it may not be reduced, is its division into 3 parts." ? Nov 6 '19 at 12:59
• maybe this link will tell you more. en.wikipedia.org/wiki/58_equal_temperament You could also try to make the test yourself, singing 2 different scales containing the same halftone - once as leading tone up to the tonic (ti-do) and once downt to the third (fa-mi) e.g. the ton B as seventh of C and as fourth of F# and then measure the difference of the pitch. The error of 1/3 tone might be smaller than to the 1/2 tone. Nov 6 '19 at 17:16
• Your answer and link suggest that division of the whole tone into 5 EQUAL parts is not what was meant by Marchetto. The link specifically states that Herlinger, the author cited in the OP, is wrong about the division being into 5 Equal parts. I think your answer would make more sense if you stated this. I couldn't work out why you had divided the whole tone into unequal parts until I read the link. Nov 6 '19 at 19:53
• @MaJoad if you divide a STRING into two parts, each part sounds an octave higher. On a guitar you can press the string down to the 12th fret, or alternatively just lightly damp the fundamental so that the octave rings out as a harmonic. The point is, if you take 2/3 of the string you get a fifth, and with 1/3 of the string you get a fifth plus an octave. In this way the G string can be made to play a D at the 7th fret for example. Repeat the process, and on the 14th fret you get an A, (2/3)x(2/3)=9/16 of the string length. You can also sound an A on the second fret at 9/8 of the string length. Nov 6 '19 at 20:05

A few preliminary responses to the OP's questions: Yes, a whole tone can be divided into any number of parts. Traditionally in the modern Western scale, we have 12 semitones per octave, where each whole tone is divided into 2 semitones. But there is no reason we can't divide musical intervals in other ways, including dividing a whole tone into 5 parts (or 3 or 4 or 10 or 200). In fact, we have a system for dividing every semitone into 100 parts, giving the unit called the cent.

And yes, a whole tone can be divided equally into two parts, but only if you use irrational numbers. Medieval music theorists, following in the tradition of Pythagoras, did not believe it was valid to tune musical intervals to irrational numbers and ratios. To tune the modern equal-tempered whole tone, you need to use a ratio involving 21/6. The modern equal-tempered semitone involves 21/12. These irrational numbers were literally viewed as irrational (i.e., defying reason) in the medieval era when it came to musical tuning. And dividing the vast majority of musical intervals precisely into two equal parts involves the square root of two, which was simply not done at this time. Hence, Marchetto "proves" that one can't divide the whole tone into two equal parts, just as would be true of most basic musical intervals for Pythagoreans.

Anyhow, on to Marchetto: The answers so far have speculated on Marchetto's tuning, but none of them seem to actually address Jan Herlinger's interpretation. Yes, there are lots of debates and conflicting interpretations among modern music theorists about what Marchetto actually meant by his talk of fives and nines, but the OP's question specifically pointed to an article by Herlinger.

And Herlinger clearly thinks Marchetto was dividing the Pythagorean whole tone (8:9 ratio, about 204 cents) into roughly five equal parts, that is, roughly 41 cents for each part. But Herlinger admits that this may have been meant as an approximation or sort of rule of thumb, as Marchetto elsewhere identifies 2/5 of a tone with the Pythagorean limma or minor semitone (243:256 ratio, about 90 cents) and his 3/5 of a tone with the Pythagorean apotome or major semitone (2048:2187, about 114 cents). And in yet another part of the treatise, Marchetto says the diatonic semitone or apotome can be viewed as a 16:17 ratio (about 105 cents), while his enharmonic semitone or limma was a 17:18 ratio (about 99 cents). Clearly there's some inconsistency in his implications. If you strictly follow a division of a 8:9 whole tone into five equal parts, they would result in unequal intervals of about 82 and 124 cents. Not quite like the traditional Pythagorean semitones, but in the same ballpark.

Herlinger goes further to suggest that the 1/5 and 4/5 tone intervals were not meant to be interpreted literally, but rather were approximations of the practice used then (and now) where, for example, leading tones are raised a bit near cadences to create tension to resolve. That would make a lot more sense from a practical musical standpoint than a literal equal division of the 8:9 whole tone into five parts. Herlinger elsewhere makes this explicit in his article "Fractional Divisions of the Whole Tone" (p. 78):

If [Marchetto's] words are taken literally, the leading tones will be much too high: [In his examples] the major sixth would measure 955 cents, the major third 455 cents, intervals closer to a minor seventh and a perfect fourth respectively. But since Marchetto intended the measurements of the two-fifths and three-fifths tone as approximations, could he not also have intended those of one-fifth and four-fifths tone as approximations? If so, he would have been described a system that showed the two traditional Pythagorean semitones as well as another pair just a bit more different in size, associated with progressions from imperfect consonances to perfect ones.

Herlinger goes on to note that such an intonation system would be particularly appropriate for the sonorities in musical compositions of the Trecento (14th-century Italian music), which used the types of chromaticism in Marchetto's examples.

Whether or not we want to view this interpretation as definitive, Herlinger wrote his entire dissertation on Marchetto, published this (and other articles) and then published an annotated translation of the Marchetto treatise. He considered Marchetto's ideas in historical context, so we might grant some weight to Herlinger's interpretation, as he has likely spent more time with this work than any other modern scholar.

It's true that viewing the 1/5 tones as somewhat flexible can result in a rough approximation to 31-tone equal temperament. And it's also true that some people have focused on the role of 9 in Marchetto's division, assuming that Marchetto intended to divide the whole tone into 9 parts (as described in a link in another answer). And here's yet a different interpretation of the 9-fold division idea. (Herlinger addresses why the Latin text probably isn't intending a 9-fold division in footnote 20.)

Suffice it to say that Marchetto's exact intention for tuning is unclear. What is clear, however, is that actual medieval and renaissance music theorists seemed to assume that Marchetto was proposing primarily a 5-fold division, not a 9-fold one. Herlinger goes into detail in both the article linked by OP in the question and in my linked Herlinger article about how Marchetto was received. Johannes Tinctoris, perhaps the most well-known music theorist of the fifteenth century, echoed Marchetto's five-fold division in his brief definitions of intervals, for example, identifying the major semitone or apotome with 3/5 tone and the minor semitone or limma with 2/5 tone.

Clearly, music theorists of the time were trying to come up with some sort of approximation to these two unequal Pythagorean semitones. Herlinger cites Goscalcus Francigena (a treatise from c. 1375) with a 3-fold division of the whole tone, also intended to approximate the Pythagorean limma and apotome. Yet elsewhere this same author gives ratios that clearly show the limma was the standard 243:256 ratio. Like Marchetto, this treatise was probably treating 2/3 and 1/3 as approximations, not necessarily to be taken literally.

For some further context, Herlinger also cites the later theorist Gaffurio (c. 1480), who clearly intends a 9-fold division of the whole tone. Gaffurio discusses a division of the whole tone into four dieses and one comma, where the comma is half of a diesis. A diesis is a generic word in tuning theory for a small interval and can be vague, but a comma in this context is pretty clear: it's referring to either the Pythagorean comma of ratio 524288:531441 (about 23.5 cents) or the syntonic comma of ratio 80:81 (about 21.5 cents), either of which would make perfect sense as about 1/9 of a whole tone.

And that's yet another clue that Herlinger is probably on the right track in his interpretation, and the links in other answers hinting at a 9-fold division are off. In Marchetto's time, there was a clearly defined name already for the ratio of about 1/9th of a whole tone: it was called a comma in a number of 13th and 14th century treatises. The linked interpretation (by Joe Monzo) tries to interpret one of Marchetto's "fifths" as a 80:81 comma, making the other four roughly twice the size (dieses?). There's simply no evidence in Marchetto that he was intending the tuning discussed by Gaffurio a century later. If he were, he likely would have used the term comma in the appropriate place, or at a minimum differentiated the relative sizes of the dieses somehow. (The other historical implication of the word comma in tuning is "the small part left over" after breaking down some larger interval. It would be the precise word Marchetto would likely have used if he meant to divide the whole tone into four "big parts" and one "smaller leftover bit," i.e., the comma.)

Given Marchetto's lack of detailed tuning instructions for his "1/5 tones" and general imprecision compared to other treatises about such a novel tuning idea, it's reasonable to assume his dieses are referring to the ancient Greek implication of that word to mean "very small interval" without necessarily a specific ratio. That said, there is a very confusing passage discussing 9, but Herlinger explains that as simply referring to the 9 in the 8:9 ratio of the whole tone, which would require a 9-fold division of the monochord to tune properly. Maybe that 9 could be interpreted differently, but I've tried to outline Herlinger's reading of Marchetto that the question asked about.

Oh, lastly, Herlinger does give an explanation of why the number nine is so important. See, for example, footnote 17: "In associating the number 9 with perfection, Marchetto stands in the mainstream of Pythagorean numerology." Nine was frequently used in Pythagorean numerology to represent a generic idea of "totality." In subsequent pages, Herlinger gives several source quotations on why nine and odd numbers in general are important to Pythagorean music theory and the whole tone.

There's a recent video from Early Music Sources about this phenomenon (fivefold division) (but in the music of Emilio de’ Cavalieri), where you can also hear it:

(wanted to post this as a comment, because it's a link, and about a different composer, but i can't)