Marchetto di Padua’s enigmatic whole-tone division

Marchetto appears to imply an interval, not degree, from his “diesis”. In Joe Monzo’s interpretation, we take the whole tone as an interval and “dieses” as degrees of the whole tone division instead of intervals. In an age of Pythagoreanism, I do not think defending quarter-tones makes much sense.

Let us analyze the quotes I copied from monz’s website:

“Because the aforesaid chromatic, diatonic, and enharmonic [species] cannot be fully treated until the whole tone is examined (since they are semitones), we shall first study the nature of the whole tone and how it is divided by numbers.”

The first critical information we glean from this statement is that the chromatic, diatonic and enharmonic are all semitones, meaning half the whole tone.

“We acknowledge therefore that the parts of itself will have to be inequalities, so that 1 is the first part; from 1 to 3, the second; from the 3 to 5, the third; from 5 to 7, the fourth; from 7 to 9, the fifth; and such 5th part is the fifth odd number of the total 9.”

As monz demonstrated:

1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 — Marchetto “part”
1 .. .. ..2 .. .. ..3 .. .. ..4 .. .. ..5 — Marchetto “diesis”

However, the word “inequalities” clearly implies that the parts are to be unequal in size. I believe, contrary to my previous assumption, that the 1-3 and 3-5 parts are smaller than 7-9 and 5-7 is even smaller. This means that Marchetto is concentrating the semitones in the center region of half the whole tone and that the 5th diesis is very large.

“Any fifth part as it is desired, may be called a diesis, whether the lowest (smallest?) or the highest (largest?) division, this is the most important division that can be obtained in singing a tone.”

Marchetto ascribes great importance to the diesis, which seems to change drastically in size. He says, “maior divisio”. It is translated here as the most important division, but I dare say he is quite literal about it. I think he means that the first, second and fifth dieses are fairly large, considering that he is talking about singers and given the previous criterion on attaining semitones.

“It is true that from three of these dieses a diatonic semitone is made. This is the larger [semitone] which is called the major apotome. It is the larger part of the tone divided in two.”

He clearly divides the whole tone into two (possibly arithmetically) and wants us to recognize the major apotome/diatonic semitone as the larger (but not the largest) semitone part composed of three dieses. He says major apatome, not apotome plainly, which means that we are talking about an interval larger than 2187:2048.

“The diatonic semitone occurs when a permutation is made from round b to square b or vice versa, whether in ascent or descent, as here [musical example]:”

STAFF2  a..G.. E..C

In the musical example above, I dare say that the interval between bb and b is none other than 15:14.

“Note that the nature of the diesis is best understood thru comparison with the chromatic semitone; much concerning the diesis will become clear as we demonstrate the nature of the chromatic semitone.”

I think he means that the diesis will reveal itself when a chromatic semitone is sung.
“The Chromatic Semitone is that which includes 4 of the 5 dieses of the whole tone, and, as said earlier, it completes a whole tone when a diesis is added to it.”

Here comes the confusing part. Remembering that the sizes of dieses are likely unequal, and that the chromatic semitone will demonstrate their nature, I believe that this must be the largest semitone at 27:25, rendering a very large fifth diesis with the ratio 25:24.

“It results when some whole tone is divided in 2 so as to color some dissonance such as a 3rd, a 6th, or a 10th striving toward some consonance.”

I understand from this statement that this interval serves as a leading-tone, by yielding 25:24 when a tone is subtracted from it!

“The first part of a tone thus divisible will be larger if the melody ascends, and is called a chroma; the part that remains is a diesis, as here: [musical example]”

STAFF1  c..c#.d | f..f#.g | G..G#.A
STAFF2  F..E..D | F..D..C | C..E..D

This perfectly gives us a chroma of 25:24 between c#-d, f#-g and G#-A!

“from the diatonic and diesis arise the chromatic.”

The fourth diesis is only 14 cents large and occuring between 15:14 and 27:25 in support of all the previous speculations.

“2 of these 5 intervals [the dieses] joined together make up the enharmonic semitone, which is the smaller.”

The diesis here is 35 cents and is the tritonic diesis at 50:49, which is half the chroma with the ratio 25:24. So the whole tone division of Marchetto becomes:

1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 — Marchetto “part”
1 .. .. ..2 .. .. ..3 .. .. ..4 .. .. ..5 — Marchetto “diesis”
50:49 .. 25:24 .. 15:14.. ..27:25.. ..9:8 — ratios of pitches
35.. .. ..70 .. .. 119 .. .. 133 .. ..204 — tones in cents
35.. .. ..35 .. .. 49.. .. ..14.. .. ..71 — consecutive int. (₵)

The size of the chromatic semitone (4 dieses) of Marchetto suggests a super-Pythagorean augmented prime (apotome), while the size of the diatonic semitone (3 dieses) suggests a 5-limit JI or meantone minor second. Confusingly enough, Marchetto calles the latter “major apotome”. If his natural diatonic scale is Pythagorean, as it seems to be, then the actual diatonic semitone is the limma. But Marchetto calls the “larger remainder” the diatonic semitone, which would produce the di-leimma (pun intended) that we have 90 cents as well as 114 cents as the diatonic semitone. To solve the issue, we must conclude that the diatonic natural scale is not a Pythagorean major, but JI or close to JI, perhaps the Rast scale suggested by Safi al-din Urmavi who lived a century prior to Marchetto:

G  1
A  9/8
B  8192/6561
C  4/3
D  3/2
E  27/16
F  16/9
G  2

In which case, the diatonic semitone becomes 2187:2048 and is the major remainder of the whole tone and the larger part of the tone divided into two (16:17:18 as Dave Keenan demonstrated). For this to be composed of three “dieses” that are not equalities, we need two quarter-tones and a comma, which arises if we divide the whole tone into 9 arithmetical parts (just as we divided it into two previously):

72/72      1/1       0 cents       0
73/72                24 cents      1    1st part/diesis
74/72      37/36     47 cents      2
75/72      25/24     71 cents      3    2nd part/diesis
76/72      19/18     94 cents      4
77/72                116 cents     5    3rd part/diesis
78/72      13/12     139 cents     6
79/72                161 cents     7    4th part/diesis
80/72      10/9      182 cents     8
81/72      9/8       204 cents     9    5th part/diesis

Notice, that we have a comma of 24 cents as the first diesis, a quarter-tone of 47 cents as the second diesis and 71 cents as the third diesis. This solves the matter of the diatonic semitone.

What about the chromatic semitone then? Marchetto says that it is the “chroma” that completes to a whole tone with the addition of the diesis, is a diatonic semitone plus a diesis, and is composed of 4 of the 5 dieses mentioned. This could mean nothing other than 161 cents given above. Though, I would have much preferred 13:12 as the chroma, but Marchetto seems to insist on taking only the primes within the number 9.

This leaves us with 25/24 as the enharmonic semitone, if we take a comma plus a quarter-tone, or 19/18 as the enharmonic semitone if we take two quarter-tones. This interpretation favours 53-EDO as a basis for “Marchettan harmony”.

Dr. Oz. (8/18/2008)