## 1.244 proportion

ES: proporción, I: proporzione, F: proportion, D: Proportio, NL: ?, DK: ?, S: ?, FI: suhde.

[Latin: proportio] Described in great detail by Gaffurius, in Practica musicae (published in Milan in 1496). In mensural notation, proportion is:

1. A ratio that expresses the relationship between the note values that follow with those that precede;
2. A ratio between the note values of a passage and the ‘normal’ relationship of note values to the metrical pulse. (A special case of the first definition.)

The most common proportions are:

• 2:1 (or simply 2), expressed by a vertical line through the mensuration sign (the origin of the alla breve time signature), or by turning the sign backwards
• 3:1 (or simply 3)
• 3:2 (sesquialtera)

To ‘cancel’ any of these, the inverse proportion is applied. Thus:

• 1:2 cancels 2:1
• 1:3 cancels 3:1
• 2:3 cancels 3:2
• and so on.

Gaffurius enumerates five basic types of major:minor proportions and their inverses:

1. Multiplex, if the major number is an exact multiple of the minor (2:1, 3:1, 4:2, 6:3); and its inverse, Submultiplex (1:2, 1:3, 2:4, 3:6)
2. Epimoria or Superparticular [orig. Epimoria seu Superparticularis], if the major number is one more than the minor (3:2, 4:3, 5:4); and its inverse, Subsuperparticular (2:3, 3:4, 4:5)
3. Superpartiens, if the major number is one less than twice the minor (5:3, 7:4, 9:5, 11:6); and its inverse, subsuperpartiens (3:5, 4:7, 5:9, 6:11)
4. Multiplexsuperparticular, if the major number is one more than twice the minor (5:2, 7:3, 9:4); and its inverse, Submultiplexsuperparticular (2:5, 3:7, 4:9)
5. Multiplexsuperpartiens, if the major number is one less than some other multiple (usually three or four) of the minor (8:3, 11:4, 14:5, 11:3); and its inverse, Submultiplexsuperpartiens (3:8, 4:11, 5:14, 3:11)

He then continues to subdivide each type in various ways. For the multiplex proportions, for example, he indicates how many times greater the major number is than the minor:

• If two times greater, the proportion is dupla. If inverted, it’s called subdupla. Examples: 2:1, 4:2, and 6:3.
• If three, tripla; and its inversion, subtripla. Example: 3:1, 6:2, and 9:3.
• If four, quadrupla; and its inversion, subquadrupla. Example: 4:1, 8:2, and 12:3

Other proportions were possible, but whether they were frequently used is another question:

• 33:9, triplasuperbipartientetertias
• 51:15, triplasuperbipartientequintas