On the Institution of Music (De institutione musica)

...the octave original: "diapason" consonance is surely born. For four to three holds a sesquitertial sesquitertia: a ratio of 4:3, where the larger number contains the smaller plus one-third of it proportion. However, three is joined to two in a sesquialter sesqualtera: a ratio of 3:2, where the larger number contains the smaller plus one-half of it comparison. And therefore, the number four compared to two is coupled by a double original: "dupla," a 2:1 ratio comparison. But the sesquitertial ratio creates the fourth original: "diatesseron" consonance; the sesquialter creates the fifth original: "diapente" consonance. The double ratio, in turn, effects the octave original: "diapason" symphony. Therefore, the fourth and the fifth together join to form one octave consonance.

Again, a tone In this context, a "whole tone" or major second cannot be divided into equal parts. Why this is so will be clear later. For now, it suffices to know only this: that a tone is never divided into two equal parts. And so that this may be most easily proved, let there be a sesquioctave sesquioctava: a ratio of 9:8, which mathematically defines a whole tone in Pythagorean tuning proportion between 8 and 9. Between these, no natural middle number falls. Therefore, let us multiply these by two: twice 8 becomes 16, and twice 9 becomes 18. Between 16 and 18, one number falls naturally, which is 17. Let these be arranged in the order: 16, 17, 18.

Therefore, 16 and 18, when compared, retain the sesquioctave proportion, and therefore the tone. But the middle number, 17, does not divide this proportion into equal parts. For when 17 is compared to 16, it contains the whole of 16 and its sixteenth part (namely, the value of one). But if the third number, which is 18, is compared to the number 17, it contains the whole of 17 and its seventeenth part. Therefore, it does not exceed the smaller number by the same parts as it is exceeded by the larger number. The seventeenth part is smaller, while the sixteenth part is larger. Yet both are called semitones; not because semitones are entirely equal halves, but because the prefix "semi-" is usually used for that which does not

reach full integrity Boethius argues that "semitone" means "incomplete tone" rather than "exactly half a tone.". But between these, one is called a major semitone and the other a minor semitone.

What a whole semitone is, or from which first smallest numbers it consists, is now more clearly explained. For what was said regarding the division of the tone does not pertain to showing the types of semitones, but rather to the fact that a tone cannot be separated into two equal divisions. The fourth original: "diatesseron", which is a consonance of voices, encompasses four notes and three intervals. Moreover, it consists of two tones and an incomplete semitone specifically the "minor semitone" or "limma".

Let the following description be set down: 192, 216, 243, 256. If the number 192 is compared to 256, a sesquitertial [4:3] proportion is made, and it will sound as a fourth consonance. But if 216 is compared to 192, it is a sesquioctave [9:8] proportion. For their difference is 24, which is the eighth part of 192. It is, therefore, a tone. Again, if 243 is compared to 216, it will be another sesquioctave [9:8] proportion. For their difference, 27, is proven to be the eighth part of 216.

There remains the space from 243 to 256, whose difference is 13; this, when multiplied by eight, does not seem to fulfill the half of 243. Therefore, it is not a [half] semitone, but less than a semitone. For a semitone would be thought of as a whole half if their difference, which is 13, multiplied eight times, could have equaled half of the number 243. Therefore, the comparison of 243 to 256 is the minor semitone.

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