The Origin of the Melodic Intervals Smaller than Consonances

Of strings which are under equal tension:

• the longer ones give lower sounds
• the shorter ones give higher sounds

Thus ‘high’ and ’low’ are the appropriate differences in harmony.

Individually, they belong separately to other individual branches of knowledge.

In Geometry:

• ‘high’ is ‘acute’ (Latin for high)
• ’low’ is ‘obtuse’

In Physics:

• ‘high’ is ’light’
• ’low’ is ‘heavy’ (Latin for low)

In other contexts:

• ‘high’ is ‘sharp’’ (Latin for high)
  • This expresses thin and penetrating
• ’low’ is heavy

But high and low linked together, and opposed to each other, are used only in musical notes.

• However, they keep some of their original meaning.

In Geometry, acute is less than obtuse.

Likewise in harmony, a high note expresses smallness, penetrating, raised, and fluttering from lightness.

In Physics, heavy things have a great weight, and light things a small. Likewise, a low note expresses largeness and lowness. Light things soar up to the heights.

Likewise in harmony, a low note is considered weighty, deeper or profound (bass). A high note is considered raised.

On the lute, the first string is the highest. It gives out a low sound was due only to its position on the instrument..

Generally:

• men are dominant over women
• grown men are dominant over boys

These give out a lower note, as if drawn from a greater depth from the throat.

Individual men, as we learn from the sense of touch, extract a lower note from a greater depth, and a higher from above.

• Those who sing lowest extend the body so that the note may emerge as deeply as possible.
• Those who sing high also stretch their necks, yet not to make their necks long, but to tighten more effectively the upper bands of their throats.

Therefore it was for these reasons that there was born in harmony the concept of raised and deep, for which we frequently use high and low.

Therefore since high and deep are at other times words referring to place, the habit of speech, following these its basic principles, also adapts to notes what properly belongs to places, that is to say intervals in Greek diastema “separations.”

It is places which are said to be at a distance, diastema, “be separated.”

Lastly, the discipline of harmony has also transferred this word to its pictorial representations or staves (which will be treated below), as they consist of high and low lines.

On that basis the geometrical sense has been restored to the word.

Therefore what were hitherto called the proportions of the strings will in future generally be called the intervals of notes which strings of unequal length give out.

For notes of the same sound, corresponding with strings of equal length and equal tensions, do not make an interval, since they are of equal pitch.

Nevertheless, in Book 5 this sense of the word “interval” will have to be avoided because interval in its astronomical sense will be frequently repeated. It refers to:

• the straight line between a planet and the Sun
• the space between different spheres

Furthermore in Chapter 3, the proportions were considered under 2 headings, either:

• individually in their own right, or
• in relation to each other with respect to their order which extended from the smaller term, or string, of some compound proportion, to the greater or longer, and the other way round.

The intervals also are considered either individually or in their own right, or in relation to each other, with respect to their harmonic position.

Thus in the continuous ordering of a number of intervals (as when every pair of adjacent intervals always has the same term in common, which is the greater term of one, and the lesser of the other which is in the direction of the lower notes) the interval which is between the lower notes is always called the inferior one, and the one which is between the higher notes the superior.

In Geometry, proportions are recognized as equal even if the terms of One are not equal to the terms of the other, and the difference between the terms of one is not equal to the difference between the terms of the other.

Thus if there are three strings in the proportions of the numbers 4:6:9, the proportion 4:6 is considered the same as 6:9 notwithstanding that both the actual terms, and also the differences 2 and 3 are unequal.

In harmony similarly all intervals between notes coming from the strings which are in the same proportion are both consid- ered equal and are also written with the same numerical mark.

Furthermore they are depicted on the stave with equal intervals of lines, so that we completely forget the inequality which there is between the differences of the various strings.

It therefore follows that we call the intervals which have a smaller proportion, smaller [in Latin “minor”] and those which have a greater proportion greater [in Latin “major”], without regard to the greatness or smallness of the corresponding terms in either case.

Therefore with these preliminary statements serving as definitions, we must now proceed to examine the differences between the intervals.