But since the numbers proposed did not afford space
[p. 343]
sufficient for the middle intervals, therefore there was a
necessity to allow larger bounds for the proportions. And
now we are to tell you what those bounds and middle
spaces are. And first, concerning the medieties (or mean
terms); of which that which equally exceeds and is exceeded by the same number is called arithmetical; the
other, which exceeds and is exceeded by the same proportional part of the extremes, is called sub-contrary. Now
the extremes and the middle of an arithmetical mediety
are 6, 9, 12. For 9 exceeds 6 as it is exceeded by 12,
that is to say, by the number three. The extremes and
middle of the sub-contrary are 6, 8, 12, where 8 exceeds
6 by 2, and 12 exceeds 8 by 4; yet 2 is equally the third
of 6, as 4 is the third of 12. So that in the arithmetical
mediety the middle exceeds and is exceeded by the same
number; but in the sub-contrary mediety, the middle term
wants of one of the extremes, and exceeds the other by
the same part of each extreme; for in the first 3 is the
third part of the mean; but in the latter 4 and 2 are third
parts each of a different extreme. Whence it is called
sub-contrary. This they also call harmonic, as being that
whose middle and extremes afford the first concords;
that is to say, between the highest and lowermost lies the
diapason, between the highest and the middle lies the diapente, and between the middle and lowermost lies the
fourth or diatessaron. For suppose the highest extreme
to be placed at nete and the lowermost at hypate, the
middle will fall upon mese, making a fifth to the uppermost extreme, but a fourth to the lowermost. So that
nete answers to 12, mese to 8, and hypate to 6.
This work is licensed under a
Creative Commons Attribution-ShareAlike 3.0 United States License.
An XML version of this text is available for download, with the additional restriction that you offer Perseus any modifications you make. Perseus provides credit for all accepted changes, storing new additions in a versioning system.