15. But since the numbers proposed did not afford space 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.