thing from a half to two thirds of a tone from the bottom in en-harmonic, anything from two thirds to 1 1/4 tones in chromatic, and anything from 1 1/4 to 1 1/2 in diatonic. Converted into cents, the tolerance-bands appear as follows:
enharmonic
50-66
50-66
367-400
chromatic
67-100
67-183
250-366
diatonic
67-100
150-233
200-250
Within these catch-all bands Aristoxenus recognizes the following specific 'shades' or 'hues' as being familiar (the figures representing the intervals in tone-units):
enharmonic
1/4
1/4
2
soft chromatic
1/3
1/3
1 5/6
hemiolic chromatic
3/8
3/8
1 3/4
tonic chromatic
1/2
1/2
1 1/2
soft diatonic
1/2
3/4
1 1/4
tense diatonic
1/2
1
1
He also mentions as being melodically proper an unlabelled form of chromatic with the intervals , and an unlabelled form of diatonic with the intervals .33
In the Euclidean Sectio Canonis we find the diatonic constructed on the same pattern as in Philolaus, with two 9 : 8 tones; this is also presupposed in Plato's Timaeus. The Sectio refers likewise to an enharmonic scale featuring the full ditone interval of 408 cents obtained by tuning from the higher note by ascending fourths and descending fifths. It is demonstrated that the small intervals below the ditone cannot be equal; but the implication is that people thought of them as being equal.34
Ptolemy cites the ratios arrived at by two other mathematicians of before his own time besides Archytas: Eratosthenes (third century BC) and Didymus (first century AD). Converted into cents, they appear thus:
Eratosthenes
Didymus
enharmonic
44
45
409
54
57
387
chromatic
89
94
315
112
71
315
diatonic
90
204
204
112
182
204
33Harm. 1. 22-7, 2. 49-52.
34Sectio Canonis 17-20 (Barker, GMW ii. 203-8): Pl. Tim. 35b-36b.
, and an unlabelled form of diatonic with the intervals 
.33