THE SOFT DIATONIC AND TENSE CHROMATIC SEMITONES

We have found the conclusion inevitable that Ptolemy's arguments, although in all probability providing trustworthy information about the employment of minor resonant intervals in second–century cithara music, fail in the case of those that lack resonant relations within the tetrachord. Thus we must raise the question whether we must leave it at that, perhaps assuming that there was at any rate not much point in standardising non–resonant intervals – or whether we can adduce arguments for specific pitches, perhaps different from those derived mathematically by Ptolemy. Indeed, at least speculations of the latter kind are possible, if only a small part of the foregoing considerations are accepted.

From the text of the Harmonics, we have extracted a number of conditions to which our tetrachords must comply:

1. (1) The soft diatonic semitone is smaller than the le îmma of 90 cents by a size that permits the unequivocal assessment of this relation under experimental conditions.

2. (2) The corresponding lower tense chromatic semitone is of roughly the same size, perhaps a bit smaller, but almost certainly not larger.

3. (3) The same chromatic semitone is significantly larger than the ‘tonic diatonic’ semitone (of 63 cents). This difference is stated without reference to experimental verification; it is regarded as common knowledge.

For coherent tests, we must assume a pitch difference of at least 10 cents. A universally recognised difference, on the other hand, cannot have involved less than 15 cents.