Smyrna gives an incomplete account of Thrasyllus of Mendes' formulation of the greater perfect system, from which the diatonic and enharmonic genera can be deduced. For the chromatic genus, however, all that is given is a 32:27 proportion of mese to lichanos. This leaves 9:8 for the pyknon, but there is no information at all about the position of the chromatic parhypate and therefore of the division of the pyknon into two semitones, though it may have been the limma of 256:243, as Boethius does later. Someone has referred to this speculative reconstructions as the traditional Pythagorean tuning of the chromatic genus: hypate parhypate lichanos mese 4:3 81:64 32:27 1:1 | 256:243 | 2187:2048 | 32:27 | -498 -408 -294 0 cents Archytas used the simpler and more consonant 9:7, which he used in all three of his genera. His chromatic division is: hypate parhypate lichanos mese 4:3 9:7 32:27 1:1 | 28:27 | 243:224 | 32:27 | -498 -435 -294 0 cents According to Ptolemy's calculations, Didymus's chromatic has only 5-limit intervals, with the smallest possible numerators and denominators. The successive intervals are all superparticular ratios: hypate parhypate lichanos mese 4:3 5:4 6:5 1:1 | 16:15 | 25:24 | 6:5 | -498 -386 -316 0 cents Byzantine music In Byzantine music the chromatic genus is the genus on which the second mode and second plagal mode are based. The "extra" mode nenano is also based on this genus. Enharmonic Aristoxenus describes the enharmonic genus (; ) as the "highest and most difficult for the senses". Historically it has been the most mysterious and controversial of the three genera. Its characteristic interval is a ditone (or major third in modern terminology), leaving the pyknon to be divided by two intervals smaller than a semitone called dieses (approximately quarter tones, though they could be calculated in a variety of ways). Because it is not easily represented by Pythagorean tuning or meantone temperament, there was much fascination with it in the Renaissance. In the modern tuning system of twelve-tone equal temperament, enharmonic refers to tones that are identical, but spelled differently. In other tuning systems, enharmonic notes, such as C and D, may be close but not identical, differing by a comma (an interval smaller than a semitone, like a diesis). Notation Modern notation for enharmonic notes requires two special symbols for raised and lowered quarter tones or half-semitones or quarter steps.