Similarly:
the ratio for the third was 9=8 9=8 = 81=64;
the ratio for the sixth was 9=8 3=2 = 27=16;
the ratio for the seventh was 3=2 9=8 9=8 = 243=128.
Some of these ratios were corrected in the so called natural just scale, that
is a scale in which the frequencies of notes are related by ratios of small
whole numbers (just intonation). This tuning system of the later ancient Greek
modes was codied by Claudius Ptolemy (about 85-165 AD) and developed by
Gioseo Zarlino (Istitutioni Harmoniche 1558). The corrections made on the
Pythagorean scale are the following: 81:64 was corrected in 5:4, 27:16 in 5:3
and 243:128 in 15:8 (see the table below).