A method of tuning a 12-tone chromatic scale by ear, put forward by the ancient Greek music-theorist Aristoxenus, in which two series of concords -- composed of perfect 4ths and 5ths -- are generated, so that the last tones generated by each series form another concord similar to all the rest.
The first series of concords is generated from the central note μηση (mese) by perfect-4ths upward and perfect 5ths downward, as follows:
the first series of concords: D# D D C# / \ C / \ C B / \ / \ Bb / \ / \ Bb A A \ / \ / G# \ / \ / G G \ / F# \ / F F E D#
The second series of concords is generated from μηση (mese) by a succession of perfect-4ths downward and perfect-5ths upward, as follows:
the second series of concords: D# D# D / C# C# / C / \ / B B / \ / Bb / \ / \ / A A / \ / \ / G# \ / \ / G# G \ / \ / F# \ / F# F \ / E E
When the notes from both series are presented together, they form a 12-tone chromatic scale:
both series of concords together: __+3__ | | | D# | D# D D | / C# / \ C# | / C / \ C / \ | / B / \ B / \ / \ / Bb / \ / \ / \ / \ Bb / A A \ / \ / / G# \ / \ / \ / \ / G# G \ / G \ / \ / F# \ / F# \ / F \ / F E E
The resulting Bb and D# interval between the final notes of each series is supposed to also sound like a concord. In true pythagorean tuning, it would actually be an augmented-3rd Bb:D# of ratio 177147:131072 = 2,3-monzo [-17 11, > = ~521.5050095 cents, and would thus sound discordant -- hence, to narrow that interval to something resembling the concordant 4th of ratio 4:3 = 2,3-monzo [2 -1, > = ~498.0449991 cents, the difference between them, the pythagorean comma of ratio 531441:524288 = 2,3-monzo [-19 12, > = ~23.46001038 cents needs to vanish somewhere. The way to do this with the least overall effect on any given interval of the 11 others, is to distribute it evenly among all 11 of them. To do this, each concord has to be altered slightly, or in modern language, "tempered": the 5ths have to be narrowed, and the 4ths have to be widened. If each concord is tempered by 1/12 of the pythagorean comma, then the final interval Bb:D# will be narrowed by 11/12 of the pythagorean comma, which makes it exactly equal in size to all of the 1/12-comma-wide 4ths. This results in 12-edo tuning.
Aristoxenus did not quantify any of his concords mathematically, insisting instead that musical intervals should be judged by ear. Therefore, it is impossible to state the mathematics of his tuning with certainty - but the inescapable conclusion is that in order to achieve 12 consonant intervals as in his tuning-by-concords scheme, the tuning must be an approximation of 12-edo.
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