Talk:Essential tempering comma

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Definition still wrong

Despite my earlier attempt to frame Gene's definition for identifying essentially tempered triads (and excluding tetrads), now I find it also fails this task.

For example, 121/120 induces essentially tempered triads. One of them is 1-11/8-5/3 with steps 11/8-11/9-6/5. Although 11/10 and 12/11 are interchangeable in the chord construction, 11/8 and 15/11 aren't since 11/8 is 11-odd-limit and 15/11 isn't, which constitutes the source of the tempering essence.

Further investigation suggests that the definition is probably a sufficient (and not necessary) condition for identifying essentially tempered chords.

My revised definition:

There are three ratios r₁, r₂, and r₃ in S such that c = r₁r₂/r₃, and
cⁿr_i is not a member of S for i = 1, 2, 3, where n is a nonzero integer.

Can anyone check it out? FloraC (talk) 14:44, 7 January 2022 (UTC) -- Updated FloraC (talk) 12:28, 11 March 2022 (UTC)