Talk:Essential tempering comma: Difference between revisions

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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.
-Despite my earlier attempt to frame it 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, the pair of 11/8 and 15/11 isn't since 11/8 is 11-odd-limit and 15/11 isn't, which constitutes the source of the tempering essence.
+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.
-How about this definition:
+Further investigation suggests that the definition is probably a sufficient (and not necessary) condition for identifying essentially tempered chords.
-# There are three intervals ''i'', ''j'', and ''k'' in S such that ''c'' = ''ij''/''k'', and
-# None of ''c<sup>n</sup>i'', ''c<sup>n</sup>j'', and ''c<sup>n</sup>k'' is a member of S, where ''n'' is a nonzero integer.
-Can anyone check it out? [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 14:44, 7 January 2022 (UTC)
+My revised definition:
+# There are three ratios ''r''<sub>1</sub>, ''r''<sub>2</sub>, and ''r''<sub>3</sub> in S such that ''c'' = ''r''<sub>1</sub>''r''<sub>2</sub>/''r''<sub>3</sub>, and
+# ''c<sup>n</sup>r<sub>i</sub>'' is not a member of S for ''i'' = 1, 2, 3, where ''n'' is a nonzero integer.
+Can anyone check it out? [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 14:44, 7 January 2022 (UTC) -- Updated [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 12:28, 11 March 2022 (UTC)