Essential tempering comma: Difference between revisions

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== Gene's derivation ==

Suppose ''S'' is a set of JI ratios ''r'' including 1 and 2 with 1 ≤ ''r'' ≤ 2 such that if ''r'' is in ''S'', so is 2/''r''. ''S'' is intended to represent a set of pitch classes defining "consonance". A JI interval ''c'' is an essential tempering comma for ''S'' if:

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# There are three ratios ''r''1, ''r''2, and ''r''3 in ''S'' such that ''c'' = ''r''1''r''2/''r''3.

This derivation is a sufficient and not necessary condition for identifying essentially tempered triads. For example, 121/120 in the 11-odd-limit ~~is rejected, but it induces essentially tempered triads~~ (→ [[biyatismic chords]])~~. One of them is 1–11/8–5/3 with steps 11/8, 11/9, 6/5~~.

This derivation is a sufficient and not necessary condition for identifying essentially tempered triads due to how aggressively it rejects commas that are too large. This does come with the benefit of discarding absurdly large commas. For example, 125/108 is a comma whose essentially tempered chord is a stack of three 6/5's closing at the octave. Assume pure octaves, it requires each 6/5 to be tuned sharper than the just value of 5/4, and is rejected, in fact leaving 128/125 as the only 5-odd-limit essential tempering comma. However, some commas are reasonably sized but still rejected, such as 121/120 in the 11-odd-limit (→ [[biyatismic chords]]).

Note also that it only identifies ''triads''. There are commas that induce essentially tempered chords whose basic forms are ''tetrads''. For example, 81/80 induces an essentially tempered tetrad (→ [[didymic chords]]), despite that any three of the components are essentially just.

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 == Gene's derivation ==
 Suppose ''S'' is a set of JI ratios ''r'' including 1 and 2 with 1 ≤ ''r'' ≤ 2 such that if ''r'' is in ''S'', so is 2/''r''. ''S'' is intended to represent a set of pitch classes defining "consonance". A JI interval ''c'' is an essential tempering comma for ''S'' if:
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 # 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>.
-This derivation is a sufficient and not necessary condition for identifying essentially tempered triads. For example, 121/120 in the 11-odd-limit is rejected, but it induces essentially tempered triads (→ [[biyatismic chords]]). One of them is 1–11/8–5/3 with steps 11/8, 11/9, 6/5.
+This derivation is a sufficient and not necessary condition for identifying essentially tempered triads due to how aggressively it rejects commas that are too large. This does come with the benefit of discarding absurdly large commas. For example, 125/108 is a comma whose essentially tempered chord is a stack of three 6/5's closing at the octave. Assume pure octaves, it requires each 6/5 to be tuned sharper than the just value of 5/4, and is rejected, in fact leaving 128/125 as the only 5-odd-limit essential tempering comma. However, some commas are reasonably sized but still rejected, such as 121/120 in the 11-odd-limit (→ [[biyatismic chords]]).
 Note also that it only identifies ''triads''. There are commas that induce essentially tempered chords whose basic forms are ''tetrads''. For example, 81/80 induces an essentially tempered tetrad (→ [[didymic chords]]), despite that any three of the components are essentially just.