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:

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:

# ''c'' is greater than 1 but less than the smallest interval between any two members of S.

# ''c'' is greater than 1 but less than the smallest interval between any two members of ''S''.

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

# 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. 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.

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.

== Examples ==

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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:
+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:
-# ''c'' is greater than 1 but less than the smallest interval between any two members of S.
+# ''c'' is greater than 1 but less than the smallest interval between any two members of ''S''.
-# 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>.
+# 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. 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.
 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.
 == Examples ==