Essential tempering comma: Difference between revisions

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The '''essential tempering commas''' are commas that induce [[Dyadic chord #Essentially tempered dyadic chords|essentially tempered ~~triads~~]].

The '''essential tempering commas''' are commas that induce [[Dyadic chord #Essentially tempered dyadic chords|essentially tempered chords]].

~~Note the definition is about~~ ''triads''. There are commas that induce essentially tempered chords whose basic forms are ''tetrads'', and these may not satisfy the definition given below. For example, 81/80 induces an essentially tempered tetrad (→ [[didymic chords]]), despite that any three of the components are essentially just.

== Gene's derivation ==

~~== Definition ==~~

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 ~~intervals~~ ''i'' including 1 and 2 with 1 ≤ ''i'' ≤ 2 such that if ''i'' is in S, so is 2/''i''. 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.

# There are three ~~intervals~~ ''i'', ''j'', and ''k'' in S such that ''c'' = ''ij''/''k''.

# 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.

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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* [[Dyadic chord #Essentially tempered dyadic chords]]

[[Category:Regular temperament theory]]

[[Category:Comma]]

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-The '''essential tempering commas''' are commas that induce [[Dyadic chord #Essentially tempered dyadic chords|essentially tempered triads]].
+The '''essential tempering commas''' are commas that induce [[Dyadic chord #Essentially tempered dyadic chords|essentially tempered chords]].
-Note the definition is about ''triads''. There are commas that induce essentially tempered chords whose basic forms are ''tetrads'', and these may not satisfy the definition given below. For example, 81/80 induces an essentially tempered tetrad (→ [[didymic chords]]), despite that any three of the components are essentially just.
+== Gene's derivation ==
-== Definition ==
+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 intervals ''i'' including 1 and 2 with 1 ≤ ''i'' ≤ 2 such that if ''i'' is in S, so is 2/''i''. 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.
-# There are three intervals ''i'', ''j'', and ''k'' in S such that ''c'' = ''ij''/''k''.
+# 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.
+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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 * [[Dyadic chord #Essentially tempered dyadic chords]]
+[[Category:Regular temperament theory]]
 [[Category:Comma]]