Tonality diamond: Difference between revisions

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== Construction ==

A generalized tonality diamond can be constructed given an equave '''E''' and ''n'' harmonics '''P1, P2, ... Pn''', sorted in increasing size ''after being equave-reduced'' so as to lie between 1 and '''E'''. (In the ''q''-odd-limit construction, the harmonics are simply the octave-reduced odd harmonics up to ''q''.) The tonality diamond then consists of the harmonics '''P1, P2, ... Pn''', their octave complements '''E/P1, E/P2, ... E/Pn''' alongside fractions of the harmonics amongst each other: '''Pi/Pj''' for every ''i'' > ''j'', and '''EPi/Pj''' for every ''i'' < ''j'' (in ~~additional~~ to the [[unison]]). If the harmonics are all linearly independent (as in the 5-odd or 7-odd limits), there are ''n(n+1)'' distinct consonances; however, if some fraction of two harmonics reduces to a different harmonic [e.g. (3/2)/(9/8) = 4/3] or is equivalent to another fraction [e.g. (15/8)/(9/8) = 5/3 = 2*(5/4)/(3/2)], this number reduces.

A generalized tonality diamond can be constructed given an equave '''E''' and ''n'' harmonics '''P1, P2, ... Pn''', sorted in increasing size ''after being equave-reduced'' so as to lie between 1 and '''E'''. (In the ''q''-odd-limit construction, the harmonics are simply the octave-reduced odd harmonics up to ''q''.) The tonality diamond then consists of the harmonics '''P1, P2, ... Pn''', their octave complements '''E/P1, E/P2, ... E/Pn''' alongside fractions of the harmonics amongst each other: '''Pi/Pj''' for every ''i'' > ''j'', and '''EPi/Pj''' for every ''i'' < ''j'' (in addition to the [[unison]]). If the harmonics are all linearly independent (as in the 5-odd or 7-odd limits), there are ''n''(''n''+1) distinct consonances; however, if some fraction of two harmonics reduces to a different harmonic [e.g. (3/2)/(9/8) = 4/3] or is equivalent to another fraction [e.g. (15/8)/(9/8) = 5/3 = 2*(5/4)/(3/2)], this number reduces.

=== Relationship to subgroups ===

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 == Construction ==
-A generalized tonality diamond can be constructed given an equave '''E''' and ''n'' harmonics '''P<sub>1</sub>, P<sub>2</sub>, ... P<sub>n</sub>''', sorted in increasing size ''after being equave-reduced'' so as to lie between 1 and '''E'''. (In the ''q''-odd-limit construction, the harmonics are simply the octave-reduced odd harmonics up to ''q''.) The tonality diamond then consists of the harmonics '''P<sub>1</sub>, P<sub>2</sub>, ... P<sub>n</sub>''', their octave complements '''E/P<sub>1</sub>, E/P<sub>2</sub>, ... E/P<sub>n</sub>''' alongside fractions of the harmonics amongst each other: '''P<sub>i</sub>/P<sub>j</sub>''' for every ''i'' > ''j'', and '''EP<sub>i</sub>/P<sub>j</sub>''' for every ''i'' < ''j'' (in additional to the [[unison]]). If the harmonics are all linearly independent (as in the 5-odd or 7-odd limits), there are ''n(n+1)'' distinct consonances; however, if some fraction of two harmonics reduces to a different harmonic [e.g. (3/2)/(9/8) = 4/3] or is equivalent to another fraction [e.g. (15/8)/(9/8) = 5/3 = 2*(5/4)/(3/2)], this number reduces.
+A generalized tonality diamond can be constructed given an equave '''E''' and ''n'' harmonics '''P<sub>1</sub>, P<sub>2</sub>, ... P<sub>n</sub>''', sorted in increasing size ''after being equave-reduced'' so as to lie between 1 and '''E'''. (In the ''q''-odd-limit construction, the harmonics are simply the octave-reduced odd harmonics up to ''q''.) The tonality diamond then consists of the harmonics '''P<sub>1</sub>, P<sub>2</sub>, ... P<sub>n</sub>''', their octave complements '''E/P<sub>1</sub>, E/P<sub>2</sub>, ... E/P<sub>n</sub>''' alongside fractions of the harmonics amongst each other: '''P<sub>i</sub>/P<sub>j</sub>''' for every ''i'' > ''j'', and '''EP<sub>i</sub>/P<sub>j</sub>''' for every ''i'' < ''j'' (in addition to the [[unison]]). If the harmonics are all linearly independent (as in the 5-odd or 7-odd limits), there are ''n''(''n''+1) distinct consonances; however, if some fraction of two harmonics reduces to a different harmonic [e.g. (3/2)/(9/8) = 4/3] or is equivalent to another fraction [e.g. (15/8)/(9/8) = 5/3 = 2*(5/4)/(3/2)], this number reduces.
 === Relationship to subgroups ===