Tonality diamond: Difference between revisions
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{{Wikipedia|Tonality diamond}} | {{Wikipedia|Tonality diamond}} | ||
The ''q''-[[odd-limit]] '''tonality diamond''' is the [[diamond function]] applied to the odd numbers from 1 to ''q'': diamond ({1, 3, 5, … , ''q''}). Another way of defining it is in terms of [[Wikipedia:Height function#Naive height|naive height] | The ''q''-[[odd-limit]] '''tonality diamond''' is the [[diamond function]] applied to the odd numbers from 1 to ''q'': diamond ({1, 3, 5, … , ''q''}). Another way of defining it is in terms of [[Wikipedia:Height function#Naive height|naive height] - the most common number theoretic height function on rational numbers: <math>H\left(\frac{n}{d}\right) = max(|n|, |d|)</math> - as all rational numbers which are the quotient of two positive odd integers ''n''/''d'' with ''H''(''n''/''d'') ≤ ''q'', [[octave-reduced]]. | ||
== 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. | |||
=== Relationship to subgroups === | |||
While, given any subgroup of [[just intonation]], a tonality diamond can be constructed from the equave and the higher primes in the subgroup, the correspondence is not one-to-one: an infinite number of possible tonality diamonds are constructible from a subgroup; for instance, the 2.3.7 subgroup would possess distinct diamonds for harmonics 3 and 7 to equave 2, and for 3 and 21 to 2, or even for 3, 7, and 9 to 2 (to say nothing of 2 and 7/4 to 3). However, any tonality diamond with rational consonances to a rational equave defines a subgroup. | |||
== Examples of scales == | == Examples of scales == | ||
Revision as of 21:49, 16 July 2024
The q-odd-limit tonality diamond is the diamond function applied to the odd numbers from 1 to q: diamond ({1, 3, 5, … , q}). Another way of defining it is in terms of [[Wikipedia:Height function#Naive height|naive height] - the most common number theoretic height function on rational numbers: [math]\displaystyle{ H\left(\frac{n}{d}\right) = max(|n|, |d|) }[/math] - as all rational numbers which are the quotient of two positive odd integers n/d with H(n/d) ≤ q, octave-reduced.
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.
Relationship to subgroups
While, given any subgroup of just intonation, a tonality diamond can be constructed from the equave and the higher primes in the subgroup, the correspondence is not one-to-one: an infinite number of possible tonality diamonds are constructible from a subgroup; for instance, the 2.3.7 subgroup would possess distinct diamonds for harmonics 3 and 7 to equave 2, and for 3 and 21 to 2, or even for 3, 7, and 9 to 2 (to say nothing of 2 and 7/4 to 3). However, any tonality diamond with rational consonances to a rational equave defines a subgroup.
Examples of scales
Music
- Modern Jazz at the Crystal Ball by Norbert Oldani in the 7-limit diamond.