Diamond function: Difference between revisions

Line 2:

== Definition ==

Given a collection of notes S, the ''diamond'' of S, '''diamond''' (S), is the set of intervals between those notes, taking the intervals in direct and inverted form, [[octave reduction|reduced to an octave]]. For instance, given the notes {1, 3, 5}, diamond ({1, 3, 5}) is {1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}). The diamond of a set is usually considered in connection with [[just intonation]], in which case S is a set of rational numbers, but it applies to any collection; for instance diamond ({0, 400, 700}) where the notes are expressed in cents, is {0, 300, 400, 500, 700, 800, 900}. The important special case where S is the set of odd integers less than or equal to an odd ''n'' is called the [[tonality diamond]], and is often taken as the set of theoretical consonances in the ''n'' [[odd limit]]. This can be justified on the grounds that these are just the intervals appearing in the "chord of nature", or ~~overtone~~ series, hence objecting to 17/16, for example, on the grounds it isn't actually very consonant doesn't take account of the fact that the integers up to 17, a "chord of nature", contain this interval.

Given a collection of notes S, the ''diamond'' of S, '''diamond''' (S), is the set of intervals between those notes, taking the intervals in direct and inverted form, [[octave reduction|reduced to an octave]]. For instance, given the notes {1, 3, 5}, diamond ({1, 3, 5}) is {1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}). The diamond of a set is usually considered in connection with [[just intonation]], in which case S is a set of rational numbers, but it applies to any collection; for instance diamond ({0, 400, 700}) where the notes are expressed in cents, is {0, 300, 400, 500, 700, 800, 900}. The important special case where S is the set of odd integers less than or equal to an odd ''n'' is called the [[tonality diamond]], and is often taken as the set of theoretical consonances in the ''n'' [[odd limit]]. This can be justified on the grounds that these are just the intervals appearing in the "chord of nature", or harmonic series, hence objecting to 17/16, for example, on the grounds it isn't actually very consonant doesn't take account of the fact that the integers up to 17, a "chord of nature", contain this interval.

The above definition is based on sets, but it is also possible to define diamonds in terms of [[Wikipedia: Multiset|multisets]], which can lead to different results. If S is a multiset, then '''diamult''' (S) is the multiset of intervals between those notes, taking the intervals in direct and inverted form, reduced to an octave. The underlying set of notes may contain more notes than the diamond of the underlying set of the multiset S.

@@ Line 2: / Line 2: @@
 == Definition ==
-Given a collection of notes S, the ''diamond'' of S, '''diamond''' (S), is the set of intervals between those notes, taking the intervals in direct and inverted form, [[octave reduction|reduced to an octave]]. For instance, given the notes {1, 3, 5}, diamond ({1, 3, 5}) is {1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}). The diamond of a set is usually considered in connection with [[just intonation]], in which case S is a set of rational numbers, but it applies to any collection; for instance diamond ({0, 400, 700}) where the notes are expressed in cents, is {0, 300, 400, 500, 700, 800, 900}. The important special case where S is the set of odd integers less than or equal to an odd ''n'' is called the [[tonality diamond]], and is often taken as the set of theoretical consonances in the ''n'' [[odd limit]]. This can be justified on the grounds that these are just the intervals appearing in the "chord of nature", or overtone series, hence objecting to 17/16, for example, on the grounds it isn't actually very consonant doesn't take account of the fact that the integers up to 17, a "chord of nature", contain this interval.
+Given a collection of notes S, the ''diamond'' of S, '''diamond''' (S), is the set of intervals between those notes, taking the intervals in direct and inverted form, [[octave reduction|reduced to an octave]]. For instance, given the notes {1, 3, 5}, diamond ({1, 3, 5}) is {1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}). The diamond of a set is usually considered in connection with [[just intonation]], in which case S is a set of rational numbers, but it applies to any collection; for instance diamond ({0, 400, 700}) where the notes are expressed in cents, is {0, 300, 400, 500, 700, 800, 900}. The important special case where S is the set of odd integers less than or equal to an odd ''n'' is called the [[tonality diamond]], and is often taken as the set of theoretical consonances in the ''n'' [[odd limit]]. This can be justified on the grounds that these are just the intervals appearing in the "chord of nature", or harmonic series, hence objecting to 17/16, for example, on the grounds it isn't actually very consonant doesn't take account of the fact that the integers up to 17, a "chord of nature", contain this interval.
 The above definition is based on sets, but it is also possible to define diamonds in terms of [[Wikipedia: Multiset|multisets]], which can lead to different results. If S is a multiset, then '''diamult''' (S) is the multiset of intervals between those notes, taking the intervals in direct and inverted form, reduced to an octave. The underlying set of notes may contain more notes than the diamond of the underlying set of the multiset S.