Structure implies multiplicity

In <a href="/facts/Diatonic_set_theory/ygunohtU">diatonic set theory</a> structure implies multiplicity is a quality of a collection or <a href="/facts/Scale_(music)/QjMlvs3u">scale</a>. For collections or scales which have this property, the interval series formed by the shortest distance around a diatonic <a href="/facts/Circle_of_fifths/0aE2lkVq">circle of fifths</a> between members of a series indicates the number of unique <a href="/facts/Interval_(music)/aFwkkgpY">interval</a> patterns (adjacently, rather than around the circle of fifths) formed by <a href="/facts/Diatonic_transposition/qrGNnycc">diatonic transpositions</a> of that series. Structure refers to the intervals in relation to the circle of fifths; multiplicity refers to the number of times each different (adjacent) interval pattern occurs. The property was first described by <a href="/facts/John_Clough/D1uR5YGG">John Clough</a> and Gerald Myerson in "Variety and Multiplicity in Diatonic Systems" (1985). (Johnson 2003, pp. 68, 151)
Structure implies multiplicity is true of the <a href="/facts/Diatonic_collection/ejXSu379">diatonic collection</a> and the <a href="/facts/Pentatonic_scale/6YuAcqld">pentatonic scale</a>, and any subset.
For example, <a href="/facts/Cardinality_equals_variety/5fHFdv45">cardinality equals variety</a> dictates that a three member diatonic subset of the C major scale, C-D-E transposed to all <a href="/facts/Scale_degree/XooEJJya">scale degrees</a> gives three interval patterns: M2-M2, M2-m2, m2-M2.

On the circle of fifths:

C G D A E B F (C)
 1 2 1 2 1 2 3

E and C are three notes apart, C and D are two notes apart, D and E two notes apart. Just as the distance around the circle of fifths between forms the interval pattern 3-2-2, M2-M2 occurs three times, M2-m2 occurs twice, and m2-M2 occurs twice.
<a href="/facts/Cardinality_equals_variety/5fHFdv45">Cardinality equals variety</a> and structure implies multiplicity are true of all collections with <a href="/facts/Myhill%2527s_property/pP3IbHTq">Myhill's property</a>.

<ul><li>Johnson, Timothy (2003). Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Key College Publishing. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-930190-80-8.</li></ul>

Structure implies multiplicity open-in-new

Structure implies multiplicity