Constant structure: Difference between revisions
clarified and expanded diatonic scale example and interval matrix discussion |
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A [[scale]] is said to | A [[scale]] is said to be a '''constant structure''' ('''CS''') if its [[interval class]]es are distinct. That is, each [[interval size]] that occurs in the scale always spans the same number of scale steps. This means that you never get something like an interval being counted as a fourth one place, and a fifth another place. | ||
If a scale is a constant structure, that scale can be mapped to an [[isomorphic keyboard]] or similar isomorphic instrument such that each chord with the same interval structure can be played using the same fingering shape. | |||
The term "constant structure" was coined by [[Erv Wilson]]. In academic music theory, constant structure is called the partitioning property, but Erv got there first. | The term "constant structure" was coined by [[Erv Wilson]]. In academic music theory, constant structure is called the partitioning property, but Erv got there first. | ||
In terms of [[Rothenberg propriety]], strictly proper scales | In terms of [[Rothenberg propriety]], strictly proper scales are constant structures, and proper but not strictly proper scales are not. Improper scales generally are. However, the [[22edo]] scale C D E vF# G ^Ab B C (<code>4-4-3-2-2-6-1</code>) has both ambiguity (C-vF# 4th equals vF#-C 5th) and contradiction (^Ab-B 2nd exceeds E-G 3rd). The contradiction makes it improper and the ambiguity makes it not a CS. | ||
To determine if a scale | To determine if a scale is a CS, all possible intervals between scale steps must be evaluated. An easy way to do this is with an [[interval matrix]], in which each entry gives the interval spanning the number of scale steps indicated by the column, beginning with step indicated by the row. In a CS scale, each interval in the matrix must appear in only one column, corresponding to the “constant” number of steps for that interval. | ||
== Examples == | == Examples == | ||
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Note the highlighted intervals that occur in more than one column. For example, 5/4 may occur as either two or three steps of the scale. Thus, this scale | Note the highlighted intervals that occur in more than one column. For example, 5/4 may occur as either two or three steps of the scale. Thus, this scale is not a constant structure. | ||
=== Diatonic scales === | === Diatonic scales === | ||
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The | The highlighted intervals, from F to B and from B to F, are the same size in 12edo: 6\12, or 600 cents. From F to B, this interval spans four steps of our diatonic scale (an “augmented fourth”); but from B to F it spans five (a “diminished fifth”). Since the same interval spans different numbers of scale steps at different points in the scale, this scale is not a constant structure. | ||
However, in other tunings of the diatonic scale, the F–B and B–F intervals may have distinct sizes. For example, [[31edo]] (meantone) tunes F–B and B–F to 15\31 (581¢) and 16\31 (619¢) respectively: | However, in other tunings of the diatonic scale, the F–B and B–F intervals may have distinct sizes. For example, [[31edo]] (meantone) tunes F–B and B–F to 15\31 (581¢) and 16\31 (619¢) respectively: | ||
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== Novel terminology == | |||
An interval that occurs in a scale is ''CS-consistent''{{idiosyncratic}} if it always subtends the same number of scale steps. A scale is thus CS if and only if all its intervals are CS-consistent. This term could be useful because someone might only care about certain primes in a subgroup being CS-consistent. | |||
== See also == | == See also == | ||
* [[Gallery of CS Scales]] | * [[Gallery of CS Scales]] | ||
* [[ | * [[Glossary of scale properties]] | ||
* [[epimorphic]] | * [[epimorphic]] | ||
* [http://tonalsoft.com/enc/c/constant-structure.aspx Constant structure] (Tonalsoft Encyclopedia) | * [http://tonalsoft.com/enc/c/constant-structure.aspx Constant structure] (Tonalsoft Encyclopedia) | ||