Constant structure: Difference between revisions

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A [[scale]] is said to ~~have~~ '''constant structure''' ('''CS''') if its [[interval class]]es are distinct. That is, each [[interval size]] that occurs in the scale always [[subtend]]s 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.

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 [[subtend]]s 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 ~~has~~ 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.

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.

In terms of [[Rothenberg propriety]], strictly proper scales ~~have CS~~, and proper but not strictly proper scales do not. Improper scales generally do. 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 CS.

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 ~~has~~ 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.

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 ==

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| 2/1

|}

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 ~~does~~ not ~~have~~ constant structure.

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 ===

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-A [[scale]] is said to have '''constant structure''' ('''CS''') if its [[interval class]]es are distinct. That is, each [[interval size]] that occurs in the scale always [[subtend]]s 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.
+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 [[subtend]]s 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 has 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.
+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.
-In terms of [[Rothenberg propriety]], strictly proper scales have CS, and proper but not strictly proper scales do not. Improper scales generally do. 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 CS.
+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 has 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.
+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 ==
@@ Line 121: / Line 121: @@
 | 2/1
 |}
-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 does not have constant structure.
+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 ===