Glossary of scale properties: Difference between revisions

(11 intermediate revisions by 2 users not shown)

Line 1:

A simplified explanation of the various properties of periodic ~~scales~~.

A simplified explanation of the various properties of [[periodic scale]]s. Also check the main [[Glossary]].

== ~~Definitions~~ ==

{{TOC Horizontal

; ~~Interval~~

| a1=[[#A|A]]

: ~~A specific musical interval~~ (e.g. ~~a major third or minor seventh~~).

| a2=[[#B|B]]

| a3=[[#C|C]]

| a4=[[#D|D]]

| a5=[[#E|E]]

| a6=[[#F|F]]

| a7=[[#G|G]]

| a8=[[#H|H]]

| a9=[[#I|I]]

| a10=[[#J|J]]

| a11=[[#K|K]]

| a12=[[#L|L]]

| a13=[[#M|M]]

| a14=[[#N|N]]

| a15=[[#O|O]]

| a16=[[#P|P]]

| a17=[[#Q|Q]]

| a18=[[#R|R]]

| a19=[[#S|S]]

| a20=[[#T|T]]

| a21=[[#U|U]]

| a22=[[#V|V]]

| a23=[[#W–Z|W–Z]]

}}

=== A ===

; [[arity]]

: The number of distinct step sizes occurring in a given scale. Arity ''disregards other properties'', such as [[rank]] or [[maximum variety]]. For example, 12edo melodic minor is a binary scale which is not rank-2 or MV2 (MOS). An ''n''-ary scale has ''n'' distinct step sizes (e.g. [[#B|binary]], [[#T|ternary]], [[#Q|quaternary]]).

; ~~Generic interval (ordinal category)~~

=== B ===

: A ~~class of intervals which fall on the same~~ scale ~~degrees. In the diatonic scale, these classes are the set~~ of ~~seconds~~, ~~the set of thirds, the set of fifths, etc~~. ~~Generic intervals can also be likened to distances between note-heads on a traditional staff~~. ~~A generic interval composed of ''k'' scale steps in any scale, diatonic or not, can be called a "''k''-~~step~~" (terminology taken from [[TAMNAMS]])~~.

; [[binary]] scale

: A scale of [[#A|arity]] 2, i.e. with two distinct step sizes.

; ~~Scale degree~~

=== C ===

~~: The number of steps~~ [[~~subtend~~]]~~ed by a generic interval from~~ the ~~tonic. In conventional music theory, degrees~~ of the ~~diatonic~~ scale ~~are usually given 1-indexed ordinal names corresponding~~ to the diatonic intervals themselves, for example the ''fifth degree'' for the degree occupied by the fifth. However, in xenharmonic music, 0-indexed names are preferred by some people for degrees of non-diatonic scales: the ''k''-degree is the degree represented by the ''k''-step generic interval from the tonic.

; [[chirality]]

: A scale is chiral if reversing the order of the steps results in a different scale up to rotation.

== ~~Properties~~ ==

; [[constant structure]] (CS)

; ~~Alternating generator~~ (AG) ~~property~~

: A scale is a constant structure if all intervals of the same size are also within the same generic interval class. A single interval cannot be a part of two classes. The 12-tone diatonic scale does not have this property, since tritones can either be augmented fourths or diminished fifths. This is referred to as the ''partitioning property'' in most academic literature.

: A scale ~~satisfies~~ the [[~~alternating~~ generator property]] if it satisfies the following equivalent properties:

; [[convex scale|convexity]]

: A scale in a [[regular temperament]] is convex if its representation on a [[harmonic lattice diagram]] forms a convex polygon.

=== D ===

; [[distributional evenness]] (DE)

: A scale with two step sizes is ''distributionally even'' if it has its two step sizes distributed as evenly as possible.

=== E ===

; [[epimorphism]]

* '''Epimorphism''': A JI scale is ''epimorphic'' if, under some val, all scale degrees are "filled," no matter which note you choose as the tonic, and successive degrees are always increasing. Without the second condition, the scale is only ''weakly epimorphic''.

* '''Epimorph val/temperament''': A val that witnesses that a JI scale is epimorphic is called the ''epimorph val'' of the scale, and a temperament supported by an epimorph val is an ''epimorph temperament''. Many low-accuracy edos and temperaments are useful as epimorph vals and temperaments, and these temperaments imply structure rather than tuning; a CS scale may be constructed as a detempering of the low-accuracy tuning implied by such a temperament.

* Example: 5-limit [[Zarlino]] is a 2.3.5 JI scale that is epimorphic under the val {{val|7 11 16}}, and the 2.3.5 temperaments [[dicot]] and [[meantone]] are both epimorph temperaments for Zarlino.

=== F ===

=== G ===

; [[generator-offset property]] (GO)

: A scale satisfies the ''generator-offset property'' if it satisfies the following equivalent properties:

* the scale can be built by stacking alternating generators, for example 7/6 and 8/7.

* the scale is generated by two chains of generators separated by a fixed interval; either both chains are of size m, or one chain has size m and the second has size m-1.

; [[~~Constant structure~~]]

=== H ===

: A ~~scale~~ is a ~~constant structure (CS) if all intervals~~ of the ~~same size are also within~~ the same generic interval class. A single interval cannot be a part of two classes. The 12-tone diatonic scale does not have this property, since tritones can either be augmented fourths or diminished fifths. This is referred to as the ''partitioning property'' in most academic literature.

=== I ===

=== J ===

=== K ===

=== L ===

=== M ===

; [[maximal evenness]]

: A [[#P|periodic]] [[#B|binary]] scale is maximally even with respect to an [[equal-step tuning]] if it is the result of rounding a smaller equal tuning to the nearest notes of the parent equal tuning with the same equave.

; [[Rothenberg propriety~~|Propriety~~]]

; [[maximum variety]] (MV)

: The maximum [[interval variety]] from all interval classes of a [[#P|periodic scale]].

; MOS

* A scale is a [[mos scale]] if there are ''no more than'' two interval sizes for each generic interval class not including the equave. A.k.a. maximum variety 2.

; Myhill's property

* A scale has ''Myhill's property'' if there are ''exactly'' two interval sizes for each interval class not including the equave. A.k.a. strict variety 2. A scale with Myhill's property is called a ''strict mos''.

=== N ===

=== O ===

=== P ===

; [[periodic scale|periodicity]]

: A scale is periodic if its [[step pattern]] repeats after a certain [[#I|interval]].

; [[pepper ambiguity]]

: The Pepper ambiguity of an [[interval]] in an [[equal-step tuning]] is the ratio of the best approximation to the second best approximation.

=== Q ===

; [[quaternary]] scale

: A scale of [[#A|arity]] 4, i.e. with four distinct step sizes.

=== R ===

; [[rank]]

: The rank of a scale is the minimum number of intervals needed to generate the entire scale. For example, the diatonic scale is a rank-2 scale because it is entirely generated by stacking fifths and octaves.

; [[Rothenberg propriety]]

* '''Propriety''': A scale is proper if there is no overlapping of generic interval classes. This means that no third is larger than a fourth, no fourth is larger than a fifth, etc.

* '''Strict propriety''': A scale is strictly proper if the generic interval classes are disjoint. Replace the word "larger" with "larger-than-or-equal-to" in the definition above. The 12-tone diatonic scale is proper, but not strictly proper.

~~; Epimorphism~~

=== S ===

* '''Epimorphism''': A JI scale is ''epimorphic'' if, under some val, all scale degrees are "filled," no matter which note you choose as the tonic, and successive degrees are always increasing. Without the second condition, the scale is only ''weakly epimorphic''.

; [[strict variety]] (SV)

* '''Epimorph val/temperament''': A val that witnesses that a JI scale is epimorphic is called the ''epimorph val'' of the scale, and a temperament supported by an epimorph val is an ''epimorph temperament''. Many low-accuracy edos and temperaments are useful as epimorph vals and temperaments, and these temperaments imply structure rather than tuning; ~~a CS scale may be constructed as a detempering of the low-accuracy tuning implied by such a temperament.~~

: The [[interval variety]] of all interval classes of a [[#P|periodic scale]], when all interval classes have the same interval variety.

* Example: 5-limit [[~~Zarlino~~]] ~~is a 2.3.5 JI scale that is epimorphic under the val {{val|7 11 16}}, and the 2.3.5 temperaments~~ [[~~dicot~~]] ~~and~~ [[~~meantone~~]] ~~are both epimorph temperaments for Zarlino~~.

; ~~Symmetry~~

; symmetry

: A scale is symmetrical if at least one mode of the scale is symmetrical. Therefore, every interval of that mode must have an inverse. These scales will always have an odd number of notes ''per period''. They may not always have an odd number of notes ''per octave'', however. The diatonic scale is symmetrical, but so is 12edo.

; ~~Myhill's property and MOS~~

=== T ===

* '''Myhill's/MOS property''': A scale ~~has Myhill's property if there are ''exactly'' two interval sizes for each (reduced) interval class not including the equave.~~ A ~~scale is a MOS scale if there are ''no more than'' two interval sizes for each generic interval class not including the equave~~. ~~This is equivalent to a scale being Myhill with a smaller equave~~. ~~Myhill's property is sometimes called "strict MOS".~~

; [[ternary]] scale

: A scale of [[#A|arity]] 3, i.e. with three distinct step sizes.

~~; Trivalence property~~

~~: Same as Myhill's property, but replace "two interval sizes"~~ with "three ~~interval~~ sizes~~." The scale formed from the notes of a dominant 7th chord (e.g. C-E-G-Bb-C) is an example of a trivalent scale~~.

~~The 12-tone diatonic scale has~~ Myhill's property, ~~and~~ is ~~also distributionally even~~.

; trivalence property

: Same as [[#M|Myhill's property]], but replace "two interval sizes" with "three interval sizes". A.k.a. strict variety 3. The scale formed from the notes of a dominant 7th chord (e.g. C-E-G-Bb-C) is an example of a trivalent scale.

The diminished scale is an MOS with a 1/4-octave period. Because there is only one interval size at the period, it does not have exactly two interval sizes per interval class. Therefore, it is MOS, but doesn't have Myhill's property.

=== U ===

=== V ===

=== W–Z ===

~~An EDO is a kind of degenerate MOS,~~ in ~~that it is distributionally even. It does not have~~ Myhill's property~~. In other words~~, ~~it has no more than two interval sizes for each generic interval class, but does not have exactly two interval sizes~~.

== Examples ==

The [[5L 2s|diatonic scale]] in 12edo has Myhill's property, and is also distributionally even.

~~; Arity, binary, ternary, ''n''-ary~~

The [[diminished scale]] is a mos with a 1/4-octave period. Because there is only one interval size at the period, it does not have exactly two interval sizes per interval class. Therefore, it is a mos, but does not have Myhill's property.

: The ~~number of distinct step sizes occurring in a given scale. Arity ''disregards other properties'', such as~~ [[~~rank]] or [[maximum variety~~]]. ~~For example~~, ~~12edo melodic minor~~ is a ~~binary scale which is~~ not ~~rank-2 or MV2 (MOS)~~.

~~; Convexity~~

An edo is a kind of degenerate mos, in that it is distributionally even. It does not have Myhill's property. In other words, it has no more than two interval sizes for each generic interval class, but does not have exactly two interval sizes.

~~; [[Maximal evenness]]~~

~~; [[Pepper ambiguity]]~~

== See also ==

* [[Glossary]]

* [[Periodic scale]] – contains mathematical definitions of several scale properties

* [[~~Periodic_scale~~|~~Periodic scale~~]]

[[Category:Terms| ]]

[[Category:Scale]]

~~[[Category:~~Todo~~:add examples]]~~

~~[[Category:Todo:~~add illustration]]

@@ Line 1: / Line 1: @@
-A simplified explanation of the various properties of periodic scales.
+A simplified explanation of the various properties of [[periodic scale]]s. Also check the main [[Glossary]].
-== Definitions ==
+{{TOC Horizontal
-; Interval
+| a1=[[#A|A]]
-: A specific musical interval (e.g. a major third or minor seventh).
+| a2=[[#B|B]]
+| a3=[[#C|C]]
+| a4=[[#D|D]]
+| a5=[[#E|E]]
+| a6=[[#F|F]]
+| a7=[[#G|G]]
+| a8=[[#H|H]]
+| a9=[[#I|I]]
+| a10=[[#J|J]]
+| a11=[[#K|K]]
+| a12=[[#L|L]]
+| a13=[[#M|M]]
+| a14=[[#N|N]]
+| a15=[[#O|O]]
+| a16=[[#P|P]]
+| a17=[[#Q|Q]]
+| a18=[[#R|R]]
+| a19=[[#S|S]]
+| a20=[[#T|T]]
+| a21=[[#U|U]]
+| a22=[[#V|V]]
+| a23=[[#W–Z|W–Z]]
+}}
+=== A ===
+; [[arity]]
+: The number of distinct step sizes occurring in a given scale. Arity ''disregards other properties'', such as [[rank]] or [[maximum variety]]. For example, 12edo melodic minor is a binary scale which is not rank-2 or MV2 (MOS). An ''n''-ary scale has ''n'' distinct step sizes (e.g. [[#B|binary]], [[#T|ternary]], [[#Q|quaternary]]).
-; Generic interval (ordinal category)
+=== B ===
-: A class of intervals which fall on the same scale degrees. In the diatonic scale, these classes are the set of seconds, the set of thirds, the set of fifths, etc. Generic intervals can also be likened to distances between note-heads on a traditional staff. A generic interval composed of ''k'' scale steps in any scale, diatonic or not, can be called a "''k''-step" (terminology taken from [[TAMNAMS]]).
+; [[binary]] scale
+: A scale of [[#A|arity]] 2, i.e. with two distinct step sizes.
-; Scale degree
+=== C ===
-: The number of steps [[subtend]]ed by a generic interval from the tonic. In conventional music theory, degrees of the diatonic scale are usually given 1-indexed ordinal names corresponding to the diatonic intervals themselves, for example the ''fifth degree'' for the degree occupied by the fifth. However, in xenharmonic music, 0-indexed names are preferred by some people for degrees of non-diatonic scales: the ''k''-degree is the degree represented by the ''k''-step generic interval from the tonic.
+; [[chirality]]
+: A scale is chiral if reversing the order of the steps results in a different scale up to rotation.
-== Properties ==
+; [[constant structure]] (CS)
-; Alternating generator (AG) property
+: A scale is a constant structure if all intervals of the same size are also within the same generic interval class. A single interval cannot be a part of two classes. The 12-tone diatonic scale does not have this property, since tritones can either be augmented fourths or diminished fifths. This is referred to as the ''partitioning property'' in most academic literature.
-: A scale satisfies the [[alternating generator property]] if it satisfies the following equivalent properties:
+; [[convex scale|convexity]]
+: A scale in a [[regular temperament]] is convex if its representation on a [[harmonic lattice diagram]] forms a convex polygon.
+=== D ===
+; [[distributional evenness]] (DE)
+: A scale with two step sizes is ''distributionally even'' if it has its two step sizes distributed as evenly as possible.
+=== E ===
+; [[epimorphism]]
+* '''Epimorphism''': A JI scale is ''epimorphic'' if, under some val, all scale degrees are "filled," no matter which note you choose as the tonic, and successive degrees are always increasing. Without the second condition, the scale is only ''weakly epimorphic''.
+* '''Epimorph val/temperament''': A val that witnesses that a JI scale is epimorphic is called the ''epimorph val'' of the scale, and a temperament supported by an epimorph val is an ''epimorph temperament''. Many low-accuracy edos and temperaments are useful as epimorph vals and temperaments, and these temperaments imply structure rather than tuning; a CS scale may be constructed as a detempering of the low-accuracy tuning implied by such a temperament.
+* Example: 5-limit [[Zarlino]] is a 2.3.5 JI scale that is epimorphic under the val {{val|7 11 16}}, and the 2.3.5 temperaments [[dicot]] and [[meantone]] are both epimorph temperaments for Zarlino.
+=== F ===
+=== G ===
+; [[generator-offset property]] (GO)
+: A scale satisfies the ''generator-offset property'' if it satisfies the following equivalent properties:
 * the scale can be built by stacking alternating generators, for example 7/6 and 8/7.
 * the scale is generated by two chains of generators separated by a fixed interval; either both chains are of size m, or one chain has size m and the second has size m-1.
-; [[Constant structure]]
+=== H ===
-: A scale is a constant structure (CS) if all intervals of the same size are also within the same generic interval class. A single interval cannot be a part of two classes. The 12-tone diatonic scale does not have this property, since tritones can either be augmented fourths or diminished fifths. This is referred to as the ''partitioning property'' in most academic literature.
+=== I ===
+=== J ===
+=== K ===
+=== L ===
+=== M ===
+; [[maximal evenness]]
+: A [[#P|periodic]] [[#B|binary]] scale is maximally even with respect to an [[equal-step tuning]] if it is the result of rounding a smaller equal tuning to the nearest notes of the parent equal tuning with the same equave.
-; [[Rothenberg propriety|Propriety]]
+; [[maximum variety]] (MV)
+: The maximum [[interval variety]] from all interval classes of a [[#P|periodic scale]].
+; MOS
+* A scale is a [[mos scale]] if there are ''no more than'' two interval sizes for each generic interval class not including the equave. A.k.a. maximum variety 2.
+; Myhill's property
+* A scale has ''Myhill's property'' if there are ''exactly'' two interval sizes for each interval class not including the equave. A.k.a. strict variety 2. A scale with Myhill's property is called a ''strict mos''.
+=== N ===
+=== O ===
+=== P ===
+; [[periodic scale|periodicity]]
+: A scale is periodic if its [[step pattern]] repeats after a certain [[#I|interval]].
+; [[pepper ambiguity]]
+: The Pepper ambiguity of an [[interval]] in an [[equal-step tuning]] is the ratio of the best approximation to the second best approximation.
+=== Q ===
+; [[quaternary]] scale
+: A scale of [[#A|arity]] 4, i.e. with four distinct step sizes.
+=== R ===
+; [[rank]]
+: The rank of a scale is the minimum number of intervals needed to generate the entire scale. For example, the diatonic scale is a rank-2 scale because it is entirely generated by stacking fifths and octaves.
+; [[Rothenberg propriety]]
 * '''Propriety''': A scale is proper if there is no overlapping of generic interval classes. This means that no third is larger than a fourth, no fourth is larger than a fifth, etc.
 * '''Strict propriety''': A scale is strictly proper if the generic interval classes are disjoint. Replace the word "larger" with "larger-than-or-equal-to" in the definition above. The 12-tone diatonic scale is proper, but not strictly proper.
-; Epimorphism
+=== S ===
-* '''Epimorphism''': A JI scale is ''epimorphic'' if, under some val, all scale degrees are "filled," no matter which note you choose as the tonic, and successive degrees are always increasing. Without the second condition, the scale is only ''weakly epimorphic''.
+; [[strict variety]] (SV)
-* '''Epimorph val/temperament''': A val that witnesses that a JI scale is epimorphic is called the ''epimorph val'' of the scale, and a temperament supported by an epimorph val is an ''epimorph temperament''. Many low-accuracy edos and temperaments are useful as epimorph vals and temperaments, and these temperaments imply structure rather than tuning; a CS scale may be constructed as a detempering of the low-accuracy tuning implied by such a temperament.
+: The [[interval variety]] of all interval classes of a [[#P|periodic scale]], when all interval classes have the same interval variety.
-* Example: 5-limit [[Zarlino]] is a 2.3.5 JI scale that is epimorphic under the val {{val|7 11 16}}, and the 2.3.5 temperaments [[dicot]] and [[meantone]] are both epimorph temperaments for Zarlino.
-; Symmetry
+; symmetry
 : A scale is symmetrical if at least one mode of the scale is symmetrical. Therefore, every interval of that mode must have an inverse. These scales will always have an odd number of notes ''per period''. They may not always have an odd number of notes ''per octave'', however. The diatonic scale is symmetrical, but so is 12edo.
-; Myhill's property and MOS
+=== T ===
-* '''Myhill's/MOS property''': A scale has Myhill's property if there are ''exactly'' two interval sizes for each (reduced) interval class not including the equave. A scale is a MOS scale if there are ''no more than'' two interval sizes for each generic interval class not including the equave. This is equivalent to a scale being Myhill with a smaller equave. Myhill's property is sometimes called "strict MOS".
+; [[ternary]] scale
+: A scale of [[#A|arity]] 3, i.e. with three distinct step sizes.
-; Trivalence property
-: Same as Myhill's property, but replace "two interval sizes" with "three interval sizes." The scale formed from the notes of a dominant 7th chord (e.g. C-E-G-Bb-C) is an example of a trivalent scale.
-The 12-tone diatonic scale has Myhill's property, and is also distributionally even.
+; trivalence property
+: Same as [[#M|Myhill's property]], but replace "two interval sizes" with "three interval sizes". A.k.a. strict variety 3. The scale formed from the notes of a dominant 7th chord (e.g. C-E-G-Bb-C) is an example of a trivalent scale.
-The diminished scale is an MOS with a 1/4-octave period. Because there is only one interval size at the period, it does not have exactly two interval sizes per interval class. Therefore, it is MOS, but doesn't have Myhill's property.
+=== U ===
+=== V ===
+=== W–Z ===
-An EDO is a kind of degenerate MOS, in that it is distributionally even. It does not have Myhill's property. In other words, it has no more than two interval sizes for each generic interval class, but does not have exactly two interval sizes.
+== Examples ==
+The [[5L 2s|diatonic scale]] in 12edo has Myhill's property, and is also distributionally even.
-; Arity, binary, ternary, ''n''-ary
+The [[diminished scale]] is a mos with a 1/4-octave period. Because there is only one interval size at the period, it does not have exactly two interval sizes per interval class. Therefore, it is a mos, but does not have Myhill's property.
-: The number of distinct step sizes occurring in a given scale. Arity ''disregards other properties'', such as [[rank]] or [[maximum variety]]. For example, 12edo melodic minor is a binary scale which is not rank-2 or MV2 (MOS).
-; Convexity
+An edo is a kind of degenerate mos, in that it is distributionally even. It does not have Myhill's property. In other words, it has no more than two interval sizes for each generic interval class, but does not have exactly two interval sizes.
-; [[Maximal evenness]]
-; [[Pepper ambiguity]]
 == See also ==
+* [[Glossary]]
+* [[Periodic scale]] – contains mathematical definitions of several scale properties
-* [[Periodic_scale|Periodic scale]]
+[[Category:Terms| ]]
 [[Category:Scale]]
-[[Category:Todo:add examples]]
+{{Todo| add illustration }}
-[[Category:Todo:add illustration]]