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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 diatonic, 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 contemporary xenharmonic scale theory, 0-indexed names are preferred 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 (alt-gen, 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 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.
* 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 has 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.
* '''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.
* '''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 ===
* '''Weak epimorphism''': A scale is weakly epimorphic if, under some val, all scale degrees are "filled," no matter which note you choose as the tonic.
; [[strict variety]] (SV)
* '''Epimorphism''': A weakly epimorphic scale is epimorphic if it keeps rising in pitch as you go to higher scale degrees – the (n + 1)st degree is higher than the nth degree.
: The [[interval variety]] of all interval classes of a [[#P|periodic scale]], when all interval classes have the same interval variety.


; 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.
: 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.


; MOS/DE/Myhill's
=== T ===
* '''[[Distributional evenness]]''': A scale is distributionally even (DE) if there are no more than two interval sizes for each generic interval class (e.g. major/minor thirds, perfect/augmented fourths, etc).
; [[ternary]] scale
* '''Myhill's property''': A scale has Myhill's property if there are exactly two interval sizes for each generic interval class (except octaves or other equivalence intervals such as tritaves).
: 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.
* '''[[Moment of symmetry]]''': Both DE and Myhill's are essentially synonymous with MOS; Myhill's property is sometimes called "strict MOS".


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/DE, 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.  


; Convexity
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.
; [[Maximal evenness]]
 
; [[Pepper ambiguity]]
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.


== See also ==
== See also ==
* [[Glossary]]
* [[Periodic scale]] – contains mathematical definitions of several scale properties


* [[Periodic_scale|Periodic scale]]
[[Category:Terms| ]]
 
[[Category:Scale]]
[[Category:Scale]]
[[Category:Todo:add examples]]
{{Todo| add illustration }}
[[Category:Todo:add illustration]]s

Latest revision as of 01:54, 20 July 2025

A simplified explanation of the various properties of periodic scales. Also check the main Glossary.

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. binary, ternary, quaternary).

B

binary scale
A scale of arity 2, i.e. with two distinct step sizes.

C

chirality
A scale is chiral if reversing the order of the steps results in a different scale up to rotation.
constant structure (CS)
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.
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 ⟨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.

H

I

J

K

L

M

maximal evenness
A periodic 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.
maximum variety (MV)
The maximum interval variety from all interval classes of a 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

periodicity
A scale is periodic if its step pattern repeats after a certain 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 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.

S

strict variety (SV)
The interval variety of all interval classes of a periodic scale, when all interval classes have the same interval variety.
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.

T

ternary scale
A scale of 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". 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.

U

V

W–Z

Examples

The diatonic scale in 12edo has Myhill's property, and is also distributionally even.

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.

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.

See also

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