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Glossary of scale properties: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
A simplified explanation of the various properties of [[periodic scale]]s. Also check the main [[Glossary]].
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:Sarzadoce|Sarzadoce]] and made on <tt>2012-03-23 15:32:21 UTC</tt>.<br>
: The original revision id was <tt>314084878</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">A simplified explanation of the various properties of periodic scales.


(provided to you by Ryan!)
{{TOC Horizontal
| a1=[[#A|A]]
| 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]]).


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


==Definitions:==  
=== C ===
; [[chirality]]
: A scale is chiral if reversing the order of the steps results in a different scale up to rotation.


**Scale Degrees:** The amount of steps subtended in an interval, (A perfect *fifth* falls on the *5th* scale degree; so does a diminished *fifth*).
; [[constant structure]] (CS)
**Interval:** A specific musical interval (e.g. a major third or minor seventh).
: 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.
**Generic Interval:** A class of intervals which fall on the same scale degrees (e.g. thirds, fifths, sixths, etc). Generic intervals can also be likened to distances between note-heads on a traditional staff.


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


==Properties:==  
=== 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.


**Constant Structure:** A scale has 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 however. The 12-tone diatonic scale does not have this property, since tritones can either be augmented fourths or diminished fifths.
=== 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.


**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...
=== F ===
* **Strict Propriety:** A scale is strictly proper if the Generic Interval classes are discrete. 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.
=== 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.


**Weakly Epimorphic:** A scale is weakly epimorphic if, under some val, all scale degrees are "filled," no matter which note you choose as the fundamental.
=== H ===
* **Epimorphic:** Something silly that basically means your scale is non-negative, or something like that. The 12-tone diatonic scale is epimorphic.
=== 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.


**Maximal Evenness:** A scale is Maximally Even if there are no more than two Interval sizes for each Generic Interval class (e.g. Major/Minor thirds, Perfect/Augmented fourths, etc). Usually when someone is talking about Maximal Evenness, they are talking about Equal Divisions of the Octave. The definition can be extended to other scales, however.
; [[maximum variety]] (MV)  
* **Myhill's Property:** A scale has Myhill's property if every Generic Interval class contains exactly two Interval sizes (bar periods/octaves). The 12-tone diatonic scale has Myhill's property, and is also Maximally Even.
: The maximum [[interval variety]] from all interval classes of a [[#P|periodic scale]].


**Trivalence Property:** Same as Myhill's property, but replace "two Interval sizes" with "three Interval sizes." If we form a scale from the notes of a Dominant 7th chord, (e.g. C-E-G-Bb-C) this scale has the Trivalence Property.
; 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.  


**Symmetrical:** 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 12-tone diatonic scale is symmetrical.</pre></div>
; Myhill's property
<h4>Original HTML content:</h4>
* 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''.
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Scale properties simplified&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A simplified explanation of the various properties of periodic scales.&lt;br /&gt;
 
&lt;br /&gt;
=== N ===
(provided to you by Ryan!)&lt;br /&gt;
=== O ===
&lt;br /&gt;
=== P ===
&lt;br /&gt;
; [[periodic scale|periodicity]]
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Definitions:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Definitions:&lt;/h2&gt;
: A scale is periodic if its [[step pattern]] repeats after a certain [[#I|interval]].
&lt;br /&gt;
 
&lt;strong&gt;Scale Degrees:&lt;/strong&gt; The amount of steps subtended in an interval, (A perfect *fifth* falls on the *5th* scale degree; so does a diminished *fifth*).&lt;br /&gt;
; [[pepper ambiguity]]
&lt;strong&gt;Interval:&lt;/strong&gt; A specific musical interval (e.g. a major third or minor seventh).&lt;br /&gt;
: The Pepper ambiguity of an [[interval]] in an [[equal-step tuning]] is the ratio of the best approximation to the second best approximation.
&lt;strong&gt;Generic Interval:&lt;/strong&gt; A class of intervals which fall on the same scale degrees (e.g. thirds, fifths, sixths, etc). Generic intervals can also be likened to distances between note-heads on a traditional staff.&lt;br /&gt;
 
&lt;br /&gt;
=== Q ===
&lt;br /&gt;
; [[quaternary]] scale
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Properties:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Properties:&lt;/h2&gt;
: A scale of [[#A|arity]] 4, i.e. with four distinct step sizes.
&lt;br /&gt;
 
&lt;strong&gt;Constant Structure:&lt;/strong&gt; A scale has 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 however. The 12-tone diatonic scale does not have this property, since tritones can either be augmented fourths or diminished fifths.&lt;br /&gt;
=== R ===
&lt;br /&gt;
; [[rank]]
&lt;strong&gt;Propriety:&lt;/strong&gt; 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...&lt;br /&gt;
: 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.
&lt;ul&gt;&lt;li&gt;&lt;strong&gt;Strict Propriety:&lt;/strong&gt; A scale is strictly proper if the Generic Interval classes are discrete. Replace the word &amp;quot;larger&amp;quot; with &amp;quot;larger-than-or-equal-to&amp;quot; in the definition above. The 12-tone diatonic scale is proper, but not strictly proper.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
 
&lt;strong&gt;Weakly Epimorphic:&lt;/strong&gt; A scale is weakly epimorphic if, under some val, all scale degrees are &amp;quot;filled,&amp;quot; no matter which note you choose as the fundamental.&lt;br /&gt;
; [[Rothenberg propriety]]
&lt;ul&gt;&lt;li&gt;&lt;strong&gt;Epimorphic:&lt;/strong&gt; Something silly that basically means your scale is non-negative, or something like that. The 12-tone diatonic scale is epimorphic.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
* '''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.
&lt;strong&gt;Maximal Evenness:&lt;/strong&gt; A scale is Maximally Even if there are no more than two Interval sizes for each Generic Interval class (e.g. Major/Minor thirds, Perfect/Augmented fourths, etc). Usually when someone is talking about Maximal Evenness, they are talking about Equal Divisions of the Octave. The definition can be extended to other scales, however.&lt;br /&gt;
* '''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.
&lt;ul&gt;&lt;li&gt;&lt;strong&gt;Myhill's Property:&lt;/strong&gt; A scale has Myhill's property if every Generic Interval class contains exactly two Interval sizes (bar periods/octaves). The 12-tone diatonic scale has Myhill's property, and is also Maximally Even.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
 
&lt;strong&gt;Trivalence Property:&lt;/strong&gt; Same as Myhill's property, but replace &amp;quot;two Interval sizes&amp;quot; with &amp;quot;three Interval sizes.&amp;quot; If we form a scale from the notes of a Dominant 7th chord, (e.g. C-E-G-Bb-C) this scale has the Trivalence Property.&lt;br /&gt;
=== S ===
&lt;br /&gt;
; [[strict variety]] (SV)
&lt;strong&gt;Symmetrical:&lt;/strong&gt; 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 &lt;u&gt;period&lt;/u&gt;. They may not always have an odd number of notes per &lt;u&gt;octave&lt;/u&gt;, however. The 12-tone diatonic scale is symmetrical.&lt;/body&gt;&lt;/html&gt;</pre></div>
: The [[interval variety]] of all interval classes of a [[#P|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 [[#A|arity]] 3, i.e. with three distinct step sizes.
 
; 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.
 
=== U ===
=== V ===
=== W–Z ===
 
== Examples ==
The [[5L 2s|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 ==
* [[Glossary]]
* [[Periodic scale]] – contains mathematical definitions of several scale properties
 
[[Category:Terms| ]]
[[Category:Scale]]
{{Todo| add illustration }}
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