Glossary of scale properties: Difference between revisions

<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!)

==Definitions==  

**Scale degree:** The amount of steps subtended in an interval. (A perfect *fifth* falls on the *5th* scale degree; so does a diminished *fifth*).

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

**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.

==Properties==  

**[[Constant Structure|Constant structure]]:** 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.

* **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.

* **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.

* **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.

**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).

**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 distributionally even.

**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.

**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.</pre></div>

<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;!-- 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;

&lt;br /&gt;

&lt;strong&gt;Scale degree:&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;

&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;!-- 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;

&lt;strong&gt;&lt;a class="wiki_link" href="/Constant%20Structure"&gt;Constant structure&lt;/a&gt;:&lt;/strong&gt; 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 &lt;em&gt;partitioning property&lt;/em&gt; in most academic literature.&lt;br /&gt;

&lt;strong&gt;Propriety&lt;/strong&gt;&lt;br /&gt;

&lt;ul&gt;&lt;li&gt;&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;/li&gt;&lt;li&gt;&lt;strong&gt;Strict Propriety:&lt;/strong&gt; A scale is strictly proper if the generic interval classes are disjoint. 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;Epimorphism&lt;/strong&gt;&lt;br /&gt;

&lt;ul&gt;&lt;li&gt;&lt;strong&gt;Weak Epimorphism:&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 tonic.&lt;/li&gt;&lt;li&gt;&lt;strong&gt;Epimorphism:&lt;/strong&gt; 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.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;

&lt;strong&gt;Distributional Evenness:&lt;/strong&gt; 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).&lt;br /&gt;

&lt;br /&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 distributionally even.&lt;br /&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; 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.&lt;br /&gt;

&lt;strong&gt;Symmetry:&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 diatonic scale is symmetrical, but so is 12edo.&lt;/body&gt;&lt;/html&gt;</pre></div>