Glossary of scale properties
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A simplified explanation of the various properties of periodic scales. (provided to you by Ryan!) ==Definitions:== **Scale Degrees:** 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:** 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. **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 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. **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. * **Epimorphic:** Something silly that basically means your scale is non-negative, or something like that. The 12-tone diatonic scale is epimorphic. **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. * **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. **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. **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.
Original HTML content:
<html><head><title>Scale properties simplified</title></head><body>A simplified explanation of the various properties of periodic scales.<br /> <br /> (provided to you by Ryan!)<br /> <br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Definitions:"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definitions:</h2> <br /> <strong>Scale Degrees:</strong> The amount of steps subtended in an interval, (A perfect *fifth* falls on the *5th* scale degree; so does a diminished *fifth*).<br /> <strong>Interval:</strong> A specific musical interval (e.g. a major third or minor seventh).<br /> <strong>Generic Interval:</strong> 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.<br /> <br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x-Properties:"></a><!-- ws:end:WikiTextHeadingRule:2 -->Properties:</h2> <br /> <strong>Constant Structure:</strong> 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.<br /> <br /> <strong>Propriety:</strong> 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...<br /> <ul><li><strong>Strict Propriety:</strong> 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.</li></ul><br /> <strong>Weakly Epimorphic:</strong> A scale is weakly epimorphic if, under some val, all scale degrees are "filled," no matter which note you choose as the fundamental.<br /> <ul><li><strong>Epimorphic:</strong> Something silly that basically means your scale is non-negative, or something like that. The 12-tone diatonic scale is epimorphic.</li></ul><br /> <strong>Maximal Evenness:</strong> 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.<br /> <ul><li><strong>Myhill's Property:</strong> 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.</li></ul><br /> <strong>Trivalence Property:</strong> 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.<br /> <br /> <strong>Symmetrical:</strong> 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 <u>period</u>. They may not always have an odd number of notes per <u>octave</u>, however. The 12-tone diatonic scale is symmetrical.</body></html>