Skip-fifteen scale: Difference between revisions

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There are actually two '''skip-fifteen scales''', a utonal and an otonal version. Both are heptatonic scales (seven tones per octave).

I (Mason Green) am not sure if anyone else has written about this type of scale before. It might be a re-discovery, but I find it interesting. [In fact, the otonal skip15 is the [[Dwarves|dwarf scale]] for the 13-limit 7edo patent val. It is epimorphic and [[~~constant_structure|~~constant structure]]. It is also the 13-limit otonal heptad, and has been listed here as [[~~oheptad|~~oheptad]], and the utonal version as [[~~uheptad|~~uheptad]].]

I (Mason Green) am not sure if anyone else has written about this type of scale before. It might be a re-discovery, but I find it interesting. [In fact, the otonal skip15 is the [[Dwarves|dwarf scale]] for the 13-limit 7edo patent val. It is epimorphic and [[constant structure]]. It is also the 13-limit otonal heptad, and has been listed here as [[oheptad]], and the utonal version as [[uheptad]].]

A skip-fifteen scale is the octave-repeating scale in just intonation consisting of all the harmonics between 8 and 16, but skipping the 15th (hence the name). The utonal version instead uses the subharmonics between 1/16 and 1/8, excluding 1/15.

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~~== ==~~

==Utonal skip15 scale==

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Although it's also possible to include the fifteenth (sub)harmonic, thus generating an octatonic scale, I believe it's better to omit it. Specifically, this helps ensure that:

~~<ul><li>~~Every interval one step wide is perceived as a (major or neutral) tone, and~~</li><li>~~Every interval two steps wide is perceived as a third, and~~</li><li>~~The 15:13 interval (which is tonally ambiguous in that it is almost halfway between a third and a whole tone) is omitted. Omitting this interval ensures a clear separation between tones and thirds.~~</li><li>~~It is possible to cycle through all the pitch classes by taking the steps two at a time. In combination with the last property, this makes a circle of (unequal) thirds. This circle of thirds is 4:5:6:7:9:11:13:16 (or its utonal counterpart), a [https://en.wikipedia.org/wiki/Fifteenth double octave]-repeating scale in which every interval is a third. There are seven different types of thirds used in the circle, each type occurring exactly once in the circle: supermajor (9:7), wide major (13:11), major (5:4), wide neutral (16:13), narrow neutral (11:9), minor (6:5), and subminor (7:6).~~</li></ul>~~

* Every interval one step wide is perceived as a (major or neutral) tone, and

* Every interval two steps wide is perceived as a third, and

* The 15:13 interval (which is tonally ambiguous in that it is almost halfway between a third and a whole tone) is omitted. Omitting this interval ensures a clear separation between tones and thirds.

* It is possible to cycle through all the pitch classes by taking the steps two at a time. In combination with the last property, this makes a circle of (unequal) thirds. This circle of thirds is 4:5:6:7:9:11:13:16 (or its utonal counterpart), a [https://en.wikipedia.org/wiki/Fifteenth double octave]-repeating scale in which every interval is a third. There are seven different types of thirds used in the circle, each type occurring exactly once in the circle: supermajor (9:7), wide major (13:11), major (5:4), wide neutral (16:13), narrow neutral (11:9), minor (6:5), and subminor (7:6).

There is also a "phantom third" (14:11) which is not part of the circle of thirds, but appears in the skip15 scales as an interval three (rather than two) steps wide. This phantom third means that neither skip15 scale is [[Rothenberg_propriety|proper]] (since it's narrower than the supermajor third, which occurs as a two-step interval in the scales).

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The skip15 scales (and the circle of thirds) can be approximated in many high edos, which are capable of distinguishing between all eight different thirds (the seven that are part of the circle, plus the "phantom" 11:14) third. Note that in order to ensure that every third has a unique approximation, it may be necessary to use a non-patent val in some cases.

[[Category:heptatonic]]

[[Category:just_intonation]]

[[Category:Dwarves]]

[[Category:7-tone scales]]

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 There are actually two '''skip-fifteen scales''', a utonal and an otonal version. Both are heptatonic scales (seven tones per octave).
-I (Mason Green) am not sure if anyone else has written about this type of scale before. It might be a re-discovery, but I find it interesting. [In fact, the otonal skip15 is the [[Dwarves|dwarf scale]] for the 13-limit 7edo patent val. It is epimorphic and [[constant_structure|constant structure]]. It is also the 13-limit otonal heptad, and has been listed here as [[oheptad|oheptad]], and the utonal version as [[uheptad|uheptad]].]
+I (Mason Green) am not sure if anyone else has written about this type of scale before. It might be a re-discovery, but I find it interesting. [In fact, the otonal skip15 is the [[Dwarves|dwarf scale]] for the 13-limit 7edo patent val. It is epimorphic and [[constant structure]]. It is also the 13-limit otonal heptad, and has been listed here as [[oheptad]], and the utonal version as [[uheptad]].]
 A skip-fifteen scale is the octave-repeating scale in just intonation consisting of all the harmonics between 8 and 16, but skipping the 15th (hence the name). The utonal version instead uses the subharmonics between 1/16 and 1/8, excluding 1/15.
@@ Line 41: / Line 41: @@
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 |}
-== ==
 ==Utonal skip15 scale==
@@ Line 85: / Line 83: @@
 Although it's also possible to include the fifteenth (sub)harmonic, thus generating an octatonic scale, I believe it's better to omit it. Specifically, this helps ensure that:
-<ul><li>Every interval one step wide is perceived as a (major or neutral) tone, and</li><li>Every interval two steps wide is perceived as a third, and</li><li>The 15:13 interval (which is tonally ambiguous in that it is almost halfway between a third and a whole tone) is omitted. Omitting this interval ensures a clear separation between tones and thirds.</li><li>It is possible to cycle through all the pitch classes by taking the steps two at a time. In combination with the last property, this makes a circle of (unequal) thirds. This circle of thirds is 4:5:6:7:9:11:13:16 (or its utonal counterpart), a [https://en.wikipedia.org/wiki/Fifteenth double octave]-repeating scale in which every interval is a third. There are seven different types of thirds used in the circle, each type occurring exactly once in the circle: supermajor (9:7), wide major (13:11), major (5:4), wide neutral (16:13), narrow neutral (11:9), minor (6:5), and subminor (7:6).</li></ul>
+* Every interval one step wide is perceived as a (major or neutral) tone, and
+* Every interval two steps wide is perceived as a third, and
+* The 15:13 interval (which is tonally ambiguous in that it is almost halfway between a third and a whole tone) is omitted. Omitting this interval ensures a clear separation between tones and thirds.
+* It is possible to cycle through all the pitch classes by taking the steps two at a time. In combination with the last property, this makes a circle of (unequal) thirds. This circle of thirds is 4:5:6:7:9:11:13:16 (or its utonal counterpart), a [https://en.wikipedia.org/wiki/Fifteenth double octave]-repeating scale in which every interval is a third. There are seven different types of thirds used in the circle, each type occurring exactly once in the circle: supermajor (9:7), wide major (13:11), major (5:4), wide neutral (16:13), narrow neutral (11:9), minor (6:5), and subminor (7:6).
 There is also a "phantom third" (14:11) which is not part of the circle of thirds, but appears in the skip15 scales as an interval three (rather than two) steps wide. This phantom third means that neither skip15 scale is [[Rothenberg_propriety|proper]] (since it's narrower than the supermajor third, which occurs as a two-step interval in the scales).
@@ Line 94: / Line 95: @@
 The skip15 scales (and the circle of thirds) can be approximated in many high edos, which are capable of distinguishing between all eight different thirds (the seven that are part of the circle, plus the "phantom" 11:14) third. Note that in order to ensure that every third has a unique approximation, it may be necessary to use a non-patent val in some cases.
 [[Category:heptatonic]]
 [[Category:just_intonation]]
+[[Category:Dwarves]]
+[[Category:7-tone scales]]