Carlos harmonic scale: Difference between revisions

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The harmonic scale can be derived as such: a ringer scale that specifically starts on a <math>2^n</math>th harmonic. To fit twelve pitches while fitting this requirement, the scale must therefore start on the 16th harmonic and end on the 32nd; in other words, it must be a subset of [[16afdo|16::32]].

Consider the 12edo [[patent val]] up to the 31-limit: '''⟨12 19 28 34 42 44 49 51 54 58 59]''' Based on this patent val, we can deduce that 12edo tempers out the [[superparticular]] ratios [[23/22]], [[26/25]], [[29/28]], and [[31/30]]. This means that we can only use one of the harmonics listed in each ratio in the scale; otherwise, "retempering" the scale will lead to two notes with the same pitch~~. Even numbers can be prioritized, since they reduce to simpler ratios when put over 16~~.

Consider the 12edo [[patent val]] up to the 31-limit: '''⟨12 19 28 34 42 44 49 51 54 58 59]''' Based on this patent val, we can deduce that 12edo tempers out the [[superparticular]] ratios [[23/22]], [[26/25]], [[29/28]], and [[31/30]]. This means that we can only use one of the harmonics listed in each ratio in the scale; otherwise, "retempering" the scale will lead to two notes with the same pitch.

Thus, we can we can remove the 23, 25, 29, and 31 from the [[16afdo|16::32]] scale to arrive at the '''16:17:18:19:20:21:22:24:26:27:28:30:32''' scale—the harmonic scale.

Even numbers can be prioritized, since they reduce to simpler ratios when put over 16. Thus, we can we can remove the 23, 25, 29, and 31 from the [[16afdo|16::32]] scale to arrive at the '''16:17:18:19:20:21:22:24:26:27:28:30:32''' scale—the harmonic scale.

== Usage and History ==

The harmonic scale ~~is typically~~ used as an alternative tuning for regular twelve-tone pianos to play spectral or [[otonal]] music. Versions of the scale are known to have been used by composers [[Ezra Sims]], [http://www.ekmelic-music.org/en/person.php?n=RH Franz Richter Herf], [[Wendy Carlos]] in her ''Beauty and the Beast'' (1986)<ref>Milano, Dominic (November 1986). [http://www.wendycarlos.com/other/PDF-Files/Kbd86Tunings*.pdf "A Many-Colored Jungle of Exotic Tunings"], ''Keyboard''.</ref> and [[Ben Johnston]] in ''Suite for Microtonal Piano'' (1978).

The harmonic scale can be used as an alternative tuning for regular twelve-tone pianos to play spectral or [[otonal]] music, although it finds more regular use merely as a unique microtonal retuning of 12edo. Versions of the scale are known to have been used by composers [[Ezra Sims]], [http://www.ekmelic-music.org/en/person.php?n=RH Franz Richter Herf], [[Wendy Carlos]] in her ''Beauty and the Beast'' (1986)<ref>Milano, Dominic (November 1986). [http://www.wendycarlos.com/other/PDF-Files/Kbd86Tunings*.pdf "A Many-Colored Jungle of Exotic Tunings"], ''Keyboard''.</ref> and [[Ben Johnston]] in ''Suite for Microtonal Piano'' (1978).

== Scala file ==

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 The harmonic scale can be derived as such: a ringer scale that specifically starts on a <math>2^n</math><sup>th</sup> harmonic. To fit twelve pitches while fitting this requirement, the scale must therefore start on the 16<sup>th</sup> harmonic and end on the 32<sup>nd</sup>; in other words, it must be a subset of [[16afdo|16::32]].
-Consider the 12edo [[patent val]] up to the 31-limit: '''⟨12 19 28 34 42 44 49 51 54 58 59]''' Based on this patent val, we can deduce that 12edo tempers out the [[superparticular]] ratios [[23/22]], [[26/25]], [[29/28]], and [[31/30]]. This means that we can only use one of the harmonics listed in each ratio in the scale; otherwise, "retempering" the scale will lead to two notes with the same pitch. Even numbers can be prioritized, since they reduce to simpler ratios when put over 16.
+Consider the 12edo [[patent val]] up to the 31-limit: '''⟨12 19 28 34 42 44 49 51 54 58 59]''' Based on this patent val, we can deduce that 12edo tempers out the [[superparticular]] ratios [[23/22]], [[26/25]], [[29/28]], and [[31/30]]. This means that we can only use one of the harmonics listed in each ratio in the scale; otherwise, "retempering" the scale will lead to two notes with the same pitch.
-Thus, we can we can remove the 23, 25, 29, and 31 from the [[16afdo|16::32]] scale to arrive at the '''16:17:18:19:20:21:22:24:26:27:28:30:32''' scale&mdash;the harmonic scale.
+Even numbers can be prioritized, since they reduce to simpler ratios when put over 16. Thus, we can we can remove the 23, 25, 29, and 31 from the [[16afdo|16::32]] scale to arrive at the '''16:17:18:19:20:21:22:24:26:27:28:30:32''' scale&mdash;the harmonic scale.
 == Usage and History ==
-The harmonic scale is typically used as an alternative tuning for regular twelve-tone pianos to play spectral or [[otonal]] music. Versions of the scale are known to have been used by composers [[Ezra Sims]], [http://www.ekmelic-music.org/en/person.php?n=RH Franz Richter Herf], [[Wendy Carlos]] in her ''Beauty and the Beast'' (1986)<ref>Milano, Dominic (November 1986). [http://www.wendycarlos.com/other/PDF-Files/Kbd86Tunings*.pdf "A Many-Colored Jungle of Exotic Tunings"], ''Keyboard''.</ref> and [[Ben Johnston]] in ''Suite for Microtonal Piano'' (1978).
+The harmonic scale can be used as an alternative tuning for regular twelve-tone pianos to play spectral or [[otonal]] music, although it finds more regular use merely as a unique microtonal retuning of 12edo. Versions of the scale are known to have been used by composers [[Ezra Sims]], [http://www.ekmelic-music.org/en/person.php?n=RH Franz Richter Herf], [[Wendy Carlos]] in her ''Beauty and the Beast'' (1986)<ref>Milano, Dominic (November 1986). [http://www.wendycarlos.com/other/PDF-Files/Kbd86Tunings*.pdf "A Many-Colored Jungle of Exotic Tunings"], ''Keyboard''.</ref> and [[Ben Johnston]] in ''Suite for Microtonal Piano'' (1978).
 == Scala file ==