Carlos harmonic scale: Difference between revisions

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~~{{todo|inline=1|merge articles|comment=Merge with [[Carlos harm]]?}}~~

The '''harmonic scale''' is a scale ~~of twelve~~ [[just intonation]] ~~pitches~~ that repeats at the [[octave]]. Its pitches are derived from the [[harmonic series]] of a single frequency up to the [[21/1|~~21st~~ harmonic]], meaning they go up to the [[19-limit]]. The harmonic scale can ~~also~~ be described as ~~an arbitrary~~ subset of [[16afdo|16~~-ODO~~]] where ~~the~~ [[23/1|23~~rd~~]], [[25/1|25~~th~~]], [[29/1|29~~th~~]], and [[31/1|31~~st~~]] ~~harmonics~~ are removed.

[[File:Carlos_Harmonic.png|thumb|Circle diagram.]]

The '''Carlos harmonic scale''' or simply '''harmonic scale''' is a twelve note scale in [[just intonation]] that repeats at the [[octave]]. Its pitches are derived from the [[harmonic series]] of a single frequency up to the [[27/1|27th harmonic]], meaning they go up to the [[27-odd-limit]], or the [[19-limit]]. The harmonic scale can be described as a subset of [[16afdo|mode 16 of the harmonic series]] where harmonics [[23/1|23]], [[25/1|25]], [[29/1|29]], and [[31/1|31]] are removed, producing a [[constant structure]]. It is also an example of including every harmonic as far out as possible while maintaining constant structure, which means it is an example of a [[ringer scale]].

{| class="wikitable"

== Interval table ==

{| class="wikitable center-all right-4"

|+ Intervals of the harmonic scale

|-

! Harmonic !! Ratio !! Decimal !! Cents !! Deviation from [[12edo~~|12-TET~~]]

! Harmonic !! Ratio !! Decimal !! Cents !! Deviation from [[12edo]]

|-

| 16 || [[1/1]] || 1.0000 || 0.000 || 0\12 ± 0.000

Line 35:

Line 36:

|}

== As a NEJI ==

== Perspectives ==

=== As a NEJI ===

The harmonic scale can be viewed as an intentionally inaccurate [[NEJI|~~12-NEJI~~]]. From [[12edo~~|12-TET~~]], the harmonic scale has a ~~total~~ error of 194.193 cents and an average error of 16.183 cents.

The harmonic scale can be viewed as an intentionally inaccurate [[NEJI|12neji]]. From [[12edo]], the harmonic scale has a sum of error of 194.193 cents and an average error of 16.183 cents.

== ~~Usage and History~~ ==

=== As a ringer scale ===

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], ~~and especially~~ [[Wendy Carlos]] in her ''Beauty ~~and~~ the Beast'' (1986) and [[Ben Johnston]] in ~~his works~~ for ~~retuned~~ piano.

The harmonic scale can be interpreted as a [[ringer scale]] detempered from [[12edo]] devised for spectralist purposes; specifically, it is devised such that the root is a 2''n''-th harmonic, allowing it to act as the "fundamental frequency" pitch class (in an octave-repeating scale). Typical [[Ringer scale #List of ringer scales|Ringer 12]] scales, however, do not have this particular focus.

The harmonic scale can be derived as such: a ringer scale that specifically starts on a 2''n''-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]]: {{val| 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 ratio]]s [[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. Thus, we can we can remove the 23, 25, 29, and 31 from the 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 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 in 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 ==

<pre>

! carlos_harm.scl

!

Carlos Harmonic & Ben Johnston's scale of 'Blues' from Suite f.micr.piano (1977) & David Beardsley's scale of 'Science Friction'

12

!

17/16

9/8

19/16

5/4

21/16

11/8

3/2

13/8

27/16

7/4

15/8

2/1

</pre>

== Music ==

; [[Xotla]]

* from ''Microtones & Garden Gnomes'' (2017)

** "Flying Dutchman's Folly" [https://xotla.bandcamp.com/track/flying-dutchmans-folly-harmonic-just-intonation Bandcamp] | [https://youtu.be/XF4cVQTHYSk?si=HHzDU0tVLYm52Kyc YouTube]

** "Bonus - Flying Dutchman's Folly Pixelized" [https://xotla.bandcamp.com/track/bonus-flying-dutchmans-folly-pixelized-harmonic-just-intonation Bandcamp]

== See also ==

* [[Carlos Alpha]]

* [[Carlos Beta]]

* [[Carlos Gamma]]

* [[Carlos super|Carlos Super Just]]

== References ==

[[Category:12-tone scales]]

[[Category:19-limit]]

[[Category:Wendy Carlos]]

[[Category:Just intonation scales]]

[[Category:Pages with Scala files]]

@@ Line 1: / Line 1: @@
-{{todo|inline=1|merge articles|comment=Merge with [[Carlos harm]]?}}
 {{Wikipedia|Harmonic scale}}
-The '''harmonic scale''' is a scale of twelve [[just intonation]] pitches that repeats at the [[octave]]. Its pitches are derived from the [[harmonic series]] of a single frequency up to the [[21/1|21<sup>st</sup> harmonic]], meaning they go up to the [[19-limit]]. The harmonic scale can also be described as an arbitrary subset of [[16afdo|16-ODO]] where the [[23/1|23<sup>rd</sup>]], [[25/1|25<sup>th</sup>]], [[29/1|29<sup>th</sup>]], and [[31/1|31<sup>st</sup>]] harmonics are removed.
+[[File:Carlos_Harmonic.png|thumb|Circle diagram.]]
+The '''Carlos harmonic scale''' or simply '''harmonic scale''' is a twelve note scale in [[just intonation]] that repeats at the [[octave]]. Its pitches are derived from the [[harmonic series]] of a single frequency up to the [[27/1|27th harmonic]], meaning they go up to the [[27-odd-limit]], or the [[19-limit]]. The harmonic scale can be described as a subset of [[16afdo|mode 16 of the harmonic series]] where harmonics [[23/1|23]], [[25/1|25]], [[29/1|29]], and [[31/1|31]] are removed, producing a [[constant structure]]. It is also an example of including every harmonic as far out as possible while maintaining constant structure, which means it is an example of a [[ringer scale]].
-{| class="wikitable"
+== Interval table ==
+{| class="wikitable center-all right-4"
 |+ Intervals of the harmonic scale
 |-
-! Harmonic !! Ratio !! Decimal !! Cents !! Deviation from [[12edo|12-TET]]
+! Harmonic !! Ratio !! Decimal !! Cents !! Deviation from [[12edo]]
 |-
 | 16 || [[1/1]] || 1.0000 || 0.000 || 0\12 ± 0.000
@@ Line 35: / Line 36: @@
 |}
-== As a NEJI ==
+== Perspectives ==
+=== As a NEJI ===
-The harmonic scale can be viewed as an intentionally inaccurate [[NEJI|12-NEJI]]. From [[12edo|12-TET]], the harmonic scale has a total error of 194.193 cents and an average error of 16.183 cents.
+The harmonic scale can be viewed as an intentionally inaccurate [[NEJI|12neji]]. From [[12edo]], the harmonic scale has a sum of error of 194.193 cents and an average error of 16.183 cents.
-== Usage and History ==
+=== As a ringer scale ===
-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], and especially [[Wendy Carlos]] in her ''Beauty and the Beast'' (1986) and [[Ben Johnston]] in his works for retuned piano.
+The harmonic scale can be interpreted as a [[ringer scale]] detempered from [[12edo]] devised for spectralist purposes; specifically, it is devised such that the root is a 2<sup>''n''</sup>-th harmonic, allowing it to act as the "fundamental frequency" pitch class (in an octave-repeating scale). Typical [[Ringer scale #List of ringer scales|Ringer 12]] scales, however, do not have this particular focus.
+The harmonic scale can be derived as such: a ringer scale that specifically starts on a 2<sup>''n''</sup>-th 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]]: {{val| 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 ratio]]s [[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. Thus, we can we can remove the 23, 25, 29, and 31 from the 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 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 in 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 ==
+<pre>
+! carlos_harm.scl
+!
+Carlos Harmonic & Ben Johnston's scale of 'Blues' from Suite f.micr.piano (1977) & David Beardsley's scale of 'Science Friction'
+!
+/16
+/8
+/16
+/4
+/16
+/8
+/2
+/8
+/16
+/4
+/8
+/1
+</pre>
+== Music ==
+; [[Xotla]]
+* from ''Microtones & Garden Gnomes'' (2017)
+** "Flying Dutchman's Folly" [https://xotla.bandcamp.com/track/flying-dutchmans-folly-harmonic-just-intonation Bandcamp] | [https://youtu.be/XF4cVQTHYSk?si=HHzDU0tVLYm52Kyc YouTube]
+** "Bonus - Flying Dutchman's Folly Pixelized" [https://xotla.bandcamp.com/track/bonus-flying-dutchmans-folly-pixelized-harmonic-just-intonation Bandcamp]
+== See also ==
+* [[Carlos Alpha]]
+* [[Carlos Beta]]
+* [[Carlos Gamma]]
+* [[Carlos super|Carlos Super Just]]
+== References ==
+<references />
+[[Category:12-tone scales]]
+[[Category:19-limit]]
+[[Category:Wendy Carlos]]
+[[Category:Just intonation scales]]
+[[Category:Pages with Scala files]]