Carlos harmonic scale: Difference between revisions
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{{Wikipedia|Harmonic scale}} | {{Wikipedia|Harmonic scale}} | ||
The '''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 [[ | [[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]]. | |||
== Interval table == | == Interval table == | ||
{| class="wikitable" | {| class="wikitable center-all right-4" | ||
|+ Intervals of the harmonic scale | |+ Intervals of the harmonic scale | ||
|- | |- | ||
! Harmonic !! Ratio !! Decimal !! Cents !! Deviation from [[12edo | ! Harmonic !! Ratio !! Decimal !! Cents !! Deviation from [[12edo]] | ||
|- | |- | ||
| 16 || [[1/1]] || 1.0000 || 0.000 || 0\12 ± 0.000 | | 16 || [[1/1]] || 1.0000 || 0.000 || 0\12 ± 0.000 | ||
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== Perspectives == | == Perspectives == | ||
=== As a NEJI === | === As a NEJI === | ||
The harmonic scale can be viewed as an intentionally inaccurate [[NEJI| | 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. | ||
=== As a | === As a ringer scale === | ||
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 < | 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 < | 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: | 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. | ||
Thus, we can we can remove the 23, 25, 29, and 31 from the | 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 | == Usage and history == | ||
The harmonic scale | 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 == | == Scala file == | ||
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! carlos_harm.scl | ! carlos_harm.scl | ||
! | ! | ||
Carlos Harmonic & | Carlos Harmonic & Ben Johnston's scale of 'Blues' from Suite f.micr.piano (1977) & David Beardsley's scale of 'Science Friction' | ||
12 | 12 | ||
! | ! | ||
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* [[Carlos super|Carlos Super Just]] | * [[Carlos super|Carlos Super Just]] | ||
==References== | == References == | ||
<references /> | <references /> | ||