Carlos harmonic scale: Difference between revisions

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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 [[21/1|21st harmonic]], meaning they go up to 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, ~~so that it's~~ [[CS]]. It's also an example of including every harmonic as far out as possible while maintaining ~~this property~~, ~~so that~~ it's an example of a [[~~Ringer~~ 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 [[21/1|21st harmonic]], meaning they go up to the [[21-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 ==

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== ~~As a NEJI~~ ==

== Perspectives ==

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.

=== As a NEJI ===

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

=== 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 <math>2^n</math>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 <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.

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 ==

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 {{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 [[21/1|21<sup>st</sup> harmonic]], meaning they go up to 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, so that it's [[CS]]. It's also an example of including every harmonic as far out as possible while maintaining this property, so that it's an example of a [[Ringer 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 [[21/1|21<sup>st</sup> harmonic]], meaning they go up to the [[21-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 ==
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 |}
-== As a NEJI ==
+== Perspectives ==
-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.
+=== As a NEJI ===
+The harmonic scale can be viewed as an intentionally inaccurate [[NEJI|12-NEJI]]. From [[12edo]], the harmonic scale has a total error of 194.193 cents and an average error of 16.183 cents.
+=== 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 <math>2^n</math><sup>th</sup> 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 <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.
+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 ==