Carlos harmonic scale - Revision history

Tonwood: /* Usage and history */

2026-05-06T04:13:36Z

Usage and history

← Older revision		Revision as of 04:13, 6 May 2026
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	== Usage and history ==		== 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 ~~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 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 ==

Yourmusic Productions: Fix math error.

2026-02-13T08:38:52Z

Fix math error.

← Older revision		Revision as of 08:38, 13 February 2026
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	{{Wikipedia\|Harmonic scale}}		{{Wikipedia\|Harmonic scale}}
	[[File:Carlos_Harmonic.png\|thumb\|Circle diagram.]]		[[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 [[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]].		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 ==

FloraC: Cleanup & style

2026-02-06T08:10:01Z

Cleanup & style

← Older revision		Revision as of 08:10, 6 February 2026
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	{{Wikipedia\|Harmonic scale}}		{{Wikipedia\|Harmonic scale}}
	[[File:Carlos_Harmonic.png\|thumb\|Circle diagram.]]		[[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 [[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]].		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 [[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 ==		== 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~~\|12-TET~~]]		! 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\|~~12-NEJI~~]]. From [[12edo]], 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.

	=== As a ~~Ringer Scale~~ ===		=== 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 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 <~~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]].		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: ~~'''⟨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.		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 ~~[[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 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~~ ==		== 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 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).
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	! carlos_harm.scl		! carlos_harm.scl
	!		!
	Carlos Harmonic &~~amp;~~ Ben Johnston's scale of 'Blues' from Suite f.micr.piano (1977) &~~amp;~~ David Beardsley's scale of 'Science Friction'		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 />

Yourmusic Productions: Add circle diagram.

2026-02-03T03:00:09Z

Add circle diagram.

← Older revision		Revision as of 03:00, 3 February 2026
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	{{Wikipedia\|Harmonic scale}}		{{Wikipedia\|Harmonic scale}}
			[[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 [[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]].		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 [[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]].

TromboneBoi9: altered Usage and History section

2025-07-02T17:18:42Z

altered Usage and History section

← Older revision		Revision as of 17:18, 2 July 2025
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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]].		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—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 ==		== 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 ==		== Scala file ==

VectorGraphics at 22:43, 24 June 2025

2025-06-24T22:43:50Z

← Older revision		Revision as of 22:43, 24 June 2025
Line 1:		Line 1:
	{{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 [[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]].		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 [[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 ==		== Interval table ==

VectorGraphics: VectorGraphics moved page Harmonic scale to Carlos harmonic scale: "harmonic scale" is extremely vague

2025-06-24T22:43:30Z

VectorGraphics moved page Harmonic scale to Carlos harmonic scale: "harmonic scale" is extremely vague

← Older revision	Revision as of 22:43, 24 June 2025
(No difference)

Hkm at 18:13, 24 June 2025

2025-06-24T18:13:13Z

← Older revision		Revision as of 18:13, 24 June 2025
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	[[Category:Wendy Carlos]]		[[Category:Wendy Carlos]]
	[[Category:Just intonation scales]]		[[Category:Just intonation scales]]
	~~[[Category:Pages with mostly numerical content]]~~
	[[Category:Pages with Scala files]]		[[Category:Pages with Scala files]]

Lériendil: Text replacement - "Category:Just intonation scales" to "Category:Just intonation scales Category:Pages with mostly numerical content"

2025-06-24T00:27:39Z

Text replacement - "Category:Just intonation scales" to "Category:Just intonation scales Category:Pages with mostly numerical content"

← Older revision		Revision as of 00:27, 24 June 2025
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	[[Category:Wendy Carlos]]		[[Category:Wendy Carlos]]
	[[Category:Just intonation scales]]		[[Category:Just intonation scales]]
			[[Category:Pages with mostly numerical content]]
	[[Category:Pages with Scala files]]		[[Category:Pages with Scala files]]

TromboneBoi9: rephrasing, added "as a ringer scale" section

2025-06-07T04:31:23Z

rephrasing, added "as a ringer scale" section

← Older revision		Revision as of 04:31, 7 June 2025
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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 [[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 ==		== 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—the harmonic scale.

	== Usage and History ==		== Usage and History ==

← Older revision		Revision as of 03:00, 3 February 2026
Line 1:		Line 1:
	{{Wikipedia\|Harmonic scale}}		{{Wikipedia\|Harmonic scale}}
			[[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 [[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]].		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 [[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]].