Just intonation - Revision history

Inthar: /* Approaches */

2026-04-02T17:25:32Z

Approaches

← Older revision		Revision as of 17:25, 2 April 2026
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	== Approaches ==		== Approaches ==
	There are many approaches to JI. The following approaches are not necessarily mutually exclusive:		There are many approaches to JI. The following approaches are not necessarily mutually exclusive:
	* Stacking-based/"geometric" JI: conceived as taking place on a JI subgroup or lattice. This approach focuses on a fixed set of intervals and chords and progressions built by stacking said intervals.		* Stacking-based/"geometric" JI: conceived as taking place on a JI subgroup or lattice. This approach focuses on a fixed set of intervals and chords that are used locally at any given time and progressions built by stacking said local intervals.
	* Free JI: not fixed to a particular set of notes. The dominant approach in Ben Johnston's JI string quartets.		* Free JI: not fixed to a particular set of notes. The dominant approach in Ben Johnston's JI string quartets.
	* Linear JI: harmonic series focused, including nejis ("near equal/equivalent JI"), which are subsets of harmonic series modes approximating a given scale. Includes primodality, Zhea Erose's JI approach.		* Linear JI: harmonic series focused, including nejis ("near equal/equivalent JI"), which are subsets of harmonic series modes approximating a given scale. Includes primodality, Zhea Erose's JI approach.

Overthink: Use {{Cat}}

2026-03-21T03:44:36Z

Use {{Cat}}

← Older revision		Revision as of 03:44, 21 March 2026
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	* [[Detempering]] an [[equal temperament]]		* [[Detempering]] an [[equal temperament]]
	* [[Constant structure]]: An often desired property for JI scales		* [[Constant structure]]: An often desired property for JI scales
	~~[[Category:~~Core knowledge]]
			{{Cat\|Core knowledge}}

Inthar: /* Approaches */

2026-02-28T14:11:52Z

Approaches

← Older revision		Revision as of 14:11, 28 February 2026
Line 9:		Line 9:
	== Approaches ==		== Approaches ==
	There are many approaches to JI. The following approaches are not necessarily mutually exclusive:		There are many approaches to JI. The following approaches are not necessarily mutually exclusive:
	* Stacking-based JI: conceived as taking place on a JI subgroup or lattice. This approach focuses on a fixed set of intervals and chords and progressions built by stacking said intervals.		* Stacking-based/"geometric" JI: conceived as taking place on a JI subgroup or lattice. This approach focuses on a fixed set of intervals and chords and progressions built by stacking said intervals.
	* Free JI: not fixed to a particular set of notes. The dominant approach in Ben Johnston's JI string quartets.		* Free JI: not fixed to a particular set of notes. The dominant approach in Ben Johnston's JI string quartets.
	* Linear JI: harmonic series focused, including nejis ("near equal/equivalent JI"), which are subsets of harmonic series modes approximating a given scale. Includes primodality, Zhea Erose's JI approach.		* Linear JI: harmonic series focused, including nejis ("near equal/equivalent JI"), which are subsets of harmonic series modes approximating a given scale. Includes primodality, Zhea Erose's JI approach.

Vector: accidentally tapped rollback | Undo revision 4364 by Vector (talk)

2026-02-28T02:59:37Z

accidentally tapped rollback | Undo revision 4364 by Vector (talk)

← Older revision		Revision as of 02:59, 28 February 2026
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	'''Just intonation''' is the set of intervals corresponding to frequency ratios between whole numbers, and the approach to musical tuning which utilizes exclusively such intervals. Just intonation can be described in terms of the harmonic series (which is the set of tones at integer multiples of a fundamental frequency), where all just intervals can be found between notes in the harmonic series. Particularly low-complexity just intervals tend to be perceived as consonant, serving as significant tonal anchors for the building of scales. These intervals include [[2/1]], [[3/1]], [[3/2]], [[4/3]], and [[5/3]]. Additionally, much of xenharmonic tuning theory tends to be built around just intonation intervals as tuning targets, as they are often easy to recognize by ear and tune to.		'''Just intonation''' is the set of intervals corresponding to frequency ratios between whole numbers, and the approach to musical tuning which utilizes exclusively such intervals. Just intonation can be described in terms of the harmonic series (which is the set of tones at integer multiples of a fundamental frequency), where all just intervals can be found between notes in the harmonic series. Particularly low-complexity just intervals tend to be perceived as consonant, serving as significant tonal anchors for the building of scales. These intervals include [[2/1]], [[3/1]], [[3/2]], [[4/3]], and [[5/3]]. An additional, more detailed list of these intervals can be found at [[List of locking intervals]]. Additionally, much of xenharmonic tuning theory tends to be built around just intonation intervals as tuning targets, as they are often easy to recognize by ear and tune to.

	In older materials, "just intonation" referred in particular to 5-[[limit]] just intonation - that is, just intonation where no frequency ratio could have a prime factor exceeding 5. (Thus, this included 3/2 and 25/24, but not, for instance, 9/7. High-complexity [[Pythagorean tuning\|Pythagorean]] intervals were generally considered separately.)		In older materials, "just intonation" referred in particular to 5-[[limit]] just intonation - that is, just intonation where no frequency ratio could have a prime factor exceeding 5. (Thus, this included 3/2 and 25/24, but not, for instance, 9/7. High-complexity [[Pythagorean tuning\|Pythagorean]] intervals were generally considered separately.)

Vector: Reverted edit by Vector (talk) to last revision by Inthar

2026-02-28T02:58:42Z

Reverted edit by Vector (talk) to last revision by Inthar

← Older revision		Revision as of 02:58, 28 February 2026
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	'''Just intonation''' is the set of intervals corresponding to frequency ratios between whole numbers, and the approach to musical tuning which utilizes exclusively such intervals. Just intonation can be described in terms of the harmonic series (which is the set of tones at integer multiples of a fundamental frequency), where all just intervals can be found between notes in the harmonic series. Particularly low-complexity just intervals tend to be perceived as consonant, serving as significant tonal anchors for the building of scales. These intervals include [[2/1]], [[3/1]], [[3/2]], [[4/3]], and [[5/3~~]]. An additional, more detailed list of these intervals can be found at [[List of locking intervals~~]]. Additionally, much of xenharmonic tuning theory tends to be built around just intonation intervals as tuning targets, as they are often easy to recognize by ear and tune to.		'''Just intonation''' is the set of intervals corresponding to frequency ratios between whole numbers, and the approach to musical tuning which utilizes exclusively such intervals. Just intonation can be described in terms of the harmonic series (which is the set of tones at integer multiples of a fundamental frequency), where all just intervals can be found between notes in the harmonic series. Particularly low-complexity just intervals tend to be perceived as consonant, serving as significant tonal anchors for the building of scales. These intervals include [[2/1]], [[3/1]], [[3/2]], [[4/3]], and [[5/3]]. Additionally, much of xenharmonic tuning theory tends to be built around just intonation intervals as tuning targets, as they are often easy to recognize by ear and tune to.

	In older materials, "just intonation" referred in particular to 5-[[limit]] just intonation - that is, just intonation where no frequency ratio could have a prime factor exceeding 5. (Thus, this included 3/2 and 25/24, but not, for instance, 9/7. High-complexity [[Pythagorean tuning\|Pythagorean]] intervals were generally considered separately.)		In older materials, "just intonation" referred in particular to 5-[[limit]] just intonation - that is, just intonation where no frequency ratio could have a prime factor exceeding 5. (Thus, this included 3/2 and 25/24, but not, for instance, 9/7. High-complexity [[Pythagorean tuning\|Pythagorean]] intervals were generally considered separately.)

Vector at 00:46, 28 February 2026

2026-02-28T00:46:25Z

← Older revision		Revision as of 00:46, 28 February 2026
Line 1:		Line 1:
	'''Just intonation''' is the set of intervals corresponding to frequency ratios between whole numbers, and the approach to musical tuning which utilizes exclusively such intervals. Just intonation can be described in terms of the harmonic series (which is the set of tones at integer multiples of a fundamental frequency), where all just intervals can be found between notes in the harmonic series. Particularly low-complexity just intervals tend to be perceived as consonant, serving as significant tonal anchors for the building of scales. These intervals include [[2/1]], [[3/1]], [[3/2]], [[4/3]], and [[5/3]]. Additionally, much of xenharmonic tuning theory tends to be built around just intonation intervals as tuning targets, as they are often easy to recognize by ear and tune to.		'''Just intonation''' is the set of intervals corresponding to frequency ratios between whole numbers, and the approach to musical tuning which utilizes exclusively such intervals. Just intonation can be described in terms of the harmonic series (which is the set of tones at integer multiples of a fundamental frequency), where all just intervals can be found between notes in the harmonic series. Particularly low-complexity just intervals tend to be perceived as consonant, serving as significant tonal anchors for the building of scales. These intervals include [[2/1]], [[3/1]], [[3/2]], [[4/3]], and [[5/3]]. An additional, more detailed list of these intervals can be found at [[List of locking intervals]]. Additionally, much of xenharmonic tuning theory tends to be built around just intonation intervals as tuning targets, as they are often easy to recognize by ear and tune to.

	In older materials, "just intonation" referred in particular to 5-[[limit]] just intonation - that is, just intonation where no frequency ratio could have a prime factor exceeding 5. (Thus, this included 3/2 and 25/24, but not, for instance, 9/7. High-complexity [[Pythagorean tuning\|Pythagorean]] intervals were generally considered separately.)		In older materials, "just intonation" referred in particular to 5-[[limit]] just intonation - that is, just intonation where no frequency ratio could have a prime factor exceeding 5. (Thus, this included 3/2 and 25/24, but not, for instance, 9/7. High-complexity [[Pythagorean tuning\|Pythagorean]] intervals were generally considered separately.)

Inthar: /* JI scale constructions and properties */

2026-02-24T03:58:34Z

JI scale constructions and properties

← Older revision		Revision as of 03:58, 24 February 2026
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	== JI scale constructions and properties ==		== JI scale constructions and properties ==
	* [[Tonality diamond]]s		* [[Tonality diamond]]s and [[cross-set]]s
	* [[Generator sequence]]s: stacking a sequence of JI generators cyclically		* [[Generator sequence]]s: stacking a sequence of JI generators cyclically
	* [[Combination product set]]s		* [[Combination product set]]s

Inthar: /* Approaches */

2026-02-24T03:52:31Z

Approaches

← Older revision		Revision as of 03:52, 24 February 2026
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	* Linear JI: harmonic series focused, including nejis ("near equal/equivalent JI"), which are subsets of harmonic series modes approximating a given scale. Includes primodality, Zhea Erose's JI approach.		* Linear JI: harmonic series focused, including nejis ("near equal/equivalent JI"), which are subsets of harmonic series modes approximating a given scale. Includes primodality, Zhea Erose's JI approach.
	** The basic scales in (octave-equivalent) linear JI are ''harmonic series modes''; for example, mode 5 or 5:6:7:8:9:10 (in shorthand, 5::10), and mode 7 is 7:8:9:10:11:12:13:14.		** The basic scales in (octave-equivalent) linear JI are ''harmonic series modes''; for example, mode 5 or 5:6:7:8:9:10 (in shorthand, 5::10), and mode 7 is 7:8:9:10:11:12:13:14.
	** ''Nejis'' are subsets of harmonic series modes that approximate a given non-JI scale; for example, 26:29:32:34:38:42:44:50:52 is a neji for the Dylathian mode of the [[oneirotonic]] scale. On this wiki we call ~~nejis~~ a "~~Dylathian~~ neji", or "13edo neji", etc. based on what scale is being approximated.		** ''Nejis'' are subsets of harmonic series modes that approximate a given non-JI scale; for example, 26:29:32:34:38:42:44:50:52 is a neji for the Dylathian mode of the [[oneirotonic]] scale. On this wiki we call a neji a "Lydian neji", or "13edo neji", etc. based on what scale is being approximated.

	== JI scale constructions and properties ==		== JI scale constructions and properties ==

Inthar: /* Approaches */

2026-02-24T03:51:38Z

Approaches

← Older revision		Revision as of 03:51, 24 February 2026
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	* Free JI: not fixed to a particular set of notes. The dominant approach in Ben Johnston's JI string quartets.		* Free JI: not fixed to a particular set of notes. The dominant approach in Ben Johnston's JI string quartets.
	* Linear JI: harmonic series focused, including nejis ("near equal/equivalent JI"), which are subsets of harmonic series modes approximating a given scale. Includes primodality, Zhea Erose's JI approach.		* Linear JI: harmonic series focused, including nejis ("near equal/equivalent JI"), which are subsets of harmonic series modes approximating a given scale. Includes primodality, Zhea Erose's JI approach.
	** The basic linear JI ~~scales~~ are ''harmonic series modes''; for example, mode 5 or 5:6:7:8:9:10 (in shorthand, 5::10), and mode 7 is 7:8:9:10:11:12:13:14.		** The basic scales in (octave-equivalent) linear JI are ''harmonic series modes''; for example, mode 5 or 5:6:7:8:9:10 (in shorthand, 5::10), and mode 7 is 7:8:9:10:11:12:13:14.
	** ''Nejis'' are subsets of harmonic series modes that approximate a given non-JI scale; for example, 26:29:32:34:38:42:44:50:52 is a neji for the Dylathian mode of the [[oneirotonic]] scale. On this wiki we call nejis a "Dylathian neji", or "13edo neji", etc. based on what scale is being approximated.		** ''Nejis'' are subsets of harmonic series modes that approximate a given non-JI scale; for example, 26:29:32:34:38:42:44:50:52 is a neji for the Dylathian mode of the [[oneirotonic]] scale. On this wiki we call nejis a "Dylathian neji", or "13edo neji", etc. based on what scale is being approximated.

Inthar: /* Approaches */

2026-02-24T03:49:37Z

Approaches

← Older revision		Revision as of 03:49, 24 February 2026
Line 9:		Line 9:
	== Approaches ==		== Approaches ==
	There are many approaches to JI. The following approaches are not necessarily mutually exclusive:		There are many approaches to JI. The following approaches are not necessarily mutually exclusive:
	* Stacking-based JI: conceived as taking place on a JI subgroup or lattice		* Stacking-based JI: conceived as taking place on a JI subgroup or lattice. This approach focuses on a fixed set of intervals and chords and progressions built by stacking said intervals.
	* Free JI: not fixed to a particular set of notes. The dominant approach in Ben Johnston's JI string quartets.		* Free JI: not fixed to a particular set of notes. The dominant approach in Ben Johnston's JI string quartets.
	* Linear JI: harmonic series focused, including nejis ("near equal/equivalent JI"), which are subsets of harmonic series modes approximating a given scale. Includes primodality, Zhea Erose's JI approach.		* Linear JI: harmonic series focused, including nejis ("near equal/equivalent JI"), which are subsets of harmonic series modes approximating a given scale. Includes primodality, Zhea Erose's JI approach.