13edo - Revision history

Kili: fix words redirect

2026-05-12T18:34:38Z

fix words redirect

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	'''13edo''', or 13 equal divisions of the octave, is the equal tuning featuring steps of (1200/13) ~= 92.308 cents, 13 of which stack to the octave 2/1. It does not approximate many small prime harmonics well at all, so [[Delta-rational chord\|DR]]-based interpretations may be preferred among 13edo users.		'''13edo''', or 13 equal divisions of the octave, is the equal tuning featuring steps of (1200/13) ~= 92.308 cents, 13 of which stack to the octave 2/1. It does not approximate many small prime harmonics well at all, so [[Delta-rational chord\|DR]]-based interpretations may be preferred among 13edo users.

	13edo's greatest melodic strength is its proximity to 12edo, whose most important effect is providing an [[oneirotonic]] (5L3s, LLsLLsLs) MOS which is a compressed diatonic. A functional ~~system for 13edo~~ oneirotonic ~~is provided below~~.		13edo's greatest melodic strength is its proximity to 12edo, whose most important effect is providing an [[oneirotonic]] (5L3s, LLsLLsLs) MOS which is a compressed diatonic. For Jaimbee and Inthar's functional harmony and method, see the [[oneirotonic]] page.

	== Tuning theory ==		== Tuning theory ==

Inthar: /* Jaimbee and Inthar's functional system for 13edo */ Move to "Oneirotonic" article

2026-05-09T18:08:48Z

Jaimbee and Inthar's functional system for 13edo: Move to "Oneirotonic" article

Show changes

Inthar: /* 130edo */

2026-05-05T03:32:44Z

130edo

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	''See [[26edo#130edo]].''		''See [[26edo#130edo]].''

			== See also ==
			* The [[Well-Tempered Triskaidecatonic Clavier]] (WT13C) project
	{{Navbox EDO}}		{{Navbox EDO}}
	{{Cat\|Edos}}		{{Cat\|Edos}}

Inthar: /* Degree 5 */

2026-05-05T03:09:43Z

Degree 5

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	===== Degree 5 =====		===== Degree 5 =====
	~~Dgree~~ 5, which has a major third on it, has the strongest tendency to go to degree 1 in oneirominor. Alternatively serves as degree 4 of the relative oneiromajor on degree 3, serving to tonicize the relative oneiromajor.		Degree 5, which has a major third on it, has the strongest tendency to go to degree 1 in oneirominor. Alternatively serves as degree 4 of the relative oneiromajor on degree 3, serving to tonicize the relative oneiromajor.

	The main cadence in oneirominor is <math>\hat{5}</math>>maj< <math>\hat{1}</math>sim or <math>\hat{5}</math>>dom7< <math>\hat{1}</math>sim because of the leading tone.		The main cadence in oneirominor is <math>\hat{5}</math>>maj< <math>\hat{1}</math>sim or <math>\hat{5}</math>>dom7< <math>\hat{1}</math>sim because of the leading tone.

Inthar: /* Degree t */

2026-05-05T03:06:21Z

Degree t

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	===== Degree t =====		===== Degree t =====
	Degree t is ~~a uniquely~~ oneirotonic function. It serves several secondary roles, being related to more central functions in oneirominor, <math>\hat 6,</math> <math>\hat 4</math> and <math>\hat 2</math>:		Degree t is an oneirotonic function with no analogue in diatonic. It serves several secondary roles, being related to more central functions in oneirominor, <math>\hat 6,</math> <math>\hat 4</math> and <math>\hat 2</math>:
	* <math>\hat 6</math>sim <math>\hat \mathrm{t}</math>sim or <math>\hat \mathrm{t}</math>sim <math>\hat 6</math>sim		* <math>\hat 6</math>sim <math>\hat \mathrm{t}</math>sim or <math>\hat \mathrm{t}</math>sim <math>\hat 6</math>sim
	* <math>\hat 4</math>sim <math>\hat \mathrm{t}</math>sim or <math>\hat \mathrm{t}</math>sim <math>\hat 4</math>sim		* <math>\hat 4</math>sim <math>\hat \mathrm{t}</math>sim or <math>\hat \mathrm{t}</math>sim <math>\hat 4</math>sim

Inthar: /* Progressions */

2026-05-05T03:04:55Z

Progressions

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	Common motions:		Common motions:
	* <math>\hat{1}</math> (>maj< or sim6) → <math>\mathrm{M}\hat{2}</math> sim6		* <math>\hat{1}</math> (>maj< or sim6) → <math>\mathrm{M}\hat{2}</math> sim6
	* <math>\hat{1}</math> (>maj< or sim6) → <math>\mathrm{P}\hat{4}</math> sim6 (when ending on 0d this sounds like diatonic V to I)		* <math>\hat{1}</math> (>maj< or sim6) → <math>\mathrm{P}\hat{4}</math> sim6 (when ending on <math>\hat{1}</math> this sounds like diatonic V to I)
	* <math>\hat{1}</math> (>maj< or sim6) → <math>\mathrm{m}\hat{6}</math> sim6		* <math>\hat{1}</math> (>maj< or sim6) → <math>\mathrm{m}\hat{6}</math> sim6
	* <math>\hat{1}</math> >maj< → <math>\mathrm{P}\hat{5}</math> >maj<, <math>\hat{1}</math> sim6 → <math>\mathrm{P}\hat{5}</math> (>maj< or sim6) (when ending on <math>\hat{1}</math> this is a "dominant to tonic" motion)		* <math>\hat{1}</math> >maj< → <math>\mathrm{P}\hat{5}</math> >maj<, <math>\hat{1}</math> sim6 → <math>\mathrm{P}\hat{5}</math> (>maj< or sim6) (when ending on <math>\hat{1}</math> this is a "dominant to tonic" motion)

Inthar: /* Functional harmony */

2026-05-05T03:03:30Z

Functional harmony

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	# oneiromajor, with tonic chord tract-major and the pitches of Dylathian and Ilarnekian modes		# oneiromajor, with tonic chord tract-major and the pitches of Dylathian and Ilarnekian modes
	# oneirominor, with tonic chord simic 6th and the pitches of Celephaisian and Ultharian modes		# oneirominor, with tonic chord simic 6th and the pitches of Celephaisian and Ultharian modes

	We'll call degrees that don't have a >maj< or sim6 chord ''dissonant degrees'' (keeping in mind that dissonance is a feature a chord has in a musical language rather than a purely psychoacoustic property).
	==== Oneiromajor (Dylathian) ====		==== Oneiromajor (Dylathian) ====
	In the below, interval names are in ADIN.		In the below, interval names are in ADIN.

Inthar: /* Degree 7 */

2026-05-05T03:02:15Z

Degree 7

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	===== Degree 7 =====		===== Degree 7 =====
	~~When not flattened, the leading~~ tone is the main function of ~~degree~~ 7. It can also be related to degrees <math>\hat 2,</math> <math>\hat \mathrm{t}</math> and <math>\hat 6</math>:		Leading tone is the main function of <math>\mathrm{M}\hat 7</math>. It can also be related to degrees <math>\hat 2,</math> <math>\hat \mathrm{t}</math> and <math>\hat 6</math>:
	* <math>\hat 2</math>sim <math>\mathrm{M}\hat 7</math>sim or <math>\mathrm{M}\hat 7</math>sim <math>\hat 2</math>sim		* <math>\hat 2</math>sim <math>\mathrm{M}\hat 7</math>sim or <math>\mathrm{M}\hat 7</math>sim <math>\hat 2</math>sim
	* <math>\hat 6</math>sim <math>\mathrm{M}\hat 7</math>sim or <math>\mathrm{M}\hat 7</math>sim <math>\hat 6</math>sim		* <math>\hat 6</math>sim <math>\mathrm{M}\hat 7</math>sim or <math>\mathrm{M}\hat 7</math>sim <math>\hat 6</math>sim

Inthar: /* Degree 7 */

2026-05-05T03:00:21Z

Degree 7

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	* <math>\hat 6</math>sim <math>\mathrm{M}\hat 7</math>sim or <math>\mathrm{M}\hat 7</math>sim <math>\hat 6</math>sim		* <math>\hat 6</math>sim <math>\mathrm{M}\hat 7</math>sim or <math>\mathrm{M}\hat 7</math>sim <math>\hat 6</math>sim
	* <math>\mathrm{M}\hat 7</math>sim <math>\hat \mathrm{t}</math>sim or <math>\hat \mathrm{t}</math>sim <math>\mathrm{M}\hat 7</math>sim		* <math>\mathrm{M}\hat 7</math>sim <math>\hat \mathrm{t}</math>sim or <math>\hat \mathrm{t}</math>sim <math>\mathrm{M}\hat 7</math>sim
	However, these are fairly weak because <math>\mathrm{M}\hat 7</math> is ~~far~~ from the tonic in generator space.		However, these are fairly weak because <math>\mathrm{M}\hat 7</math> is fairly distant from the tonic in generator space.

	The 7th degree may be flattened to give it a stronger functional relationship to the tonic or other degrees; the flattened 7th degree (relative to the Celephaisian mode) provides a <math>\mathrm{P}\hat{5}</math> "dominant" for the degree <math>\hat{3}</math>.		The 7th degree may be flattened to give it a stronger functional relationship to the tonic or other degrees; the flattened 7th degree (relative to the Celephaisian mode) provides a <math>\mathrm{P}\hat{5}</math> "dominant" for the degree <math>\hat{3}</math>.

Inthar: /* Degree 7 */

2026-05-05T03:00:06Z

Degree 7

← Older revision		Revision as of 03:00, 5 May 2026
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	* <math>\hat 6</math>sim <math>\mathrm{M}\hat 7</math>sim or <math>\mathrm{M}\hat 7</math>sim <math>\hat 6</math>sim		* <math>\hat 6</math>sim <math>\mathrm{M}\hat 7</math>sim or <math>\mathrm{M}\hat 7</math>sim <math>\hat 6</math>sim
	* <math>\mathrm{M}\hat 7</math>sim <math>\hat \mathrm{t}</math>sim or <math>\hat \mathrm{t}</math>sim <math>\mathrm{M}\hat 7</math>sim		* <math>\mathrm{M}\hat 7</math>sim <math>\hat \mathrm{t}</math>sim or <math>\hat \mathrm{t}</math>sim <math>\mathrm{M}\hat 7</math>sim
	However, these are fairly weak because ~~the 7th degree~~ is far from the tonic in generator space.		However, these are fairly weak because <math>\mathrm{M}\hat 7</math> is far from the tonic in generator space.

	The 7th degree may be flattened to give it a stronger functional relationship to the tonic or other degrees; the flattened 7th degree (relative to the Celephaisian mode) provides a <math>\mathrm{P}\hat{5}</math> "dominant" for the degree <math>\hat{3}</math>.		The 7th degree may be flattened to give it a stronger functional relationship to the tonic or other degrees; the flattened 7th degree (relative to the Celephaisian mode) provides a <math>\mathrm{P}\hat{5}</math> "dominant" for the degree <math>\hat{3}</math>.