Latitude - Revision history

Lériendil at 13:40, 5 September 2025

2025-09-05T13:40:36Z

← Older revision		Revision as of 13:40, 5 September 2025
Line 105:		Line 105:
	\|(Tendo)-neutral third		\|(Tendo)-neutral third
	\|(Arto)-neutral third		\|(Arto)-neutral third
	\|Neutral thirds		\|[[Neutral (interval quality)\|Neutral thirds]]
	\| rowspan="7" \|Third		\| rowspan="7" \|Third
	\|-		\|-
Line 111:		Line 111:
	\|Submajor third		\|Submajor third
	\|Supraminor third		\|Supraminor third
	\|Intraclassical thirds		\|[[Submajor and supraminor\|Intraclassical thirds]]
	\|-		\|-
	\|±7 to 12°		\|±7 to 12°
	\|Classical major third		\|Classical major third
	\|Classical minor third		\|Classical minor third
	\|Classical thirds		\|[[Pental major and minor\|Classical thirds]]
	\|-		\|-
	\|±12 to 16°		\|±12 to 16°
	\|Pythagorean major third		\|Pythagorean major third
	\|Pythagorean minor third		\|Pythagorean minor third
	\|Pythagorean thirds		\|[[Novamajor and novaminor\|Pythagorean thirds]]
	\|-		\|-
	\|±16 to 20°		\|±16 to 20°
	\|Neogothic major third		\|Neogothic major third
	\|Neogothic minor third		\|Neogothic minor third
	\|Neogothic thirds		\|[[Neogothic major and minor\|Neogothic thirds]]
	\|-		\|-
	\|±20 to 24°		\|±20 to 24°
	\|Septimal major third, supermajor third		\|Septimal major third, supermajor third
	\|Septimal minor third, subminor third		\|Septimal minor third, subminor third
	\|Septimal thirds		\|[[Supermajor and subminor\|Septimal thirds]]
	\|-		\|-
	\|±24 to 28°		\|±24 to 28°
	\|Tendo third, ultramajor third		\|Tendo third, ultramajor third
	\|Arto third, inframinor third		\|Arto third, inframinor third
	\|Tridecimal thirds/[[Interseptimal interval\|interseptimals]]		\|[[Ultramajor and inframinor\|Tridecimal thirds]]/[[Interseptimal interval\|interseptimals]]
	\|-		\|-
	\|±28 to 32°		\|±28 to 32°

VectorGraphics at 04:26, 2 September 2025

2025-09-02T04:26:47Z

← Older revision		Revision as of 04:26, 2 September 2025
Line 168:		Line 168:
	\|}		\|}

	Diatonic and antidiatonic fifths can also be categorized by the latitude of the medials they generate:		Diatonic and antidiatonic fifths can also be categorized by the latitude of the medials they generate. Other than neutral, meantone, and neogothic, the diatonic fifth ranges are formed from the quality of the major third they generate plus "pyth".
	{\| class="wikitable"		{\| class="wikitable"
	\|Third type		\|Third type

VectorGraphics at 04:23, 2 September 2025

2025-09-02T04:23:26Z

← Older revision		Revision as of 04:23, 2 September 2025
Line 211:		Line 211:
	\|-		\|-
	\|Intraclassical		\|Intraclassical
	\|~~Flattone~~		\|Subpyth
	\|689-693.4		\|689-693.4
	\|26edo		\|26edo
Line 231:		Line 231:
	\|-		\|-
	\|Septimal		\|Septimal
	\|~~Archy~~		\|Superpyth
	\|708.2-712.8		\|708.2-712.8
	\|22edo		\|22edo

Lériendil at 15:28, 22 July 2025

2025-07-22T15:28:11Z

← Older revision		Revision as of 15:28, 22 July 2025
Line 11:		Line 11:
	[[File:Imagemeeeeeeenememe.png\|thumb\|A depiction of the medial intervals of fifth-bounded triads by latitude.]]		[[File:Imagemeeeeeeenememe.png\|thumb\|A depiction of the medial intervals of fifth-bounded triads by latitude.]]

	[[File:~~Jorjor wel~~.png\|thumb\|An analogous depiction for harmony bounded by a tempered ~[[7/3]] in [[tritave]] systems; note that categorical names are based on [[4L 5s (3/1-equivalent)]] with 1-indexing.]]		[[File:Tritaveland latitude.png\|thumb\|An analogous depiction for harmony bounded by a tempered ~[[7/3]] in [[tritave]] systems; note that categorical names are based on [[4L 5s (3/1-equivalent)]] with 1-indexing.]]

	Here is a table of the latitude of medials up to ±30° in various EDOs, with respect to their fifth as the axis:		Here is a table of the latitude of medials up to ±30° in various EDOs, with respect to their fifth as the axis:

ArrowHead294 at 23:57, 21 March 2025

2025-03-21T23:57:04Z

← Older revision		Revision as of 23:57, 21 March 2025
Line 1:		Line 1:
	{{~~Breadcrumb~~\|Interval region}}		{{main\|Interval region}}

	'''Latitude''' is a measure of an [[interval]]'s size in relation to a (possibly tempered) [[perfect fifth\|fifth]], or another interval (the "axis of polarity" or just "axis") serving the same function in another harmonic system. An interval as defined by its latitude may be called a "medial" and always has an axis complement. Latitude ranges from 90 degrees Minor (-90°) to 90 degrees Major (90°), corresponding to 180edA where A is the axis. "Low-latitude" means close to 0 degrees, "high-latitude" means close to ±90 degrees.		'''Latitude''' is a measure of an [[interval]]'s size in relation to a (possibly tempered) [[perfect fifth\|fifth]], or another interval (the "axis of polarity" or just "axis") serving the same function in another harmonic system. An interval as defined by its latitude may be called a "medial" and always has an axis complement. Latitude ranges from 90 degrees Minor (-90°) to 90 degrees Major (90°), corresponding to 180edA where A is the axis. "Low-latitude" means close to 0 degrees, "high-latitude" means close to ±90 degrees.

VectorGraphics: Deformalization (as in, got rid of the reference to "formal thirds")

2025-03-18T11:19:27Z

Deformalization (as in, got rid of the reference to "formal thirds")

← Older revision		Revision as of 11:19, 18 March 2025
Line 9:		Line 9:
	Medial pairs with a higher latitude than ±45° tend to sound more dissonant against the axis or root than lower-latitude medials, and extremely low-latitude interval pairs may not sound distinct from each other. Higher latitude enables "cross-tonality", where both intervals in the pair can be played at once in the same chord, as in suspended or [[extraclassical tonality\|arto/tendo]] chords.		Medial pairs with a higher latitude than ±45° tend to sound more dissonant against the axis or root than lower-latitude medials, and extremely low-latitude interval pairs may not sound distinct from each other. Higher latitude enables "cross-tonality", where both intervals in the pair can be played at once in the same chord, as in suspended or [[extraclassical tonality\|arto/tendo]] chords.

	[[File:Imagemeeeeeeenememe.png\|thumb\|A depiction of the medial intervals ~~("formal thirds")~~ of fifth-bounded triads by latitude.]]		[[File:Imagemeeeeeeenememe.png\|thumb\|A depiction of the medial intervals of fifth-bounded triads by latitude.]]

	[[File:Jorjor wel.png\|thumb\|An analogous depiction for harmony bounded by a tempered ~[[7/3]] in [[tritave]] systems; note that categorical names are based on [[4L 5s (3/1-equivalent)]] with 1-indexing.]]		[[File:Jorjor wel.png\|thumb\|An analogous depiction for harmony bounded by a tempered ~[[7/3]] in [[tritave]] systems; note that categorical names are based on [[4L 5s (3/1-equivalent)]] with 1-indexing.]]

Lériendil at 11:53, 8 March 2025

2025-03-08T11:53:46Z

← Older revision		Revision as of 11:53, 8 March 2025
Line 3:		Line 3:
	'''Latitude''' is a measure of an [[interval]]'s size in relation to a (possibly tempered) [[perfect fifth\|fifth]], or another interval (the "axis of polarity" or just "axis") serving the same function in another harmonic system. An interval as defined by its latitude may be called a "medial" and always has an axis complement. Latitude ranges from 90 degrees Minor (-90°) to 90 degrees Major (90°), corresponding to 180edA where A is the axis. "Low-latitude" means close to 0 degrees, "high-latitude" means close to ±90 degrees.		'''Latitude''' is a measure of an [[interval]]'s size in relation to a (possibly tempered) [[perfect fifth\|fifth]], or another interval (the "axis of polarity" or just "axis") serving the same function in another harmonic system. An interval as defined by its latitude may be called a "medial" and always has an axis complement. Latitude ranges from 90 degrees Minor (-90°) to 90 degrees Major (90°), corresponding to 180edA where A is the axis. "Low-latitude" means close to 0 degrees, "high-latitude" means close to ±90 degrees.

	Latitude allows the characterization of, i.e. different "flavors" of thirds, with respect to a fifth that might be tempered, as opposed to in terms of absolute interval ranges that may be misaligned with the intervals' harmonic function. For example, one might describe 3\13, 4\13, and 5\13 in 13edo based on their size as a [[minor third\|subminor third]], a [[major third\|submajor third]], and an ultramajor third, but that doesn't reflect their function in triadic harmony in 13edo. Using latitude, one can see that their latitudes are -22.5°, 0°, and 22.5°, and from that they may be assigned the labels subminor, [[neutral third\|neutral]], and supermajor, which better reflect their role.		Latitude allows the characterization of, i.e. different "flavors" of thirds, with respect to a fifth that might be tempered, as opposed to in terms of absolute interval ranges that may be misaligned with the intervals' harmonic function. For example, one might describe 3\13, 4\13, and 5\13 in [[13edo]] based on their size as a [[minor third\|subminor third]], a [[major third\|submajor third]], and an ultramajor third, but that doesn't reflect their function in [[triadic harmony]] in 13edo. Using latitude, one can see that their latitudes are -22.5°, 0°, and 22.5°, and from that they may be assigned the labels subminor, [[neutral third\|neutral]], and supermajor, which better reflect their role.

	They additionally allow the generalization of the triadic concept to "axis" intervals other than the fifth. For example, if one were to make a harmonic system using 5/3 as the "axis" rather than 3/2, chords built within a range of 5/3 would contain 3:4:5 and 12:15:20, and the medial intervals of the major third and perfect fourth would have a latitude of ±11.4°		They additionally allow the generalization of the triadic concept to "axis" intervals other than the fifth. For example, if one were to make a harmonic system using [[5/3]] as the "axis" rather than [[3/2]], chords built within a range of 5/3 would contain 3:4:5 and 12:15:20, and the medial intervals of the major third and perfect fourth would have a latitude of ±11.4°

	Medial pairs with a higher latitude than ±45° tend to sound more dissonant against the axis or root than lower-latitude medials, and extremely low-latitude interval pairs may not sound distinct from each other. Higher latitude enables "cross-tonality", where both intervals in the pair can be played at once in the same chord, as in suspended or [[extraclassical tonality\|arto/tendo]] chords.		Medial pairs with a higher latitude than ±45° tend to sound more dissonant against the axis or root than lower-latitude medials, and extremely low-latitude interval pairs may not sound distinct from each other. Higher latitude enables "cross-tonality", where both intervals in the pair can be played at once in the same chord, as in suspended or [[extraclassical tonality\|arto/tendo]] chords.

Lériendil at 00:36, 8 March 2025

2025-03-08T00:36:26Z

← Older revision		Revision as of 00:36, 8 March 2025
Line 9:		Line 9:
	Medial pairs with a higher latitude than ±45° tend to sound more dissonant against the axis or root than lower-latitude medials, and extremely low-latitude interval pairs may not sound distinct from each other. Higher latitude enables "cross-tonality", where both intervals in the pair can be played at once in the same chord, as in suspended or [[extraclassical tonality\|arto/tendo]] chords.		Medial pairs with a higher latitude than ±45° tend to sound more dissonant against the axis or root than lower-latitude medials, and extremely low-latitude interval pairs may not sound distinct from each other. Higher latitude enables "cross-tonality", where both intervals in the pair can be played at once in the same chord, as in suspended or [[extraclassical tonality\|arto/tendo]] chords.

	[[File:Imagemeeeeeeenememe.png\|thumb\|A depiction of the medial intervals ("formal thirds") of fifth-bounded triads by latitude]]		[[File:Imagemeeeeeeenememe.png\|thumb\|A depiction of the medial intervals ("formal thirds") of fifth-bounded triads by latitude.]]

			[[File:Jorjor wel.png\|thumb\|An analogous depiction for harmony bounded by a tempered ~[[7/3]] in [[tritave]] systems; note that categorical names are based on [[4L 5s (3/1-equivalent)]] with 1-indexing.]]

	Here is a table of the latitude of medials up to ±30° in various EDOs, with respect to their fifth as the axis:		Here is a table of the latitude of medials up to ±30° in various EDOs, with respect to their fifth as the axis:

Lériendil at 00:31, 8 March 2025

2025-03-08T00:31:50Z

← Older revision		Revision as of 00:31, 8 March 2025
Line 1:		Line 1:
	{{Breadcrumb\|Interval region}}		{{Breadcrumb\|Interval region}}
	~~[[File:Imagemeeeeeeenememe.png\|thumb\|Image]]~~
	'''Latitude''' is a measure of an [[interval]]'s size in relation to a (possibly tempered) [[perfect fifth\|fifth]], or another interval (the "axis of polarity" or just "axis") serving the same function in another harmonic system. An interval as defined by its latitude may be called a "medial" and always has an axis complement. Latitude ranges from 90 degrees Minor (-90°) to 90 degrees Major (90°), corresponding to 180edA where A is the axis. "Low-latitude" means close to 0 degrees, "high-latitude" means close to ±90 degrees.		'''Latitude''' is a measure of an [[interval]]'s size in relation to a (possibly tempered) [[perfect fifth\|fifth]], or another interval (the "axis of polarity" or just "axis") serving the same function in another harmonic system. An interval as defined by its latitude may be called a "medial" and always has an axis complement. Latitude ranges from 90 degrees Minor (-90°) to 90 degrees Major (90°), corresponding to 180edA where A is the axis. "Low-latitude" means close to 0 degrees, "high-latitude" means close to ±90 degrees.

Line 8:		Line 8:

	Medial pairs with a higher latitude than ±45° tend to sound more dissonant against the axis or root than lower-latitude medials, and extremely low-latitude interval pairs may not sound distinct from each other. Higher latitude enables "cross-tonality", where both intervals in the pair can be played at once in the same chord, as in suspended or [[extraclassical tonality\|arto/tendo]] chords.		Medial pairs with a higher latitude than ±45° tend to sound more dissonant against the axis or root than lower-latitude medials, and extremely low-latitude interval pairs may not sound distinct from each other. Higher latitude enables "cross-tonality", where both intervals in the pair can be played at once in the same chord, as in suspended or [[extraclassical tonality\|arto/tendo]] chords.

			[[File:Imagemeeeeeeenememe.png\|thumb\|A depiction of the medial intervals ("formal thirds") of fifth-bounded triads by latitude]]

	Here is a table of the latitude of medials up to ±30° in various EDOs, with respect to their fifth as the axis:		Here is a table of the latitude of medials up to ±30° in various EDOs, with respect to their fifth as the axis:

VectorGraphics at 00:26, 8 March 2025

2025-03-08T00:26:29Z

← Older revision		Revision as of 00:26, 8 March 2025
Line 1:		Line 1:
	{{Breadcrumb\|Interval region}}		{{Breadcrumb\|Interval region}}
			[[File:Imagemeeeeeeenememe.png\|thumb\|Image]]
	'''Latitude''' is a measure of an [[interval]]'s size in relation to a (possibly tempered) [[perfect fifth\|fifth]], or another interval (the "axis of polarity" or just "axis") serving the same function in another harmonic system. An interval as defined by its latitude may be called a "medial" and always has an axis complement. Latitude ranges from 90 degrees Minor (-90°) to 90 degrees Major (90°), corresponding to 180edA where A is the axis. "Low-latitude" means close to 0 degrees, "high-latitude" means close to ±90 degrees.		'''Latitude''' is a measure of an [[interval]]'s size in relation to a (possibly tempered) [[perfect fifth\|fifth]], or another interval (the "axis of polarity" or just "axis") serving the same function in another harmonic system. An interval as defined by its latitude may be called a "medial" and always has an axis complement. Latitude ranges from 90 degrees Minor (-90°) to 90 degrees Major (90°), corresponding to 180edA where A is the axis. "Low-latitude" means close to 0 degrees, "high-latitude" means close to ±90 degrees.

← Older revision		Revision as of 11:19, 18 March 2025
Line 9:		Line 9:
	Medial pairs with a higher latitude than ±45° tend to sound more dissonant against the axis or root than lower-latitude medials, and extremely low-latitude interval pairs may not sound distinct from each other. Higher latitude enables "cross-tonality", where both intervals in the pair can be played at once in the same chord, as in suspended or [[extraclassical tonality\|arto/tendo]] chords.		Medial pairs with a higher latitude than ±45° tend to sound more dissonant against the axis or root than lower-latitude medials, and extremely low-latitude interval pairs may not sound distinct from each other. Higher latitude enables "cross-tonality", where both intervals in the pair can be played at once in the same chord, as in suspended or [[extraclassical tonality\|arto/tendo]] chords.

	[[File:Imagemeeeeeeenememe.png\|thumb\|A depiction of the medial intervals ~~("formal thirds")~~ of fifth-bounded triads by latitude.]]		[[File:Imagemeeeeeeenememe.png\|thumb\|A depiction of the medial intervals of fifth-bounded triads by latitude.]]

	[[File:Jorjor wel.png\|thumb\|An analogous depiction for harmony bounded by a tempered ~[[7/3]] in [[tritave]] systems; note that categorical names are based on [[4L 5s (3/1-equivalent)]] with 1-indexing.]]		[[File:Jorjor wel.png\|thumb\|An analogous depiction for harmony bounded by a tempered ~[[7/3]] in [[tritave]] systems; note that categorical names are based on [[4L 5s (3/1-equivalent)]] with 1-indexing.]]

← Older revision		Revision as of 00:26, 8 March 2025
Line 1:		Line 1:
	{{Breadcrumb\|Interval region}}		{{Breadcrumb\|Interval region}}
			[[File:Imagemeeeeeeenememe.png\|thumb\|Image]]
	'''Latitude''' is a measure of an [[interval]]'s size in relation to a (possibly tempered) [[perfect fifth\|fifth]], or another interval (the "axis of polarity" or just "axis") serving the same function in another harmonic system. An interval as defined by its latitude may be called a "medial" and always has an axis complement. Latitude ranges from 90 degrees Minor (-90°) to 90 degrees Major (90°), corresponding to 180edA where A is the axis. "Low-latitude" means close to 0 degrees, "high-latitude" means close to ±90 degrees.		'''Latitude''' is a measure of an [[interval]]'s size in relation to a (possibly tempered) [[perfect fifth\|fifth]], or another interval (the "axis of polarity" or just "axis") serving the same function in another harmonic system. An interval as defined by its latitude may be called a "medial" and always has an axis complement. Latitude ranges from 90 degrees Minor (-90°) to 90 degrees Major (90°), corresponding to 180edA where A is the axis. "Low-latitude" means close to 0 degrees, "high-latitude" means close to ±90 degrees.