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[[File:Imagemef.png|thumb|1305x1305px|The spectrum of mediants]]
{{main|Interval region}}
A mediant is an interval defined by being a specified portion of a fifth, and that can be used in a chord with that fifth to produce a certain quality.  A mediant always has a fifth complement, and the portion of the fifth that they differ by is the contrastiveness of that pair of mediants. For example, the contrastiveness of the classical major and minor thirds is 0.1, and for the septimal thirds it's about 0.24.


'''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.


Mediant names can be assigned based on contrastiveness, as follows:
The latitude can be calculated using the formula ''ℓ = (s - a/2)/(a/2)*90°'', where ''s'' is the size of the interval in cents and ''a'' is the size of the axis in cents. For example, to calculate the latitude of the classic major third 5/4 relative to the perfect fifth 3/2, one first calculates the cent sizes of 5/4 (''log<sub>2</sub>(5/4)*1200 = 386.31 ¢'') and of 3/2 (''log<sub>2</sub>(3/2)*1200 = 701.955 ¢).'' Plugging 386.31 ¢ for ''s'' and 701.955 ¢ for ''a'' into the formula, we get ''ℓ = (386.31 - 701.955/2)/(701.955/2)*90° = (386.31 - 350.9775)/(350.9775)*90° = 9.06°.''
{| class="wikitable"
 
|+
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.
!Contrastiveness
 
!Major name
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°
!Minor name
 
!General name
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.
!Type
|-
|0% to 2.9%
|(Tendo)-neutral third
|(Arto)-neutral third
|Neutral thirds
| rowspan="7" |Third
|-
|2.9% to 7.1%
|Submajor third
|Supraminor third
|Intraclassical thirds
|-
|7.1% to 12.9%
|Classical major third
|Classical minor third
|Classical thirds
|-
|12.9% to 17.1%
|Pythagorean major third
|Pythagorean minor third
|Pythagorean thirds
|-
|17.1% to 21.4%
|Neogothic major third
|Neogothic minor third
|Neogothic thirds
|-
|21.4% to 25.7%
|Septimal major third, supermajor third
|Septimal minor third, subminor third
|Septimal thirds
|-
|25.7% to 30%
|Tendo third, ultramajor third
|Arto third, inframinor third
|Tridecimal thirds/[[Interseptimal interval|interseptimals]]
|-
|30% to 35.7%
|Major paraslendric
|Minor paraslendric
|Paraslendrics
| rowspan="6" |Second/fourth
|-
|35.7% to 40%
|Supraslendric
|Subslendric
|Extraslendrics
|-
|40% to 44.3%
|Major suspended
|Minor suspended
|Suspendeds
|-
|44.3% to 47.1%
|Suprasuspended
|Subsuspended
|Extrasuspendeds
|-
|47.1% to 52.9%
|Major paratetracot
|Minor paratetracot
|Paratetracots
|-
|>52.9%
| colspan="3" | -
|}


[[File:Imagemeeeeeeenememe.png|thumb|A depiction of the medial intervals of fifth-bounded triads by latitude.]]


Pairs of mediants with higher contrastiveness than 50% tend to sound more dissonant against the fifth or root than lower-contrastiveness mediants, and extremely low-contrastiveness mediants may not sound distinct from each other. Higher contrastiveness enables "cross-tonality", where both mediants in the pair can be played at once in the same chord, as in suspended or arto/tendo chords.
[[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 contrastiveness of mediants up to 50% in various EDOs:
Here is a table of the latitude of medials up to ±30° in various EDOs, with respect to their fifth as the axis:
{| class="wikitable"
{| class="wikitable"
|+
!Fifth size in edosteps
!Fifth size in edosteps
!Edos
!Edos
!Contrastiveness of mediant pairs
!Latitude (±n°)
|-
|-
|3
|3
|5
|5
|33.3%
|30
|-
|-
|4
|4
|7
|7
|'''0%,''' 50%
|0
|-
|-
|5
|5
|9
|9
|'''20%'''
|18
|-
|-
|6
|6
|10
|10
|'''0%,''' 33.3%
|0, 30
|-
|-
|7
|7
|12
|12
|'''14.3%,''' 42.9%
|12.8
|-
|-
|8
|8
|13, 14
|13, 14
|'''0%, 25%,''' 50%
|0, 22.5
|-
|-
|9
|9
|15, 16
|15, 16
|'''11.1%,''' 33.3%
|10, 30
|-
|-
|10
|10
|17, 18
|17, 18
|'''0%, 20%,''' 40%
|0, 18
|-
|-
|11
|11
|18, 19
|18, 19
|'''9.1%, 27.3%,''' 45.5%
|8.2, 24.5
|-
|12
|20, 21
|'''0.0%, 16.7%,''' 33.3%, 50%
|-
|-
|13
|13
|22, 23
|22, 23
|'''7.7%, 23.1%,''' 38.5%
|6.9, 20.8
|-
|-
|14
|14
|23, 24, 25
|23, 24, 25
|'''0%, 14.3%, 28.6%,''' 42.9%
|0, 12.8, 25.7
|-
|-
|15
|15
|25, 26, 27
|25, 26, 27
|'''6.7%, 20%,''' 33.3%, 46.7%
|6, 18, 30
|-
|-
|16
|16
|26, 27, 28
|26, 27, 28
|'''0%, 12.5%, 25%,''' 37.5%, 50%
|0, 11.3, 22.5
|-
|-
|17
|17
|28, 29, 30
|28, 29, 30
|'''5.9%, 17.6%, 29.4%,''' 41.2%
|5.3, 15.9, 26.5
|-
|-
|18
|18
|30, 31, 32
|30, 31, 32
|'''11.1%, 22.2%,''' 33.3%, 44.4%
|0, 10, 20, 30
|-
|19
|31, 32, 33, 34
|'''5.3%, 15.8%, 26.3%,''' 36.8%, 47.4%
|-
|-
|20
|20
|33, 34, 35, 36
|33, 34, 35, 36
|'''0%, 10%, 20%, 30%,''' 40%, 50%
|0, 9, 18, 27
|-
|24
|39, 40, 41, 42, 43
|0, 7.5, 15, 22.5, 30
|-
|31
|51, 52, 53, 54, 55, 56
|2.9, 8.7, 14.5, 20.3, 26.1
|}
 
== Latitude-based interval regions ==
Names can be given to intervals based on latitude. A table is provided here for fifths:
{| class="wikitable"
|+
!Latitude
!Major name
!Minor name
!General name
!Type
|-
|-
|21
|±0 to 3°
|35, 36, 37, 38
|(Tendo)-neutral third
|'''4.8%, 14.3%, 23.8%,''' 33.3%, 42.9%
|(Arto)-neutral third
|[[Neutral (interval quality)|Neutral thirds]]
| rowspan="7" |Third
|-
|±3 to 7°
|Submajor third
|Supraminor third
|[[Submajor and supraminor|Intraclassical thirds]]
|-
|±7 to 12°
|Classical major third
|Classical minor third
|[[Pental major and minor|Classical thirds]]
|-
|-
|22
|±12 to 16°
|36, 37, 38, 39
|Pythagorean major third
|'''9.1%, 18.2%, 27.3%,''' 36.4%, 45.5%
|Pythagorean minor third
|[[Novamajor and novaminor|Pythagorean thirds]]
|-
|-
|23
|±16 to 20°
|38, 39, 40, 41
|Neogothic major third
|'''4.3%, 13%, 21.7%,''' 30.4%, 39.1%, 47.8%
|Neogothic minor third
|[[Neogothic major and minor|Neogothic thirds]]
|-
|-
|24
|±20 to 24°
|39, 40, 41, 42, 43
|Septimal major third, supermajor third
|'''0.0%, 8.3%, 16.7%, 25%,''' 33.3%, 41.7%, 50%
|Septimal minor third, subminor third
|[[Supermajor and subminor|Septimal thirds]]
|-
|-
|25
|±24 to 28°
|41, 42, 43, 44, 45
|Tendo third, ultramajor third
|'''4%, 12%, 20%, 28%,''' 36%, 44%
|Arto third, inframinor third
|[[Ultramajor and inframinor|Tridecimal thirds]]/[[Interseptimal interval|interseptimals]]
|-
|-
|26
|±28 to 32°
|43, 44, 45, 46, 47
|Major paraslendric
|'''7.7%, 15.4%, 23.1%,''' 30.8%, 38.5%, 46.2%
|Minor paraslendric
|Paraslendrics
| rowspan="6" |Second/fourth
|-
|-
|27
|±32 to 36°
|44, 45, 46, 47, 48
|Supraslendric
|'''3.7%, 11.1%, 18.5%, 25.9%,''' 33.3%, 40.7%, 48.1%
|Subslendric
|Extraslendrics
|-
|-
|28
|±36 to 40°
|46, 47, 48, 49, 50
|Major suspended
|'''0%, 7.1%, 14.3%, 21.4%, 28.6%,''' 35.7%, 42.9%
|Minor suspended
|Suspendeds
|-
|-
|29
|±40 to 44°
|48, 49, 50, 51, 52
|Suprasuspended
|'''3.4%, 10.3%, 17.2%, 24.1%,''' 31.0%, 37.9%, 44.8%
|Subsuspended
|Extrasuspendeds
|-
|-
|30
|±44 to 48°
|49, 50, 51, 52, 53, 54
|Major paratetracot
|'''6.7%, 13.3%, 20%,''' 26.7%, 33.3%, 40% 46.7%
|Minor paratetracot
|Paratetracots
|-
|-
|31
|Beyond ±48°
|51, 52, 53, 54, 55, 56
| colspan="3" | -
|'''3.2%, 9.7%, 16.1%, 22.6%, 29.0%,''' 35.5%, 41.9%, 48.4%
|}
|}
Diatonic and antidiatonic fifths can also be categorized by the contrastiveness of the mediants 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
Line 213: Line 179:
|Interseptimal
|Interseptimal
|Inframedio
|Inframedio
|654.6-661.4
|654.6-660.5
|11edo
|11edo
|-
|-
|Septimal
|Septimal
|Avila
|Avila
|661.4-665.3
|660.5-664.6
|29edo
|29edo
|-
|-
|Neogothic
|Neogothic
|Pelogic
|Pelogic
|665.3-669.3
|664.6-668.7
|9edo
|9edo
|-
|-
|Pythagorean
|Pythagorean
|Mediocratic
|Mediocratic
|669.3-673.3
|668.7-672.9
|25edo
|25edo
|-
|-
|Classical
|Classical
|Mavila
|Mavila
|673.3-678.7
|672.9-678.2
|16edo
|16edo
|-
|-
|Intraclassical
|Intraclassical
|Sharpmavila
|Sharpmavila
|678.8-682.9
|678.2-682.5
|30edo
|30edo
|-
|-
|Neutral
|Neutral
|Neutral
|Neutral
|682.9-688.5
|682.5-689
|7edo
|7edo
|-
|-
|Intraclassical
|Intraclassical
|Flattone
|Subpyth
|688.5-692.8
|689-693.4
|26edo
|26edo
|-
|-
|Classical
|Classical
|Meantone
|Meantone
|692.8-698.5
|693.4-699
|19edo
|19edo
|-
|-
|Pythagorean
|Pythagorean
|Pythagorean
|Pythagorean
|698.5-702.9
|699-703.6
|12edo
|12edo
|-
|-
|Neogothic
|Neogothic
|Neogothic
|Neogothic
|702.9-707.3
|703.6-708.2
|17edo
|17edo
|-
|-
|Septimal
|Septimal
|Archy
|Superpyth
|707.4-711.9
|708.2-712.8
|22edo
|22edo
|-
|-
|Interseptimal
|Interseptimal
|Ultrapyth
|Ultrapyth
|711.9-720
|712.8-720
|5edo
|5edo
|}
|}
== Triads ==
[[Just intonation|Just]] [[triad]]s containing arithmetically progressed frequency divisions - that is, [[isoharmonic]] triads - are always "major"; how major it is can be characterized by latitude.
{| class="wikitable"
|+
!Name
!Triad
!Latitude of medial
|-
|
|1:3:5
|32.8°
|-
|
|1:2:3
|23.5°
|-
|BP "wide" triad
|3:5:7
|18.5°
|-
|
|2:3:4
|15.3°
|-
|BP "narrow" triad
|5:7:9
|13°
|-
|Inverted major triad
|3:4:5
|11.4°
|-
|Mintaka triad
|7:9:11
|10.1°
|-
|Major triad
|4:5:6
|9.1°
|-
|
|9:11:13
|8.2°
|-
|(Septimal) diminished triad
|5:6:7
|7.5°
|-
|
|11:13:15
|7°
|-
|Quartal triad
|6:7:8
|6.5°
|-
|
|13:15:17
|6°
|-
|
|7:8:9
|5.6°
|-
|
|15:17:19
|5.3°
|-
|
|8:9:10
|5°
|}
[[Category:Terms]]
Retrieved from "https://en.xen.wiki/w/User:VectorGraphics/Latitude"