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{{Breadcrumb|Interval region}}
{{Breadcrumb|Interval region}}


''This page is about the way of dividing fifths into "thirds". For the use of "mediant" in the context of general mathematics and just intonation, see [[Mediant (just intonation)]].''[[File:Imagemef.png|thumb|1305x1305px|The spectrum of mediants]]
[[File:Imagemef.png|thumb|1305x1305px|The spectrum of medials]]
   
   
'''"Mediant"''' is a term devised by Vector to refer to an interval as 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 10%, and for the septimal thirds it's about 24%.  
'''Latitude''' is a measure of an interval's size in relation to a (possibly tempered) fifth, or another interval (the "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 subminor third, a major 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, 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°


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 [[Arto and tendo theory|arto/tendo]] chords.


Here is a table of the latitude of medials up to ±45° in various EDOs, with respect to their fifth as the axis:


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 and tendo theory|arto/tendo]] chords.


Here is a table of the contrastiveness of mediants up to 50% in various EDOs:
(rescue table from earlier revision)
{| class="wikitable"
== Latitude-based interval regions ==
|+
Names can be given to intervals based on latitude. A table is provided here for fifths:
!Fifth size in edosteps
!Edos
!Contrastiveness of mediant pairs
|-
|3
|5
|33.3%
|-
|4
|7
|'''0%,''' 50%
|-
|5
|9
|'''20%'''
|-
|6
|10
|'''0%,''' 33.3%
|-
|7
|12
|'''14.3%,''' 42.9%
|-
|8
|13, 14
|'''0%, 25%,''' 50%
|-
|9
|15, 16
|'''11.1%,''' 33.3%
|-
|10
|17, 18
|'''0%, 20%,''' 40%
|-
|11
|18, 19
|'''9.1%, 27.3%,''' 45.5%
|-
|12
|20, 21
|'''0.0%, 16.7%,''' 33.3%, 50%
|-
|13
|22, 23
|'''7.7%, 23.1%,''' 38.5%
|-
|14
|23, 24, 25
|'''0%, 14.3%, 28.6%,''' 42.9%
|-
|15
|25, 26, 27
|'''6.7%, 20%,''' 33.3%, 46.7%
|-
|16
|26, 27, 28
|'''0%, 12.5%, 25%,''' 37.5%, 50%
|-
|17
|28, 29, 30
|'''5.9%, 17.6%, 29.4%,''' 41.2%
|-
|18
|30, 31, 32
|'''11.1%, 22.2%,''' 33.3%, 44.4%
|-
|19
|31, 32, 33, 34
|'''5.3%, 15.8%, 26.3%,''' 36.8%, 47.4%
|-
|20
|33, 34, 35, 36
|'''0%, 10%, 20%, 30%,''' 40%, 50%
|-
|21
|35, 36, 37, 38
|'''4.8%, 14.3%, 23.8%,''' 33.3%, 42.9%
|-
|22
|36, 37, 38, 39
|'''9.1%, 18.2%, 27.3%,''' 36.4%, 45.5%
|-
|23
|38, 39, 40, 41
|'''4.3%, 13%, 21.7%,''' 30.4%, 39.1%, 47.8%
|-
|24
|39, 40, 41, 42, 43
|'''0.0%, 8.3%, 16.7%, 25%,''' 33.3%, 41.7%, 50%
|-
|25
|41, 42, 43, 44, 45
|'''4%, 12%, 20%, 28%,''' 36%, 44%
|-
|26
|43, 44, 45, 46, 47
|'''7.7%, 15.4%, 23.1%,''' 30.8%, 38.5%, 46.2%
|-
|27
|44, 45, 46, 47, 48
|'''3.7%, 11.1%, 18.5%, 25.9%,''' 33.3%, 40.7%, 48.1%
|-
|28
|46, 47, 48, 49, 50
|'''0%, 7.1%, 14.3%, 21.4%, 28.6%,''' 35.7%, 42.9%
|-
|29
|48, 49, 50, 51, 52
|'''3.4%, 10.3%, 17.2%, 24.1%,''' 31.0%, 37.9%, 44.8%
|-
|30
|49, 50, 51, 52, 53, 54
|'''6.7%, 13.3%, 20%,''' 26.7%, 33.3%, 40% 46.7%
|-
|31
|51, 52, 53, 54, 55, 56
|'''3.2%, 9.7%, 16.1%, 22.6%, 29.0%,''' 35.5%, 41.9%, 48.4%
|}
 
== Vector's nomenclature for mediants ==
Mediant names can be assigned based on contrastiveness, as follows:
{| class="wikitable"
{| class="wikitable"
|+
|+
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|}
|}


 
Diatonic and antidiatonic fifths can also be categorized by the contrastiveness of the medials they generate:
Diatonic and antidiatonic fifths can also be categorized by the contrastiveness of the mediants they generate:
{| class="wikitable"
{| class="wikitable"
|Third type
|Third type
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[[Category:Terms]]
[[Category:Terms]]
== Superparticular triads ==
A superparticuar triad, such as 3:4:5 or 4:5:6, is always "major"; how major it is can be characterized by latitude.
Retrieved from "https://en.xen.wiki/w/User:VectorGraphics/Latitude"