Extraclassical tonality: Difference between revisions

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== Arto and tendo triads ==

Arto and Tendo triads are the same shape as major and minor triads, but with a more extreme difference between the thirds. This can be characterized in terms of [[Mediant (tonality)|mediants]], where the contrastiveness of the arto and tendo mediants is higher than that of minor and major. An arto triad 26:30:39 results by flatting the third of a minor triad by approximately one [[Quartertone|quarter tone]], producing an inframinor third of ~250 [[cents]]. The Tendo triad results by sharping the third in a major triad by approximately one quarter tone, producing an ultramajor third of ~450 cents. The Tendo chord 10:13:15, being of a simpler ratio, is considered to be more consonant than the arto chord which is quite complex. Also, 10:13:15 tendo chord is only slightly more dissonant than the minor chord 10:12:15. Although they appear to be the same, arto and tendo chords are not seven limit, and tuning them to 14:18:21 and 6:7:9 may compromise the cross-tonal nature of the chords; using the septimal second and subfourth may make the intervals too wide/narrow to function as thirds.

Arto and Tendo triads are the same shape as major and minor triads, but with a more extreme difference between the thirds. This can be characterized in terms of [[Mediant (tonality)|mediants]], where the contrastiveness of the arto and tendo mediants is higher than that of minor and major. An arto triad 26:30:39 results by flatting the third of a minor triad by approximately one [[Quartertone|quarter tone]], producing an inframinor third of ~250 [[cents]]. The Tendo triad results by sharping the third in a major triad by approximately one quarter tone, producing an ultramajor third of ~450 cents. The Tendo chord 10:13:15, being of a simpler ratio, is considered to be more consonant than the arto chord which is quite complex. Also, 10:13:15 tendo chord is only slightly more dissonant than the minor chord 10:12:15.

More generally, thirds of roughly 250 and 450 cents, or [[User:VectorGraphics/The spectrum of mediants|mediants]] of a fifth with a contrastiveness of 25-30%, may be considered arto and tendo thirds.

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[[24edo]] allows both arto/tendo and minor/major chords to be used.

== Tunings ==

The canonical tunings of the arto and tendo thirds are 15/13 and 13/10. However, several tuning options are available.

* The thirds could be tuned to the 7-limit intervals 7/6 and 9/7. This slightly compromises cross-tonality with a submajor second in between the intervals instead of a wider major second, however it makes the ratios simpler when played individually.

* Alternatively, they may be tuned to the 7-limit intervals 8/7 and 21/16, at the cost of perhaps not sounding like thirds.

* In 24edo, the thirds can be tuned to 250c and 450c.

* In slendric systems, especially those with a flat fifth such as 16edo, the arto and tendo thirds may be widened to 1\3edf and 2\3edf.

* Any pair of [[List of root-3rd-P5 triads in JI|JI mediants]] with a contrastiveness about 25-30%.

* In terms of [[Fractional monzo|fmonzos]], the arto and tendo thirds may be tuned as 1\2edf ± 1\2ed(9/8).

[[Category:15-odd-limit]]

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 == Arto and tendo triads ==
-Arto and Tendo triads are the same shape as major and minor triads, but with a more extreme difference between the thirds. This can be characterized in terms of [[Mediant (tonality)|mediants]], where the contrastiveness of the arto and tendo mediants is higher than that of minor and major. An arto triad 26:30:39 results by flatting the third of a minor triad by approximately one [[Quartertone|quarter tone]], producing an inframinor third of ~250 [[cents]]. The Tendo triad results by sharping the third in a major triad by approximately one quarter tone, producing an ultramajor third of ~450 cents. The Tendo chord 10:13:15, being of a simpler ratio, is considered to be more consonant than the arto chord which is quite complex. Also, 10:13:15 tendo chord is only slightly more dissonant than the minor chord 10:12:15. Although they appear to be the same, arto and tendo chords are not seven limit, and tuning them to 14:18:21 and 6:7:9 may compromise the cross-tonal nature of the chords; using the septimal second and subfourth may make the intervals too wide/narrow to function as thirds.
+Arto and Tendo triads are the same shape as major and minor triads, but with a more extreme difference between the thirds. This can be characterized in terms of [[Mediant (tonality)|mediants]], where the contrastiveness of the arto and tendo mediants is higher than that of minor and major. An arto triad 26:30:39 results by flatting the third of a minor triad by approximately one [[Quartertone|quarter tone]], producing an inframinor third of ~250 [[cents]]. The Tendo triad results by sharping the third in a major triad by approximately one quarter tone, producing an ultramajor third of ~450 cents. The Tendo chord 10:13:15, being of a simpler ratio, is considered to be more consonant than the arto chord which is quite complex. Also, 10:13:15 tendo chord is only slightly more dissonant than the minor chord 10:12:15.
 More generally, thirds of roughly 250 and 450 cents, or [[User:VectorGraphics/The spectrum of mediants|mediants]] of a fifth with a contrastiveness of 25-30%, may be considered arto and tendo thirds.
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 [[24edo]] allows both arto/tendo and minor/major chords to be used.
+== Tunings ==
+The canonical tunings of the arto and tendo thirds are 15/13 and 13/10. However, several tuning options are available.
+* The thirds could be tuned to the 7-limit intervals 7/6 and 9/7. This slightly compromises cross-tonality with a submajor second in between the intervals instead of a wider major second, however it makes the ratios simpler when played individually.
+* Alternatively, they may be tuned to the 7-limit intervals 8/7 and 21/16, at the cost of perhaps not sounding like thirds.
+* In 24edo, the thirds can be tuned to 250c and 450c.
+* In slendric systems, especially those with a flat fifth such as 16edo, the arto and tendo thirds may be widened to 1\3edf and 2\3edf.
+* Any pair of [[List of root-3rd-P5 triads in JI|JI mediants]] with a contrastiveness about 25-30%.
+* In terms of [[Fractional monzo|fmonzos]], the arto and tendo thirds may be tuned as 1\2edf ± 1\2ed(9/8).
 [[Category:15-odd-limit]]