Armodue harmony: Difference between revisions

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=Reasons why one would want to use 16edo=

==16edo as ~~based on the 5th and 7th harmonic~~==

==16edo as a 2.(3).5.7 system vs. 12edo as a 2.3.5(.7) system==

The 12-form, which has been standard for several centuries (such as 12edo, [[Pythagorean]][12], and [[Meantone]][12]) is based on [[3/1]], the third harmonic of the overtone series, which forms a perfect twelfth (octave-reducible to a perfect fifth [[3/2]]) with the first harmonic or fundamental. Also, the circle of fifths is based on the perfect fifth and hence on the same frequency ratio 3/2.

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Being exactly half of 16 eka, the interval of 8 eka is its own complement; this feature makes it particularly uneasy and unstable - just like the tritone among the dodecatonic tempered system.

When the interval of 8 eka is present in a chord, the entire chord becomes unstable; on the other hand, even the most dissonant chord comes out static and stabilized in the absence of that interval.

When the interval of 8 eka is present in a chord, the entire chord becomes unstable; on the other hand, even the most dissonant chord comes out static and stabilized in the absence of that interval. This plays interestingly with the basic 2.5.7 chord of 4:5:7, as no matter what, any inversion of the chord will contain this unstable interval due to the [[jubilic]] temperament. Instead, it suggests that some other chord avoiding 7:5 might be a more harmonically stable base for Armodue theory, such as dyads of 4:5 and 4:7 played independently, perhaps with the flat fifth as a supporting interval.

The interval of 8 eka is the first axis of symmetry in Armodue: indeed in this system we proceed with progressive splittings of the tenth: 16 eka, 8 eka, 4 eka, 2 eka and 1 eka. As the tritone of the tempered system, the interval of 8 eka is an unstable dissonance.

@@ Line 12: / Line 12: @@
 =Reasons why one would want to use 16edo=
-==16edo as based on the 5th and 7th harmonic==
+==16edo as a 2.(3).5.7 system vs. 12edo as a 2.3.5(.7) system==
 The 12-form, which has been standard for several centuries (such as 12edo, [[Pythagorean]][12], and [[Meantone]][12]) is based on [[3/1]], the third harmonic of the overtone series, which forms a perfect twelfth (octave-reducible to a perfect fifth [[3/2]]) with the first harmonic or fundamental. Also, the circle of fifths is based on the perfect fifth and hence on the same frequency ratio 3/2.
@@ Line 120: / Line 120: @@
 Being exactly half of 16 eka, the interval of 8 eka is its own complement; this feature makes it particularly uneasy and unstable - just like the tritone among the dodecatonic tempered system.
-When the interval of 8 eka is present in a chord, the entire chord becomes unstable; on the other hand, even the most dissonant chord comes out static and stabilized in the absence of that interval.
+When the interval of 8 eka is present in a chord, the entire chord becomes unstable; on the other hand, even the most dissonant chord comes out static and stabilized in the absence of that interval. This plays interestingly with the basic 2.5.7 chord of 4:5:7, as no matter what, any inversion of the chord will contain this unstable interval due to the [[jubilic]] temperament. Instead, it suggests that some other chord avoiding 7:5 might be a more harmonically stable base for Armodue theory, such as dyads of 4:5 and 4:7 played independently, perhaps with the flat fifth as a supporting interval.
 The interval of 8 eka is the first axis of symmetry in Armodue: indeed in this system we proceed with progressive splittings of the tenth: 16 eka, 8 eka, 4 eka, 2 eka and 1 eka. As the tritone of the tempered system, the interval of 8 eka is an unstable dissonance.