Armodue harmony: Difference between revisions

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==The supremacy of the fifth and the seventh harmonic in Armodue==

The twelve note system that has been ruling for several centuries is based on the third harmonic of the overtone series, which forms a perfect twelfth (octave-reducible to a perfect fifth) with the first harmonic or fundamental. Also, the ~~pythagorean tradition - the cycle~~ of fifths - is based on the perfect fifth and hence on the same frequency ratio 3:2.

The twelve note system that has been ruling 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) 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.

But, if in ~~the twelve note system~~ the pitch of the third harmonic - hence the perfect fourth and the perfect fifth - are almost perfectly respected (the tempered fourth and fifth differ only ~~of the fiftieth part of a semitone~~ from ~~the natural fifth and fourth~~), this cannot be said about the odd harmonics (even harmonics are not counted because they are ~~simply duplicates in the~~ octave of odd harmonics) immediately above the third one: the fifth and seventh harmonic.

But, if in 12edo, the pitch of the third harmonic - hence the [[4/3|perfect fourth]] and the [[perfect fifth]] - are almost perfectly respected (the tempered fourth and fifth differ by only 2 cents from JI), this cannot be said about the odd harmonics (even harmonics are not counted because they are octave-reducible to odd harmonics) immediately above the third one: the fifth and seventh harmonic.

In the tempered system the fifth and the seventh harmonic appear as major third and minor seventh intervals formed to the fundamental (equating all intervals to their octave-reduced form for simplicity), but the sizes of the tempered major third and minor seventh do not match the sizes of the respective natural intervals.

In the tempered system the fifth and the seventh harmonic appear as [[major third]] and [[minor seventh]] intervals formed to the fundamental (equating all intervals to their octave-reduced form for simplicity), but the sizes of the tempered major third and minor seventh do not match the sizes of the respective natural intervals.

In Armodue (see [[16edo|16edo]]), in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness.

For this reason, especially important in Armodue are the intervals of ~~five eka~~ (corresponding to the interval ratio ~~given by the fifth harmonic~~) and ~~the interval of~~ 13 ~~eka~~ (corresponding to the interval ratio ~~subsisting with the seventh harmonic~~). The circle of fifths which is the base of ~~the dodecatonic system~~ is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the [[7/4|seventh]] harmonic ~~(7/4 Ratio)~~.

For this reason, especially important in Armodue are the intervals of 5\16edo (corresponding to the interval ratio 5/4) and 13\16edo (corresponding to the interval ratio 7/4). The circle of fifths which is the base of 12edo is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the [[7/4|seventh]] harmonic.

==The triple mean of the double diagonal / side of the square==

From a philosophical point of view, the system of twelve notes was justified in the past by this mathematical property: the arithmetic mean and the harmonic mean of the octave (interval ratio 2:1) correspond to the perfect fifth (ratio 3:2) and the perfect fourth (ratio 4:3), while the geometric mean divides the octave exatcly into two tritone intervals (ratio: square root of 2).

Analogously, the philosophical foundation of Armodue and ~~esadecafonia~~ can be shown by calculating the three means of frequency geometrically equivalent to the ratio between the double of the diagonal (square root of 2 multiplied by 2) and the side length (of measure: 1) of a square.

Analogously, the philosophical foundation of Armodue and 16edo can be shown by calculating the three means of frequency geometrically equivalent to the ratio between the double of the diagonal (square root of 2 multiplied by 2) and the side length (of measure: 1) of a square.

The arithmetic mean is exactly ~~the interval of nine eka~~, the geometric mean exactly ~~twelve eka~~ and finally the harmonic mean exactly equal to ~~fifteen eka~~.

The arithmetic mean is exactly 9\16edo, the geometric mean exactly 12\16edo and finally the harmonic mean exactly equal to 15\16edo.

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==1 eka and 15 eka==

The interval of one eka, the degree of the chromatic scale of Armodue equal to 3/4 of a semitone (75 cents), is the smallest interval of the system and is very close to the chromatic semitone postulated by Zarlino in his natural scale (based on simple ratios) and accounts for 70 cents.

The interval of one eka, or 1\16edo, the degree of the chromatic scale of Armodue equal to 3/4 of a semitone (75 cents), is the smallest interval of the system and is very close to the chromatic semitone postulated by Zarlino in his natural scale (based on simple ratios) and accounts for 70 cents.

This property of the eka makes it particularly euphonious and familiar to the ear: the eka is perceived as a natural interval not less than a semitone of the dodecatonic scale. In a free melodic improvisation, chromatic successions of consecutive ekas sound much like chromatic successions of semitones. Therefore, all the harmonic techniques inherent in chromaticism can be applied in Armodue considering the eka as equivalent to a tempered semitone. The complement of one eka is the interval of 15 eka, comparable to a slightly enlarged major seventh of the dodecatonic system. The small size of the eka also makes it appropriate to evoke oriental sounds and atmospheres. The small intervals of 1 eka, 2 eka and 3 eka in Armodue lend themselves magnificiently to the design of melodies and scales of exquisite modal and arabic flavour.

@@ Line 11: / Line 11: @@
 ==The supremacy of the fifth and the seventh harmonic in Armodue==
-The twelve note system that has been ruling for several centuries is based on the third harmonic of the overtone series, which forms a perfect twelfth (octave-reducible to a perfect fifth) with the first harmonic or fundamental. Also, the pythagorean tradition - the cycle of fifths - is based on the perfect fifth and hence on the same frequency ratio 3:2.
+The twelve note system that has been ruling 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) 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.
-But, if in the twelve note system the pitch of the third harmonic - hence the perfect fourth and the perfect fifth - are almost perfectly respected (the tempered fourth and fifth differ only of the fiftieth part of a semitone from the natural fifth and fourth), this cannot be said about the odd harmonics (even harmonics are not counted because they are simply duplicates in the octave of odd harmonics) immediately above the third one: the fifth and seventh harmonic.
+But, if in 12edo, the pitch of the third harmonic - hence the [[4/3|perfect fourth]] and the [[perfect fifth]] - are almost perfectly respected (the tempered fourth and fifth differ by only 2 cents from JI), this cannot be said about the odd harmonics (even harmonics are not counted because they are octave-reducible to odd harmonics) immediately above the third one: the fifth and seventh harmonic.
-In the tempered system the fifth and the seventh harmonic appear as major third and minor seventh intervals formed to the fundamental (equating all intervals to their octave-reduced form for simplicity), but the sizes of the tempered major third and minor seventh do not match the sizes of the respective natural intervals.
+In the tempered system the fifth and the seventh harmonic appear as [[major third]] and [[minor seventh]] intervals formed to the fundamental (equating all intervals to their octave-reduced form for simplicity), but the sizes of the tempered major third and minor seventh do not match the sizes of the respective natural intervals.
 In Armodue (see [[16edo|16edo]]), in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness.
-For this reason, especially important in Armodue are the intervals of five eka (corresponding to the interval ratio given by the fifth harmonic) and the interval of 13 eka (corresponding to the interval ratio subsisting with the seventh harmonic). The circle of fifths which is the base of the dodecatonic system is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the [[7/4|seventh]] harmonic (7/4 Ratio).
+For this reason, especially important in Armodue are the intervals of 5\16edo (corresponding to the interval ratio 5/4) and 13\16edo (corresponding to the interval ratio 7/4). The circle of fifths which is the base of 12edo is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the [[7/4|seventh]] harmonic.
 ==The triple mean of the double diagonal / side of the square==
 From a philosophical point of view, the system of twelve notes was justified in the past by this mathematical property: the arithmetic mean and the harmonic mean of the octave (interval ratio 2:1) correspond to the perfect fifth (ratio 3:2) and the perfect fourth (ratio 4:3), while the geometric mean divides the octave exatcly into two tritone intervals (ratio: square root of 2).
-Analogously, the philosophical foundation of Armodue and esadecafonia can be shown by calculating the three means of frequency geometrically equivalent to the ratio between the double of the diagonal (square root of 2 multiplied by 2) and the side length (of measure: 1) of a square.
+Analogously, the philosophical foundation of Armodue and 16edo can be shown by calculating the three means of frequency geometrically equivalent to the ratio between the double of the diagonal (square root of 2 multiplied by 2) and the side length (of measure: 1) of a square.
-The arithmetic mean is exactly the interval of nine eka, the geometric mean exactly twelve eka and finally the harmonic mean exactly equal to fifteen eka.
+The arithmetic mean is exactly 9\16edo, the geometric mean exactly 12\16edo and finally the harmonic mean exactly equal to 15\16edo.
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 ==1 eka and 15 eka==
-The interval of one eka, the degree of the chromatic scale of Armodue equal to 3/4 of a semitone (75 cents), is the smallest interval of the system and is very close to the chromatic semitone postulated by Zarlino in his natural scale (based on simple ratios) and accounts for 70 cents.
+The interval of one eka, or 1\16edo, the degree of the chromatic scale of Armodue equal to 3/4 of a semitone (75 cents), is the smallest interval of the system and is very close to the chromatic semitone postulated by Zarlino in his natural scale (based on simple ratios) and accounts for 70 cents.
 This property of the eka makes it particularly euphonious and familiar to the ear: the eka is perceived as a natural interval not less than a semitone of the dodecatonic scale. In a free melodic improvisation, chromatic successions of consecutive ekas sound much like chromatic successions of semitones. Therefore, all the harmonic techniques inherent in chromaticism can be applied in Armodue considering the eka as equivalent to a tempered semitone. The complement of one eka is the interval of 15 eka, comparable to a slightly enlarged major seventh of the dodecatonic system. The small size of the eka also makes it appropriate to evoke oriental sounds and atmospheres. The small intervals of 1 eka, 2 eka and 3 eka in Armodue lend themselves magnificiently to the design of melodies and scales of exquisite modal and arabic flavour.