Armodue harmony: Difference between revisions
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<span style="font-size: 140%;">Armodue: basic elements of harmony</span> | <span style="font-size: 140%;">Armodue: basic elements of harmony</span> | ||
This is a translation of an article by Luca Attanasio, with added contributions by Xenharmonic Wiki contributors. Original page in italian: [http://www.armodue.com/armonia.htm http://www.armodue.com/armonia.htm] | |||
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For terminology see the [[Armodue_theory|Armodue overview page]]. | For terminology see the [[Armodue_theory|Armodue overview page]]. | ||
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= | =Reasons why one would want to use 16edo= | ||
==The supremacy of the fifth and the seventh harmonic in Armodue== | ==Harmonic standpoint: The supremacy of the fifth and the seventh harmonic in Armodue== | ||
The | 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. | ||
In 12edo, the the third harmonic - hence the [[4/3|perfect fourth]] and the [[perfect fifth]] - is extremely accurate (the tempered fourth and fifth differ by only 2 cents from JI), but 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 | 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). However, while the major third interval is decently well-represented (but not sufficient in the eyes of some xenharmonicists), the minor seventh is "out of tune" with 7/4 enough to have a distinctly different sound. | ||
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. | 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 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. | This can be seen as an approach to tuning that takes advantage of the idea that simpler ratios can be functionally approximated with greater error: by choosing a tuning with greater error on lower primes as opposed to higher ones, one can create a much more consistent feeling than if the highest errors are on higher primes. In essence, 16edo's 3/1, 5/1, and 7/1 are backwards from 12edo's, with 7 being nearly perfect, 5 being decent, and 3 being distinctly out-of-tune. | ||
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 (or equivalently, 3 eka), emphasizing the priority of the fifth and the [[7/4|seventh]] harmonic. Notably, neither of these intervals close the circle early like 12edo's major third does, nor do they reach each other in a small number of steps like 12edo's perfect fifth does for its major third, meaning that 16edo can be seen as a more "pure" 2.5.7 system than 12edo as a 2.3.5 system. | |||
==The triple mean of the double diagonal / side of the square== | ==Philosophical standpoint: 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). | 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). | ||
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Armodue consists of sixteen types of intervals, which can be grouped two by two (by complementarity: each two intervals are the reverse of the other and add up to the Tenth (interval sum of 16 eka)) in eight categories that will be to be analysed individually: | Armodue consists of sixteen types of intervals, which can be grouped two by two (by complementarity: each two intervals are the reverse of the other and add up to the Tenth (interval sum of 16 eka)) in eight categories that will be to be analysed individually: | ||
1 eka - 15 eka | 1 eka - 15 eka (semitone, major seventh) | ||
2 eka - 14 eka | 2 eka - 14 eka (neutral tone, dominant seventh) | ||
3 eka - 13 eka | 3 eka - 13 eka (whole tone, harmonic seventh) | ||
4 eka - 12 eka | 4 eka - 12 eka (minor third, major sixth) | ||
5 eka - 11 eka | 5 eka - 11 eka (major third, minor sixth) | ||
6 eka - 10 eka | 6 eka - 10 eka (subfourth, superfifth) | ||
7 eka - 9 eka | 7 eka - 9 eka (perfect fourth, perfect fifth) | ||
8 eka | 8 eka (tritone) | ||
==1 eka and 15 eka== | ==1 eka and 15 eka== | ||
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The intervals corresponding to the perfect fourth and and the perfect fifth in Armodue are the intervals of 7 and 9 eka, the first one quantifies in a slightly sharpened fourth (525 cents), the second in a slightly flattened fifth (675 cents). | The intervals corresponding to the perfect fourth and and the perfect fifth in Armodue are the intervals of 7 and 9 eka, the first one quantifies in a slightly sharpened fourth (525 cents), the second in a slightly flattened fifth (675 cents). | ||
These Armodue intervals, however, are incompatible with the concept of the cycle of the fifth that is on the base of the dodecatonic system. Especially with the intervals of 7 and 9 eka, Armodue shows an entirely new and different system, compared to the ruling dodecatonic system. | These Armodue intervals, however, are incompatible with the concept of the cycle of the fifth that is on the base of the dodecatonic system. Especially with the intervals of 7 and 9 eka, Armodue shows an entirely new and different system, compared to the ruling dodecatonic system. Specifically, these intervals generate the antidiatonic and then armotonic scale, with the latter being the main scale used in Armodue theory. If they are incorrectly treated as dodecatonic fifths, it leads to absurdities such as negative semitones and minor thirds being larger than major thirds! | ||
The Armodue "fourth" and "fifth" are euphonious and absolutely consonant, but in a radically different range from those of the dodecatonic system. It is suggested to use the intervals of 7 and 9 eka with caution, just to highlight the special significance that the "fourth" and "fifth" acquire in Armodue and not mislead the western ear deeply accustomed to the cycle of fifths. | The Armodue "fourth" and "fifth" are euphonious and absolutely consonant, but in a radically different range from those of the dodecatonic system. It is suggested to use the intervals of 7 and 9 eka with caution, just to highlight the special significance that the "fourth" and "fifth" acquire in Armodue and not mislead the western ear deeply accustomed to the cycle of fifths. | ||
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==8 eka== | ==8 eka== | ||
The interval of 8 eka divides the Tenth of Armodue in half, just as the tritone halves the typical octave. Indeed 8 eka correspond exactly to three wholetones, hence to an augmented fourth or diminished fifth. | The interval of 8 eka divides the Tenth of Armodue in half, just as the tritone halves the typical octave; in fact, the interval is the same as the 12edo tritone. Indeed 8 eka correspond exactly to three wholetones, hence to an augmented fourth or diminished fifth. | ||
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. | 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. | ||