Armodue harmony: Difference between revisions
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=Reasons why one would want to use 16edo= | =Reasons why one would want to use 16edo= | ||
== | ==16edo as based on the 5th and 7th harmonic== | ||
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. | 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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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. | 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. | ||
== | ==Tuning characteristics of 16edo== | ||
Many brass instruments rely on both binary keying patterns and the harmonic series to play the full range of notes; since 16edo finds its first good odd harmonic at 7/1, it is then fortuitous that, as a power of two, 16edo fits perfectly within 4 keys on such an instrument. Also, being a power of two, 16edo is easy to tune by taking lots of square roots, which are the only form of root that is constructible geometrically. Relatedly, various expressions involving the square root of 2 result in ratios that are close to notes of 16edo. | |||
This is analogous to the fact that various expressions involving the octave itself correspond closely to notes of 12edo: the arithmetic mean and the harmonic mean of the octave [[2/1]] correspond to the perfect fifth [[3/2]] and the perfect fourth [[4/3]], while the geometric mean divides the octave exactly into two "semioctave" ([[2edo|1\2edo]], sqrt(2)) tritone intervals. | |||
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=16edo intervals= | |||
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==Qualitative categories of intervals== | ==Qualitative categories of intervals== | ||
Armodue consists of | Armodue consists of 16 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 octave) in eight categories that will be to be analysed individually: | ||
1 eka - 15 eka ( | 1 eka - 15 eka (eka, major seventh) | ||
2 eka - 14 eka (neutral tone, dominant seventh) | 2 eka - 14 eka (neutral tone, dominant seventh) | ||
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==1 eka and 15 eka== | ==1 eka and 15 eka== | ||
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) | The interval of one eka, the "minor 1-step" 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) which is roughly 70 cents. According to some theories, 70c is the optimal distance from the tonic for a leading tone: for example, a resolution from 1130c to the octave is strong and directed; a smaller diesis (i.e. 1170-1200c) or a larger one (i.e. 1050-1200c) results in a weaker resolution. | ||
Because of this, the eka sounds 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, the "major 8-step", comparable to a slightly enlarged major seventh of the dodecatonic system. 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 flavour. | |||
In Armodue the intervals of 1 and 15 eka eka are considered harsh dissonances and as such should be used with caution in chords. However, all rules may be applied that already govern the treatment of harsh dissonances in the dodecatonic system. | In Armodue the intervals of 1 and 15 eka eka are considered harsh dissonances and as such should be used with caution in chords. However, all rules may be applied that already govern the treatment of harsh dissonances in the dodecatonic system. | ||
==2 eka and 14 eka== | ==2 eka and 14 eka== | ||
The interval of 2 eka corresponds to 3/2 of a tempered semitone (quantifies in 150 cents) and is found exactly between the eleventh and twelfth harmonic of the overtone series. It is the interval that is obtained by dividing the Tenth of Armodue (the classic Octave of 2/1) into eight equal parts, to form the [[8edo|8-equal tempered]] scale. Since only harmonics of higher number come close to it, it sounds particularly unnatural to the ear. This makes it suitable for geometric constructions where symmetry and artificialness prevail, for example in fractal sound structures. | The interval of 2 eka, the "major 1-step", corresponds to 3/2 of a tempered semitone (quantifies in 150 cents) and is found exactly between the eleventh and twelfth harmonic of the overtone series. It is the interval that is obtained by dividing the Tenth of Armodue (the classic Octave of 2/1) into eight equal parts, to form the [[8edo|8-equal tempered]] scale. Since only harmonics of higher number come close to it, it sounds particularly unnatural to the ear. This makes it suitable for geometric constructions where symmetry and artificialness prevail, for example in fractal sound structures. | ||
In the dodecatonic system, the octave (1200 cents) divides up into an augmented fourth and a diminished fifth (600 cents), the tritone thus obtained can be divided into two minor thirds (300 cents), but the minor third may not further be subdivided into two parts. It is here that, where the possibilities of the dodecatonic system end, the possibilities of Armodue start, and we can continue in progressive subdivisions: the minor third, redefined as four eka (300 cents), is two eka plus two eka (150 cents); in turn, two eka is made up of two intervals of one eka each (75 cents). These algebraic/geometric properties of the initially considered interval of two eka make it particularly suitable for symmetric harmonic constructions - hence "speculative harmony". In practice, you can build speculative chords using only intervals of 16, 8, 4 or 2 eka between a voice and the adjacent one. One or more notes thus obtained can later be altered one eka up or down; in this way, a harmonic construction that is rigidly squared first gains a new harmonic coloration particularly significant in the context. | In the dodecatonic system, the octave (1200 cents) divides up into an augmented fourth and a diminished fifth (600 cents), the tritone thus obtained can be divided into two minor thirds (300 cents), but the minor third may not further be subdivided into two parts. It is here that, where the possibilities of the dodecatonic system end, the possibilities of Armodue start, and we can continue in progressive subdivisions: the minor third, redefined as four eka (300 cents), is two eka plus two eka (150 cents); in turn, two eka is made up of two intervals of one eka each (75 cents). These algebraic/geometric properties of the initially considered interval of two eka make it particularly suitable for symmetric harmonic constructions - hence "speculative harmony". In practice, you can build speculative chords using only intervals of 16, 8, 4 or 2 eka between a voice and the adjacent one. One or more notes thus obtained can later be altered one eka up or down; in this way, a harmonic construction that is rigidly squared first gains a new harmonic coloration particularly significant in the context. | ||
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This mediation between tone and semitone that is realized in Armodue can be much exploited in compositional technique, the principle can be to blur the contours of the "tone" (3 eka) and "semitone" (1 eka) replacing them both with the neutral and ambiguous interval 2 eka. Or, in the opposite direction, you can initially enunciate sound agglomerations where there are intervals of 2 eka, which are then replaced - "colouring" the harmonies and melodies - with intervals of 1 and 3 eka (just at will of the composer). | This mediation between tone and semitone that is realized in Armodue can be much exploited in compositional technique, the principle can be to blur the contours of the "tone" (3 eka) and "semitone" (1 eka) replacing them both with the neutral and ambiguous interval 2 eka. Or, in the opposite direction, you can initially enunciate sound agglomerations where there are intervals of 2 eka, which are then replaced - "colouring" the harmonies and melodies - with intervals of 1 and 3 eka (just at will of the composer). | ||
The interval of 2 eka and its complement of 14 eka are defined as neutral dissonances of Armodue. | The interval of 2 eka and its complement of 14 eka (the "minor 8-step") are defined as neutral dissonances of Armodue. | ||
==3 eka and 13 eka== | ==3 eka and 13 eka== | ||
The interval of 3 eka corresponds to the "wholetone" of Armodue (it is slightly wider than the tempered wholetone). This interval is particularly pleasing to the ear because it is very close to the natural tone that is formed with the seventh and the eighth harmonic (the tempered wholetone, by comparison, sounds less natural to the ear because it is formed with higher harmonics: the eighth and ninth). If you build scales using successions of the "wholetone" of Armodue, or proceeding for jumps of 3 eka, you get particularly evocative sounds - of vague pentatonic flavor. | The interval of 3 eka corresponds to the "wholetone" of Armodue (it is slightly wider than the tempered wholetone). It is the "minor 2-step". This interval is particularly pleasing to the ear because it is very close to the natural tone that is formed with the seventh and the eighth harmonic (the tempered wholetone, by comparison, sounds less natural to the ear because it is formed with higher harmonics: the eighth and ninth). If you build scales using successions of the "wholetone" of Armodue, or proceeding for jumps of 3 eka, you get particularly evocative sounds - of vague pentatonic flavor. | ||
The complement of the interval of 3 eka is the interval of 13 eka, which has a huge importance in Armodue as it corresponds to the [[7/4|natural minor seventh]] - the interval given by the ratio of the fourth harmonic with the seventh harmonic. Who has delved deeper into harmony topics is aware of how much a dominant seventh chord (example: C-E-G-Bb) played in 12-tone equal temperament differs from the corresponding natural chord found by overlapping the fourth, the fifth, the sixth and the seventh harmonic. This is due mainly to the non-negligible difference in pitch of the tempered minor seventh (1000 cents) and the seventh harmonic (968,83 cents)(in this example the Bb). In Armodue, the minor seventh is returned to its natural pitch, the interval of 13 eka is perceived as very natural and euphonious (975 cents). | The complement of the interval of 3 eka is the interval of 13 eka, the "major 7-step", which has a huge importance in Armodue as it corresponds to the [[7/4|natural minor seventh]] - the interval given by the ratio of the fourth harmonic with the seventh harmonic. Who has delved deeper into harmony topics is aware of how much a dominant seventh chord (example: C-E-G-Bb) played in 12-tone equal temperament differs from the corresponding natural chord found by overlapping the fourth, the fifth, the sixth and the seventh harmonic. This is due mainly to the non-negligible difference in pitch of the tempered minor seventh (1000 cents) and the seventh harmonic (968,83 cents)(in this example the Bb). In Armodue, the minor seventh is returned to its natural pitch, the interval of 13 eka is perceived as very natural and euphonious (975 cents). | ||
From a philosophical point of view, with the two discussed intervals, Armodue performs squaring the circle; within a "square", rigidly geometric structure (the division into sixteen steps), it places two "round", vague and exotic intervals as the natural wholetone and the natural minor seventh (in 3 and 13 eka). This latter consideration may prove as a very interesting opportunity for a composer. | From a philosophical point of view, with the two discussed intervals, Armodue performs squaring the circle; within a "square", rigidly geometric structure (the division into sixteen steps), it places two "round", vague and exotic intervals as the natural wholetone and the natural minor seventh (in 3 and 13 eka). This latter consideration may prove as a very interesting opportunity for a composer. | ||
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==4 eka and 12 eka== | ==4 eka and 12 eka== | ||
With the intervals of 4 and 12 eka we have two intervals that are very popular and familiar to the ear translated into Armodue. Indeed 4 eka and 12 eka correspond exactly to the minor third and the major sixth of the dodecatonic system. Therefore, there is the evident possibility of evoking major and minor triads with Armodue (the minor triad is created in stacking 4 eka and 5 eka on a base tone, the major triad stacking 5 eka and 4 eka). The perfect equivalence of the two considered intervals in the dodecatonic system and Armodue is a crucial point in the inevitable interaction that the ear of a western listener will establish between the two different tempered systems. In fact, listening to the intervals of 4 and 12 eka, the ear will immediately associate these Armodue intervals to two already familiar ones (the minor third and the major sixth). For this reason, many of the other intervals present - in an Armodue environment - in a context to those of 4 and 12 eka are likely to be felt by the ear as abnormal and unknown. The composers will give much attention every time they use one of these two intervals, trying to predict the reactions of an ear used to the dodecatonic system. | With the intervals of 4 and 12 eka we have two intervals that are very popular and familiar to the ear translated into Armodue. Indeed 4 eka (the "major 2-step") and 12 eka (the "minor 7-step") correspond exactly to the minor third and the major sixth of the dodecatonic system. Therefore, there is the evident possibility of evoking major and minor triads with Armodue (the minor triad is created in stacking 4 eka and 5 eka on a base tone, the major triad stacking 5 eka and 4 eka). The perfect equivalence of the two considered intervals in the dodecatonic system and Armodue is a crucial point in the inevitable interaction that the ear of a western listener will establish between the two different tempered systems. In fact, listening to the intervals of 4 and 12 eka, the ear will immediately associate these Armodue intervals to two already familiar ones (the minor third and the major sixth). For this reason, many of the other intervals present - in an Armodue environment - in a context to those of 4 and 12 eka are likely to be felt by the ear as abnormal and unknown. The composers will give much attention every time they use one of these two intervals, trying to predict the reactions of an ear used to the dodecatonic system. | ||
In a particular and interesting case, a composer could also decide not to use the intervals giving the color to the harmonies - minor and major thirds and sixths transferred in Armodue: intervals of 4, 5, 11 and 12 eka. Excluding these four types of intervals in the texture of the chords, the ear probably will realize at once that it is in a new and unknown musical environment. | In a particular and interesting case, a composer could also decide not to use the intervals giving the color to the harmonies - minor and major thirds and sixths transferred in Armodue: intervals of 4, 5, 11 and 12 eka. Excluding these four types of intervals in the texture of the chords, the ear probably will realize at once that it is in a new and unknown musical environment. | ||
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==5 eka and 11 eka== | ==5 eka and 11 eka== | ||
The interval of 5 eka is close to the natural major third that appears as the ratio between the fourth and fifth harmonic of the overtone series. The major third of the dodecatonic system (400 cents) is not so close to the natural major third (386,31 cents) as the interval of 5 eka (375 cents). | The interval of 5 eka (the "minor 3-step") is close to the natural major third that appears as the ratio between the fourth and fifth harmonic of the overtone series. The major third of the dodecatonic system (400 cents) is not so close to the natural major third (386,31 cents) as the interval of 5 eka (375 cents). | ||
The complement of 5 eka is the interval of 11 eka, very close to the natural minor sixth that occurs between the fifth and eighth harmonic. Since the intervals of 5 and 11 eka are associated to the tempered third and sixth, the same considerations hold that were made in the previous paragraph about the intervals of 4 and 12 eka. They too are classified as sweet consonances. | The complement of 5 eka is the interval of 11 eka (the "major 6-step"), very close to the natural minor sixth that occurs between the fifth and eighth harmonic. Since the intervals of 5 and 11 eka are associated to the tempered third and sixth, the same considerations hold that were made in the previous paragraph about the intervals of 4 and 12 eka. They too are classified as sweet consonances. | ||
==6 eka and 10 eka== | ==6 eka and 10 eka== | ||
The interval of 6 eka is particularly striking, with its complement of 10 eka. 6 eka is located exactly at the point of equidistance between the major third and the perfect fourth of the tempered system, while 10 eka is the average between a perfect fifth and a minor sixth. | The interval of 6 eka is particularly striking, with its complement of 10 eka. 6 eka (a "major 3-step") is located exactly at the point of equidistance between the major third and the perfect fourth of the tempered system, while 10 eka (a "minor 6-step") is the average between a perfect fifth and a minor sixth. | ||
This property constitutes the most interesting point of the intervals examined so far. The interval of 6 eka joins the concise color of the major third with the propulsion and the dynamism of the perfect fourth, the interval of 10 eka combines the twilight character of the minor sixth with the staticity and the balance of the perfect fifth. | This property constitutes the most interesting point of the intervals examined so far. The interval of 6 eka joins the concise color of the major third with the propulsion and the dynamism of the perfect fourth, the interval of 10 eka combines the twilight character of the minor sixth with the staticity and the balance of the perfect fifth. | ||
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==7 eka and 9 eka== | ==7 eka and 9 eka== | ||
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 "perfect 4-step" and "perfect 5-step" respectively), 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. 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! | 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! | ||
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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; 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. | The interval of 8 eka, the "augmented 4-step" or "diminished 5-step", 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. | ||