Armodue armonia
Armodue: basic elements of harmony
This is a translation of an article by Luca Attanasio. Original page in italian: http://www.armodue.com/armonia.htm
For terminology see the Armodue overview page.
Chapter 1:
Two theses supporting the system
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.
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.
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), 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 seventh harmonic (7/4 Ratio).
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.
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.
Chapter 2:
The interval table
Qualitative categories of intervals
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
2 eka - 14 eka
3 eka - 13 eka
4 eka - 12 eka
5 eka - 11 eka
6 eka - 10 eka
7 eka - 9 eka
8 eka
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.
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.
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
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 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.
The octatonic scale obtained by stepping through intervals of two consecutive eka has a very peculiar sound: from its lack of gravitation, it could be compared to the wholetone scale so much used by Debussy, but the interval of two eka is perceived by the western ear as closer to a semitone than to a tempered wholetone. The end result could be described as a scale about halfway between the chromatic and the wholetone scale in the dodecatonic system. It can be very effective to use the 8-equal scale in chords and other types of scales in the melody, or vice versa: the introduction of different scale systems where the octatonic scale is primarily used adds variety to the harmonic-melodic architecture. Finally, it can be very fruitful in the composition with Armodue to recall that you have two types of "semitones" at your disposal: a "small semitone" (one eka) and a "big semitone" (two eka), next to the "wholetone" of Armodue (three eka). This in fact allows variations and potential completely absent in the dodecatonic system, where there is one - and only one - kind of semitone. For example, you can replace in a composition all intervals of one eka by intervals of two eka and vice versa, varying the type of "semitone small or big" perceived. A melody could fluctuate continually varying intervals in one and two eka, according to the principle of microvariation or according to techniques of musical minimalism (stubborn repetitions with small changes introduced). One very interesting thing a composer must keep in mind is the ambivalence of the 2 eka interval: this interval is exactly halfway between the wholetone and the tempered semitone.
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.
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 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 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.
The intervals of 3 and 13 eka are among the most suggestive intervals in Armodue and should be classified as sweet dissonances, as the tempered major second and minor seventh.
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.
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.
Moreover, the exclusion of minor and major thirds and sixths in the dodecatonic system has already been successfully tested by several composers; from an aesthetic point of view, the lack of colour and the openness of sounds that are feeled then are particularly significant and expressive.
Another interval of Armodue perfectly equivalent to an interval of the dodecatonic system is the one of 8 eka, whose width is found to be equal to that of the tritone. However, the restless nature of that interval makes it unique and not comparable to the intervals just treated.
The intervals of 4 and 12 eka belong without doubt - being of equivalent size - to the same category that includes the minor third and the major sixth: that of sweet consonances.
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 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.
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.
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.
In the tempered system, thirds and sixths give color to the chords, while the base tones of the same chords are concatenated in jumps of perfect fourths and fifths. In Armodue, the ambiguity of the "neutral" intervals that mediate between third and fourth as between fifth and sixth makes them suitable for use both in the construction of the chords and in the concatenation XXX of the base tones.
Like the intervals of 2 and 14 eka (which are also "neutral": the first halfway between minor and major second and the second equidistant from minor and major seventh), the intervals of 6 and 10 eka are unfamiliar to the western ear (excluding those accustomed to the quarter-tone).
The intervals of 6 and 10 eka are classified as neutral consonances.
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).
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.
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.
As for the classification, 7 and 9 eka belong to the open consonances.
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.
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.
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.
Gradation of harmonic tensions
When a chord is followed by another chord, we must carefully evaluate the consonance/dissonance quality of every interval contained - in addition to dealing with the fluidity of the movement of the voices. If different chords must be put into a sequence, we recommend to let them proceed so that there is a gradation in the distribution of the interval tensions.
The progression order (forward or backwards) will be as follows:
Open consonances - neutral consonances - sweet consonances - sweet dissonances - neutral dissonances - harsh dissonances - unstable dissonances.
If, for example, a progression is to be built containing a chord "A", where sweet dissonances prevail, a chord "B", where open consonances prevail, and a "C", with prevalence of neutral consonances, possible progressions are B - C - A (order of growing tension) or A - C - B (order of decreasing tension).
Arrangement of tones in the chords
Chords can be realized in any way, placing the notes that constitute it more or less distributed (narrow or wide) on a higher or a lower register (more or less clarity), in pyramid form or pyramid upside down ( more or less harmonic force) or in another form, with doubles, repetitions and omissions of one or more parties (emphasis on the intervals involved in the doubled or tripled notes).
Compound intervals (i.e. one or tenths added to the the simple interval), in chords with more widely distributed notes, come out clearer and more resonant than the corresponding simple (octave-reduced) intervals in the case of sweet or neutral consonances, more powerful in the case of open consonances - while the intervals of dissonant character, when appearing as compound intervals, lose much of their sharp character and gain brilliance.
When in a chord the intervals between adjacent notes decrease from the bass to the melody note, we have the pyramid form, which increases the clarity and resonance of the chord (since the ear associates this to the series of the harmonics); the contrary happens stacking the intervals in increasing size (upside down pyramid form).
If we want to emphasize the dissonant quality of a chord, we can simply double or triple the notes that form the dissonant intervals, and we will proceed similarly with the notes that form consonances to increase the consonance of the chord.
Chapter 3:
Creating scales with Armodue: modal systems
Modal systems based on tetrachords and pentachords
There are many systems how to generate various scales in Armodue. The most important ones, however, are based on the formation of tetrachords or pentachords and their subsequent unions. By analogy with the tetrachords (modes) of the interval system that was already developed in ancient Greek music, the tenth of Armodue (the tempered octave) is divided into two intervals of seven eka each (seven eka have a width close to five semitones or a perfect fourth of the tempered system): a lower interval of seven eka and an upper interval of also seven eka.
The two intervals constituted that way will limit the tetrachords or pentachords that will be constituted inside of them and will be disjoint with a central interval of two eka between - to sum up to 16 eka or the entire scope of the Tenth of Armodue.
Here is the distribution pattern of the Tenth 1~1:
1, 1#, 2, 2#, 3, 3#, 4, 5 - 5# - 6, 6#, 7, 7#, 8, 8#, 9, 1
The first interval of 7 eka, the lower one, spans the scope of notes between '1' and '5', the second interval of 7 eka, the upper one, includes notes from '6' to '1'.
Between the two intervals there is a disjunction of 2 eka (note '5' to '6'). Once the two limiting intervals are established as hinges or fixed structures, we proceed to "find notes" within the two intervals. In practice, the task is to partition the scalar distances between notes '1' ~ '5' and '6' ~ '1' in various ways. 7 eka can, for example, be partitioned into 3 + 3 + 1 eka obtaining the lower tetrachord '1', '2#', '4', '5' and/or the upper tetrachord '6', '7#', '9', '1'.
On the other hand, seven eka can be arranged in a pentachord whose formula (interval structure) is 2 + 2 + 2 + 1 eka. In this latter case, we will have a lower pentachord '1', '2', '3', '4', '5' and/or an upper pentachord '6', '7', '8', '9', '1'. The lower and the upper interval do not necessarily have to by divided in the same way; each lower tetrachord/pentachord can be combined with any tetrachord/pentachord type for the upper half, to create a big variety of heptatonic (type tetrachord/tetrachord), octatonic (type tetrachord/pentachord) or nonatonic (type pentachord/pentachord) scales.
Here is the table of all possible tetrachord/pentachord formulas (excluding formulas that contain conscutive successions of 1 eka, for their excessive chromatic quality) across a total of seven eka:
TETRACHORDS
Tetrachordal interval partition: 1 + 2 + 4 eka
1, 2, 4
1, 4, 2
2, 1, 4
2, 4, 1
4, 1, 2
4, 2, 1
Tetrachordal interval partition: 1+ 3 + 3 eka
1, 3, 3
3, 1, 3
3, 3, 1
Tetrachordal interval partition: 2 + 2 + 3 eka
2, 2, 3
2, 3, 2
3, 2, 2
Tetrachordal interval partition: 1 + 1 + 5 eka
1, 5, 1
PENTACHORDS
Pentachordal interval partition: 1 + 2 + 2 + 2 eka
1, 2, 2, 2
2, 1, 2, 2
2, 2, 1, 2
2, 2, 2, 1
Pentachordal interval partition: 1 + 1 + 2 + 3 eka
1, 2, 1, 3
1, 3, 1, 2
2, 1, 3, 1
3, 1, 2, 1
1, 2, 3, 1
1, 3, 2, 1
Overall, we have 13 types of tetrachords and 10 types of pentachords at our disposition to create scales; the number of different realizable scales is therefore 23 (23 = 13 + 10) squared, that is something like 529 different scales (in the context of scales that are organized into two limiting intervals of seven eka with 2 eka between).
Each of these scales is transposable to any key (the possibilities are: 8464 scales; the product of 529 types of scales * 16 possible tonics).
If for example we combine the tetrachord with formula 2, 2, 3 with the pentachord formula with 2,1,3,1 get the eight-note (octatonic) scale formed by the following notes (starting with note '1'):
'1', '2', '3', '5' - '6', '7', '7#', '9', '1'.
To build scales with full awareness, we must be familiar with the listed 23 types of tetrachords and pentachords and thoroughly study their sound and quality. Only when we have fully mastered the structures at the base of the scales - precisely the tetrachords and pentachords - we can proceed to their mutual combination in the formation of scales.
Modal systems based on hexachords
The tetrachords and pentachords treated so far are focused on the characteristic interval of 7eka, the Armodue equivalent of the Diatessaron in the greek modal system (the perfect fourth, five tempered semitones wide).
In Armodue, however, there is another very significant interval that deserves to function as Pivotal interval: that of 13 eka - whose width is very close to the natural minor seventh (the frequency ratio between the seventh and the fourth harmonic of the overtone series [7/4]).
Assuming the interval of 13 eka as delimiting interval (that includes the notes '1' ~ '8' of the standard Tenth '1' to '1'), the following partitions with hexachords stand out as important (many other divisions are possible, of course):
Hexachordal interval partition: 1 + 3 + 3 + 3 + 3 eka
1, 3, 3, 3, 3
3, 1, 3, 3, 3
3, 3, 1, 3, 3
3, 3, 3, 1, 3
3, 3, 3, 3, 1
For example, taking the formula 3, 1, 3, 3, 3 we get a scale formed by the notes: '1', '2#', '3', '5', '6#', '8' to which we can add the note '9' as leading tone.
Other modal systems and various considerations on the scales in Armodue
The combination possibilities of trichords, tetrachords, pentachords, hexachords and heptachords sum up to a very large number: at least in theory, different types of intervals (5, 6, 7, 8, 9, 10 or more eka) can function as Pivotal intervals whose interior is organized in trichords, tetrachords etc. - in turn freely combined to form scales.
Listing all the possibilities is beyond the scope of this brief treatise. At least in the initial phase, however, it will be advisable to use few types of scales, maybe with frequent tonal modulations (transposing the limited number of chosen scales to different tonics).
The listener should acquire some familiarity with few well-chosen tetrachords, pentachords, etc., rather than getting disoriented by a continuous change of intervallic structures.
A peculiarity of Armodue: the symmetrical "neo-diminished" scales
A very particular sound in the Armodue environment is achieved by the symmetrical scale with the structure 1 + 3 + 1 + 3 + 1 + 3 + 1 + 3 eka or its variant 3 + 1 + 3 + 1 + 3 + 1 + 3 + 1 eka.
These two scales strongly evoke the sound of the "diminished" scales of the dodecatonic tempered system with their alternating tone-semitone pattern (widely used in jazz and blues): c - c# - d# - e - f# g - a - a#- - c (symmetrical scale of type semitone-tone) or c - d - eb - f - gb - g# - a - h - c (symmetrical scale of type tone-semitone). In Armodue, these scales are named for their special quality "neo-diminished" scales. (Actually, they are scales of diminished temperament, in 12edo as well as in 16edo!)
| Diminished ocatatonic scale in 12edo | Diminished octatonic scale in 16edo |
Modes of limited transposition
"Modes of limited transposition" incorporates and expands - applying it in Armodue - the modal system designed by Olivier Messiaen ("Modes à transpositions limitées").
The basic idea of Messiaen is to divide the octave equally in two, three, four or six equal intervals (two tritones, three major thirds , four minors thirds or six wholetones) and then structure these intervals with further internal division in groups or modules (inserting notes in a specific order).
The particularity is to make the last note of one module coincide with the first note of the next group.
For example, dividing the octave in three equal parts: c - e; e - ab; as - c, Messiaen structures his third mode by establishing the form: tone - semitone - tone and thus the succession of notes (cylce of 6 eka iterated 8 times):
c - d - eb - e - f# - g - ab - bb - b - c
The notes marked in bold c, e, ab delimit the three parts that make up the octave and constitute at the same time the first and last note of each module of the type tone - semitone - semitone.
In Armodue, the Tenth of sixteen eka can be divided into 2, 4, or 8 parts.
But apart from these subdivisions of the Tenth, the modal system of Messiaen can be extended to all intervals with sizes between 3 and 8 eka.
If, for example, we take the interval of 6 eka (2 tones plus a quartertone in the 12edo system) we obtain the following sequence of notes:
'1', '4', '7#', '2', '5#', '8#', '3', '6#', '1'.
As we can ascertain, over the range of three Tenths (equivalent to three 12-edo octaves) the circle closes, and from the starting note '1' we arrive again at note '1' three Tenths higher, by successive steps of 6 eka.
In strictly mathematical terms, the least common multiple of 16 eka (the size of the Tenth) and 6 eka (the iterated interval) is 48 eka (corresponding to the size of three Tenths).
Every interval of 6 eka - iterated eight times in the three Tenths range - can be organized inside in various ways; for example in modules of 3 + 2 + 1 eka, in which case we obtain the following scalar system of 24 notes:
'1', '2#', '3#', '4'; ('4'), '6', '7', '7#'; ('7#'), '9', '1#', '2'; ('2'), '3#', '5', '5#'; ('5#'), '7', '8', '8#'; ('8#'), '1#', '2#', '3'; ('3'), '5', '6', '6#'; ('6#'), '8', '9', '1'.
[note: the repeated notes in brackets are meant to highlight how the first and the last note of every group coincide.]
All the sixteen notes appear at least once inside the scale shown.
So, it's essential to remember the context of membership of the notes in respect to the three Tenths.
Supposing - for simplicity - to play in a three Tenths register, we'll have eight notes per Tenth at our disposal for the creation of the chord, melody and counterpoint texture.
Specifically, the notes:
'1', '2#', '3#', '4', '6', '7', '7#', '9' in the first Tenth, the lowest;
'1#', '2', '3#', '5', '5#', '7', '8', '8#' in the second Tenth, the central;
'1#', '2#', '3', '5', '6', '6#', '8', '9' in the third Tenth, the highest.
Note that if we iterate the original defining interval of 6 eka starting from note '1#' instead of '1', we obtain the eight Pivotal notes missing in the sequence that starts from note '1':
'1#', '5', '8', '2#', '6', '9', '3#', '7', '1#'.
To the notes found that way we can apply the same division according to the 3-2-1 eka module, obtaining the same 24 notes scale, but raised 1 eka.
Chapter 4:
"Geometric" harmonic constructions with Armodue
The composer Scriabin has based much of his compositions on "nucleopolar accordances" or "chord centers" (Klangzentrum).
To understand the nature of such accordances we should refer to specular reflections of light in a crystal, according to a strict central symmetry.
An image that comes in mind about is the eight-pointed 'star' that sometimes appears in a photo area when the lens faces a rather intense light source.
Scriabin's idea is to render {paint?} this optical phenomenon through the sounds.
Hence the formation of a center-note (central light source) from which depart upper interval-rays and relative higher notes and, by symmetry or reflection law, lower reflected rays-intervals with equal amplitude but with reversed direction and the corresponding notes lower than the central note.
The central note is thus more properly called nucleopolar note, a real mirror where the lower intervals and low notes are reflected in the higher intervals and high notes.
The resulting chord - built in perfect central symmetry or nucleopolarity - constitutes the fundamental sound system to which we will refer in the texture of the melodies and architecture of the chords.
The modulation principle, in this environment, is to be meant as minimal and gradual variation of the size of one or several intervals that are 'pivoted' on the chord's nucleopolar note, or - more drastically - to be interpreted as a radical and brusque transformation of the same intervals.
In Armodue, nucleopolar harmony cans find a wider and more perfect application, compared to the twelve-tone system. In fact, a sixteen notes scale seems already to contain implicitly the central symmetry principle (sixteen is the perfect square of a perfect square -4- and corresponds to the mandala archetype as conceived in almost all cultures and traditions of the world: a wheel with eight or sixteen spokes).
Let's assume, for example, the note '5' as nucleopolar. The note '5' will work as a mirror in which lower and higher intervals and notes will reflect.
If we elect the three intevals of 6, 8 and 13 eka, it will generate the following sound structure (read the notes from left to right in an orderly proceeding from low to high):
'7' - '9' - '1#' - '5' - '8' - '9' - '3'.
As we can verify, from the note '5' the two mirrored 6 eka intervals spread (defined by notes '1#' - '5' and '5' - '8'), the two intervals of 8 eka (defined by notes '9' - '5' and '5' - '9') and the two intevals of 13 eka (defined by notes '7' - '5' and '5' - '3').
We have thus obtained three notes lower than the nucleopolar note '5' (the notes '7', '9' and '1#') and three higher than the same note '5' (thenotes '8', '9', '3')
The sound structure obtained this way can be used as is in the construction of melodies and chords, or it can be liable to undergo microvariations and be modulated.
For example, we can just bring a slight variation - hence an asymmetry - that momentarily but in a significant way destabilizes the strict "crystallinity" of the sound raising the note '1#' for bend one eka (replace it by note 2):
'7' - '9' - '2' - '5' - '8' - '9' - '3'.
Later, we can restore the perfect equilibrium and symmetry of the sound structure altering in equal sense and contrary motion the note 8 that balances the note 2 (the 8 so goes down - in diametrally opposite way - of one eka and so is replaced by note '7#'):
'7' - '9' - '2' - '5' - '7#' - '9' - '3'.
The shown microvariation principle can obviously be applied also to more than a single note simultaneously.
In other circunstances, we can integrate the original structure with the addition of new intervals and notes or change the structure radically by employing different generator intervals.
Chapter 5:
"Elastic" chords
With "elastic chords" I mean all those chords constructed in a ways that all parts are equally spaced, except one or two intervals that are variable.
An example will illustrate the principle better. If you construct a chord putting six intervals on top of each other each of which is 7 eka in size, you get an equally-spaced chord composed of the notes:
'1' - '5' - '8#' - '3#' - '7#' - '2#' - '6#'.
If you now provide that one of the intervals does not have to be necessarily 7 eka but may vary at will, you get what I call an "elastic" chord: that is a chord where all the "rings" (intervals) are rigid (the equidistant intervals) except for one elastic "ring" likely to be shortened or extended.
In the example chord, if you choose the second interval to be the one with the elastic propertiy, the original chord can vary and become for example:
'1' - '5' - '8' - '3' - '7' - '2' - '6'.
(shortening the original interval '5'-'8#' to 6 eka instead of 7) or
'1' - '5' - '1#' - '5#' - '9' - '4' - '8'.
(extending the original interval '5'-'8#' to 10 eka instead of 7).
There arise many effective progressions of chords varied by the selection of one or two "elastic" intervals.