Yahoo Tuning Groups Ultimate Backup tuning-math The Unidala

I have posted "The Unidala" in the folder "J_Gill" at the files page for this group. It is a way of depicting the relationship between the linear progression of integers giving birth to a geometric spiral of musical intervals, focusing on the twelve possible major triads of a 12 tone JI system utilizing 4:5:6, 8:10:12, and 12:15:18 intervals.

Note that the integer spacing depicting movements of 4 half steps between the root and the Maj 3 is identical to the integer spacing depicting movements of 3 half steps between the Maj 3 and the 5 notes, for each of the twelve triad represented.

The pattern spirals outward with a scale factor of 1 for F-A-C (4:5:6), a scale factor of 2 (or twice the distance around the spiral) for C-E-G (8:10:12), and a scale factor of 3 (three times the distance around the spiral) for G-B-D (12:15:18).

The spiral continues outward, corresponding to the notes of the major triads in certain locations of the web-like structure, and NOT corresponding to any notes of a twelve tone sytem in other of its locations, to a total of 72 locations total (before the triads begin to repeat). The location numbered 7, followed by increases of integer multiples of 12 (19 31, 43, 55, 67) represent what I call "null locations", simply bridging a numerical gap between (for instance) C as location 6 as well as C as location 8, and D as location 18, as well as D as location 20. Beginning after null location 19, the depiction of the major triads repeats as the spiral grows utilizing integer spacings identical to the C-E-G (8:10:12) and G-B-D (12:15:18) patterns.

Following integer location 18 (D as the 5th of a triad) the correspondence between linear growth by integers and geometric growth of JI musical intervals breaks down.
At the left and right sides of the Unidala spiral, two independent "cyclic" whole-tone scales are formed. On the left side, from the 5th (or G) travelling DOWNWARD in pitch in whole-step movements (from 7, 5, 3, 1, 11, to 9) numbered movements.
On the right side, from the 1/1 (or C) travelling UPWARD in pitch in whole-step movements (from 0, 2, 4, 6, 8, to10). These counter-rotating polygonal (hexagonal, in this case) structures, which also represent TWO consecutive movements of 7 half-steps (up or down), are key structures for translation from the Chromatic domain to the Melodic (circles of Fourths/Fifths) domain, as well as to the Harmonic domains.

The integers 1 thru 8 generate the six JI ratios 6/5, 5/4, 4/3, 3/2, 8/5, and 5/3,
corresponding to a scale which includes the Min 3, Maj 3, 4, 5, Min 6, and Maj 6 intervals, centered around the 1/1 interval, are described by a matrix notated by (the amazing Joe) Monzo's "JustMusic" notation as follows:

3^(-1) 5^(1) 3^(0) 5^(1) OR A E
I I I I
3^(-1) 5^(0) 3^(0) 5^(0) 3^(1) 5^(0) F C G
I I I I
3^(0) 5^(-1) 3^(1) 5(-1) Ab Eb

where a syntonic bridge of 81/80 exists between C and Ab, G and Eb, and a syntonic bridge of 80/81 exists between F and A, C and E.

The integers 8 thru 16 generate the 4 JI ratios of 16/15 and 9/8, and their octave -reduced inversions generate the JI ratios 9/5 and 15/8, respectively. Finally, the
Flat 5 interval of Gb is generated by (15/16)(3/2) = 45/32, yielding the JI scale made up of the (octave reduced) intervals of 16/15, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 9/5, and 15/8, in addition to 1/1. A syntonic bridege between 9/8 and its octave-reduced inversion of 9/5 constitutes the only deviation from harmonic-subharmonic symmetry around the Flat 5 interval, or more generally, the interval representing the mid-point in movement from 1/1 to 2/1 in even numbered scales.

The corresponding matrix for all twelve notes (in the manner presented above) is:

3^(-1) 5^(1) 3^(0) 5^(1) 3^(1) 5^(1) 3^(2) 5^(1)
I I I I
3^(-1) 5^(0) 3^(0) 5^(0) + 3^(1) 5^(0) 3^(2) 5^(0)
I I I I
3^(-1) 5^(-1) 3^(0) 5^(-1) 3^(1) 5^(-1) 3^(2) 5^(-1)

A E B Gb
I I I I
F C + G D
I I I I
Db Ab Eb Bb

The above is identical (in structure) to AJ Ellis' "Harmonic Duodene", and includes the ratio of 9/5 as noted by Monzo as being utilized by Ramos and incorporated into the identical 12 note "periodicity block" by De Caus, and also comprises the 12 core intervals centered (between the notes) in the even numbered 22-note Indian Struti scale structure as described in Monzo's recent paper on the subject of Strutis.

It is interesting to note that, as a result of a syntonic comma in the 3^(2) 5^(-1) position of the matrix shown above, the ratio 9/5 (as opposed to the ratio 16/9) constitutes symmetry (albeit around the space between notes) of a structure
which preserves contours (rows) of successive powers of 3^(n) as well as contours (columns) of successive powers of 5^(n), despite its asymmetry around 3^(0) 5^(0), or 1/1. Perhaps, while all roads may well lead TO 1/1, or 3^(0), at least some roads may, in their subdominant/dominant/root (or, more generally, tension and resolution)
pathways, engender the direction of resolution at or around 1/1 via structures centered around the dissonant center (as in the "One Footed Bride"). Thus, some roads may "originate" from a structural center, so to speak, of "3^(1/2) 5^(0)", or the space BETWEEN the intervals. While I am NOT proposing the use of the SQR (3),
or 1.73205080757, by any means, it appears to this author that symmetry around the CENTER point of movement from 1/1 to 2/1 in even numbered scales of the degree 12, 18, 24, 30, ... etc. may, with or without commatic effects (such as the transformation of what would, upon first blush of the "One Footed Bride" appear,
for the sake of geometric symmetry, as 16/9 instead of 9/5) represent a more meaningful axis of symmetry from which to analyze tonal patterns and movements which, eventually, as Monzo states, lead to 1/1.

It seems likely to me, though unproven that these concepts may, with additional rigor, be applied generally to all scale degrees beginning with 12 and increasing by the number 6 (ie 12, 18, 24, 30...) which, in fact, have a center of symmetry in MOVEMENT BETWEEN tones, as opposed to a specific tonal center of symmetry
characteristic of scales of odd-numbered degree.

I would be very interested in thoughts and comments regarding such a scale as described above, including any comments as to the practical, rather than purely theoretical (as my thoughts are), application of the use of 9/5 as the Minor 7 interval in conjunction with the use of 9/8 as the Major 2 interval in 12 tone scales, and/or your thoughts/speculations regarding the extension of the scale degree to the higher numbers indicated above in this posting. Requests for clarifications are welcome.

Best Regards, J Gill

The Unidala

🔗JGill99@imajis.com

🔗Paul Erlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

🔗monz <joemonz@yahoo.com>