time passes … I’ll be writing more, still developing ideas in music theory as it relates to intonation. Update 9/2015 I’m troubled by how much google has crept in here … that is a very scary operation
You saw it here first! A triple flat F double sharp used to get from F major (tuned high) to D major (tuned lows) in a gross chromatic fugue, finale of String Quintet (HGI 0589, 2015). No guarantees that it will survive the edit/review … I don’t really want a hostile ‘cellist.
The markings are described elsewhere in this blog, but briefly the heavy down triangle a comma lower than the tonic stack-of-pure-fifths, the open up triangle a comma higher, the double down triangle two commas lower. The circle+cross, the light up arrow, and the light down arrow are in fifth relations among themselves, toward the dominant (up) and toward the subdominant (down).
New recording, correcting Bb A leading to third chord:
As suggested by comments from Gene Ward Smith, one can replace my arbitrary 2-digit id numbers with a 3 member vector of prime factors (the number of 2s, the number of 3s, and the number of 5s).
And so, one can label the stacks in the 3-unit-vectors that show number of prime factors, as long as one identifies the starting point. From A440, the central stack is the [0,0,0] stack; the octave higher the [1,0,0] stack; the octave lower is the [-1,0,0] stack. The major thirds in relation to the [0,0,0] stack form a [-2, 0, 1] stack; the minor thirds in relation to the [0,0,0] stack form a [1, 1, -1] stack. In relation to the octave higher [1,0,0] stack the major thirds are [-1,0,1] and the minor thirds are [2, 1, -1]. In relation to the octave lower, the major thirds are [-3, 0, 1] and the minor thirds are [0, 1, -1]. In relation to the central stack, the two-major-third group that I’ve called “thirds of secondary dominants” are [-4, 0, 2] and the two-minor-third group, rarely used, are [2, 2, -2]. The very rarely used three-major-third stack would be [-6, 0, 3] and the almost never used three-minor-third stack would be [3, 3, -3]. Then a path from one note to another begins (?always, for convenience?) as a movement of fifths up or down the beginning stack, followed by the necessary octave shifts to different ‘stack groups’, followed by whatever major third or minor third movement is necessary.
In the following discussion I refer to octaves using the Helmholtz standard markings, where a’ is the same as the USA standard A4, the note A with a frequency of 440 cps. This is in order to use number suffixes as lattice identifiers (“names”) without confusion.
Taking A=440, a’, as the center, the generator, and the base of a stack of pure
fifths, name it A40. That is, assign it to a lattice identified with the suffix 40. The name given to the octave higher A=880, a”, becomes A50; the name for the octave lower, A=220, a, becomes A30.
The harmonic fifth above A=440 is in a higher Helmholtz octave, E=660, e”, but remains named in reference to the original A40, so is identified as E40. The fifth below A=440, D=293.33, d’, is identified as D40.
The preceding preliminary discussion is intended to clarify that the id numbers do not match octave numbers as in the USA standard.
The ’30’ and the ’40’ Pythagorean stacks of pure fifths, which form the backbones for two lattices identified as ’30’ and ’40’, are given below. Each note is given an approximate frequency and its Hemholtz octave marking. The center note of the 30 stack is A=220cps, an octave lower than the center note of the 40 stack (A=440cps). The 50 stack, not illustrated, is an octave higher than the 40 stack, and there are further octave transpositions above (60, 70, and so on) and below (20, 10, and so on, and ignoring for the moment the arithmetical problems the 00 stack might pose).
E#30 etc E#4 etc A#30 = A#(3k+) =a#'''' A#4 = A#(7k+) = a#v D#30 = D#(2k+) =d#'''' D#4 = D#(5k+) = d#v G#30 = G#(1k+) =g#''' G#4 = G#(3k+) = g#'''' C#30 = C#(1k+) =c#''' C#4 = C#(2k+) = c#'''' F#30 = F#(74x) =f#'' F#4 = F#(1k+) = f#''' B 30 = B (495) =b' B 4 = B (990) = b'' E 30 = E (330) =e' E 4 = E (660) = e'' --- A 30 = A (220) =a --- --- A 4 = A (440) = a' --- D 30 = D (293) =d D 4 = D (293) = d' G 30 = G (196) =G G 4 = G (196) = g C 30 = C (12x) =C C 4 = C (12x) = c F 30 = F (08x) =F, F 4 = F (08x) = F Bb30 = Bb(05x) =Bb, Bb4 = Bb(05x) = Bb, Eb30 = Eb(03x) =Eb,, Eb4 = Eb(03x) = Eb, Ab30 = Ab(02x) =Ab,,, Ab4 = Ab(02x) = Ab,, Db30 etc Db4 etc
The ’31’, ’41’ and ’51’ stacks of pure major thirds in relation to each of
the stack of fifths:
Fx31 fx'''' | Fx41 fxv | Fx51 fxv' D#30 d#'''' | D#40 d#v | D#50 d#v' B#31 b#''' | B#41 b#''' | B#51 b#v G#30 g#''' | G#40 g#''' | G#50 g#v E#31 e#''' | E#41 e#''' | E#51 e#v C#30 c#''' | C#40 c#''' | C#50 c#v A#31 a#'' | A#41 a#''' | A#51 a#'''' F#30 f#'' | F#40 f#''' | F#50 f#'''' D#31 d#'' | D#41 d#''' | D#51 d#'''' B 30 b' | B 40 b'' | B 50 b''' G#31 g#' | G#41 g#'' | G#51 g#''' E 30 e' | E 40 e'' | E 50 e''' C#31 c#' | C#41 c#'' | C#51 c#''' A 30 a | A 40 a' | A 50 a'' F#31 f# | F#41 f#' | F#51 f#'' D 30 d | D 40 d' | D 50 d'' B 31 B | B 41 b | B 51 b' G 30 G | G 40 g | G 50 g' E 31 E | E 41 e | E 51 e' C 30 C | C 40 c | C 50 c' A 31 A, | A 41 A | A 51 a F 30 F, | F 40 F | F 50 f D 31 D, | D 41 D | D 51 d Bb30 Bb,, | Bb40 Bb, | Bb50 Bb G 31 G,, | G 41 G, | G 51 G Eb30 Eb,, | Eb40 Eb, | Eb50 Eb C 31 C,, | C 41 C, | C 51 C Ab30 Ab,,, | Ab40 Ab,, | Ab50 Ab, F 31 F,,, | F 41 F,, | F 51 F, Db30 Db,,, | Db40 Db,, | Db50 Db, Bb31 Bb,,,, | Bb41 Bb,,, | Bb51 Bb,, Gb30 Gb,,,, | Gb40 Gb,,, | Gb50 Gb,, Eb31 Eb,,,, | Eb41 Eb,,, | Eb51 Eb,, Cb30 Cb,,,, | Cb40 Cb,,, | Cb50 Cb,,
To get from c to d, remembering the Hemholtz notation indicates the USA 3 octave, as a Pythagorean whole step, then, is not a simple two-step jump up the 40 stack: one must take into account the octave. One must go from C40 to D30, and the three shortest paths are C40-G40—D40-D30, C40-G40-G30-D30 and C40-C30-G30-D30. These paths have all have three steps, not two. Tracking the intervals by ratio for the first path, we have stepped 3/2 (up a fifth), 3/2 (up a fifth) and 1/2 (down an octave). Multiplying the ratios gives us (3*3*1) / (2*2*2) = 9/8, or a large, Pythagorean, whole step.
Further, the ‘quality’ of the steps within any one of the paths differ, though each path is equivalent. Each path requires one step of ‘octave quality’ and two steps of ‘fifth quality’.
If we scan the three partial lattices above for another d we find D51 to be apparently the same note. As we follow the new paths, however, we will find D51 to be a small whole step from C40, an interval of 10/9 ratio as opposed to the larger 9/8 ratio.
So to get from c to d as a small whole step, is a different set of paths: we must arrive at D51 from C40, and, remembering in these partial lattices the only interval steps are the octave, the fifth, and the major third, we find that two of the several equivalent paths are C40-F40-Bb40-D41-D51 and C40-C50-F50-Bb50-D51. Each path requires one step of ‘octave quality’, two steps of ‘fifth quality’ and one step of ‘major third’ quality. Tracking the intervals by ratio for the first path, we have stepped 2/3 (down a fifth) 2/3 (down a fifth) 5/4 (up a major third) and 2/1 (up an octave). Multiplying the ratios gives us (2*2*5*2) / (3*3*4*1) = 40 / 36 = 10/9, a small whole step.
And, naturally, when we add the next harmonic interval, the minor third of ratio 6/5, to the set of possible lattice steps, we can replace the combined one step of ‘fifth quality’ and one step of ‘major third quality’ with a single step of ‘minor third quality’. Without redrawing the lattices we can imagine the new path as C40-F40-D41-D51, and by ratios 2/3 (down a fifth) 5/6 (down a minor third) and 2/1 (up an octave). Multiplying the ratios gives us (2*5*2) / (3*6*1) = 20/18 = 10/9.
Lattice with pure thirds above and below:
Edf+ Bdf+ Fb + Cb + Gb + Db + Ab + Eb + Bb + F + C + G + D + A + E + B / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / * Cb * Gb * Db * Ab * Eb * Bb * F * C * G * D * A * E * B * F# * C# * G# * / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B#
Lattice with pure thirds above and below, twice removed:
(++) ddf adf edf bdf fb cb gb db ab eb bb f c g / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ Edf+ Bdf+ Fb + Cb + Gb + Db + Ab + Eb + Bb + F + C + G + D + A + E + B / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / * Cb * Gb * Db * Ab * Eb * Bb * F * C * G * D * A * E * B * F# * C# * G# * / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ c g d a e b f# c# g# d# a# e# b# fx cx (--)
Same lattice as above, with approximate tuning in cents with respect to 12EDO
based on C:
(++) ddf adf edf bdf fb cb gb db ab eb bb f c g +20 +22 +24 +26 +28 +30 +32 +34 +36 +38 +40 +42 +44 +48 / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ Edf+ Bdf+ Fb + Cb + Gb + Db + Ab + Eb + Bb + F + C + G + D + A + E + B +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +24 +26 +28 +30 / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / * Cb * Gb * Db * Ab * Eb * Bb * F * C * G * D * A * E * B * F# * C# * G# * -14 -12 -10 -8 -6 -4 -2 0 +2 +4 +6 +8 +10 +12 +14 +16 / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# -28 -26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ c g d a e b f# c# g# d# a# e# b# fx cx (--) -44 -42 -40 -38 -36 -34 -32 -30 -28 -26 -24 -22 -20 -18 -16
I would like to name two paths forward: “intonalistic serialism” and “intonalistic pitch class sets”.
For the first, I can imagine a new serialism where a) there are more than 12 notes, b) certain intervals are allowed (those that can be tuned) and others disallowed (those that can’t be tuned). The question of the fifth to minor seventh (untunable as used in traditional dominant-tonic harmony) is left for later.
For the second, it would be great to develop some of the tools used in manipulation 12-member pitch class sets (and sets with fewer members) with regard to the tuning of the individual elements, and perhaps extend to sets of greater numbers. As a first step, perhaps, the 8 note diatonic set of tonic, low supertonic, high supertonic, mediant, subdominant, dominant, submediant, leading tone. Then the 10 note set of tonic-dominant harmony. usw.