User:FloraC/Fokker analysis of rank-3 scales: Difference between revisions
m -typo |
Modal chroma > residual chroma, redefine residual stability accordingly |
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<math>\displaystyle L - l = S - s</math> | <math>\displaystyle L - l = S - s</math> | ||
Two chromas are to be named: the | Two chromas are to be named: the residual chroma | ||
<math>\displaystyle | <math>\displaystyle C_r = L - S = l - s</math> | ||
and the domal chroma | and the domal chroma | ||
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<math>\displaystyle C_d = L - l = S - s</math> | <math>\displaystyle C_d = L - l = S - s</math> | ||
These form the chroma basis {''C''<sub> | These form the chroma basis {''C''<sub>r</sub>, ''C''<sub>d</sub>} of the rank-3 Fokker block. The residual chroma is assumed to be always larger than the domal chroma – since the chroma basis is essentially orderless, no Fokker block is missed or duplicate by the assumption. | ||
Clearly, a scale step can also have | Clearly, a scale step can also have | ||
<math>\displaystyle | <math>\displaystyle | ||
C_r + C_d = L - s \\ | |||
C_r - C_d = l - S | |||
</math> | </math> | ||
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<math>\displaystyle | <math>\displaystyle | ||
C_1 = | C_1 = C_r + C_d \\ | ||
C_2 = | C_2 = C_r \\ | ||
C_3 = | C_3 = C_r - C_d \\ | ||
C_4 = C_d | C_4 = C_d | ||
</math> | </math> | ||
== Modal | == Modal, Domal and Residual Rotation == | ||
Every Fokker block can be represented by a product word of mos patterns. For rank-3, it comprises two patterns. Using the notation introduced above, we notice the rotation of one pattern changes the nominal of the scale step, and the other changes the case. We will call the former the nominal pattern, and the latter the case pattern. | Every Fokker block can be represented by a product word of mos patterns. For rank-3, it comprises two patterns. Using the notation introduced above, we notice the rotation of one pattern changes the nominal of the scale step, and the other changes the case. We will call the former the nominal pattern, and the latter the case pattern. | ||
Modal rotation refers to the rotation of scale which keeps the overall pattern. In terms of product words, modal rotation rotates both patterns in the same direction and magnitude. | Modal rotation refers to the rotation of scale which keeps the overall pattern. In terms of product words, modal rotation rotates both patterns in the same direction and magnitude. | ||
Domal rotation | Domal rotation is a rotation independent of the mode. With mode labeled by brightness, domal rotation changes the scale while keeping the mode label. There is only one operation that does it: it only rotates the case pattern. Residual rotation does exactly the orthogonal to domal rotation: it only rotates the nominal pattern. | ||
For even higher-rank scales, the concepts can be generalized to a modal rotation and a sequence of domal rotations. For example, a rank-4 scale has a modal rotation, a primary domal rotation, and a secondary domal rotation. The modal rotation rotates all patterns, whereas each domal rotation rotates the corresponding pattern and all the subsequent patterns alike. | It follows that modal rotation is the composition of domal rotation and residual rotation. | ||
For even higher-rank scales, the concepts can be generalized to a modal rotation and a sequence of domal rotations. The residual rotation is any composition of a higher-level rotation and the inverse of a lower-level rotation. For example, a rank-4 scale has a modal rotation, a primary domal rotation, and a secondary domal rotation, with three residual rotations between any two of them. The modal rotation rotates all patterns, whereas each domal rotation rotates the corresponding pattern and all the subsequent patterns alike. | |||
== Illustration == | == Illustration == | ||
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The 2-step clearly shows us that the four chromas involved are 2187/2048, 135/128, 25/24 and 81/80, so we find | The 2-step clearly shows us that the four chromas involved are 2187/2048, 135/128, 25/24 and 81/80, so we find | ||
<math>\displaystyle | <math>\displaystyle C_r = 135/128, C_d = 81/80</math> | ||
Therefore, we find | Therefore, we find | ||
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|} | |} | ||
The chromas are 135/128, 25/24, and 81/80. Since there are only three sizes, how can we know which is ''C''<sub> | The chromas are 135/128, 25/24, and 81/80. Since there are only three sizes, how can we know which is ''C''<sub>r</sub> and which is ''C''<sub>d</sub>? It turns out that every possible assignment works as an actual Fokker block. A scale with this property is what they call a wakalix. | ||
; Fokker block 1 | ; Fokker block 1 | ||
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First we may assume | First we may assume | ||
<math>\displaystyle | <math>\displaystyle C_r = 135/128, C_d = 81/80</math> | ||
It is the same as Aura's diatonic scale. Therefore, we find | It is the same as Aura's diatonic scale. Therefore, we find | ||
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; Fokker block 2 | ; Fokker block 2 | ||
<math>\displaystyle | <math>\displaystyle C_r = 25/24, C_d = 81/80</math> | ||
Therefore, we find | |||
<math>\displaystyle L = 9/8, l = 10/9, S = 27/25, s = 16/15</math> | <math>\displaystyle L = 9/8, l = 10/9, S = 27/25, s = 16/15</math> | ||
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; Fokker block 3 | ; Fokker block 3 | ||
<math>\displaystyle | <math>\displaystyle C_r = 135/128, C_d = 25/24</math> | ||
Therefore, we find | Therefore, we find | ||
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The different part is that still four distinct chromas are involved. They are 7168/6561, 784/729, 343/324, 64/63. The set defines a unique chroma basis. It is | The different part is that still four distinct chromas are involved. They are 7168/6561, 784/729, 343/324, 64/63. The set defines a unique chroma basis. It is | ||
<math>\displaystyle | <math>\displaystyle C_r = 784/729, C_d = 64/63</math> | ||
Therefore, we find | Therefore, we find | ||
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|} | |} | ||
However, the step size relations do not satisfy the property of rank-3 Fokker blocks. Specifically, the 3-steps are 729/640, 6/5, 5/4 and 320/243, and we have (6/5)/(729/640) = (320/243)/(5/4). Therefore, we find the chromas ''C''<sub> | However, the step size relations do not satisfy the property of rank-3 Fokker blocks. Specifically, the 3-steps are 729/640, 6/5, 5/4 and 320/243, and we have (6/5)/(729/640) = (320/243)/(5/4). Therefore, we find the chromas ''C''<sub>r</sub> = 800/729 and ''C''<sub>d</sub> = 256/243. In the 5-steps, they are 27/20, 45/32, 64/45, 40/27. This corresponds to a different set of chromas: ''C''<sub>r</sub> = 256/243 and ''C''<sub>d</sub> = 25/24. No Fokker block has different set of chromas. | ||
Indeed, blackdye cannot be decomposed into the product word of two mosses. | Indeed, blackdye cannot be decomposed into the product word of two mosses. | ||
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For rank-3 scales, the assumption is that the domal chroma is smaller than the modal chroma, so we may let go of an extension of propriety, but stability is really a different story. | For rank-3 scales, the assumption is that the domal chroma is smaller than the modal chroma, so we may let go of an extension of propriety, but stability is really a different story. | ||
The modal–domal stability can be considered in two ways. First, there is the stability concerning the entire pitch spectrum, defined as the portion of the pitch spectrum which is not covered by the interval differences of each class on domal rotations. We will dub it the domal stability. Second, there is the stability concerning the modal coverage, defined as the portion of the pitch spectrum which is not covered by the interval differences of each class on | The modal–domal stability can be considered in two ways. First, there is the stability concerning the entire pitch spectrum, defined as the portion of the pitch spectrum which is not covered by the interval differences of each class on domal rotations. We will dub it the domal stability. Second, there is the stability concerning the modal coverage, defined as the portion of the pitch spectrum which is not covered by the interval differences of each class on residual rotations. It is equivalent to the sum of the portion which is not covered by those on modal rotations and the portion which is covered by those on domal rotations. We will dub it the residual stability. The following identity holds: | ||
<math>\displaystyle | <math>\displaystyle \text{modal stability} = \text{domal stability} + \text{residual stability} - 1</math> | ||
\text{ | |||
</math> | |||
It can be generalized to other ranks as follows. A rank-''r'' scale has ''m''-stability, where ''m'' is a natural number. The 0-stability is the modal stability. For ''m'' ≥ 1, the ''m''-stability is defined as the portion of the pitch spectrum which is not covered by the interval differences of each class on ''m''-th-level domal rotations. The (''m''<sub>1</sub>, ''m''<sub>2</sub>)-residual stability is defined as the | It can be generalized to other ranks as follows. A rank-''r'' scale has ''m''-stability, where ''m'' is a natural number. The 0-stability is the modal stability. For ''m'' ≥ 1, the ''m''-stability is defined as the portion of the pitch spectrum which is not covered by the interval differences of each class on ''m''-th-level domal rotations. The (''m''<sub>1</sub>, ''m''<sub>2</sub>)-residual stability is defined as the sum of the portion which is not covered by the interval differences of each class on ''m''<sub>1</sub>-th-level domal rotations and the portion which is covered by those on ''m''<sub>2</sub>-th-level domal rotations. | ||
The ''m''-stability is unity for ''m'' ≥ ''r'', meaning that those levels of domal rotations do not exist. | The ''m''-stability is unity for ''m'' ≥ ''r'', meaning that those levels of domal rotations do not exist. | ||
The generalized stability can also be measured for virtually anything (for example, a maqam) as long as a particular layer of abstraction is marked. To measure the stability of a maqam, all the ajnas involved and all the tunings must be specified. Let us consider maqam rast in 24edo tuning – just to demonstrate, assume only jins rast, nahawand and upper rast are used. Say we want to measure the stability caused by the switch of ajnas. Then the quartertone difference between jins nahawand and jins rast is the only variation. The stability at this level is therefore 23/24. | The generalized stability can also be measured for virtually anything (for example, a maqam) as long as a particular layer of abstraction is marked out. To measure the stability of a maqam, all the ajnas involved and all the tunings must be specified. Let us consider maqam rast in 24edo tuning – just to demonstrate, assume only jins rast, nahawand and upper rast are used. Say we want to measure the stability caused by the switch of ajnas. Then the quartertone difference between jins nahawand and jins rast is the only variation. The stability at this level is therefore 23/24. | ||
== Release Notes == | == Release Notes == | ||
© 2022 Flora Canou | © 2022 Flora Canou | ||
Version | Version Stable 0 | ||
This work is licensed under the [https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Attribution-ShareAlike 4.0 International License]. | This work is licensed under the [https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Attribution-ShareAlike 4.0 International License]. | ||