User:FloraC/Fokker analysis of rank-3 scales
Fokker analysis refers to the analysis of scales as Fokker blocks. In this essay, Fokker analysis of rank-3 scales is illustrated and the related issues are discussed in later chapters.
Rank-1 scales i.e. equal-step scales and rank-2 scales i.e. mos scales have been well explored. The scene of rank-3 scales has been a mess. Fragmented definitions and theories are all over the place. It will benefit from Fokker analysis very prominently.
Notation
For rank-3 Fokker blocks we denote the steps, sorted by size, as L, l, S, s. Let us bear in mind that they satisfy
[math]\displaystyle{ \displaystyle L - l = S - s }[/math]
Two chromas are to be named: the modal chroma
[math]\displaystyle{ \displaystyle C_m = L - S = l - s }[/math]
and the domal chroma
[math]\displaystyle{ \displaystyle C_d = L - l = S - s }[/math]
These form the chroma basis {Cm, Cd} of the rank-3 Fokker block. The modal 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
[math]\displaystyle{ \displaystyle C_m + C_d = L - s \\ C_m - C_d = l - S }[/math]
In general, the four distinct chromas in a rank-3 Fokker block, sorted by size, are
[math]\displaystyle{ \displaystyle C_1 = C_m + C_d \\ C_2 = C_m \\ C_3 = C_m - C_d \\ C_4 = C_d }[/math]
Modal Rotation and Domal 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.
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 does exactly the orthogonal. 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.
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.
Illustration
The process of Fokker analysis will be illustrated with four examples. Make sure to go through all of them because some issues will be explained in the course.
Example 1: Aura's Diatonic Scale
Aura's diatonic scale is a very typical rank-3 Fokker block. The steps are:
9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1
Hence, we see the three types of steps: 9/8, 10/9 and 16/15. Only looking at the step sizes does not reveal much, but if we look at all the classes i.e. the interval matrix formed by modal rotations, we find the maximum variety is 4.
| Steps | Intervals |
|---|---|
| 1-step | 16/15, 10/9, 9/8 |
| 2-step | 32/27, 6/5, 5/4, 81/64 |
| 3-step | 4/3, 27/20, 45/32 |
| 4-step | 64/45, 40/27, 3/2 |
| 5-step | 128/81, 8/5, 5/3, 27/16 |
| 6-step | 16/9, 9/5, 15/8 |
| 7-step | 2/1 |
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{ \displaystyle C_m = 135/128, C_d = 81/80 }[/math]
Therefore, we find
[math]\displaystyle{ \displaystyle L = 9/8, l = 10/9, S = 16/15, s = 256/243 }[/math]
The pattern is then
[math]\displaystyle{ \displaystyle \text{LlSLLlS} }[/math]
It can be decomposed into the product of two diatonic scales: LLsLLLs × LsLLLsL. From here, we see the scale in question is in
[math]\displaystyle{ \displaystyle \text{mode 5|1, dome 3|3} }[/math]
Example 2: Zarlino
A more interesting example is zarlino. We will soon find out how it can be viewed as multiple Fokker blocks. The steps are:
9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1
The interval matrix formed by modal rotations shows it has a maximum variety of 3.
| Steps | Intervals |
|---|---|
| 1-step | 16/15, 10/9, 9/8 |
| 2-step | 32/27, 6/5, 5/4 |
| 3-step | 4/3, 27/20, 45/32 |
| 4-step | 64/45, 40/27, 3/2 |
| 5-step | 8/5, 5/3, 27/16 |
| 6-step | 16/9, 9/5, 15/8 |
| 7-step | 2/1 |
The chromas are 135/128, 25/24, and 81/80. Since there are only three sizes, how can we know which is Cm and which is Cd? 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
First we may assume
[math]\displaystyle{ \displaystyle C_m = 135/128, C_d = 81/80 }[/math]
It is the same as Aura's diatonic scale. Therefore, we find
[math]\displaystyle{ \displaystyle L = 9/8, l = 10/9, S = 16/15, s = 256/243 }[/math]
The pattern is then
[math]\displaystyle{ \displaystyle \text{LlSLlLS} }[/math]
It can be decomposed into the product of two diatonic scales: LLsLLLs × LsLLsLL. From here, we see the scale in question is in
[math]\displaystyle{ \displaystyle \text{mode 5|1, dome 2|4} }[/math]
- Fokker block 2
[math]\displaystyle{ \displaystyle C_m = 25/24, C_d = 81/80 }[/math]
We find
[math]\displaystyle{ \displaystyle L = 9/8, l = 10/9, S = 27/25, s = 16/15 }[/math]
The pattern is then
[math]\displaystyle{ \displaystyle \text{LlsLlLs} }[/math]
It can be decomposed into the product of a diatonic scale and a mosh scale: LLsLLLs × LssLsLs. It is in
[math]\displaystyle{ \displaystyle \text{mode 5|1, dome 4|2} }[/math]
- Fokker block 3
[math]\displaystyle{ \displaystyle C_m = 135/128, C_d = 25/24 }[/math]
Therefore, we find
[math]\displaystyle{ \displaystyle L = 75/64, l = 9/8, S = 10/9, s = 16/15 }[/math]
The pattern is then
[math]\displaystyle{ \displaystyle \text{lSslSls} }[/math]
It can be decomposed into the product of a mosh scale and an antidiatonic scale: LssLsLs × sLssLss. It is in
[math]\displaystyle{ \displaystyle \text{mode 4|2, dome 4|2} }[/math]
Example 3: Pattern "abcba"
The pattern of "abcba" is quite special. While it has a maximum variety of 3, unlike zarlino, it corresponds to a unique Fokker block. The mechanic will be revealed right here. The following steps will be used:
9/8, 9/7, 14/9, 16/9, 2/1
The interval matrix formed by modal rotations shows it has a maximum variety of 3.
| Steps | Intervals |
|---|---|
| 1-step | 9/8, 8/7, 98/81 |
| 2-step | 9/7, 81/64, 112/81 |
| 3-step | 81/56, 128/81, 14/9 |
| 4-step | 81/49, 7/4, 16/9 |
| 5-step | 2/1 |
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{ \displaystyle C_d = 784/729, C_m = 64/63 }[/math]
Therefore, we find
[math]\displaystyle{ \displaystyle L = 896/729, l = 98/81, S = 8/7, s = 9/8 }[/math]
The pattern is then
[math]\displaystyle{ \displaystyle \text{sSlSs} }[/math]
It can be decomposed into the product of an antimanic scale and a pentic scale: ssLss × sLsLs. It is in
[math]\displaystyle{ \displaystyle \text{mode 3|3, dome 3|3} }[/math]
Example 4: Blackdye
Finally, as a counterexmaple, blackdye is not a Fokker block. Here we try analysing the 5-limit JI tuning of blackdye. The steps are:
81/80, 9/8, 6/5, 4/3, 27/20, 3/2, 8/5, 16/9, 9/5, 2/1
Again, looking at the interval matrix formed by modal rotations, we find the maximum variety is 4, so it seems to suggest a rank-3 Fokker block.
| Steps | Intervals |
|---|---|
| 1-step | 81/80, 16/15, 10/9 |
| 2-step | 9/8, 32/27 |
| 3-step | 729/640, 6/5, 5/4, 320/243 |
| 4-step | 81/64, 4/3 |
| 5-step | 27/20, 45/32, 64/45, 40/27 |
| 6-step | 3/2, 128/81 |
| 7-step | 243/160, 8/5, 5/3, 1280/729 |
| 8-step | 27/16, 16/9 |
| 9-step | 9/5, 15/8, 160/81 |
| 10-step | 2/1 |
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 Cm = 800/729 and Cd = 256/243. In the 5-steps, they are 27/20, 45/32, 64/45, 40/27. This corresponds to a different set of chromas: Cm = 256/243 and Cd = 25/24. No Fokker block has different set of chromas.
Indeed, blackdye cannot be decomposed into the product word of two mosses.
Paradigm Shift
Over the years, the scale has been treated as a fixed structure up to modal rotation, yet this essay presents it as a different entity – it is a multilayer mechanism with variability.
This kind of variability is notably featured by Arabic maqamat. A maqam is not a scale in the traditional sense, but a composite, with building blocks of ajnas flexibly glued together. That is not to say scales should be framed like maqamat, but nonetheless, some inspirations can be drawn.
Specifically, a Fokker block can be viewed as a particular facet of a Fokker arena, defined by the chroma basis. A thorough exploration of such a scale thus includes both modal and domal rotation. The domes in an arena have been described as "disjoint" from each other, but here we observe it as another abstract layer of the same scale, an orthogonal one to the modes.
This novel view of scales entails some changes to the related measures, as discussed below.
Propriety and Stability Measures of Fokker blocks
Rothenberg propriety is a modal propriety. Likewise, Rothenberg stability is a modal stability. Lumma's refinements take the same principles. They work for scales in the traditional sense but not for what we have here because they disregard domal rotations.
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 domal rotations but is covered by those on modal rotations. We will dub it the residual stability. The following identity holds:
[math]\displaystyle{ \displaystyle \text{domal stability} = \text{modal stability} + \text{residual stability} }[/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 (m1, m2)-residual stability is defined as the portion of the pitch spectrum which is not covered by the interval differences of each class on m2-th-level domal rotations but is covered by those on m1-th-level domal rotations.
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.
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