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. So 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 L - l = S - s[/math]

Two chromas are to be named: the residual chroma

[math]\displaystyle C_\text{r} = L - S = l - s[/math]

and the domal chroma

[math]\displaystyle C_\text{d} = L - l = S - s[/math]

These form the chroma basis {C_r, C_d} 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

[math]\displaystyle C_\text{r} + C_\text{d} = L - s \\ C_\text{r} - C_\text{d} = l - S [/math]

In general, the four distinct chromas in a rank-3 Fokker block, sorted by size, are

[math]\displaystyle C_1 = C_\text{r} + C_\text{d} \\ C_2 = C_\text{r} \\ C_3 = C_\text{r} - C_\text{d} \\ C_4 = C_\text{d} [/math]

except that C_d can be larger than C_r − C_d.

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.

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 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.

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

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: Bilawal

The bilawal scale is Aura's preferred variant of diatonic scale. It 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.

Interval matrix of bilawal

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 C_\text{r} = 135/128, C_\text{d} = 81/80[/math]

Therefore, we find

[math]\displaystyle L = 9/8, l = 10/9, S = 16/15, s = 256/243[/math]

The pattern is then

[math]\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 \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.

Interval matrix of zarlino

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 C_r and which is C_d? It turns out that every possible assignment works as an actual Fokker block. A scale with this property is what they call a pairwise mos.

Fokker block 1

First we may assume

[math]\displaystyle C_\text{r} = 135/128, C_\text{d} = 81/80[/math]

It is the same as bilawal. Therefore, we find

[math]\displaystyle L = 9/8, l = 10/9, S = 16/15, s = 256/243[/math]

The pattern is then

[math]\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 \text{mode 5|1, dome 2|4}[/math]

Fokker block 2

[math]\displaystyle C_\text{r} = 25/24, C_\text{d} = 81/80[/math]

Therefore, we find

[math]\displaystyle L = 9/8, l = 10/9, S = 27/25, s = 16/15[/math]

The pattern is then

[math]\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 \text{mode 5|1, dome 4|2}[/math]

Fokker block 3

[math]\displaystyle C_\text{r} = 135/128, C_\text{d} = 25/24[/math]

Therefore, we find

[math]\displaystyle L = 75/64, l = 9/8, S = 10/9, s = 16/15[/math]

The pattern is then

[math]\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 \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.

Interval matrix of pattern "abcba"

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 C_\text{r} = 784/729, C_\text{d} = 64/63[/math]

Therefore, we find

[math]\displaystyle L = 896/729, l = 98/81, S = 8/7, s = 9/8[/math]

The pattern is then

[math]\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 \text{mode 2|2, dome 2|2}[/math]

Example 4: Blackdye

Finally, as a counterexample, blackdye is not a rank-3 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.

Interval matrix of blackdye

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 chroma basis composed of C_r = 800/729 and C_d = 256/243. Meanwhile, the 5-steps are 27/20, 45/32, 64/45, 40/27. This corresponds to a different chroma basis: C_r = 256/243 and C_d = 25/24. No Fokker block has different chroma bases.

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 and associated melody patterns flexibly put 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. Although the domes in an arena have been described as "disjoint" from each other, here we observe it as another abstract layer of the same scale, an independent one of 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. Carl 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, since it has been assumed that the domal chroma is smaller than the residual chroma, and thus smaller than their sum, we may let go of an extension of propriety. Stability is a different story, though. Even before diving into rank-3 stability measures, another modification on the existing definition is presented here.

The modal stability is defined as unity minus the sum of chroma variations in the scale on modal rotations. It should be easy to notice this definition is identical to Lumma stability when the scale is Rothenberg proper. The different part is, when the scale is improper, any overlap of the pitch spectrum is counted separately, yielding a lower result.

On top of that, stability can be considered in three distinct ways. First, there is the stability concerning domal rotations, described 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 residual rotations, described as the portion of the pitch spectrum which is not covered by the interval differences of each class on residual rotations. We will dub it the residual stability. Finally, there is the stability concerning both, described as the portion of the pitch spectrum which is not covered by the interval differences of each class on domal and residual rotations. We will dub it the total stability.

If we denote the stabilities σ_d, σ_r, and σ_t, respectively, they can be derived by

[math]\displaystyle \sigma_\text{d} = 1 - (n - n_\text{p})C_d \\ \sigma_\text{r} = 1 - (n - n_\text{p})C_r \\ \sigma_\text{t} = 1 - (n - n_\text{p})(C_r + C_d) [/math]

where n is the number of tones and n_p is the number of periods per equave. Hence, the following identity holds:

[math]\displaystyle \sigma_\text{t} = \sigma_\text{d} + \sigma_\text{r} - 1[/math]

The modal stability and the total stability are distinct in that the sum of the domal and residual chromas need not be covered on modal rotations, since the modal rotation keeps a linked mode–dome relation. For the same reason, modal stability must be said regarding the current mode–dome relation, so it is awkward to work with given the paradigm shift proposed above. For rank-2 Fokker blocks, considered as a degenerate case of rank-3, it is not a problem – domal stability becomes unity as domal variation is zero, and both residual stability and total stability are the same as modal stability. For rank-3, total stability is expected to be used in place of modal stability.

They can be generalized to higher ranks as follows. A Fokker block has S-free stability as well as S-linked stability, where S is a set enumerating the levels of rotations – for linked stability, how the levels are related must be specified. The total stability is the all-level free stability. The modal stability is the all-level linked stability. The two types collapse into one if S is a single level. The stability is assumed to be unity if S is empty, or a level that does 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 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.

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