User:FloraC/Fokker analysis of rank-3 scales: Difference between revisions

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 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.
+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 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. 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.
+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 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. 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.
+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 the assumption is that the domal chroma is smaller than the modal chroma, we may let go of an extension of propriety, but stability is really a different story.
+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–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:
+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.
-<math>\displaystyle \text{modal stability} = \text{domal stability} + \text{residual stability} - 1</math>
+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''.
-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.
+If we denote the stabilities ''σ''<sub>d</sub>, ''σ''<sub>r</sub>, and ''σ''<sub>t</sub>, respectively, the stabilities can be derived by
-The ''m''-stability is unity for ''m'' ≥ ''r'', meaning that those levels of domal rotations do not exist.
+<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''<sub>p</sub> 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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 © 2022 Flora Canou
-Version Stable 0
+Version Stable 1
 This work is licensed under the [https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Attribution-ShareAlike 4.0 International License].