User:FloraC/Fokker analysis of rank-3 scales: Difference between revisions
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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''. | 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''<sub>d</sub>, ''σ''<sub>r</sub>, and ''σ''<sub>t</sub>, respectively, | If we denote the stabilities ''σ''<sub>d</sub>, ''σ''<sub>r</sub>, and ''σ''<sub>t</sub>, respectively, they can be derived by | ||
<math>\displaystyle | <math>\displaystyle | ||
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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. | 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. | 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 == | ||