User:FloraC/Fokker analysis of rank-3 scales: Difference between revisions
Some final wording improvements |
m Something happened that broke my otherwise nicely typeset math |
||
| (4 intermediate revisions by 2 users not shown) | |||
| Line 8: | Line 8: | ||
<math>\displaystyle L - l = S - s</math> | <math>\displaystyle L - l = S - s</math> | ||
Two chromas are to be named: the residual chroma | Two chromas are to be named: the ''residual chroma'' | ||
<math>\displaystyle C_\text{r} = L - S = l - s</math> | <math>\displaystyle C_\text{r} = L - S = l - s</math> | ||
and the domal chroma | and the ''domal chroma'' | ||
<math>\displaystyle C_\text{d} = L - l = S - s</math> | <math>\displaystyle C_\text{d} = L - l = S - s</math> | ||
| Line 21: | Line 21: | ||
<math>\displaystyle | <math>\displaystyle | ||
C_\text{r} + C_\text{d} = L - s \\ | \begin{align} | ||
C_\text{r} - C_\text{d} = l - S | C_\text{r} + C_\text{d} &= L - s \\ | ||
C_\text{r} - C_\text{d} &= l - S | |||
\end{align} | |||
</math> | </math> | ||
| Line 28: | Line 30: | ||
<math>\displaystyle | <math>\displaystyle | ||
C_1 = C_\text{r} + C_\text{d} \\ | \begin{align} | ||
C_2 = C_\text{r} \\ | C_1 &= C_\text{r} + C_\text{d} \\ | ||
C_3 = C_\text{r} - C_\text{d} \\ | C_2 &= C_\text{r} \\ | ||
C_4 = C_\text{d} | C_3 &= C_\text{r} - C_\text{d} \\ | ||
C_4 &= C_\text{d} | |||
\end{align} | |||
</math> | </math> | ||
except that ''C''<sub>d</sub> can be larger than ''C''<sub>r</sub> − ''C''<sub>d</sub>. | |||
== Modal, Domal and Residual Rotation == | == 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 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. | ''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. | It follows that modal rotation is the composition of domal rotation and residual rotation. | ||
| Line 58: | Line 64: | ||
|+ Interval matrix of bilawal | |+ Interval matrix of bilawal | ||
|- | |- | ||
! Steps !! Intervals | ! Steps<ref>The ''n-step'' is zero-indexed. </ref> !! Intervals | ||
|- | |- | ||
| 1-step || 16/15, 10/9, 9/8 | | 1-step || 16/15, 10/9, 9/8 | ||
| Line 210: | Line 216: | ||
=== Example 4: Blackdye === | === Example 4: Blackdye === | ||
Finally, as a | 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: | ||
<pre>81/80, 9/8, 6/5, 4/3, 27/20, 3/2, 8/5, 16/9, 9/5, 2/1</pre> | <pre>81/80, 9/8, 6/5, 4/3, 27/20, 3/2, 8/5, 16/9, 9/5, 2/1</pre> | ||
| Line 249: | Line 255: | ||
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. | 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. | 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<ref>Precisely, "[the maqam] is a system of scales, habitual melodic phrases, modulation possibilities, ornamentation techniques and aesthetic conventions that together form a rich melodic framework and artistic tradition." Johnny Farraj. MaqamWorld. </ref>. 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. | 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. | ||
| Line 266: | Line 272: | ||
If we denote the stabilities ''σ''<sub>d</sub>, ''σ''<sub>r</sub>, and ''σ''<sub>t</sub>, respectively, they can be derived by | 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 | ||
\sigma_\text{d} = 1 - (n - n_\text{p})C_d \\ | \begin{align} | ||
\sigma_\text{r} = 1 - (n - n_\text{p})C_r \\ | \sigma_\text{d} &= 1 - (n - n_\text{p})C_d \\ | ||
\sigma_\text{t} = 1 - (n - n_\text{p})(C_r + C_d) | \sigma_\text{r} &= 1 - (n - n_\text{p})C_r \\ | ||
\sigma_\text{t} &= 1 - (n - n_\text{p})(C_r + C_d) | |||
\end{align} | |||
</math> | </math> | ||
| Line 281: | Line 289: | ||
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. | ||
== Notes == | |||
<references/> | |||
== Release Notes == | == Release Notes == | ||
© 2022 Flora Canou | © 2022 Flora Canou | ||
Version Stable | Version Stable 3 | ||
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]. | ||