User:FloraC/Fokker analysis of rank-3 scales - Revision history

FloraC: Something happened that broke my otherwise nicely typeset math

2025-07-07T14:42:18Z

Something happened that broke my otherwise nicely typeset math

← Older revision		Revision as of 14:42, 7 July 2025
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	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>

Sintel: /* Notation */ fix some newlines that were broken since mathjax update

2025-07-03T13:06:10Z

Notation: fix some newlines that were broken since mathjax update

← Older revision		Revision as of 13:06, 3 July 2025
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	<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>

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	<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>.		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 ==

FloraC: Fix an overlooked case

2022-07-01T18:26:35Z

Fix an overlooked case

← Older revision		Revision as of 18:26, 1 July 2022
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	C_4 = C_\text{d}		C_4 = C_\text{d}
	</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 ==
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	© 2022 Flora Canou		© 2022 Flora Canou

	Version Stable 2		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].

FloraC: Italicize neologisms; +notes

2022-06-07T18:46:50Z

Italicize neologisms; +notes

← Older revision		Revision as of 18:46, 7 June 2022
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	<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>
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	== 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.
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	\|+ 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
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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.		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.
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	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 ==

FloraC: -typo

2022-05-19T20:01:20Z

-typo

← Older revision		Revision as of 20:01, 19 May 2022
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	=== Example 4: Blackdye ===		=== Example 4: Blackdye ===
	Finally, as a ~~counterexmaple~~, blackdye is ''not'' a rank-3 Fokker block. Here we try analysing the 5-limit JI tuning of blackdye. The steps are:		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>

FloraC: Some final wording improvements

2022-05-19T19:58:04Z

Some final wording improvements

← Older revision		Revision as of 19:58, 19 May 2022
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	=== Example 1: Bilawal ===		=== Example 1: Bilawal ===
	The bilawal scale is Aura's preferred diatonic scale. It is a very typical rank-3 Fokker block. The steps are:		The bilawal scale is Aura's preferred variant of diatonic scale. It is a very typical rank-3 Fokker block. The steps are:

	<pre>9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1</pre>		<pre>9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1</pre>
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	=== Example 4: Blackdye ===		=== 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:		Finally, as a counterexmaple, 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>
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	\|}		\|}

	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~~ ''C''<sub>r</sub> = 800/729 and ''C''<sub>d</sub> = 256/243. In the 5-steps~~, they~~ are 27/20, 45/32, 64/45, 40/27. This corresponds to a different ~~set of chromas~~: ''C''<sub>r</sub> = 256/243 and ''C''<sub>d</sub> = 25/24. No Fokker block has different ~~sets of chromas~~.		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''<sub>r</sub> = 800/729 and ''C''<sub>d</sub> = 256/243. Meanwhile, the 5-steps are 27/20, 45/32, 64/45, 40/27. This corresponds to a different chroma basis: ''C''<sub>r</sub> = 256/243 and ''C''<sub>d</sub> = 25/24. No Fokker block has different chroma bases.

	Indeed, blackdye cannot be decomposed into the product word of two mosses.		Indeed, blackdye cannot be decomposed into the product word of two mosses.
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	© 2022 Flora Canou		© 2022 Flora Canou

	Version Stable 1		Version Stable 2

	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].

FloraC: I should refuse to use that term

2022-05-19T18:27:19Z

I should refuse to use that term

← Older revision		Revision as of 18:27, 19 May 2022
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	\|}		\|}

	The chromas are 135/128, 25/24, and 81/80. Since there are only three sizes, how can we know which is ''C''<sub>r</sub> and which is ''C''<sub>d</sub>? It turns out that every possible assignment works as an actual Fokker block. A scale with this property is what they call a ~~wakalix~~.		The chromas are 135/128, 25/24, and 81/80. Since there are only three sizes, how can we know which is ''C''<sub>r</sub> and which is ''C''<sub>d</sub>? 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		; Fokker block 1

FloraC: /* Illustration */ correction

2022-05-18T18:23:14Z

Illustration: correction

← Older revision		Revision as of 18:23, 18 May 2022
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	It can be decomposed into the product of an antimanic scale and a pentic scale: ssLss × sLsLs. It is in		It can be decomposed into the product of an antimanic scale and a pentic scale: ssLss × sLsLs. It is in

	<math>\displaystyle \text{mode 3\|3, dome 3\|3}</math>		<math>\displaystyle \text{mode 2\|2, dome 2\|2}</math>

	=== Example 4: Blackdye ===		=== Example 4: Blackdye ===
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	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 ''C''<sub>r</sub> = 800/729 and ''C''<sub>d</sub> = 256/243. In the 5-steps, they are 27/20, 45/32, 64/45, 40/27. This corresponds to a different set of chromas: ''C''<sub>r</sub> = 256/243 and ''C''<sub>d</sub> = 25/24. No Fokker block has different sets of chromas.		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 ''C''<sub>r</sub> = 800/729 and ''C''<sub>d</sub> = 256/243. In the 5-steps, they are 27/20, 45/32, 64/45, 40/27. This corresponds to a different set of chromas: ''C''<sub>r</sub> = 256/243 and ''C''<sub>d</sub> = 25/24. No Fokker block has different sets of chromas.

	Indeed, blackdye cannot be decomposed into the product word of two mosses.		Indeed, blackdye cannot be decomposed into the product word of two mosses.

	== Paradigm Shift ==		== Paradigm Shift ==

FloraC at 18:10, 18 May 2022

2022-05-18T18:10:06Z

← Older revision		Revision as of 18:10, 18 May 2022
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	<math>\displaystyle C_\text{r} = 135/128, C_\text{d} = 81/80</math>		<math>\displaystyle C_\text{r} = 135/128, C_\text{d} = 81/80</math>

	It is the same as ~~Aura's diatonic scale~~. Therefore, we find		It is the same as bilawal. Therefore, we find

	<math>\displaystyle L = 9/8, l = 10/9, S = 16/15, s = 256/243</math>		<math>\displaystyle L = 9/8, l = 10/9, S = 16/15, s = 256/243</math>
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	\|}		\|}

	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 ''C''<sub>r</sub> = 800/729 and ''C''<sub>d</sub> = 256/243. In the 5-steps, they are 27/20, 45/32, 64/45, 40/27. This corresponds to a different set of chromas: ''C''<sub>r</sub> = 256/243 and ''C''<sub>d</sub> = 25/24. No Fokker block has different ~~set~~ of chromas.		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 ''C''<sub>r</sub> = 800/729 and ''C''<sub>d</sub> = 256/243. In the 5-steps, they are 27/20, 45/32, 64/45, 40/27. This corresponds to a different set of chromas: ''C''<sub>r</sub> = 256/243 and ''C''<sub>d</sub> = 25/24. No Fokker block has different sets of chromas.

	Indeed, blackdye cannot be decomposed into the product word of two mosses.		Indeed, blackdye cannot be decomposed into the product word of two mosses.

FloraC: /* Paradigm Shift */ clarification based on feedback

2022-05-18T15:03:55Z

Paradigm Shift: clarification based on feedback

← Older revision		Revision as of 15:03, 18 May 2022
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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.		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 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. 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 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.

	This novel view of scales entails some changes to the related measures, as discussed below.		This novel view of scales entails some changes to the related measures, as discussed below.

	== Propriety and Stability Measures of Fokker Blocks ==		== Propriety and Stability Measures of Fokker Blocks ==