Pinetone: Difference between revisions

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Consider instead the [[just]] scale: 10/9 6/5 4/3 3/2 5/3 9/5 2/1, a just Dorian scale. This scale has 1 large step of [[9/8]], 4 medium steps of [[10/9]], and 2 small steps of [[27/25]], with step pattern MsMLMsM (mode 0). It can be represented with [[Step pattern|step signature]] and step mapping 1L 4M 2s = (9/8, 10/9, 27/25). This is our [[just]] Porcutone diatonic. If we temper out the difference between L and M, we get LsLLLsL, [[Meantone]][7] mode 0: Dorian; if we temper out instead the difference between [[10/9]] and [[27/25]], we get sssLsss, [[Porcupine]][7] mode 0, which is referred to as ''symmetric minor''. In this way, the [[just]] Porcutone diatonic represents both [[Porcupine]][7] and [[Meantone]][7]. To name this mode of the Porcutone diatonic, we simply add the mode names together, prefixing the [[Porcupine]][7] functional mode names introduced in Table 1., with the [[Meantone]] diatonic mode names referenced in Table 2., so mode 0 of the Porcutone diatonic is called ''Dorian symmetric minor''. We continue this naming process with the other 6 modes to arrive at the modes shown in Table 3.

Tables 1. and 2. show the modes of [[Porcupine]][7], and [[Meantone]][7], respectively, in the [[5-limit]]. Given that intervals of tempered scales represent more than a single [[Just intonation|JI]] interval each, modes are described in their ''JI pre-image,'' the simplest [[Just intonation|JI]] ratios each interval above the tonic represents. Along with the step pattern and mode number, the modes' [[UDP]] are shown. A mode's [[Modal UDP notation|UDP]] shows the number of generators in the direction the brighten the intervals of scale, followed the number of generators in the direction that darkens it, (followed by the number of periods per octave, if it is not one. In this case the scale repeats at the octave, so P = 1, and is not shown). Table 3. shows the modes of the [[5-limit]] Porcutone diatonic, along with the name and [[step pattern]] of the corresponding [[Porcupine]][7] and [[Meantone]][7] modes, which can be arrived from their corresponding Porcutone modes by tempering out the [[Porcupine]] and [[Meantone]] [[comma]]<nowiki/>s respectively.

{| class="wikitable"

|+Table 1. Modes of 5-limit Porcupine[7]

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Unlike the Porcutone diatonic, and chromatic scales, the Porcutone octatonic is chiral, and is therefore not a step-nested scale. As we can see, it is more complex than the Porcutone diatonic. The Porcutone pentatonic and diatonic scales is also wakalix / PWF, and it can be seen that the Porcutone octatonic is more complex than the Porcutone pentatonic as well. It is left as an excercise for the reader to determine the complexity of the porcutone chromatic, and compare that to the porcutone octatonic.

== Summary for xen-math nerds ==

The Porcutone system is built via step nesting from the 5-limit minor seventh tetrad: 6/5 3/2 9/5 2/1. It's a 12-note rank-3 [[Meantone]][12] x [[Ripple]][12] [[Fokker block]], a [[step-nested scale]] that also tempers to [[Porcupine]][8], comprising a diatonic [[Meantone]][7]-[[Porcupine]][7]-[[Dicot]][7] [[wakalix]] / 3-[[Step-nested scale|SNS]] on the white keys, and a pentatonic [[Meantone]][5]-[[Father]][5]-[[Bug]][5] [[wakalix]] on the 'black' keys.

The Porcutone system is built via step nesting from the 5-limit minor seventh tetrad: 6/5 3/2 9/5 2/1. It's bounds are the set of temperings of the step-nested children of the 4-note SNS tetrad 6/5 3/2 9/5 2/1.

The Porcutone chromatic is a 12-note rank-3 [[Meantone]][12] x [[Ripple]][12] [[Fokker block]], a [[step-nested scale]] that also tempers to [[Porcupine]][8], comprising a diatonic [[Meantone]][7]-[[Porcupine]][7]-[[Dicot]][7] [[wakalix]] / 3-[[Step-nested scale|SNS]] on the white keys, and a pentatonic [[Meantone]][5]-[[Father]][5]-[[Bug]][5] [[wakalix]] on the 'black' keys.

For the accompanying mapping for the Lumatone keyboard the G♯ / A♭ key is coloured pink (and the remaining chromatic keys blue), and along with the white keys makes a [[Porcupine]][8] / [[Father]][8] [[Fokker block]] (any colours could be chosen instead of white, pink, and blue).

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 Consider instead the [[just]] scale: 10/9 6/5 4/3 3/2 5/3 9/5 2/1, a just Dorian scale. This scale has 1 large step of [[9/8]], 4 medium steps of [[10/9]], and 2 small steps of [[27/25]], with step pattern MsMLMsM (mode 0). It can be represented with [[Step pattern|step signature]] and step mapping 1L 4M 2s = (9/8, 10/9, 27/25). This is our [[just]] Porcutone diatonic. If we temper out the difference between L and M, we get LsLLLsL, [[Meantone]][7] mode 0: Dorian; if we temper out instead the difference between [[10/9]] and [[27/25]], we get sssLsss, [[Porcupine]][7] mode 0, which is referred to as ''symmetric minor''. In this way, the [[just]] Porcutone diatonic represents both [[Porcupine]][7] and [[Meantone]][7]. To name this mode of the Porcutone diatonic, we simply add the mode names together, prefixing the [[Porcupine]][7] functional mode names introduced in Table 1., with the [[Meantone]] diatonic mode names referenced in Table 2., so mode 0 of the Porcutone diatonic is called ''Dorian symmetric minor''. We continue this naming process with the other 6 modes to arrive at the modes shown in Table 3.
 Tables 1. and 2. show the modes of [[Porcupine]][7], and [[Meantone]][7], respectively, in the [[5-limit]]. Given that intervals of tempered scales represent more than a single [[Just intonation|JI]] interval each, modes are described in their ''JI pre-image,'' the simplest [[Just intonation|JI]] ratios each interval above the tonic represents. Along with the step pattern and mode number, the modes' [[UDP]] are shown. A mode's [[Modal UDP notation|UDP]] shows the number of generators in the direction the brighten the intervals of scale, followed the number of generators in the direction that darkens it, (followed by the number of periods per octave, if it is not one. In this case the scale repeats at the octave, so P = 1, and is not shown). Table 3. shows the modes of the [[5-limit]] Porcutone diatonic, along with the name and [[step pattern]] of the corresponding [[Porcupine]][7] and [[Meantone]][7] modes, which can be arrived from their corresponding Porcutone modes by tempering out the [[Porcupine]] and [[Meantone]] [[comma]]<nowiki/>s respectively.
 {| class="wikitable"
 |+Table 1. Modes of 5-limit Porcupine[7]
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 Unlike the Porcutone diatonic, and chromatic scales, the Porcutone octatonic is chiral, and is therefore not a step-nested scale. As we can see, it is more complex than the Porcutone diatonic. The Porcutone pentatonic and diatonic scales is also wakalix / PWF, and it can be seen that the Porcutone octatonic is more complex than the Porcutone pentatonic as well. It is left as an excercise for the reader to determine the complexity of the porcutone chromatic, and compare that to the porcutone octatonic.
 == Summary for xen-math nerds ==
-The Porcutone system is built via step nesting from the 5-limit minor seventh tetrad: 6/5 3/2 9/5 2/1. It's a 12-note rank-3 [[Meantone]][12] x [[Ripple]][12] [[Fokker block]], a [[step-nested scale]] that also tempers to [[Porcupine]][8], comprising a diatonic [[Meantone]][7]-[[Porcupine]][7]-[[Dicot]][7] [[wakalix]] / 3-[[Step-nested scale|SNS]] on the white keys, and a pentatonic [[Meantone]][5]-[[Father]][5]-[[Bug]][5] [[wakalix]] on the 'black' keys.
+The Porcutone system is built via step nesting from the 5-limit minor seventh tetrad: 6/5 3/2 9/5 2/1. It's bounds are the set of temperings of the step-nested children of the 4-note SNS tetrad 6/5 3/2 9/5 2/1.
+The Porcutone chromatic is a 12-note rank-3 [[Meantone]][12] x [[Ripple]][12] [[Fokker block]], a [[step-nested scale]] that also tempers to [[Porcupine]][8], comprising a diatonic [[Meantone]][7]-[[Porcupine]][7]-[[Dicot]][7] [[wakalix]] / 3-[[Step-nested scale|SNS]] on the white keys, and a pentatonic [[Meantone]][5]-[[Father]][5]-[[Bug]][5] [[wakalix]] on the 'black' keys.
 For the accompanying mapping for the Lumatone keyboard the G♯ / A♭ key is coloured pink (and the remaining chromatic keys blue), and along with the white keys makes a [[Porcupine]][8] / [[Father]][8] [[Fokker block]] (any colours could be chosen instead of white, pink, and blue).