Pinetone: Difference between revisions

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We are familiar with the [[Zarlino]]/Ptolemy [[just]] major scale: 9/8 5/4 4/3 3/2 5/3 15/8 2/1. This scale has 3 large steps of [[9/8]], 2 medium steps of [[10/9]], and 2 small steps of [[16/15]], with [[step pattern]] LMsLMLs. If we temper out the difference between L and M, we get LLsLLLs, which, as mode 2 of [[Meantone]][7] is the familiar Ionian/major mode.

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]] Pinetone 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]] Pinetone diatonic represents both [[Porcupine]][7] and [[Meantone]][7]. To name this mode of the Pinetone 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 Pinetone 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.

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]] Pinetone 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]] Pinetone diatonic represents both [[Porcupine]][7] and [[Meantone]][7]. To name this mode of the Pinetone diatonic, we simply add the mode names together, prefixing the [[Porcupine]][7] functional mode names introduced in Table 2.1., with the [[Meantone]] diatonic mode names referenced in Table 2.2., so mode 0 of the Pinetone diatonic is called ''Dorian symmetric minor''. We continue this naming process with the other 6 modes to arrive at the modes shown in Table 2.3.

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]] Pinetone diatonic, along with the name and [[step pattern]] of the corresponding [[Porcupine]][7] and [[Meantone]][7] modes, which can be arrived from their corresponding Pinetone modes by tempering out the [[Porcupine]] and [[Meantone]] [[comma]]<nowiki/>s respectively.

Tables 2.1. and 2.2. 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 2.3. shows the modes of the [[5-limit]] Pinetone diatonic, along with the name and [[step pattern]] of the corresponding [[Porcupine]][7] and [[Meantone]][7] modes, which can be arrived from their corresponding Pinetone modes by tempering out the [[Porcupine]] and [[Meantone]] [[comma]]<nowiki/>s respectively.

In Tables 1.-4., Modes marked with '*' have a consonant triad on their root.

In Tables 2.1.-2.4., Modes marked with '*' have a consonant triad on their root.

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|+Table 2.1. Modes of 5-limit Porcupine[7]

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 We are familiar with the [[Zarlino]]/Ptolemy [[just]] major scale: 9/8 5/4 4/3 3/2 5/3 15/8 2/1. This scale has 3 large steps of [[9/8]], 2 medium steps of [[10/9]], and 2 small steps of [[16/15]], with [[step pattern]] LMsLMLs. If we temper out the difference between L and M, we get LLsLLLs, which, as mode 2 of [[Meantone]][7] is the familiar Ionian/major mode.
-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]] Pinetone 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]] Pinetone diatonic represents both [[Porcupine]][7] and [[Meantone]][7]. To name this mode of the Pinetone 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 Pinetone 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.
+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]] Pinetone 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]] Pinetone diatonic represents both [[Porcupine]][7] and [[Meantone]][7]. To name this mode of the Pinetone diatonic, we simply add the mode names together, prefixing the [[Porcupine]][7] functional mode names introduced in Table 2.1., with the [[Meantone]] diatonic mode names referenced in Table 2.2., so mode 0 of the Pinetone diatonic is called ''Dorian symmetric minor''. We continue this naming process with the other 6 modes to arrive at the modes shown in Table 2.3.
-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]] Pinetone diatonic, along with the name and [[step pattern]] of the corresponding [[Porcupine]][7] and [[Meantone]][7] modes, which can be arrived from their corresponding Pinetone modes by tempering out the [[Porcupine]] and [[Meantone]] [[comma]]<nowiki/>s respectively.
+Tables 2.1. and 2.2. 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 2.3. shows the modes of the [[5-limit]] Pinetone diatonic, along with the name and [[step pattern]] of the corresponding [[Porcupine]][7] and [[Meantone]][7] modes, which can be arrived from their corresponding Pinetone modes by tempering out the [[Porcupine]] and [[Meantone]] [[comma]]<nowiki/>s respectively.
-In Tables 1.-4., Modes marked with '*' have a consonant triad on their root.
+In Tables 2.1.-2.4., Modes marked with '*' have a consonant triad on their root.
 {| class="wikitable"
 |+Table 2.1. Modes of 5-limit Porcupine[7]