Pinetone: Difference between revisions

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The diatonic scale has a [[Step pattern|step signature]] of [[5L 2s]], meaning it has 5 large steps and 2 small step arranged in the [[step pattern]] LsLLLsL (represented in mode 0, Dorian mode). In [[Meantone]][7], the large step represents both [[9/8]] and [[10/9]], the major and minor tones (''tempering out'' the [[81/80]] interval that separates them) hence the name "Meantone". The small step represents [[16/15]] and [[27/25]] (which differ again by [[81/80]]). We write this in the form of a [[Step pattern|step signature]] and ''step mapping'' as [[5L 2s]] = (9/8~10/9, 16/15~27/25). [[Porcupine]][7] instead has a [[Step pattern|step signature]] and step mapping [[1L 6s]] = (~9/8, 10/9~27/25), hence the difference between [[10/9]] and [[27/25]], i.e., [[250/243]], is tempered out. In mode 0 it has [[step pattern]] sssLsss. [[81/80]] is called the [[Meantone comma]], and [[250/243]] is called the [[Porcupine]] comma.

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.

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.

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 The diatonic scale has a [[Step pattern|step signature]] of [[5L 2s]], meaning it has 5 large steps and 2 small step arranged in the [[step pattern]] LsLLLsL (represented in mode 0, Dorian mode). In [[Meantone]][7], the large step represents both [[9/8]] and [[10/9]], the major and minor tones (''tempering out'' the [[81/80]] interval that separates them) hence the name "Meantone". The small step represents [[16/15]] and [[27/25]] (which differ again by [[81/80]]). We write this in the form of a [[Step pattern|step signature]] and ''step mapping'' as [[5L 2s]] = (9/8~10/9, 16/15~27/25). [[Porcupine]][7] instead has a [[Step pattern|step signature]] and step mapping [[1L 6s]] = (~9/8, 10/9~27/25), hence the difference between [[10/9]] and [[27/25]], i.e., [[250/243]], is tempered out. In mode 0 it has [[step pattern]] sssLsss. [[81/80]] is called the [[Meantone comma]], and [[250/243]] is called the [[Porcupine]] comma.
-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.
+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.