Pinetone: Difference between revisions

Line 10:

If you have a [[Lumatone]], you can use the standard Bosanquet mapping for 12edo. The white keys are the porcutone diatonic, a cross between the meantone diatonic scale and Porcupine[7], and then black keys give the porcutone pentatonic, which approximates the just intonation pentatonic scale 9/8 5/4 3/2 5/3 2/1. I've chosen to colour the G#/Ab key pink, and the other chromatic keys blue, because I'm a proud trans woman and a big nerd. You can use any colours, but I find it helps to colour the G#/Ab key a different colour since that's the one chromatic key used along with the diatonic keys to make the porcutone octatonic.

== How it works ==

The diatonic scale has a step signature of [[5L 2s]], meaning it has 5 large steps and 2 small step arranged in the step pattern LsLLLsL (represent 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]] 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 as [[5L 2s]] = (9/8~10/9, 16/15~27/25). Porcupine[7] instead has step step signature and step mapping [[1L 6s]] = (~9/8, 10/9~27/25), hence the difference between 10/9 and 27/25, [[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, the mode 2 of Meantone[7], 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 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 porcutone diatonic mode 0 is said to be the [[product word]] of Meantone[7] mode 0 and Porcupine[7] mode 0, i.e., LsLLLsL *sssLsss -> MsMLMsM, where L*L -> L, L*s -> M, and s*s -> s. So the porcutone diatonic is the product of Meantone[7] and Porcupine[7].

To name this mode of the porcutone diatonic, we simply add the mode names together, prefixing the Porcupine[7] functional mode name (which I am introducing here) with the meantone diatonic mode name, so mode 0 of the porcutone diatonic is called Dorian symmetric minor. We continue this process with the other 6 modes:

{| class="wikitable"

|+Modes of the just porcutone diatonic

!Mode number

!Mode in JI

!Step pattern

!Meantone[7]

!Diatonic mode

!Porcupine[7]

!Porcupine[7] mode

!Porcutone diatonic mode

|-

|3

|10/9 5/4 25/18 3/2 5/3 50/27 2/1

|MLMsMMs

|LLLsLLs

|Lydian

|sLsssss

|Dark major

|Lydian dark major

|-

|2

|9/8 5/4 27/20 3/2 5/3 9/5 2/1

|LMsMMsM

|LLsLLsL

|Mixolydian

|Lssssss

|Bright major

|Mixolydian bright minor

|-

|1

|10/9 100/81 4/3 40/27 5/3 50/27 2/1

|MMsMLMs

|LLsLLLs

|Ionian

|ssssLss

|Bright diminished

|Ionian bright diminished

|-

|0

|10/9 6/5 4/3 3/2 5/3 9/5 2/1

|MsMLMsM

|LsLLLsL

|Dorian

|sssLsss

|Symmetric minor

|Dorian symmetric minor

|-

| -1

|27/25 6/5 27/20 3/2 81/50 9/5 2/1

|sMLMsMM

|sLLLsLL

|Phrygian

|ssLssss

|Bright minor

|Phrygian bright minor

|-

| -2

|10/9 6/5 4/3 40/27 8/5 16/9 2/1

|MsMMsML

|LsLLsLL

|Aeolian

|ssssssL

|Magical seventh

|Aeolian magical seventh

|-

| -3

|27/25 6/5 4/3 36/25 8/5 9/5 2/1

|sMMsMLM

|sLLsLLL

|Locrian

|sssssLs

|Dark diminished

|Locrian dark diminished

|}

The minor tone small step of Porcupine[7] can also represent the neutral seconds 11/10 and 12/11, since 10/9*11/10*12/11 = 4/3, and 4/3 is subtended by 3 small steps of Porcupine[7], tempering out both [[100/99]] and [[121/120]]. 11/8 is easily reached in Porcupine[7] as a major 4th, subtended by 2 small steps and 1 large step. The small step of Porcupine[7] represents all of 10/9, 11/10, 12/11 and 27/25, in order of largest to smallest. In the porcutone diatonic, the small step is 27/25 and the medium step is 10/9. We can access our 11-limit harmonies in porcutone by tempering out [[100/99]], which separates 10/9 from 11/10, as well as 27/25 from 12/11. This leads to step signature and step mapping 1L 4M 2s = (9/8~25/22, 10/9~11/10, 27/25~12/11). Since [[100/99]] is called the [[Ptolemisma]], we can call the resulting scale the ptolemismic porcutone diatonic.

The modes of the ptolemismic porcutone diatonic are shown below in their simplest JI pre-image (the simplest JI ratios each interval above the tonic represents), and in cents, in an optimized tuning called [[TE tuning]].

{| class="wikitable"

|+Modes of the ptolemismic porcutone diatonic

!Porcutone diatonic mode

!Step pattern

!Mode as simplest JI pre-image

!Mode in cents

|-

|Lydian dark major

|mLmsmms

|~ 10/9 5/4 11/8 3/2 5/3 11/6 2/1

|174.055 383.834 557.888 704.524 878.579 1052.633 1199.269

|-

|Mixolydian bright minor

|Lmsmmsm

|~ 9/8 5/4 15/11 3/2 5/3 9/5 2/1

|209.779 383.834 530.469 704.524 878.579 1025.214 1199.269

|-

|Ionian bright diminished

|mmsmLms

|~ 10/9 11/9 4/3 22/15 5/3 11/6 2/1

|174.055 348.110 494.745 668.800 878.579 1052.633 1199.269

|-

|Dorian symmetric minor

|msmLmsm

|~ 10/9 6/5 4/3 3/2 5/3 9/5 2/1

|174.055 320.690 494.745 704.524 878.579 1025.214 1199.269

|-

|Phrygian bright minor

|smLmsmm

|~ 12/11 6/5 15/11 3/2 18/11 9/5 2/1

|146.635 320.690 530.469 704.524 851.159 1025.214 1199.269

|-

|Aeolian magical seventh

|msmmsmL

|~ 10/9 6/5 4/3 22/15 8/5 16/9 2/1

|174.055 320.690 494.745 668.800 815.435 989.490 1199.269

|-

|Locrian dark diminished

|smmsmLm

|~ 12/11 6/5 4/3 16/11 8/5 9/5 2/1

|146.635 320.690 494.745 641.380 815.435 1025.214 1199.269

|}

We see 11/8 as the 4th in Lydian dark major. In Meantone[7] this is an augmented fourth, so whereas 11/8 is represented by the major 4th in Porcupine (L+ 2*s), it is represented by the augmented fourth of Meantone[7] (3*L). The meantone extension representing 11/8 with an augmented fourth is call Meanenneadecal, referencing the fact that it is most at home in [[19edo]].

For the 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.

@@ Line 10: / Line 10: @@
 If you have a [[Lumatone]], you can use the standard Bosanquet mapping for 12edo. The white keys are the porcutone diatonic, a cross between the meantone diatonic scale and Porcupine[7], and then black keys give the porcutone pentatonic, which approximates the just intonation pentatonic scale 9/8 5/4 3/2 5/3 2/1. I've chosen to colour the G#/Ab key pink, and the other chromatic keys blue, because I'm a proud trans woman and a big nerd. You can use any colours, but I find it helps to colour the G#/Ab key a different colour since that's the one chromatic key used along with the diatonic keys to make the porcutone octatonic.
+== How it works ==
+The diatonic scale has a step signature of [[5L 2s]], meaning it has 5 large steps and 2 small step arranged in the step pattern LsLLLsL (represent 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]] 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 as [[5L 2s]] = (9/8~10/9, 16/15~27/25). Porcupine[7] instead has step step signature and step mapping [[1L 6s]] = (~9/8, 10/9~27/25), hence the difference between 10/9 and 27/25, [[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, the mode 2 of Meantone[7], 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 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 porcutone diatonic mode 0 is said to be the [[product word]] of Meantone[7] mode 0 and Porcupine[7] mode 0, i.e., LsLLLsL *sssLsss -> MsMLMsM, where L*L -> L, L*s -> M, and s*s -> s. So the porcutone diatonic is the product of Meantone[7] and Porcupine[7].
+To name this mode of the porcutone diatonic, we simply add the mode names together, prefixing the Porcupine[7] functional mode name (which I am introducing here) with the meantone diatonic mode name, so mode 0 of the porcutone diatonic is called Dorian symmetric minor. We continue this process with the other 6 modes:
+{| class="wikitable"
+|+Modes of the just porcutone diatonic
+!Mode number
+!Mode in JI
+!Step pattern
+!Meantone[7]
+!Diatonic mode
+!Porcupine[7]
+!Porcupine[7] mode
+!Porcutone diatonic mode
+|-
+|3
+|10/9 5/4 25/18 3/2 5/3 50/27 2/1
+|MLMsMMs
+|LLLsLLs
+|Lydian
+|sLsssss
+|Dark major
+|Lydian dark major
+|-
+|2
+|9/8 5/4 27/20 3/2 5/3 9/5 2/1
+|LMsMMsM
+|LLsLLsL
+|Mixolydian
+|Lssssss
+|Bright major
+|Mixolydian bright minor
+|-
+|1
+|10/9 100/81 4/3 40/27 5/3 50/27 2/1
+|MMsMLMs
+|LLsLLLs
+|Ionian
+|ssssLss
+|Bright diminished
+|Ionian bright diminished
+|-
+|0
+|10/9 6/5 4/3 3/2 5/3 9/5 2/1
+|MsMLMsM
+|LsLLLsL
+|Dorian
+|sssLsss
+|Symmetric minor
+|Dorian symmetric minor
+|-
+| -1
+|27/25 6/5 27/20 3/2 81/50 9/5 2/1
+|sMLMsMM
+|sLLLsLL
+|Phrygian
+|ssLssss
+|Bright minor
+|Phrygian bright minor
+|-
+| -2
+|10/9 6/5 4/3 40/27 8/5 16/9 2/1
+|MsMMsML
+|LsLLsLL
+|Aeolian
+|ssssssL
+|Magical seventh
+|Aeolian magical seventh
+|-
+| -3
+|27/25 6/5 4/3 36/25 8/5 9/5 2/1
+|sMMsMLM
+|sLLsLLL
+|Locrian
+|sssssLs
+|Dark diminished
+|Locrian dark diminished
+|}
+The minor tone small step of Porcupine[7] can also represent the neutral seconds 11/10 and 12/11, since 10/9*11/10*12/11 = 4/3, and 4/3 is subtended by 3 small steps of Porcupine[7], tempering out both [[100/99]] and [[121/120]]. 11/8 is easily reached in Porcupine[7] as a major 4th, subtended by 2 small steps and 1 large step. The small step of Porcupine[7] represents all of 10/9, 11/10, 12/11 and 27/25, in order of largest to smallest. In the porcutone diatonic, the small step is 27/25 and the medium step is 10/9. We can access our 11-limit harmonies in porcutone by tempering out [[100/99]], which separates 10/9 from 11/10, as well as 27/25 from 12/11. This leads to step signature and step mapping 1L 4M 2s = (9/8~25/22, 10/9~11/10, 27/25~12/11). Since [[100/99]] is called the [[Ptolemisma]], we can call the resulting scale the ptolemismic porcutone diatonic.
+The modes of the ptolemismic porcutone diatonic are shown below in their simplest JI pre-image (the simplest JI ratios each interval above the tonic represents), and in cents, in an optimized tuning called [[TE tuning]].
+{| class="wikitable"
+|+Modes of the ptolemismic porcutone diatonic
+!Porcutone diatonic mode
+!Step pattern
+!Mode as simplest JI pre-image
+!Mode in cents
+|-
+|Lydian dark major
+|mLmsmms
+|~ 10/9 5/4 11/8 3/2 5/3 11/6 2/1
+|174.055 383.834 557.888 704.524 878.579 1052.633 1199.269
+|-
+|Mixolydian bright minor
+|Lmsmmsm
+|~ 9/8 5/4 15/11 3/2 5/3 9/5 2/1
+|209.779 383.834 530.469 704.524 878.579 1025.214 1199.269
+|-
+|Ionian bright diminished
+|mmsmLms
+|~ 10/9 11/9 4/3 22/15 5/3 11/6 2/1
+|174.055 348.110 494.745 668.800 878.579 1052.633 1199.269
+|-
+|Dorian symmetric minor
+|msmLmsm
+|~ 10/9 6/5 4/3 3/2 5/3 9/5 2/1
+|174.055 320.690 494.745 704.524 878.579 1025.214 1199.269
+|-
+|Phrygian bright minor
+|smLmsmm
+|~ 12/11 6/5 15/11 3/2 18/11 9/5 2/1
+|146.635 320.690 530.469 704.524 851.159 1025.214 1199.269
+|-
+|Aeolian magical seventh
+|msmmsmL
+|~ 10/9 6/5 4/3 22/15 8/5 16/9 2/1
+|174.055 320.690 494.745 668.800 815.435 989.490 1199.269
+|-
+|Locrian dark diminished
+|smmsmLm
+|~ 12/11 6/5 4/3 16/11 8/5 9/5 2/1
+|146.635 320.690 494.745 641.380 815.435 1025.214 1199.269
+|}
+We see 11/8 as the 4th in Lydian dark major. In Meantone[7] this is an augmented fourth, so whereas 11/8 is represented by the major 4th in Porcupine (L+ 2*s), it is represented by the augmented fourth of Meantone[7] (3*L). The meantone extension representing 11/8 with an augmented fourth is call Meanenneadecal, referencing the fact that it is most at home in [[19edo]].
 For the 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.