Pinetone: Difference between revisions

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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.

== 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.

For the accompanying mapping for the Lumatone keyboard the G# / Ab 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).

~~== Porcutone diatonic ==~~

The Porcutone diatonic is a [[wakalix]] (pairwise well-formed scale) and a [[step-nested scale]]: A detempering of [[Meantone]][7] and [[Porcupine]][7], (and also of [[Dicot]][7]), a [[Fokker block]] with [[Unison vector|unison vectors]] of [[81/80]] and [[250/243]] (and [[25/24]]) has 1 large step of 9/8 (L x L), 3 medium steps of 10/9 (L x s), and 3 small steps of 27/25 (s x s).

~~Warning, math, skip to where it says 'math over' if you're not into it:~~ The Porcutone diatonic is a [[wakalix]] (pairwise well-formed scale) and a [[step-nested scale]]: A detempering of [[Meantone]][7] and [[Porcupine]][7], (and also of [[Dicot]][7]), a [[Fokker block]] with [[Unison vector|unison vectors]] of [[81/80]] and [[250/243]] (and [[25/24]]) has 1 large step of 9/8 (L x L), 3 medium steps of 10/9 (L x s), and 3 small steps of 27/25 (s x s)

~~The six modes of the just porcutone diatonic are:~~

* Mode 3: 10/9 5/4 25/18 3/2 5/3 50/27 2/1 = Meantone[7] 6|0 x Porcupine[7] 5|1 = LLLsLLs x sLsssss = MLMsMMs = Lydian x dark major = Lydian dark major

* Mode 2: 9/8 5/4 27/20 3/2 5/3 9/5 2/1 = Meantone[7] 4|2 x Porcupine[7] 6|0 = LLsLLsL x Lssssss = LMsMMsM = Mixolydian x bright major = Mixolydian bright major

* Mode 1: 10/9 100/81 4/3 40/27 5/3 50/27 2/1 = Meantone[7] 5|1 x Porcupine[7] 2|4 = LLsLLLs x ssssLss = MMsMLMs = Ionian x bright diminished = Ionian bright diminished

* Mode 0: 10/9 6/5 4/3 3/2 5/3 9/5 2/1 = Meantone[7] 3|3 x Porcupine[7] 3|3 = LsLLLsL x sssLsss = MsMLMsM = Dorian x symmetric minor = Dorian symmetric minor

* Mode -1: 27/25 6/5 27/20 3/2 81/50 9/5 2/1 = Meantone[7] 1|5 x Porcupine[7] 4|2 = sLLLsLL x ssLssss = sMLMsMM = Phrygian x bright minor = Phrygian bright minor

* Mode -2: 10/9 6/5 4/3 40/27 8/5 16/9 2/1 = Meantone[7] 2|4 x Porcupine[7] 0|6 = LsLLsLL x ssssssL = MsMMsML = Aeolian x magical seventh mode = Aeolian magical seventh

* Mode -3: 27/25 6/5 4/3 36/25 8/5 9/5 2/1 = Meantone[7] 0|6 x Porcupine[7] 1|5 = sLLsLLL x sssssLs = sMMsMLs = Locrian x dark diminished = Locrian dark diminished

~~If we temper so that 10/9 ~ 11/10, and 27/25 ~ 12/11 (and 9/8 ~ 25/22) ie., tempering out [[100/99]], we get the Ptolemismic Porcutone diatonic with modes:~~

* Lydian dark major: ~ 10/9 5/4 11/8 3/2 5/3 11/6 2/1

* Mixolydian bright major: ~ 9/8 5/4 15/11 3/2 5/3 9/5 2/1

* Ionian bright diminished: ~ 10/9 11/9 4/3 22/15 5/3 11/6 2/1

* Dorian symmetric minor: ~ 10/9 6/5 4/3 3/2 5/3 9/5 2/1

* Phrygian bright minor: ~ 12/11 6/5 15/11 3/2 18/11 9/5 2/1

* Aeolian magical seventh: ~ 10/9 6/5 4/3 22/15 8/5 16/9 2/1

* Locrian dark diminished: ~ 12/11 6/5 4/3 16/11 8/5 9/5 2/1

[http://x31eq.com/cgi-bin/rt.cgi?ets=1ce_3p_4p&limit=2_3_5_11 Tuned to TE] (math over, here's tuning of the modes in cents, and their names, which combine the diatonic mode names with the functional Porcupine[7] mode names)

* Lydian dark major: 174.055 383.834 557.888 704.524 878.579 1052.633 1199.269

* Mixolydian bright major: 209.779 383.834 530.469 704.524 878.579 1025.214 1199.269

* Ionian bright diminished: 174.055 348.11 494.745 668.8 878.579 1052.633 1199.269

* Dorian symmetric minor: 174.055 320.69 494.745 704.524 878.579 1025.214 1199.269

* Phrygian bright minor: 146.635 320.69 530.469 704.524 851.159 1025.214 1199.269

* Aeolian magical seventh: 174.055 320.69 494.745 668.8 815.435 989.49 1199.269

* Locrian dark diminished: 146.635 320.69 494.745 641.38 815.435 1025.214 1199.269

The 11th harmonic as a Porcupine M4 is found at Meantone P4, tempering out 33/32, leading to [[Meanenneadecal]], where 33/32 is tempered out, hence the Ptotemismic Porcupine diatonic is a detempering of [[Meanenneadecal]].

== Porcutone chromatic and Porcutone octatonic ==

@@ Line 140: / Line 140: @@
 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.
+== 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.
 For the accompanying mapping for the Lumatone keyboard the G# / Ab 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).
-== Porcutone diatonic ==
+The Porcutone diatonic is a [[wakalix]] (pairwise well-formed scale) and a [[step-nested scale]]: A detempering of [[Meantone]][7] and [[Porcupine]][7], (and also of [[Dicot]][7]), a [[Fokker block]] with [[Unison vector|unison vectors]] of [[81/80]] and [[250/243]] (and [[25/24]]) has 1 large step of 9/8 (L x L), 3 medium steps of 10/9 (L x s), and 3 small steps of 27/25 (s x s).
-Warning, math, skip to where it says 'math over' if you're not into it: The Porcutone diatonic is a [[wakalix]] (pairwise well-formed scale) and a [[step-nested scale]]: A detempering of [[Meantone]][7] and [[Porcupine]][7], (and also of [[Dicot]][7]), a [[Fokker block]] with [[Unison vector|unison vectors]] of [[81/80]] and [[250/243]] (and [[25/24]]) has 1 large step of 9/8 (L x L), 3 medium steps of 10/9 (L x s), and 3 small steps of 27/25 (s x s)
-The six modes of the just porcutone diatonic are:
-* Mode 3: 10/9 5/4 25/18 3/2 5/3 50/27 2/1 = Meantone[7] 6|0 x Porcupine[7] 5|1 = LLLsLLs x sLsssss = MLMsMMs = Lydian x dark major = Lydian dark major
-* Mode 2: 9/8 5/4 27/20 3/2 5/3 9/5 2/1 = Meantone[7] 4|2 x Porcupine[7] 6|0 = LLsLLsL x Lssssss = LMsMMsM = Mixolydian x bright major = Mixolydian bright major
-* Mode 1: 10/9 100/81 4/3 40/27 5/3 50/27 2/1 = Meantone[7] 5|1 x Porcupine[7] 2|4 = LLsLLLs x ssssLss = MMsMLMs = Ionian x bright diminished = Ionian bright diminished
-* Mode 0: 10/9 6/5 4/3 3/2 5/3 9/5 2/1 = Meantone[7] 3|3 x Porcupine[7] 3|3 = LsLLLsL x sssLsss = MsMLMsM = Dorian x symmetric minor = Dorian symmetric minor
-* Mode -1: 27/25 6/5 27/20 3/2 81/50 9/5 2/1 = Meantone[7] 1|5 x Porcupine[7] 4|2 = sLLLsLL x ssLssss = sMLMsMM = Phrygian x bright minor = Phrygian bright minor
-* Mode -2: 10/9 6/5 4/3 40/27 8/5 16/9 2/1 = Meantone[7] 2|4 x Porcupine[7] 0|6 = LsLLsLL x ssssssL = MsMMsML = Aeolian x magical seventh mode = Aeolian magical seventh
-* Mode -3: 27/25 6/5 4/3 36/25 8/5 9/5 2/1 = Meantone[7] 0|6 x Porcupine[7] 1|5 = sLLsLLL x sssssLs = sMMsMLs = Locrian x dark diminished = Locrian dark diminished
-If we temper so that 10/9 ~ 11/10, and 27/25 ~ 12/11 (and 9/8 ~ 25/22) ie., tempering out [[100/99]], we get the Ptolemismic Porcutone diatonic with modes:
-* Lydian dark major: ~ 10/9 5/4 11/8 3/2 5/3 11/6 2/1
-* Mixolydian bright major: ~ 9/8 5/4 15/11 3/2 5/3 9/5 2/1
-* Ionian bright diminished: ~ 10/9 11/9 4/3 22/15 5/3 11/6 2/1
-* Dorian symmetric minor:  ~ 10/9 6/5 4/3 3/2 5/3 9/5 2/1
-* Phrygian bright minor: ~ 12/11 6/5 15/11 3/2 18/11 9/5 2/1
-* Aeolian magical seventh: ~ 10/9 6/5 4/3 22/15 8/5 16/9 2/1
-* Locrian dark diminished: ~ 12/11 6/5 4/3 16/11 8/5 9/5 2/1
-[http://x31eq.com/cgi-bin/rt.cgi?ets=1ce_3p_4p&limit=2_3_5_11 Tuned to TE] (math over, here's tuning of the modes in cents, and their names, which combine the diatonic mode names with the functional Porcupine[7] mode names)
-* Lydian dark major: 174.055 383.834 557.888 704.524 878.579 1052.633 1199.269
-* Mixolydian bright major: 209.779 383.834 530.469 704.524 878.579 1025.214 1199.269
-* Ionian bright diminished: 174.055 348.11 494.745 668.8 878.579 1052.633 1199.269
-* Dorian symmetric minor: 174.055 320.69 494.745 704.524 878.579 1025.214 1199.269
-* Phrygian bright minor: 146.635 320.69 530.469 704.524 851.159 1025.214 1199.269
-* Aeolian magical seventh: 174.055 320.69 494.745 668.8 815.435 989.49 1199.269
-* Locrian dark diminished: 146.635 320.69 494.745 641.38 815.435 1025.214 1199.269
-The 11th harmonic as a Porcupine M4 is found at Meantone P4, tempering out 33/32, leading to [[Meanenneadecal]], where 33/32 is tempered out, hence the Ptotemismic Porcupine diatonic is a detempering of [[Meanenneadecal]].
 == Porcutone chromatic and Porcutone octatonic ==