Pinetone: Difference between revisions

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mLm(sm)Lm(sm)Lm(sm)Lm -> mLmsmLmsmLmsmLm, which we later introduce as Pinetone-15.

==== Pinetone diminished heptatonic ====

Starting with the 5-limit diminished tetrad 6/5 36/25 5/3 2/1, putting 10/9 above each note takes us to the Pinetone diminished octatonic 10/9 6/5 4/3 36/25 8/5 5/3 50/27 2/1. The 5-limit diminished tetrad has 3 large steps of 6/5, and one small step of 125/108. If we only put 10/9 above the large steps, we get the scale 10/9 6/5 4/3 36/25 5/3 50/27 2/1, with step pattern msmsLms, comprising 3 small steps of 27/25, 3 medium steps of 10/9, and one large step of 125/108. Tempering the small and medium steps together gives us the scale sssssLss, Porcupine[7]. Tempering L=m gives us LsLsLLs, Dicot[7], and tempering s=L gives us sLsLLsL as Sixix[7]. Tempering out 100/99 leads to simplest JI pre-image 10/9 6/5 4/3 16/11 5/3 11/6 2/1, and additionally tempering out 144/143 to 10/9 6/5 4/3 13/9 5/3 11/6 2/1. The Pinetone diminished heptatonic's large step tempers to 55/48 under 2.3.5.11 Ptolemismic, and to 15/13 under 2.3.5.11.13 Ptolemismic.

==Summary for xen-math nerds==

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Replacing the A with an A♭ instead, we get the modes of the Pinetone harmonic major scale. Starting on D we get the mode:

174.055 320.69 494.745 641.38 878.579 1025.214 1199.269 as the notes D E F G A♭ B C D, representing 10/9 6/5 4/3 36/25~13/9 5/3 9/5 2/1

174.055 320.69 494.745 641.38 878.579 1025.214 1199.269 as the notes D E F G A♭ B C D, representing 10/9 6/5 4/3 36/25~13/9 5/3 9/5 2/1.

Which has modes:

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*smsLsmm Phrygian ♭4 symmetric minor starting on E

*smmsmsL Locrian magical ♭♭7 starting on B

We can see that this scale differs from the Pinetone diminished heptatonic by only a single note 9/5 instead of 50/27. The augmented step of the Pinetone harmonic minor and major scales is the same as of the Pinetone diminished heptatonic, representing 15/13 when 325/324 is tempered out (the difference between 100/99 and 144/143).

==Pinetone hyperchromatic scales==

@@ Line 2,380: / Line 2,380: @@
 mLm(sm)Lm(sm)Lm(sm)Lm -> mLmsmLmsmLmsmLm, which we later introduce as Pinetone-15.
+==== Pinetone diminished heptatonic ====
+Starting with the 5-limit diminished tetrad 6/5 36/25 5/3 2/1, putting 10/9 above each note takes us to the Pinetone diminished octatonic 10/9 6/5 4/3 36/25 8/5 5/3 50/27 2/1. The 5-limit diminished tetrad has 3 large steps of 6/5, and one small step of 125/108. If we only put 10/9 above the large steps, we get the scale 10/9 6/5 4/3 36/25 5/3 50/27 2/1, with step pattern msmsLms, comprising 3 small steps of 27/25, 3 medium steps of 10/9, and one large step of 125/108. Tempering the small and medium steps together gives us the scale sssssLss, Porcupine[7]. Tempering L=m gives us LsLsLLs, Dicot[7], and tempering s=L gives us sLsLLsL as Sixix[7]. Tempering out 100/99 leads to simplest JI pre-image 10/9 6/5 4/3 16/11 5/3 11/6 2/1, and additionally tempering out 144/143 to 10/9 6/5 4/3 13/9 5/3 11/6 2/1. The Pinetone diminished heptatonic's large step tempers to 55/48 under 2.3.5.11 Ptolemismic, and to 15/13 under 2.3.5.11.13 Ptolemismic.
 ==Summary for xen-math nerds==
@@ Line 2,417: / Line 2,420: @@
 Replacing the A with an A♭ instead, we get the modes of the Pinetone harmonic major scale. Starting on D we get the mode:
-.055 320.69 494.745 641.38 878.579 1025.214 1199.269 as the notes D E F G A♭ B C D, representing 10/9 6/5 4/3 36/25~13/9 5/3 9/5 2/1
+.055 320.69 494.745 641.38 878.579 1025.214 1199.269 as the notes D E F G A♭ B C D, representing 10/9 6/5 4/3 36/25~13/9 5/3 9/5 2/1.
 Which has modes:
@@ Line 2,428: / Line 2,431: @@
 *smsLsmm Phrygian ♭4 symmetric minor starting on E
 *smmsmsL Locrian magical ♭♭7 starting on B
+We can see that this scale differs from the Pinetone diminished heptatonic by only a single note 9/5 instead of 50/27. The augmented step of the Pinetone harmonic minor and major scales is the same as of the Pinetone diminished heptatonic, representing 15/13 when 325/324 is tempered out (the difference between 100/99 and 144/143).
 ==Pinetone hyperchromatic scales==

Pinetone: Difference between revisions

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