Pinetone: Difference between revisions

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 L 1m 4s = (27/25~12/11~13/12, 25/24~33/32~27/26, 81/80~250/243~55/54~121/120~40/39) = (142.6653, 66.6782, 33.3391), which we note is very similar to 2.3.5.11.13 Ptolemismic.
+== Pinetone melodic major and minor ==
+As well as our natural keys, diatonic scales are available in the Pinetone chromatic in 5 other keys (not 11, since we do not temper out the [[Pythagorean comma]]), i.e., in mode -3 the following diatonic scales are avilable:
+D E F G A B C
+D E F♯ G A B C
+D E F♯ G A B C♯
+D E F♯ G♯ A B C♯
+D♯ E F♯ G♯ A  B C♯
+D♯ E F♯ G♯ A♯  B C♯
+in mode 3 the following additional 5 diatonic scales are available:
+D E F G A B♭ C
+D E♭ F G A B♭ C
+D E♭ F G A♭ B♭ C
+D♭ E♭ F G A♭ B♭ C
+D♭ E♭ F G♭ A♭ B♭ C
+Each of these 11 scales are different (in terms of the intervals they contain). We will explore the D E F♯ G A B C and D E F G A B♭ C, which we will call the Pinetone melodic major and minor respectively.
+The scale D E F G A B C has step pattern MsMLMsM, which tempers to sssLsss under Porcupine (M=s) and LsLLLsL under Meantone (M=L).
+Raising an F to an F♯ replaces ~6/5 with ~11/9, i.e., it raises the F by ~55/54, which is tempered out in Porcupine, so the scale D E F♯ G A B C tempers to sssLsss under Porcupine as before, but to LLsLLsL under Meantone (Mixolydian mode rather than Dorian as before), so will call the mode approximating the ratios 10/9 11/9 4/3 3/2 5/3 9/5 2/1 Mixolydian symmetric minor.
+{| class="wikitable"
+|+Table 4.1. Modes of the Ptolemismic Pinetone melodic major
+!Mode number
+!Pinetone diatonic mode
+!Step pattern
+!Mode as simplest JI pre-image
+|-
+|3
+|Ionian bright major*
+|Lmsmmms
+|~ 9/8 5/4 15/11 3/2 5/3 11/6 2/1
+|-
+|2
+|Lydian bright diminished
+|mmmsLms
+|~ 10/9 11/9 110/81 22/15 5/3 11/6 2/1
+|-
+|1
+|Aeolian bright minor*
+|msLmsmm
+|~ 10/9 6/5 15/11 3/2 18/11 9/5 2/1
+|-
+| 0
+|Mixolydian symmetric minor
+|mmsLmsm
+|~ 10/9 11/9 4/3 3/2 5/3 9/5 2/1
+|-
+| -1
+|Locian dark major
+|sLmsmmm
+|~ 12/11 27/22 15/11 81/55 18/11 9/5 2/1
+|-
+| -2
+|Dorian magical seventh
+|msmmmsL
+|~ 10/9 6/5 4/3 22/15 44/27 16/9 2/1
+|-
+| -3
+|Phrygian dark diminished
+|smmmsLm
+|~ 12/11 6/5 4/3 22/15 8/5 9/5 2/1
+|}
+Similarly, lowering B to B♭ lowers by ~55/54, leading to the Aeolian symmetric minor shown in Table 4.2, which I consider to be a really beautiful minor minor mode.
+{| class="wikitable"
+|+Table 4.2. Modes of the Ptolemismic Pinetone melodic minor
+!Mode number
+!Pinetone diatonic mode
+!Step pattern
+!Mode as simplest JI pre-image
+|-
+|3
+|Ionian dark major*
+|mLsmmms
+|~ 10/9 5/4 15/11 3/2 5/3 11/6 2/1
+|-
+|2
+|Dorian bright major
+|Lsmmmsm
+|~ 9/8 27/22 15/11 3/2 5/3 9/5 2/1
+|-
+|1
+|Lydian dark diminished
+|mmmsmLs
+|~ 10/9 11/9 110/81 22/15 44/27 11/6 2/1
+|-
+| 0
+|Aeolian symmetric minor*
+|msmLsmm
+|~ 10/9 6/5 4/3 3/2 18/11 9/5 2/1
+|-
+| -1
+|Mixolydian bright diminished
+|mmsmLsm
+|~ 10/9 11/9 4/3 22/15 5/3 9/5 2/1
+|-
+| -2
+|Locrian bright minor
+|smLsmmm
+|~ 12/11 6/5 15/11 81/55 11/8 9/5 2/1
+|-
+| -3
+|Phrygian magical seventh
+|smmmsmL
+|~ 12/11 6/5 4/3 22/15 8/5 16/9 2/1
+|}
+Note that the Pinetone melodic major includes a neutral triad 18:22:27 on the root of the Mixolydian symmetric minor, and the Pinetone melodic minor includes it's inverse on the root of the Dorian bright major, a flavour of triad not available in the Pinetone diatonic. Note also though the only one major and one minor triad are available in each of these scales, as opposed to the two each available in the Pinetone diatonic, hence the labelling of these scales as for melodic purposes.
 ==Pinetone octatonic scales==