Pinetone: Difference between revisions

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We know the (meantone) pentatonic scale to be a subset of the (meantone) diatonic scale. Similarly, the porcutone pentatonic is a subset of the porcutone diatonic. We also know that adding a (meantone) pentatonic to a (meantone) diatonic leads to a (meantone) chromatic, i.e., diatonic on white keys + pentatonic on black keys. We can do this with porcutone.

=== Porcutone pentatonic ===

Using the familiar Bosanquet 12-note keyboard mapping (the preset for 12edo), we set the porcutone diatonic scale to the white keys, starting on D. We than add, on F♯/G♭, the porcutone penatonic as a set of 5 chromatic keys. There are two options for the chromatic keys, either all sharps or all flats. All sharps makes the porcutone harmonic minor available, and all flats makes the porcutone harmonic major available. These scales will be discussed below. In either case, in the just tuning, the chromatic keys give the scale 9/8 5/4 3/2 5/3 2/1, starting from F♯/G♭, tuned to 100/81 (F♯) or 162/125 (G♭) from D. This scale has step pattern msLsL, with step signature and step mapping 2L 1m 2s = (6/5, 9/8, 10/9). The same scale is also available as G-A-B-D-E.

We are familiar with this scale as the just pentatonic. If we temper m and s together, we get Meantone[5]: ssLsL. If we temper m and L together instead we get a scale called Father[5], tempering out the diatonic semitone 16/15. This mode of Father[5] has step pattern LsLsL. Keep the connection to Father[5] in the back of your minds for now, we'll come back to it. The porcutone pentatonic is also a subset of the porcutone diatonic, since Meantone[5] is a subset of Meantone[7]. It is available as G-A-B-D-E. The scale F-G-A-C-D is a mode of the inverse of G-A-B-D-E. The porcutone pentatonic is [[Chirality|''chiral'']] (i.e., it is not symmetric, unlike the porcutone diatonic and procutone chromatic scales, which are ''achiral''). There is a pair of porcutone pentatonic scales, the right handed porcutone pentatonic, 9/8 5/4 3/2 5/3 2/1 in JI, it's mirror inverse the left handed porcutone pentatonic, 10/9 5/4 3/2 5/3 2/1 in JI, which tempers to ssLsL in Meantone, but to sLLsL in Father. F-G-A-C-D gives a left handed porcutone pentatonic, while G-A-B-D-E, F♯-G♯-A♯-C♯-D♯, and G♭-A♭-B♭-D♭-E♭ are right handed porcutone pentatonics.

Adding the right handed porcutone ~~chromatic~~ (on F♯/G♭) to the just porcutone ~~chromatic~~, a 12-note mirror-symmetric scale with step signature and step mapping of 7L 1m 4s = (27/25, 25/24, 250/243) = (133.2376c, 70.6724c, 49.1661c), i.e., 7 large steps of what was the small step of the just porcutone diatonic, 1 medium step of the chromatic semitone 25/24, the distance between 6/5 and 5/4, and 4 small steps of 250/243, the porcupine comma, that separates 10/9 from 27/25. For the all sharps scale, we set mode -3 on D (for all flats we set mode 3 on D): 250/243 10/9 6/5 100/81 4/3 25/18 3/2 125/81 5/3 9/5 50/27 2/1, with step pattern sLLsLmLsLLsL.

=== Porcutone chromatic ===

Adding the right handed porcutone pentatonic (on F♯/G♭) to the just porcutone diatonic, a 12-note mirror-symmetric scale with step signature and step mapping of 7L 1m 4s = (27/25, 25/24, 250/243) = (133.2376c, 70.6724c, 49.1661c), i.e., 7 large steps of what was the small step of the just porcutone diatonic, 1 medium step of the chromatic semitone 25/24, the distance between 6/5 and 5/4, and 4 small steps of 250/243, the porcupine comma, that separates 10/9 from 27/25. For the all sharps scale, we set mode -3 on D (for all flats we set mode 3 on D): 250/243 10/9 6/5 100/81 4/3 25/18 3/2 125/81 5/3 9/5 50/27 2/1, with step pattern sLLsLmLsLLsL.

The now familiar Meantone comma of 81/80 separates the medium step (25/24) from the small step (250/243), so our porcutone chromatic is a ''detempering'' of Meantone[12], the meantone chromatic scale, just like how the porcutone diatonic is a detempering of Meantone[7], the meantone diatonic scale.

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 We know the (meantone) pentatonic scale to be a subset of the (meantone) diatonic scale. Similarly, the porcutone pentatonic is a subset of the porcutone diatonic. We also know that adding a (meantone) pentatonic to a (meantone) diatonic leads to a (meantone) chromatic, i.e., diatonic on white keys + pentatonic on black keys. We can do this with porcutone.
+=== Porcutone pentatonic ===
 Using the familiar Bosanquet 12-note keyboard mapping (the preset for 12edo), we set the porcutone diatonic scale to the white keys, starting on D. We than add, on F♯/G♭, the porcutone penatonic as a set of 5 chromatic keys. There are two options for the chromatic keys, either all sharps or all flats. All sharps makes the porcutone harmonic minor available, and all flats makes the porcutone harmonic major available. These scales will be discussed below. In either case, in the just tuning, the chromatic keys give the scale 9/8 5/4 3/2 5/3 2/1, starting from F♯/G♭, tuned to 100/81 (F♯) or 162/125 (G♭) from D. This scale has step pattern msLsL, with step signature and step mapping 2L 1m 2s = (6/5, 9/8, 10/9). The same scale is also available as G-A-B-D-E.
 We are familiar with this scale as the just pentatonic. If we temper m and s together, we get Meantone[5]: ssLsL. If we temper m and L together instead we get a scale called Father[5], tempering out the diatonic semitone 16/15. This mode of Father[5] has step pattern LsLsL. Keep the connection to Father[5] in the back of your minds for now, we'll come back to it. The porcutone pentatonic is also a subset of the porcutone diatonic, since Meantone[5] is a subset of Meantone[7]. It is available as G-A-B-D-E. The scale F-G-A-C-D is a mode of the inverse of G-A-B-D-E. The porcutone pentatonic is [[Chirality|''chiral'']] (i.e., it is not symmetric, unlike the porcutone diatonic and procutone chromatic scales, which are ''achiral''). There is a pair of porcutone pentatonic scales, the right handed porcutone pentatonic, 9/8 5/4 3/2 5/3 2/1 in JI, it's mirror inverse the left handed porcutone pentatonic, 10/9 5/4 3/2 5/3 2/1 in JI, which tempers to ssLsL in Meantone, but to sLLsL in Father. F-G-A-C-D gives a left handed porcutone pentatonic, while G-A-B-D-E, F♯-G♯-A♯-C♯-D♯, and G♭-A♭-B♭-D♭-E♭ are right handed porcutone pentatonics.
-Adding the right handed porcutone chromatic (on F♯/G♭) to the just porcutone chromatic, a 12-note mirror-symmetric scale with step signature and step mapping of 7L 1m 4s = (27/25, 25/24, 250/243) = (133.2376c, 70.6724c, 49.1661c), i.e., 7 large steps of what was the small step of the just porcutone diatonic, 1 medium step of the chromatic semitone 25/24, the distance between 6/5 and 5/4, and 4 small steps of 250/243, the porcupine comma, that separates 10/9 from 27/25. For the all sharps scale, we set mode -3 on D (for all flats we set mode 3 on D): 250/243 10/9 6/5 100/81 4/3 25/18 3/2 125/81 5/3 9/5 50/27 2/1, with step pattern sLLsLmLsLLsL.
+=== Porcutone chromatic ===
+Adding the right handed porcutone pentatonic (on F♯/G♭) to the just porcutone diatonic, a 12-note mirror-symmetric scale with step signature and step mapping of 7L 1m 4s = (27/25, 25/24, 250/243) = (133.2376c, 70.6724c, 49.1661c), i.e., 7 large steps of what was the small step of the just porcutone diatonic, 1 medium step of the chromatic semitone 25/24, the distance between 6/5 and 5/4, and 4 small steps of 250/243, the porcupine comma, that separates 10/9 from 27/25. For the all sharps scale, we set mode -3 on D (for all flats we set mode 3 on D): 250/243 10/9 6/5 100/81 4/3 25/18 3/2 125/81 5/3 9/5 50/27 2/1, with step pattern sLLsLmLsLLsL.
 The now familiar Meantone comma of 81/80 separates the medium step (25/24) from the small step (250/243), so our porcutone chromatic is a ''detempering'' of Meantone[12], the meantone chromatic scale, just like how the porcutone diatonic is a detempering of Meantone[7], the meantone diatonic scale.