Pinetone: Difference between revisions
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== The | == The porcutone pentatonic and the porcutone chromatic == | ||
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 | 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 pernatonic 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). 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. | ||
Adding these notes leads 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. | Adding these notes leads 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. | ||
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41edo with 1200.2039c octave: 7L 1m 4s = (5, 2, 1) = (146.3663c, 58.5465c, 29.2733c) | 41edo with 1200.2039c octave: 7L 1m 4s = (5, 2, 1) = (146.3663c, 58.5465c, 29.2733c) | ||
== The | == The porcutone octatonic == | ||
The porcupine comma is the small step of the scale, so tempering the porcutone chromatic scale to porcupine leads from 7L 1m 4s = (27/25, 25/24, 250/243) to 7L 1s = (10/9~27/25, 25/24~81/80), which is Porcupine[8]! The Porcupine[7] scale has its large step between G and A, so the eighth note of Porcupine[8] is either G♯ or A♭, adding another small step of Porcupine[7] below A (for G♯) or above G (A♭). Mode -3 or mode 3 of the porcutone chromatic scale, respectively, are set to D so that this is preserved in The Porcutone System. This leads to the porcutone octatonic scales: D E F G G♯/A♭ A B C. In Just intonation: 10/9 6/5 4/3 25/18 3/2 5/3 9/5 2/1 with G♯, or 10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1 with A♭. This scale has 4 large steps of 10/9, 3 medium steps of 27/25, and 1 small step of 25/24. It is not mirror-symmetric, or equivalentely, it is ''[[Chirality|chiral]]'' so it cannot be uniquely defined with a step signature like Meantone[7], Porcupine[7], Porcupine[8], Meantone[12], and the porcutone diatonic (the Zarlio/Ptolemy just major scale is also not mirror symmetric). Scales that can be uniquely defined by a step signature are called ''step-nested scales''. More on that later. The mirror inverse of any mode of the porcutone octatonic with G♯ is a mode of the porcutone octatonic with A♭. The porcutone octatonic with G♯ is called the left handed porcupine octatonic, and the porcutone octatonic with A♭ is called the right handed porcupine octatonic (see [[chirality]]). | The porcupine comma is the small step of the scale, so tempering the porcutone chromatic scale to porcupine leads from 7L 1m 4s = (27/25, 25/24, 250/243) to 7L 1s = (10/9~27/25, 25/24~81/80), which is Porcupine[8]! The Porcupine[7] scale has its large step between G and A, so the eighth note of Porcupine[8] is either G♯ or A♭, adding another small step of Porcupine[7] below A (for G♯) or above G (A♭). Mode -3 or mode 3 of the porcutone chromatic scale, respectively, are set to D so that this is preserved in The Porcutone System. This leads to the porcutone octatonic scales: D E F G G♯/A♭ A B C. In Just intonation: 10/9 6/5 4/3 25/18 3/2 5/3 9/5 2/1 with G♯, or 10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1 with A♭. This scale has 4 large steps of 10/9, 3 medium steps of 27/25, and 1 small step of 25/24. It is not mirror-symmetric, or equivalentely, it is ''[[Chirality|chiral]]'' so it cannot be uniquely defined with a step signature like Meantone[7], Porcupine[7], Porcupine[8], Meantone[12], and the porcutone diatonic (the Zarlio/Ptolemy just major scale is also not mirror symmetric). Scales that can be uniquely defined by a step signature are called ''step-nested scales''. More on that later. The mirror inverse of any mode of the porcutone octatonic with G♯ is a mode of the porcutone octatonic with A♭. The porcutone octatonic with G♯ is called the left handed porcupine octatonic, and the porcutone octatonic with A♭ is called the right handed porcupine octatonic (see [[chirality]]). | ||