Pinetone: Difference between revisions
→Pinetone melodic major and minor: re-wrote this section and added some note names to some tables from the preceding section |
→Other diatonic subsets of the Pinetone chromatic scale: added another table |
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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, and use the mode to name the scale as a whole. | 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, and use the mode to name the scale as a whole. | ||
To calculate the mode numbers for Tables 4.2-4.5, the mode numbers of their temperings to Porcupine and to Meantone were added, ordered, and renumbered. | To calculate the mode numbers for Tables 4.2-4.5, the mode numbers of their temperings to Porcupine and to Meantone were added, ordered, and renumbered. When two modes are tied, small changes in tuning will affect their order. Given that Ptolemismic Porcupine is more accurate than Ptolemismic Meantone, the Porcupine mode number is weighted more heavily to determine the mode order in the event of any ties. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Table 4.2. Modes of the Ptolemismic Pinetone Mixolydian symmetric minor | |+Table 4.2. Modes of the Ptolemismic Pinetone Mixolydian symmetric minor | ||
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|} | |} | ||
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. | 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. | ||
{| class="wikitable" | |||
|+Table 4.3. Modes of the Ptolemismic Pinetone Ionian symmetric minor | |||
!Mode number | |||
!Pinetone diatonic mode | |||
! colspan="2" |Root note | |||
!Step pattern | |||
!Mode as simplest JI pre-image | |||
|- | |||
|3 | |||
|Lydian bright major* | |||
|G | |||
|A♭ | |||
|Lmmsmms | |||
|~ 9/8 5/4 11/8 3/2 5/3 11/6 2/1 | |||
|- | |||
|2 | |||
|Ionian symmetric minor | |||
|D | |||
|E♭ | |||
|mmsLmms | |||
|~ 10/9 11/9 4/3 3/2 5/3 11/6 2/1 | |||
|- | |||
| 1 | |||
|Dorian bright minor* | |||
|E | |||
|F | |||
|msLmmsm | |||
|~ 10/9 6/5 15/11 3/2 5/3 9/5 2/1 | |||
|- | |||
| 0 | |||
|Locrian dark major | |||
|F♯ | |||
|G | |||
|sLmmsmm | |||
|~ 12/11 27/22 15/11 3/2 18/11 9/5 2/1 | |||
|- | |||
| -1 | |||
|Mixolydian magical seventh | |||
|A | |||
|B♭ | |||
|mmsmmsL | |||
|~ 10/9 11/9 4/3 22/15 44/27 16/9 2/1 | |||
|- | |||
| -2 | |||
|Phrygian bright diminished | |||
|C♯ | |||
|D | |||
|smmsLmm | |||
|~ 12/11 6/5 4/3 16/11 18/11 10/9 2/1 | |||
|- | |||
| -3 | |||
|Aeolian dark diminished | |||
|B | |||
|C | |||
|msmmsLm | |||
|~ 10/9 6/5 4/3 22/15 8/5 9/5 2/1 | |||
|} | |||
==Pinetone octatonic scales== | ==Pinetone octatonic scales== | ||
The Porcupine comma is the small step of the Pinetone chromatic, so tempering the Pinetone 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 Pinetone chromatic scale, respectively, are set to D so that this is preserved in Pinetone. This leads to the Pinetone 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 Pinetone diatonic (the Zarlino/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 Pinetone octatonic with G♯ is called the Pinetone major-harmonic octatonic, and the Pinetone octatonic with A♭ is called the Pinetone minor-harmonic octatonic. These names will make sense to the reader after further reading on these scales and the chords they contain, and on the Pintone diminished octatonic introduced below. The mirror inverse of any mode of the Pinetone major-harmonic octatonic is a mode of the Pinetone minor-harmonic octatonic (see [[chirality]]). This is true similarly of the familiar harmonic minor and harmonic major scales. | The Porcupine comma is the small step of the Pinetone chromatic, so tempering the Pinetone 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 Pinetone chromatic scale, respectively, are set to D so that this is preserved in Pinetone. This leads to the Pinetone 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 Pinetone diatonic (the Zarlino/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 Pinetone octatonic with G♯ is called the Pinetone major-harmonic octatonic, and the Pinetone octatonic with A♭ is called the Pinetone minor-harmonic octatonic. These names will make sense to the reader after further reading on these scales and the chords they contain, and on the Pintone diminished octatonic introduced below. The mirror inverse of any mode of the Pinetone major-harmonic octatonic is a mode of the Pinetone minor-harmonic octatonic (see [[chirality]]). This is true similarly of the familiar harmonic minor and harmonic major scales. | ||