Pinetone: Difference between revisions
Added rewrite of Porcutone chromatic section, started separate Porcutone octatonic section |
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Additionally available in porcutone are a set of octatonic modes with their own Porcupine functional harmony, that combine [[Porcupine]][8] with the [[oneirotonic]] modes that are gaining popularity at the moment. | Additionally available in porcutone are a set of octatonic modes with their own Porcupine functional harmony, that combine [[Porcupine]][8] with the [[oneirotonic]] modes that are gaining popularity at the moment. | ||
If you have a [[Lumatone]], you can use the standard Bosanquet mapping for 12edo. The white keys are the porcutone diatonic, a cross between the meantone diatonic scale and Porcupine[7], and then black keys give the porcutone pentatonic, which approximates the just intonation pentatonic scale 9/8 5/4 3/2 5/3 2/1. I've chosen to colour the | If you have a [[Lumatone]], you can use the standard Bosanquet mapping for 12edo. The white keys are the porcutone diatonic, a cross between the meantone diatonic scale and Porcupine[7], and then black keys give the porcutone pentatonic, which approximates the just intonation pentatonic scale 9/8 5/4 3/2 5/3 2/1. I've chosen to colour the G♯/Ab key pink, and the other chromatic keys blue, because I'm a proud trans woman and a big nerd. You can use any colours, but I find it helps to colour the G♯/A♭ key a different colour since that's the one chromatic key used along with the diatonic keys to make the porcutone octatonic. | ||
== How it works == | == How it works - Porcutone diatonic == | ||
The diatonic scale has a step signature of [[5L 2s]], meaning it has 5 large steps and 2 small step arranged in the step pattern LsLLLsL (represent in mode 0, Dorian mode). In Meantone[7], the large step represents both 9/8 and 10/9, the major and minor tones (''tempering out'' the [[81/80]] that separates them) hence the name "Meantone". The small step represents 16/15 and 27/25 (which differ again by [[81/80]]). We write this as [[5L 2s]] = (9/8~10/9, 16/15~27/25). Porcupine[7] instead has step step signature and step mapping [[1L 6s]] = (~9/8, 10/9~27/25), hence the difference between 10/9 and 27/25, [[250/243]], is tempered out. In mode 0 it has step pattern sssLsss. [[81/80]] is called the Meantone comma, and [[250/243]] is called the Porcupine comma. | The diatonic scale has a step signature of [[5L 2s]], meaning it has 5 large steps and 2 small step arranged in the step pattern LsLLLsL (represent in mode 0, Dorian mode). In Meantone[7], the large step represents both 9/8 and 10/9, the major and minor tones (''tempering out'' the [[81/80]] that separates them) hence the name "Meantone". The small step represents 16/15 and 27/25 (which differ again by [[81/80]]). We write this as [[5L 2s]] = (9/8~10/9, 16/15~27/25). Porcupine[7] instead has step step signature and step mapping [[1L 6s]] = (~9/8, 10/9~27/25), hence the difference between 10/9 and 27/25, [[250/243]], is tempered out. In mode 0 it has step pattern sssLsss. [[81/80]] is called the Meantone comma, and [[250/243]] is called the Porcupine comma. | ||
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|Locrian dark diminished | |Locrian dark diminished | ||
|} | |} | ||
Like Meantone[7] and Porcupine[7], and unlike the Ptolemy/Zarlino just major scale, the porcutone diatonic scale is ''mirror symmetric'', meaning that the mirror inverse of any mode of the scale is also a mode of the scale, i.e., if we trace the steps of the mode from the top instead of from the bottom. This is reflected with the mode numbers. The mirror inverse of mode 3, the brightest mode, is mode -3, the darkest mode, and mode 0 is itself a symmetric mode, hence 'symmetric' in the mode name. We may already know this - that the Dorian mode of the familiar diatonic scale is symmetric, and the mirror inverse of the Lydian mode is the Locrian mode. | |||
The minor tone small step of Porcupine[7] can also represent the neutral seconds 11/10 and 12/11, since 10/9*11/10*12/11 = 4/3, and 4/3 is subtended by 3 small steps of Porcupine[7], tempering out both [[100/99]] and [[121/120]]. 11/8 is easily reached in Porcupine[7] as a major 4th, subtended by 2 small steps and 1 large step. The small step of Porcupine[7] represents all of 10/9, 11/10, 12/11 and 27/25, in order of largest to smallest. In the porcutone diatonic, the small step is 27/25 and the medium step is 10/9. We can access our 11-limit harmonies in porcutone by tempering out [[100/99]], which separates 10/9 from 11/10, as well as 27/25 from 12/11. This leads to step signature and step mapping 1L 4M 2s = (9/8~25/22, 10/9~11/10, 27/25~12/11). Since [[100/99]] is called the [[Ptolemisma]], we can call the resulting scale the ptolemismic porcutone diatonic. | The minor tone small step of Porcupine[7] can also represent the neutral seconds 11/10 and 12/11, since 10/9*11/10*12/11 = 4/3, and 4/3 is subtended by 3 small steps of Porcupine[7], tempering out both [[100/99]] and [[121/120]]. 11/8 is easily reached in Porcupine[7] as a major 4th, subtended by 2 small steps and 1 large step. The small step of Porcupine[7] represents all of 10/9, 11/10, 12/11 and 27/25, in order of largest to smallest. In the porcutone diatonic, the small step is 27/25 and the medium step is 10/9. We can access our 11-limit harmonies in porcutone by tempering out [[100/99]], which separates 10/9 from 11/10, as well as 27/25 from 12/11. This leads to step signature and step mapping 1L 4M 2s = (9/8~25/22, 10/9~11/10, 27/25~12/11). Since [[100/99]] is called the [[Ptolemisma]], we can call the resulting scale the ptolemismic porcutone diatonic. | ||
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41edo: 1L 4m 2s = (7, 6, 5) = (204.8780c, 175.6098c, 146.3415c) | 41edo: 1L 4m 2s = (7, 6, 5) = (204.8780c, 175.6098c, 146.3415c) | ||
We might also relax the tuning of the octave to optimize the tuning of the scale as a whole, leading to the following TE tunings of the scales | |||
27edo with 1195.1825c octave: 1L 4m 2s = (5, 4, 3) = (221.3301c, 177.0641c, 132.7981c) | |||
34edo with 1198.2070c octave: 1L 4m 2s = (6, 5, 4) = (211.4483c, 176.2069c, 140.9655c) | |||
41edo with 1200.2039c octave: 1L 4m 2s = (7, 6, 5) = (204.9129c, 175.6396c, 146.3663c) | |||
The table below show the sizes, interval names, ratios approximated, tuning, and occurence of all intervals of the ptolemismic porcutone diatonic scale within an octave, tuned to TE tuning. | |||
{| class="wikitable" | {| class="wikitable" | ||
|+Intervals of the porcutone diatonic | |+Intervals of the porcutone diatonic | ||
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2 | 2 | ||
|} | |} | ||
== 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 then add 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. | |||
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. | |||
The ptolemismic porcutone chromatic has a step signature, mapping, and TE tuning of 7L 1m 4s = (27/25~12/11, 25/24~33/32, 250/243~55/54~121/120) = (146.6352c, 63.1434c, 27.4197c). | |||
Mode -3 approximates the JI ratios: 55/54 10/9 6/5 11/9 4/3 11/8 3/2 55/36 5/3 9/5 11/6 2/1. | |||
The TE tuning in cents is: 27.420 174.055 320.690 348.110 494.745 557.888 704.524 731.943 878.579 1025.214 1052.633 1199.269 | |||
Mode 3, the mirror inverse of mode -3, approximates the JI ratios: 12/11 10/9 6/5 72/55 4/3 16/11 3/2 18/11 5/3 9/5 108/55 2/1. | |||
The TE tuning in cents is: 146.636 174.055 320.690 467.326 494.745 641.381 704.524 851.159 878.579 1025.214 1171.849 1199.269 | |||
The ptolemismic porcutone chromatic scale is distinctly xenharmonic, and yet is related to the familiar chromatic scale. | |||
As with the porcutone diatonic, tuning the porcutone chromatic to 19edo or 31edo collapses it to the Meantone[12] (Meanenneadecal[12]) chromatic scale. Tuning it to 15edo, 22edo, or 29edo collapses it to Porcupine[8]. Step patterns, mappings and sizes for tunings to 27edo, 34edo, and 41edo are as follows: | |||
27edo: 7L 1m 4s = (3, 2, 1) = (133.3333c, 88.8889c, 44.4444c) | |||
34edo: 7L 1m 4s = (4, 2, 1) = (141.1765c, 70.5882c, 35.2941c) | |||
41edo: 7L 1m 4s = (5, 2, 1) = (146.3415c, 58.5366c, 29.2683c) | |||
And allowing octave stretch, the tuning may be optimized via TE tuning to: | |||
27edo with 1195.1825c octave: 7L 1m 4s = (3, 2, 1) = | |||
34edo with 1198.2070c octave: 7L 1m 4s = (4, 2, 1) = | |||
41edo with 1200.2039c octave: 7L 1m 4s = (5, 2, 1) = | |||
== 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. | |||
== Summary for xen-math nerds == | == Summary for xen-math nerds == | ||