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[http://x31eq.com/cgi-bin/rt.cgi?ets=7%261ce%264p&limit=2.3.5.11 7L 1m 4s = (27/25~12/11, 25/24~33/32, 250/243~55/54~121/120) = (146.63528c, 63.14327c, 27.41960c)].  
[http://x31eq.com/cgi-bin/rt.cgi?ets=7%261ce%264p&limit=2.3.5.11 7L 1m 4s = (27/25~12/11, 25/24~33/32, 250/243~55/54~121/120) = (146.63528c, 63.14327c, 27.41960c)].  


== The porcutone octatonic ==
== Porcutone octatonic scales ==
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]]).  


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Maybe you have a Lumatone, and you're wondering, ok so you can either have sharps or flats? Por queno los dos?
Maybe you have a Lumatone, and you're wondering, ok so you can either have sharps or flats? Por queno los dos?


The resulting scales don't have nice properties like the others, but it's helpful to have both sharps and flats. The concept is simple. We add the porcutone pentatonics both on G♭, and on F♯, resulting in a 17-note scale containing both the right and left handed porcutone octatonics.
Indeed we can have both!


The resulting step pattern of the scales is s(L-s)sLs(L-s)sm(L-m)ms(L-s)sLs(L-s)s. Note there are 5 different step sizes.
From the porcutone chromatic with sharps (mode -3), we add another porcutone diatonic scale, mode 0 starting on D♭, leading to the left-handed porcutone hyperchromatic scale, with step pattern, sLsLssLsmLssLsLssLs.  


We might want to add in another two notes for E♯ or F♭ and B♯  or C♭. We'll call L-s 'D' (for large diminished) and L-m 'd' (for small diminished), resulting in 19-note scales with 4 step sizes:
Or, from the porcutone chromatic with flats (mode 3), we add another porcutone diatonic scale, mode 0 starting on D♯, leading to the right-handed porcutone hyperchromatic scale, with step pattern, sLssLsLssLmsLssLsLs.


sDsDssDsmdmsDsDssDs with E♯ and B♯, or sDssDsDsmdmsDssDsDs with F♭ and C♭, or sDsDssDsmdmsDssDsDs with E♯ and C♭.
If 81/80 were additionally tempered out, these scales would temper to Flattone[19], reflected in their layout on the lumatone. These scale comprises 7 large steps approximating 117/110 (the difference between the large and small steps of the porcutone chromtic), the medium step of the porcutone chromatic, approximating 25/24 and 33/32, and 11 small steps, the same as the small step of the pocutone chromatic, approximating 250/243, 55/54, 121/120, and 40/39.


We note that the the interval D-F♭ approximates 121/108, and is very close to 9/8. Accordingly we temper out the difference between 121/108 and 9/8, 243/242, leading to Tetracot temperament, where 2*s = m (and D = d + s).
We note that sLss, the interval from D to E♯, for example, is very near 9/8, and that sLsL, the interval from D to F♭, for an example, is very near 32/27. If we recognize these approximates, we additionally temper out 243/242, or 352/351, leading to Tetracot temperament, in which case the large step approximates 16/15. This also adds 81/80 to the list of intervals approximated by the small step. Adding an additional small step above G, for the left handed hyperchromatic, or below A, for the right handed hyperchromatic, would give us Tetracot[20].  


D is the largest step, at 111.924c, followed by d at 76.009c, then by m (66.766c) and s (30.852c).
In 2.3.5.11.13 Tetracot, the left handed porcutone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 32/27 6/5 11/9 13/10 4/3 11/8 22/15 3/2 20/13 18/11 5/3 16/9 9/5 11/6 39/20 2/1, and the right handed porcutone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 9/8 6/5 11/9 13/10 4/3 15/11 16/11 3/2 20/13 18/11 5/3 27/16 9/5 11/6 39/20 2/1.
 
We will call these scales the left handed, right handed, and symmetric porcutone hyperchromatic scales respectively.
 
Given that D-E♯ is 206.744c, very close to 9/8, but maps to 121/108, we can't really avoid tempering out 243/242, tuning to Tetracot
 
The right handed porcutone hyperchromatic approximates the JI ratios:
 
40/39 12/11 10/9 32/27 6/5 11/9 13/10 4/3 11/8 16/11 3/2 20/13 18/11 5/3 9/5 11/6 39/20 2/1, with a TE tuning of:
 
33.116 142.775 175.892 287.815 318.667 351.784 461.443 494.559 561.325 637.334 704.101 737.217 846.992 879.992 991.916 1022.768 1055.884 1165.543 1198.660
 
The left handed porcutone hyperchromatic approximates the JI ratios:
 
40/39 12/11 10/9 9/8 6/5 11/9 13/10 4/3 11/8 16/11 3/2 20/13 18/11 5/3 27/16 9/5 11/6 39/20 2/1, with a TE tuning of:
 
33.116 142.775 175.892 206.744 318.667 351.784 461.443 494.559 561.325 637.334 704.101 737.217 846.992 879.992 910.845 1022.768 1055.884 1165.543 1198.660
 
The symmetric porcutone hyperchromatic approximates the JI ratios:
 
40/39 12/11 10/9 32/27 6/5 11/9 13/10 4/3 11/8 16/11 3/2 20/13 18/11 5/3 27/16 9/5 11/6 39/20 2/1, with a TE tuning of:
 
33.116 142.775 175.892 287.815 318.667 351.784 461.443 494.559 561.325 637.334 704.101 737.217 846.992 879.992 910.815 1022.768 1055.884 1165.543 1198.660


Tuned to TE 2.3.5.11.13 Tetracot, the left handed porcutone hyperchromatic in cents is


and the right handed porcutone hyperchromatic in cents is
== Comma pump ==
== Comma pump ==
We can't use our circle of fifths (Meantone comma pump) or our Porcupine comma pumps here, as both 81/80 and 250/243 are observed. In the ptolemismic tuning we temper out 100/99 which we can can pump with chord progressions such as  
We can't use our circle of fifths (Meantone comma pump) or our Porcupine comma pumps here, as both 81/80 and 250/243 are observed. In the ptolemismic tuning we temper out 100/99 which we can can pump with chord progressions such as  
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