Pinetone

Are you interested in microtonal music with wild and wacky harmonies but want some familiarity to guide you? Heard about this Porcupine thing but not sure how to get 12 notes of it? Wish you had something like Porcupine but more accurate or with more interesting scales? Introducing The Porcutone System. The scales you know and love, with a new-age quirky spin. The perfect mix of consonant and dissonant harmonies, familiar and newfangled. Try it on your keyboard straight away (if you can return your keyboard using scale files, grab this one! Copy the text into notepad and save as a .scl file).

The porcutone system combines Porcupine – arguably the best way to add the 11th harmonic to major and minor harmonies in a seven-note scale – with with Meantone – the system underpinning most common practice music from the last several hundred years, so all the same scales (diatonic, harmonic minor, pentatonic, chromatic, etc.) are still available, just with a new Porcupine spin, and the 11th harmonic!

While there aren't as many consonant major and minor triads as we are used to, they are more consonant in Porcutone.

As opposed to in 12edo, each key is distinctly different in porcutone, both a blessing and a curse.

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 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.

The 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.

We are familiar with the Zarlino/Ptolemy just major scale: 9/8 5/4 4/3 3/2 5/3 15/8 2/1. This scale has 3 large steps of 9/8, 2 medium steps of 10/9, and 2 small steps of 16/15, with step pattern LMsLMLs. If we temper out the difference between L and M, we get LLsLLLs, the mode 2 of Meantone[7], the familiar Ionian/major mode.

Consider instead the just scale: 10/9 6/5 4/3 3/2 5/3 9/5 2/1, a just Dorian scale. This scale has 1 large step of 9/8, 4 medium steps of 10/9, and 2 small steps of 27/25, with step pattern MsMLMsM (mode 0). It can be represented with step signature and step mapping 1L 4M 2s = (9/8, 10/9, 27/25). This is our just porcutone diatonic.If we temper out the difference between L and M, we get LsLLLsL, Meantone[7] mode 0: Dorian; if we temper out instead the difference between 10/9 and 27/25, we get sssLsss, Porcupine[7] mode 0, which is referred to as symmetric minor. In this way, the just porcutone diatonic represents both Porcupine[7] and Meantone[7].

To name this mode of the porcutone diatonic, we simply add the mode names together, prefixing the Porcupine[7] functional mode name (which I am introducing here) with the meantone diatonic mode name, so mode 0 of the porcutone diatonic is called Dorian symmetric minor. We continue this process with the other 6 modes:

Modes of the just porcutone diatonic

Mode number	Mode in JI	Step pattern	Meantone[7]	Diatonic mode	Porcupine[7]	Porcupine[7] mode	Porcutone diatonic mode
3	10/9 5/4 25/18 3/2 5/3 50/27 2/1	MLMsMMs	LLLsLLs	Lydian	sLsssss	Dark major	Lydian dark major
2	9/8 5/4 27/20 3/2 5/3 9/5 2/1	LMsMMsM	LLsLLsL	Mixolydian	Lssssss	Bright major	Mixolydian bright minor
1	10/9 100/81 4/3 40/27 5/3 50/27 2/1	MMsMLMs	LLsLLLs	Ionian	ssssLss	Bright diminished	Ionian bright diminished
0	10/9 6/5 4/3 3/2 5/3 9/5 2/1	MsMLMsM	LsLLLsL	Dorian	sssLsss	Symmetric minor	Dorian symmetric minor
-1	27/25 6/5 27/20 3/2 81/50 9/5 2/1	sMLMsMM	sLLLsLL	Phrygian	ssLssss	Bright minor	Phrygian bright minor
-2	10/9 6/5 4/3 40/27 8/5 16/9 2/1	MsMMsML	LsLLsLL	Aeolian	ssssssL	Magical seventh	Aeolian magical seventh
-3	27/25 6/5 4/3 36/25 8/5 9/5 2/1	sMMsMLM	sLLsLLL	Locrian	sssssLs	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.

Something to note - the Meantone diatonic scale is generated by the perfect fifth, 3/2, which means that it can be formed by stacking perfect fifths on top of each other, i.e., F-C-G-D-A-E, and all the notes are connected by perfect fifths. Porcupine[7], on the other hand, is generated by 10/9, so all notes are connected by a chain of 10/9s, i.e., A-B-C-D-E-F-G, where the large step of 9/8 then separates G from A. The Zarlino/Ptolemy just major scale 9/8 5/4 4/3 3/2 5/3 15/8 2/1 can be built of two parallel chains of 3/2, i.e., 4/3-2/1-3/2-9/8, 5/3-5/4-15/8. Accordingly it is a generator-offset scale. If the scale is on C, then D-A is not a 3/2 perfect fifth, but a wolf fifth of 40/27. The porcutone diatonic is not a generator offset scale. Setting the scale to the naturals, D E F G A B C D, 3/2 perfect fifths are available above D, E, F, and C, so there are 1 fewer 3/2 perfect fifths in the porcutone diatonic scale than in the Zarlino/Ptolemy just major scale, and two fewer than in the typical diatonic scale. Porcupine[7] also has 3/2 fifths only above D, E, F, and G. It is because 3/2 perfect fifths are available above D, E, F, and G in both Meantone[7] and Porcupine[7] that they are available above D, E, F, and G in the 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.

The modes of the ptolemismic porcutone diatonic are shown below in their simplest JI pre-image (the simplest JI ratios each interval above the tonic represents), and in cents, in an optimized tuning called TE tuning.

Modes of the ptolemismic porcutone diatonic

Porcutone diatonic mode	Step pattern	Mode as simplest JI pre-image	Mode in cents
Lydian dark major	mLmsmms	~ 10/9 5/4 11/8 3/2 5/3 11/6 2/1	174.055 383.834 557.888 704.524 878.579 1052.633 1199.269
Mixolydian bright minor	Lmsmmsm	~ 9/8 5/4 15/11 3/2 5/3 9/5 2/1	209.779 383.834 530.469 704.524 878.579 1025.214 1199.269
Ionian bright diminished	mmsmLms	~ 10/9 11/9 4/3 22/15 5/3 11/6 2/1	174.055 348.110 494.745 668.800 878.579 1052.633 1199.269
Dorian symmetric minor	msmLmsm	~ 10/9 6/5 4/3 3/2 5/3 9/5 2/1	174.055 320.690 494.745 704.524 878.579 1025.214 1199.269
Phrygian bright minor	smLmsmm	~ 12/11 6/5 15/11 3/2 18/11 9/5 2/1	146.635 320.690 530.469 704.524 851.159 1025.214 1199.269
Aeolian magical seventh	msmmsmL	~ 10/9 6/5 4/3 22/15 8/5 16/9 2/1	174.055 320.690 494.745 668.800 815.435 989.490 1199.269
Locrian dark diminished	smmsmLm	~ 12/11 6/5 4/3 16/11 8/5 9/5 2/1	146.635 320.690 494.745 641.380 815.435 1025.214 1199.269

We see 11/8 as the 4th in Lydian dark major. In Meantone[7] this is an augmented fourth. The meantone extension representing 11/8 with an augmented fourth is called Meanenneadecal, referencing the fact that it is most at home in 19edo. Tuning the scale to 19edo (or 12edo) will collapse it into a Meanenneadecal[7] diatonic scale. Similarly, tuning the scale to 15edo, 22edo, or 29edo will collapse it to Porcupine[7] scale. 27edo, 34edo, and 41edo are good tunings for the porcutone diatonic if tuning to an edo is desired.

27edo: 1L 4m 2s = (5, 4, 3) = (222.2222c, 177.7778c, 133.3333c)

34edo: 1L 4m 2s = (6, 5, 4) = (211.7647c, 176.4706c, 141.1765c)

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)

For comparison, the TE tuning has step signature and mapping

TE ptolemismic: 1L 4m 2s = (209.77855c, 174.05488c, 146.63528c)

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.

Intervals of the porcutone diatonic

Interval class	sizes	Meantone[7] names	Porcupine[7] names	Porcutone diatonic names	JI ratios approximated	size in cents (TE)	Occurence
1-step	s m L	minor 2nd major 2nd major 2nd	minor 2nd minor 2nd major 2nd	small 2nd, minor 2nd medium 2nd, major minor 2nd large 2nd, major 2nd	27/25, 12/11 10/9, 11/10 9/8, 25/22	146.635 174.055 209.779	2 4 1
2-step	m + s m + m L + m	minor 3rd major 3rd major 3rd	minor 3rd minor 3rd major 3rd	small 3rd, minor 3rd medium 3rd, major minor 3rd large 3rd, major third	6/5, 40/33 100/81, 11/9 5/4, 33/20	320.690 348.110 383.834	4 1 2
3-step	2m + s L + m + s L + 2m	perfect 4th perfect 4th augmented 4th	minor 4th major 4th major 4th	small 4th, minor 4th medium 4th, minor major 4th large 4th, major 4th	4/3, 33/25 27/20, 15/11 25/18, 11/8	494.745 530.469 557.888	4 2 1
4-step	2m + 2s 3m + s L + 2m + s	diminished 5th perfect 5th perfect 5th	minor 5th minor 5th major 5th	small 5th, minor 5th medium 5th, major minor 5th large 5th, major 5th	36/25, 16/11 40/27, 22/15 3/2, 50/33	641.380 668.800 704.524	1 2 4
5-step	3m + 2s L + 2m + 2s L + 3m + s	minor 6th minor 6th major 6th	minor 6th major 6th major 6th	small 6th, minor 6th medium 6th, minor major 6th large 6th, major 6th	8/5, 40/33 81/50, 18/11 5/3, 33/20	815.435 851.159 878.579	2 1 4
6-step	4m + 2s L + 3m + 2s L + 4m + s	minor 7th minor 7th major 7th	minor 7th major 7th major 7th	small 7th, minor 7th medium 7th, minor major 7th large 7th, major 7th	16/9, 44/25 9/5, 20/11 11/6, 50/27	989.490 1025.241 1052.633	1 4 2

The porcutone pentatonic and the porcutone chromatic

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.

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 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 Gb-Ab-Bb-Db-Eb 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.

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.

Mode -3 has 3/2 perfect fifths available above D, D♯, E, F, F♯, G, G♯, and C♯.

Mode 3 has 3/2 perfect fifths available above D, E♭, E, F, G♭, G, A♭, and D♭.

Mode -3 has 4:5:6 major triads available above E, F, F♯, G, and G♯

Mode 3 has 4:5:6 major triads available above E♭, E, F, G♭, and G

The following tables show the (3, 4) and (4, 3) triads available of mode 3 and mode -3 of the porcutone chromatic scale:

(3, 4) and (4, 3) triads of the porcutone chromatic mode -3

Note	Triad class	Triad in meantone	Triad in porcupine	Porcutone triad name	JI triads approximated	Triads in cents
D	(3, 4) (4, 3)	minor major	minor minor	minor major minor	10:12:15 18:22:27	320.690, 704.524 348.110, 704.524
D♯	(3, 4) (4, 3)	minor dim 4	minor min 4	minor dim min 4	10:12:15 10:13:15	320.690, 704.524 467.325, 704.524
E	(3, 4) (4, 3)	minor major	minor major	minor major	10:12:15 4:5:6	320.690, 704.524 383.834, 704.524
F	(3, 4) (4, 3)	aug 2 major	maj 2 major	aug maj 2 major	(14:16:21 or 6:7:9) 4:5:6	237.198, 704.524 383.834, 704.524
F♯	(3, 4) (4, 3)	minor major	major major	minor major major	22:27:33 4:5:6	356.414, 704.524 383.834, 704.524
G	(3, 4) (4, 3)	aug 2 major	maj 2 major	aug maj 2 major	(14:16:21 or 6:7:9) 4:5:6	237.198, 704.524 383.834, 704.524
G♯	(3, 4) (4, 3)	minor dim 4	diminished min 4	minor diminished dim min 4	15:18:22 10:13:15	320.690, 668.800 467.325, 704.524
A	(3, 4) (4, 3)	minor major	diminished diminished	minor diminished major diminished	15:18:22 27:33:40	320.690, 668.800 348.110, 668.800
A♯	(3, 4) (4, 3)	minor dim 6 dim 4 dim 6	minor (sub) min 6 min 4 sub minor 6	minor (sub) dim min 6 dim min 4 (sub) dim min 6	40:48:63 80:105:126	320.690, 788.015 467.325, 788.015
B	(3, 4) (4, 3)	minor major	diminished diminished	minor diminished major diminished	15:18:22 27:33:40	320.690, 668.800 348.110, 668.800
C	(3, 4) (4, 3)	aug 2 major	diminished min 2 diminished	aug min 2 diminished major diminished	135:150:198 27:33:40	201.475, 668.800 348.110, 668.800
C♯	(3, 4) (4, 3)	minor dim 4	minor min 4	minor dim min 4	10:12:15 10:13:15	320.690, 704.524 467.325, 704.524

As with the porcutone diatonic, tuning the porcutone chromatic to 19edo collapses it to the Meantone[12] (Flattone[12]) chromatic scale. Tuning it to 15edo, 22edo, or 29edo collapses it to Porcupine[8]. Step signatures, 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) (Aug maj 2 is 7/6)

34edo: 7L 1m 4s = (4, 2, 1) = (141.1765c, 70.5882c, 35.2941c) (Aug maj 2 is 7/6 and 8/7)

41edo: 7L 1m 4s = (5, 2, 1) = (146.3415c, 58.5366c, 29.2683c) (Aug maj 2 is 8/7)

104edo: 7L 1m 4s = (13, 5, 2) = (150c, 57.6923c, 23.0769c) (Aug maj 2 is 8/7)

And allowing octave stretch, the tuning may be optimized via TE tuning to:

27edo with 1195.1825c octave: 7L 1m 4s = (3, 2, 1) = (132.7981c, 88.5320c, 44.2660c)

34edo with 1198.2070c octave: 7L 1m 4s = (4, 2, 1) = (140.9655c, 70.4828c, 35.2414c)

41edo with 1200.2039c octave: 7L 1m 4s = (5, 2, 1) = (146.3663c, 58.5465c, 29.2733c)

For comparison, the TE step signature and sizes for the ptolemismic porcupine chromatic is

7L 1m 4s = (146.6352c, 63.1434c, 27.4197c).

and the TE step signature and sizes for the supermagic porcupine chromatic is

7L 1m 4s = (149.5159c, 58.8799c, 23.6254c)

I like to keep the small step large enough to sound like a melodic interval, so I either tune to TE ptolemismic, or to 41edo if an equal tuning is desired.

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 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).

On my Lumatone I chose to colour the G♯/A♭ pink, and the rest of the chromatic notes blue, so the porcutone octatonic is on the white and pink keys, while there's a porcutone diatonic on the white keys and a porcutone pentatonic on the blue and pink keys.

If we temper out the difference between the large and medium steps, we reduce the scale to Porcupine[8]. As we discussed above, Porcupine is generated by the interval 10/9. The table below introduces a set of functional mode names for Porcupine[8]. Along with the step pattern and mode number, the modes' UDP is show in the table. The UDP show the number of generators in the direction the brighten the intervals of scale, followed the number of generators in the direction that darkens it, (followed by the number of periods per octave, if it is not one. In this case the scale repeats at the octave, so P = 1, and is not shown). Instead of building chords by stacking thirds (2-step intervals), in octatonic scales we can build major and minor triads by stacking 3-step intervals! Instead of diminished, we get modes with two large fourths making a quartal chord: Accordingly we call these modes 'quartal'. When we stack 3-step intervals of 8-note scales out minor triads come in first inversion, and our major triads come in second inversion, as the 3-step intervals of octatonic scales include 5/4 and 4/3. Hence the brightest modes are quartal, and the darkest are minor.

Porcupine[8] functional mode names

Mode number	Step pattern	UDP	Mode name
4	LLLLLLLs	7|0	Bright quartal
3	LLLLLLsL	6|1	Dark quartal
2	LLLLLsLL	5|2	Bright major
1	LLLLsLLL	4|3	Middle major
-1	LLLsLLLL	3|4	Dark major
-2	LLsLLLLL	2|5	Bright minor
-3	LsLLLLLL	1|6	Middle minor
-4	sLLLLLLL	0|7	Dark minor

We get Father[8], instead, if we temper out the difference (16/15) between the large step and the small step. Recall that the porcupine pentatonic reduces to Father[5], a subset of Father[8]. Father scales are generated by an interval representing both 5/4 and 4/3 (the 3-step interval of 8-note scales). The modes of Father[8] have names in use already, as an oneirotonic. These are shown in the table below with the mode number, step patter, and UDP.

Father[8] oneirotonic mode names

Mode number	Step pattern	UDP	Mode name
4	LLsLLsLs	7|0	Dylathian (də-LA(H)TH-iən)
3	LLsLsLLs	6|1	Illarnekian (ill-ar-NEK-iən)
2	LsLLsLLs	5|2	Celephaïsian (kel-ə-FAY-zhən)
1	LsLLsLsL	4|3	Ultharian (ul-THA(I)R-iən)
-1	LsLsLLsL	3|4	Mnarian (mə-NA(I)R-iən)
-2	sLLsLLsL	2|5	Kadathian (kə-DA(H)TH-iən)
-3	sLLsLsLL	1|6	Hlanithian (lə-NITH-iən)
-4	sLsLLsLL	0|7	Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn"

For our modes of the left handed and right handed porcupine octatonic scales we prefix the functional mode names for Porcupine[8], with the oneirotonic mode names associated with Father[8]. Like in the tables of modes of the porcutone diatonic, the modes are listed in order of brightest, with the brightest mode at the top, and the darkest mode at the bottom.

modes of the left handed just porcutone octatonic

Mode in JI	Step pattern	Porcupine[8] step pattern and UDP	Porcupine[8] mode	Father[8] step pattern and UDP	Oneirotonic mode	Porcutone octatonic mode
10/9 6/5 4/3 40/27 8/5 16/9 50/27 2/1	LMLLMLsM	LLLLLLsL 6|1	Dark quartal	LsLLsLLs 5|2	Celephaïsian	Celephaïsian dark quartal
27/25 6/5 162/125 36/25 8/5 216/125 48/25 2/1	MLMLLMLs	LLLLLLLs 7|0	Bright quartal	sLsLLsLL 0|7	Sarnathian	Sarnathian bright quartal
10/9 100/81 4/3 40/27 125/81 5/3 50/27 2/1	LLMLsMLM	LLLLsLLL 4|3	Middle major	LLsLLsLs 7|0	Dylathian	Dylathian middle major
27/25 6/5 4/3 36/25 8/5 5/3 9/5 2/1	MLLMLsML	LLLLLsLL 5|2	Bright major	sLLsLLsL 2|5	Kadathian	Kadathian bright major
10/9 6/5 4/3 25/18 3/2 5/3 9/5 2/1	LMLsMLML	LLLsLLLL 3|4	Dark major	LsLLsLsL 4|3	Ultharian	Ultharian dark major
10/9 125/108 5/4 25/18 3/2 5/3 50/27 2/1	LsMLMLLM	LsLLLLLL 1|6	Middle minor	LLsLsLLs 6|1	Illarnekian	Illarnekian middle minor
27/25 6/5 5/4 27/20 3/2 81/50 9/5 2/1	MLsMLMLL	LLsLLLLL 2|5	Bright minor	sLLsLsLL 1|6	Hlanithian	Hlanithian bright minor
25/24 9/8 5/4 27/20 3/2 5/3 9/5 2/1	sMLMLLML	sLLLLLLL 0|7	Dark minor	LsLsLLsL 3|4	Mnarian	Mnarian dark minor

modes of the right handed just porcutone octatonic

Mode in JI	Step pattern	Porcupine[8] step pattern and UDP	Porcupine[8] mode	Father[8] step pattern and UDP	Oneirotonic mode	Porcutone octatonic mode
10/9 6/5 4/3 40/27 8/5 16/9 50/27 2/1	LMLLMLMs	LLLLLLLs 7|0	Bright quartal	LsLLsLsL 4|3	Ultharian	Ultharian bright quartal
27/25 6/5 162/125 36/25 8/5 216/125 48/25 2/1	LLMLMsLM	LLLLLsLL 5|2	Bright major	LLsLsLLs 6|1	Illarnekian	Illarnekian bright major
10/9 100/81 4/3 40/27 125/81 5/3 50/27 2/1	MLLMLMsL	LLLLLLsL 6|1	Dark quartal	sLLsLsLL 1|6	Hlanithian	Hlanithian dark quartal
27/25 6/5 4/3 36/25 8/5 5/3 9/5 2/1	LMLMsLML	LLLLsLLL 4|3	Middle major	LsLsLLsL 3|4	Mnarian	Mnarian middle major
10/9 6/5 4/3 25/18 3/2 5/3 9/5 2/1	LMsLMLLM	LLsLLLLL 2|5	Bright minor	LsLLsLLs 5|2	Celephaïsian	Celephaïsian bright minor
10/9 125/108 5/4 25/18 3/2 5/3 50/27 2/1	MLMsLMLL	LLLsLLLL 3|4	Middle minor	sLsLLsLL 0|7	Sarnathian	Sarnathian dark major
27/25 6/5 5/4 27/20 3/2 81/50 9/5 2/1	sLMLLMLM	sLLLLLLL 0|7	Bright minor	LLsLLsLs 7|0	Dylathian	Dylathian dark minor
25/24 9/8 5/4 27/20 3/2 5/3 9/5 2/1	MsLMLLML	LsLLLLLL 1|6	Dark minor	sLLsLLsL 2|5	Kadathian	Kadathian middle minor

Note that the darkest mode of the LH octatonic is the brightest mode of the RH octatonic, etc.

Tempering out 100/99, the large step (174.05488c) represents 10/9~11/10, the medium step (146.63528c) represents 27/25~12/11, and the small step (63.14327c) represents 25/24~33/32. The following tables display the JI intervals approximated by the modes of the ptolemismic porcutone octatonic scales, along with the scale steps in cents.

Modes of the left handed ptolemismic porcutone octatonic

Porcutone ocatonic mode	Step pattern	Mode as simplest JI pre-image	Mode in cents
Celephaïsian dark quartal	LMLLMLsM	~ 10/9 6/5 4/3 22/15 8/5 16/9 11/6 2/1	174.055 320.690 494.745 668.800 815.435 989.490 1052.633 1199.269
Sarnathian bright quartal	MLMLLMLs	~ 12/11 6/5 72/55 16/11 8/5 96/55 48/25 2/1	146.635 383.834 467.325 641.380 815.435 962.070 1136.127 1199.269
Dylathian middle major	LLMLsMLM	~ 10/9 11/9 4/3 22/15 55/36 5/3 11/6 2/1	174.055 348.110 494.745 668.800 731.943 878.579 1052.633 1199.269
Kadathian bright major	MLLMLsML	~ 12/11 6/5 4/3 16/11 8/5 5/3 9/5 2/1	146.635 320.690 494.745 641.380 815.435 878.579 1025.214 1199.269
Ultharian dark major	LMLsMLML	~ 10/9 6/5 4/3 11/8 3/2 5/3 9/5 2/1	174.055 320.690 494.745 557.888 704.524 878.579 1025.214 1199.269
Illarnekian middle minor	LsMLMLLM	~ 10/9 55/48 5/4 11/8 3/2 5/3 11/6 2/1	174.055 237.198 383.834 557.888 704.524 878.579 1052.633 1199.269
Hlanithian bright minor	MLsMLMLL	~ 12/11 6/5 5/4 15/11 3/2 18/11 9/5 2/1	146.635 320.690 383.834 530.469 704.524 851.159 1025.214 1199.269
Mnarian dark minor	sMLMLLML	~ 25/24 9/8 5/4 15/11 3/2 5/3 9/5 2/1	63.143 209.779 383.834 530.469 704.524 878.579 1025.214 1199.269

Modes of the right handed ptolemismic porcutone octatonic

Porcutone ocatonic mode	Step pattern	Mode as simplest JI pre-image	Mode in cents
Ultharian bright quartal	LMLLMLMs	~ 10/9 6/5 4/3 22/15 8/5 16/9 48/25 2/1	174.055 320.690 494.745 668.800 815.435 989.490 1136.127 1199.269
Illarnekian bright major	LLMLMsLM	~ 10/9 11/9 4/3 22/15 8/5 5/3 11/6 2/1	174.055 348.110 494.745 668.800 815.435 878.579 1052.633 1199.269
Hlanithian dark quartal	MLLMLMsL	~ 12/11 6/5 4/3 16/11 8/5 96/55 10/9 2/1	146.635 320.690 494.745 641.380 815.435 962.070 1025.214 1199.269
Mnarian middle major	LMLMsLML	~ 10/9 6/5 4/3 16/11 3/2 5/3 9/5 2/1	174.055 320.690 494.745 641.380 704.524 878.579 1025.214 1199.269
Celephaïsian bright minor	LMsLMLLM	~ 10/9 6/5 5/4 11/8 3/2 5/3 11/6 2/1	174.055 320.690 383.834 557.888 704.524 878.579 1052.633 1199.269
Sarnathian dark major	MLMsLMLL	~ 12/11 6/5 72/55 15/11 3/2 18/11 9/5 2/1	146.635 320.690 467.325 530.469 704.524 851.159 1025.214 1199.269
Dylathian dark minor	sLMLLMLM	~ 25/24 55/48 5/4 11/8 55/36 5/3 11/6 2/1	63.143 237.198 383.834 557.888 731.943 878.579 1052.633 1199.269
Kadathian middle minor	MsLMLLML	~ 12/11 9/8 5/4 15/11 3/2 5/3 9/5 2/1	146.635 209.779 383.834 530.469 704.524 878.579 1025.214 1199.269

Summary for xen-math nerds

The Porcutone system is built via step nesting from the 5-limit minor seventh tetrad: 6/5 3/2 9/5 2/1. It's a 12-note rank-3 Meantone[12] x Ripple[12] Fokker block, a step-nested scale that also tempers to Porcupine[8], comprising a diatonic Meantone[7]-Porcupine[7]-Dicot[7] wakalix / 3-SNS on the white keys, and a pentatonic Meantone[5]-Father[5]-Bug[5] wakalix on the 'black' keys.

For the accompanying mapping for the Lumatone keyboard the G# / Ab key is coloured pink (and the remaining chromatic keys blue), and along with the white keys makes a Porcupine[8] / Father[8] Fokker block (any colours could be chosen instead of white, pink, and blue).

The Porcutone diatonic is a wakalix (pairwise well-formed scale) and a step-nested scale: A detempering of Meantone[7] and Porcupine[7], (and also of Dicot[7]), a Fokker block with unison vectors of 81/80 and 250/243 (and 25/24) has 1 large step of 9/8 (L x L), 3 medium steps of 10/9 (L x s), and 3 small steps of 27/25 (s x s).

Porcutone harmonic minor and harmonic major

Additionally, we have another set of Porcupine[7] modes contained in the Porcutone octatonic: Replacing the G with the G# changes the mode of the Porcupine[7] scale represented, and replaces diatonic with harmonic minor modes for the Meantone[7] scale represented, now a MODMOS.

On D we get the scale:

174.055 320.69 557.888 704.524 878.579 1025.214 1199.269 on the notes D E F G# A B C D

We get the following 7 modes of porcutone harmonic minor scale:

• Lsmsmms Lydian ♯2 bright major
• mmsLsms Ionian ♯5 symmetric minor
• msLsmsm Ukranian dorian bright minor
• sLsmsmm Phyrgian dominant dark major
• msmmsLs harmonic minor dark diminished
• smmsLsm Locrian ♮6 bright diminished
• smsmmsL altered diminished magical seventh

Using an Ab instead, we get the scale:

174.055 320.69 494.745 641.38 878.579 1025.214 1199.269

Which has porcutone harmonic major modes:

• Lsmmsms Lydian Augmented ♯2 bright major
• msLsmms Lydian ♭3 bright minor
• sLsmmsm Mixolydian ♭2 dark major
• mmsmsLs harmonic major bright diminished
• msmsLsm Dorian ♭5 dark diminished
• smsLsmm Phrygian ♭4 symmetric minor
• smmsmsL Locrian magical ♭♭7

Tunings

We could tune the scale in many different ways. The TE tuning given above consists of 7 large steps of 146.6352c, 1 medium step of 63.1434c, and 4 small steps of 27.4197c.

We could instead tune to POTE no-7 ptolemismic, resulting in a very similar 7L 1m 4s = (146.7247c, 63.1818c, 27.4363c).

For reference, the 5-limit JI tuning of (27/25, 25/24, 250/243) is equal to (133.2376c, 70.6724c, 49.1661c). There are also least squares and minimax. I hope to figure those out.

We could also tune to edos. Tuning to 15edo, 22edo or 29edo collapses the scale to a Porcupine[8] scale, and tuning to 19edo or 31edo tempers the scale to a Meantone[12] scale. We can retain three step sizes if we tune to 27edo (using 27e), 34edo, or to 41edo.

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)

Chords

Porcutone diatonic

Since the scale is built from 6/5 3/2 9/5 2/1, it is the most readily available tetrad, containing a 4:5:6 major triad and a 10:12:15 minor triad. To tonal harmony we can use tertian chords in the diatonic scale, leading to:

• D minor 10:12:15
• E minor 10:12:15
• F major 4:5:6
• G major 4:5:6
• A porcupine diminished / meantone minor 15:18:22
• B diminished 25:30:36
• C porcupine diminished / meantone major (has a neutral third) 27:33:40

Tertian tetrads:

• D minor 7 10:12:15:18
• E minor 7 10:12:15:18
• F major 7 but it's actually a major neutral 7 chord 12:15:18:22
• G porcupine major 7 / meantone dominant 7 20:25:30:36
• A porcupine half-dim 7 / meantone minor 7 45:54:66:80
• B half diminished 7 25:30:36:45
• C porcupine half-dim 7 / meantone major 7 (has a neutral third) 27:33:40:50

9 chords:

• D 10:12:15:18:22
• E 33:40:50:60:72
• F 36:45:54:66:80
• G 20:25:30:36:45
• A 45:54:66:80:100
• B 25:30:36:45:54
• C 27:33:40:50:60

11 chords:

• D 30:36:45:54:66:80
• E 33:40:50:60:72:90
• F 36:45:54:66:80:100
• G 20:25:30:36:45:54
• A 45:54:66:80:100:120
• B 25:30:36:45:54:66
• C 27:33:40:50:60:72

13 chords:

• D 30:36:45:54:66:80:100
• E 33:40:50:60:72:90:108
• F 36:45:54:66:80:100:120
• G 20:25:30:36:45:54:66
• A 45:54:66:80:100:120
• B 25:30:36:45:54:66:80
• C 27:33:40:50:60:72:90

Quartal chords:

• D-G-C 15:20:27
• E-A-D 11:15:20
• F-B-E 24:33:44
• G-C-F 11:15:20
• A-D-G 9:12:16
• B-E-A 15:20:27
• C-F-B 6:8:11

D-G-C-F 15:20:27:36

D-G-C-F-B 30:40:54:72:99

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

D-F-A-C -> F-A-C-E -> E-G-B-D -> D-F-A-C

D-F-A -> F-B-E -> (E-G-B) -> G-B-D -> D-F-A