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? 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#/Ab 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

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 porcutone diatonic mode 0 is said to be the product word of Meantone[7] mode 0 and Porcupine[7] mode 0, i.e., LsLLLsL *sssLsss -> MsMLMsM, where L*L -> L, L*s -> M, and s*s -> s. So the porcutone diatonic is the product of Meantone[7] and Porcupine[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

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, so whereas 11/8 is represented by the major 4th in Porcupine (L+ 2*s), it is represented by the augmented fourth of Meantone[7] (3*L). The meantone extension representing 11/8 with an augmented fourth is call Meanenneadecal, referencing the fact that it is most at home in 19edo.

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 chromatic and Porcutone octatonic

If we put the small step into every medium and large step, we get the Porcutone chromatic, which is a detempering of Meantone[12]. (It’s also a detempering of a MODMOS of Diminished[12], and of Ripple[12]).

The just Porcutone chromatic has 7 large steps of 27/25, 1 medium step of 25/24, and 4 small steps of the porcupine comma, 250/243, hence it also tempers to Porcupine[8].

Tempering out 100/99, the Ptolemismic Porcutone chromatic has 7 large steps of 12/11~27/25, 1 medium step of 25/24~33/32, and 7 small steps of 250/243~55/54 (Here for TE steps)

Porcupine[7] has generator chain G-F-E-D-C-B-A. Porcupine[8] adds one note to the generator chain. Using Porcupine[7] note names, that’s either Ab or G#.

If we use a Bosanquet mapping on a keyboard using, we can map the porcutone diatonic to 7 white keys and the porcutone chromatic to 7 white keys and 5 chromatic keys. We colour the chromatic keys blue, apart from G#, which we colour pink, so that the white and pink keys make a porcutone octatonic scale, a detempered Porcupine[8]. This gives us a Meantone gamut of F-A#, and we also get a porcutone pentatonic on the blue and pink keys – F#-G#-A#-C#-D#.

Starting from D, the white keys gives us a Dorian symmetric minor scale, the white and pink keys gives us the just porcutone octatonic: 10/9 6/5 4/3 11/8 3/2 5/3 9/5 2/1, and the white, pink, and blue keys gives the just porcutone chromatic mode -3:

250/243 10/9 6/5 100/81 4/3 25/18 3/2 125/81 5/3 9/5 50/27 2/1 as D D# E F F# G G# A A# B C C#: Meantone[7] mode 3|8.

Tempering out 100/99, our Ptolemismic porcutone octatonic and chromatic are

~ 10/9 6/5 4/3 11/8 3/2 5/3 9/5 2/1 as D E F G G# A B C D

~ 55/54 10/9 6/5 11/9 4/3 11/8 3/2 55/36 5/3 9/5 11/6 2/1 as D D# E F F# G G# A A# B C C# D.

Or in cents: 174.055 320.69 494.745 557.888 704.524 878.579 1025.214 1199.269

27.42 174.055 320.69 348.11 494.745 557.888 704.524 731.943 878.579 1025.214 1052.633 1199.269.

Just porcutone octatonic: 4 large steps of 10/9, 3 medium of 27/25 and 1 small step of 25/24. It also tempers to a MODMOS of Diminished[8], and to Father[8].

Let’s introduce functional mode names for Porcupine[8]:

• Mode 4: LLLLLLLs – Bright quartal
• Mode 3: LLLLLLsL – Dark quartal
• Mode 2: LLLLLsLL – Bright major
• Mode 1: LLLLsLLL – middle major
• Mode -1: LLLsLLLL – dark major
• Mode -2: LLsLLLLL – bright minor
• Mode -3: LsLLLLLL – middle minor
• Mode -4: sLLLLLLL – dark minor

For our porcutone octatonic mode names, we can prefix these with the oneirotonic mode names, since it tempers to Father[8].

Using a G# instead of an Ab, we get the following modes for porcutone octatonic a:

• Mode 4a: LMLLMLsM -> Porcupine[8]: LLLLLLsL 6|1, Father[8]: LsLLsLLs 4|3 -> Celephaïsian dark quartal
• Mode 3a: MLMLLMLs -> Porcupine[8]: LLLLLLLs 7|0, Father[8]: sLsLLsLL 0|7 -> Sarnathian bright quartal
• Mode 2a: LLMLsMLM -> Porcupine[8]: LLLLsLLL 4|3, Father[8]: LLsLLsLs 7|0 -> Dylathian middle major
• Mode 1a: MLLMLsML -> Porcupine[8]: LLLLLsLL 5|2, Father[8]: sLLsLLsL 2|5 -> Kadathian bright major
• Mode -1a: LMLsMLML -> Porcupine[8]: LLLsLLLL 3|4, Father[8]: LsLLsLsL 5|2 -> Ultharian dark major
• Mode -2a: LsMLMLLM -> Porcupine[8]: LsLLLLLL 1|6, Father[8]: LLsLsLLs 6|1 -> Illarnekian middle minor
• Mode -3a: MLsMLMLL -> Porcupine[8]: LLsLLLLL 2|5, Father[8]: sLLsLsLL 1|6 -> Hlanithian bright minor
• Mode -4a: sMLMLLML -> Porcupine[8]: sLLLLLLL 0|7, Father[8]: LsLsLLsL 3|4 -> Mnarian dark minor

We could have chosen to include Ab instead of G# in the porcutone octatonic, which would result in the inverse of everything above, i.e., a chromatic gamut of Gb-B and inverses of the 8 porcutone octatonic modes resulting a different set of modes.

Porcutone octatonic b:

~ 10/9 6/5 4/3 16/11 3/2 5/3 9/5 2/1 as D E F G Ab A B C D

~ 12/11 10/9 6/5 72/55 4/3 16/11 3/2 18/11 5/3 9/5 108/55 2/1 as D Eb E F Gb G Ab A Bb B C Db D.

Porcutone octatonic b: 174.055 320.69 494.745 641.38 704.524 878.579 1025.214 1199.269

Porcutone Chromatic (Gb-B): 146.635 174.055 320.69 467.325 494.745 641.38 704.524 851.159 878.579 1025.214 1171.849 1199.269

• Mode 4b: LMLLMLMs -> Porcupine[8]: LLLLLLLs 7|0, Father[8]: LsLLsLsL 4|3 -> Ultharian bright quartal
• Mode 3b: LLMLMsLM -> Porcupine[8]: LLLLLsLL 5|2, Father[8]: LLsLsLLs 6|1 -> Illarnekian bright major
• Mode 2b: MLLMLMsL -> Porcupine[8]: LLLLLLsL 6|1, Father[8]: sLLsLsLL 1|6 -> Hlanithian dark quartal
• Mode 1b: LMLMsLML -> Porcupine[8]: LLLLsLLL 4|3, Father[8]: LsLsLLsL 2|5 -> Kadathian middle major
• Mode -1b: LMsLMLLM -> Porcupine[8]: LLsLLLLL 2|5, Father[8]: LsLLsLLs 5|2 -> Celephaïsian bright minor
• Mode -2b: MLMsLMLL -> Porcupine[8]: LLLsLLLL 3|4, Father[8]: sLsLLsLL 0|7 -> Sarnathian dark major
• Mode -3b: LsMLLMLM -> Porcupine[8]: sLLLLLLL 0|7, Father[8]: LLsLLsLs 7|0 -> Dylathian dark minor
• Mode -4b: MsLMLLML -> Porcupine[8]: LsLLLLLL 1|6, Father[8]: sLLsLLsL 3|4 -> Mnarian middle minor

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:

• Mode 3: Lsmsmms Lydian ♯2 bright major
• Mode 2: mmsLsms Ionian ♯5 symmetric minor
• Mode 1: msLsmsm Ukranian dorian bright minor
• Mode 0: sLsmsmm Phyrgian dominant dark major
• Mode -1: msmmsLs harmonic minor dark diminished
• Mode -2: smmsLsm Locrian ♮6 bright diminished
• Mode -3: 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:

• Mode 3: Lsmmsms Lydian Augmented ♯2 bright major
• Mode 2: msLsmms Lydian ♭3 bright minor
• Mode 1: sLsmmsm Mixolydian ♭2 dark major
• Mode 0: mmsmsLs harmonic major bright diminished
• Mode -1: msmsLsm Dorian ♭5 dark diminished
• Mode -2: smsLsmm Phrygian ♭4 symmetric minor
• Mode -3: smmsmsL Locrian magical ♭♭7

Porcutone pentatonic

Ok we’re almost done:

We just have our major and minor pentatonics left!

On F# the major pentatonic is 209.779 383.834 704.524 878.579 1199.269

~ 9/8 5/4 3/2 5/3 2/1 msLsL. Tempers to ssLsL for Meantone[5], LsLsL for Father[5], and sLLLL for Bug[5].

The same scale is also available on G.

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