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 porcupine systems combines Porcupine 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!

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

Each key is now distinctly different, 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.

For the 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).

Porcutone diatonic

Warning, math, skip to where it says 'math over' if you're not into it: 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)

The six modes of the just porcutone diatonic are:

• Mode -3: 27/25 6/5 4/3 36/25 8/5 9/5 2/1 = Meantone[7] 0|6 x Porcupine[7] 1|5 = sLLsLLL x sssssLs = sMMsMLs = Locrian x dark diminished = Locrian dark diminished
• Mode -2: 10/9 6/5 4/3 40/27 8/5 16/9 2/1 = Meantone[7] 2|4 x Porcupine[7] 0|6 = LsLLsLL x ssssssL = MsMMsML = Aeolian x magical seventh mode = Aeolian magical seventh
• Mode -1: 27/25 6/5 27/20 3/2 81/50 9/5 2/1 = Meantone[7] 1|5 x Porcupine[7] 4|2 = sLLLsLL x ssLssss = sMLMsMM = Phrygian x bright minor = Phrygian bright minor
• Mode 0: 10/9 6/5 4/3 3/2 5/3 9/5 2/1 = Meantone[7] 3|3 x Porcupine[7] 3|3 = LsLLLsL x sssLsss = MsMLMsM = Dorian x symmetric minor = Dorian symmetric minor
• Mode 1: 10/9 100/81 4/3 40/27 5/3 50/27 2/1 = Meantone[7] 5|1 x Porcupine[7] 2|4 = LLsLLLs x ssssLss = MMsMLMs = Ionian x bright diminished = Ionian bright diminished
• Mode 2: 9/8 5/4 27/20 3/2 5/3 9/5 2/1 = Meantone[7] 4|2 x Porcupine[7] 6|0 = LLsLLsL x Lssssss = LMsMMsM = Mixolydian x bright major = Mixolydian bright major
• Mode 3: 10/9 5/4 25/18 3/2 5/3 50/27 2/1 = Meantone[7] 6|0 x Porcupine[7] 5|1 = LLLsLLs x sLsssss = MLMsMMs = Lydian x dark major = Lydian dark major

If we temper so that 10/9 ~ 11/10, and 27/25 ~ 12/11 (and 9/8 ~ 25/22) ie., tempering out 100/99, we get the Ptolemismic Porcutone diatonic with modes:

• Locrian dark diminished: ~ 12/11 6/5 4/3 16/11 8/5 9/5 2/1
• Aeolian magical seventh: ~ 10/9 6/5 4/3 22/15 8/5 16/9 2/1
• Phrygian bright minor: ~ 12/11 6/5 15/11 3/2 18/11 9/5 2/1
• Dorian symmetric minor: ~ 10/9 6/5 4/3 3/2 5/3 9/5 2/1
• Ionian bright diminished: ~ 10/9 11/9 4/3 22/15 5/3 11/6 2/1
• Mixolydian bright major: ~ 9/8 5/4 15/11 3/2 5/3 9/5 2/1
• Lydian dark major: ~ 10/9 5/4 11/8 3/2 5/3 11/6 2/1

Tuned to TE (math over, here's tuning of the modes in cents, and their names, which combine the diatonic mode names with the functional Porcupine[7] mode names)

• Locrian dark diminished: 146.635 320.69 494.745 641.38 815.435 1025.214 1199.269
• Aeolian magical seventh: 174.055 320.69 494.745 668.8 815.435 989.49 1199.269
• Phrygian bright minor: 146.635 320.69 530.469 704.524 851.159 1025.214 1199.269
• Dorian symmetric minor: 174.055 320.69 494.745 704.524 878.579 1025.214 1199.269
• Ionian bright diminished: 174.055 348.11 494.745 668.8 878.579 1052.633 1199.269
• Mixolydian bright major: 209.779 383.834 530.469 704.524 878.579 1025.214 1199.269
• Lydian dark major: 174.055 383.834 557.888 704.524 878.579 1052.633 1199.269

The 11th harmonic as a Porcupine M4 is found at Meantone P4, tempering out 33/32, leading to Meanenneadecal, where 33/32 is tempered out, hence the Ptotemismic Porcupine diatonic is a detempering of Meanenneadecal.

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

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

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: smsmmsL altered diminished magical seventh
• Mode -2: smmsLsm Locrian natural 6 bright diminished
• Mode -2: msmmsLs harmonic minor dark diminished
• Mode 0: sLsmsmm Phyrgian dominant dark major
• Mode -1: msLsmsm Ukranian dorian bright minor
• Mode -2: mmsLsms Ionian #5 symmetric minor
• Mode -3: Lsmsmms Lydian #2 bright major

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: smmsmsL Locrian magical bb7
• Mode -2: smsLsmm Phrygian b4 symmetric minor
• Mode -1: msmsLsm Dorian b5 dark diminished
• Mode 0: mmsmsLs harmonic major bright diminished
• Mode 1: sLsmmsm Mixolydian b2 dark major
• Mode 2: msLsmms Lydian b3 bright minor
• Mode 3: Lsmmsms bright major

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