Pinetone

Built via step nesting from the 5-limit minor seventh tetrad: 6/5 3/2 9/5 2/1, the Porcutone system is 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

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:

• 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