Pinetone
The Porcutone system!
A 12-note rank 3 Meantone[12] Ripple[12] Fokker Block / step-nestd scale that also tempers to Porcupine[8], comprising a diatonic Meantone-Porcupine-Dicot wakalix / 3-SNS on the white keys, and a pentatonic Meantone-Father-Bug wakalix on the 'black' keys. 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.
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 = Lochrian x dark diminished = Lochrian 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 dark minor = Dorian dark 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:
Lochrian 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: Lochrian 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. 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 We’ll use porpupine[8] UDP mode names to name our porcutone octatonic modes, for now: 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]: LLLLLLLs – Bright quartal LLLLLLsL – Dark quartal LLLLLsLL – Bright major LLLLsLLL – middle major LLLsLLLL – dark major LLsLLLLL – bright minor LsLLLLLL – middle minor 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 -> Celephaïsian 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 -> Ultharian 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
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: smsmmsL altered diminished magical seventh smmsLsm Lochrian natural 6 bright diminished msmmsLs harmonic minor dark diminished sLsmsmm Phyrgian dominant dark major msLsmsm Ukranian dorian bright minor mmsLsms Ionian #5 symmetric minor 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:
smmsmsL Lochrian magical bb7 smsLsmm Phrygian b4 symmetric minor msmsLsmDorian b5 dark diminished mmsmsLs harmonic major bright diminished sLsmmsm Mixolydian b2 dark major msLsmms Lydian b3 bright minor Lsmmsms bright major
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.