Pinetone
Introduction
Definition: Pinetone is a system of related rank-3 microtonal scales: pentatonic, diatonic, octatonic, chromatic, and hyperchromatic.
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 Pinetone. 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 retune your keyboard using Scala files, grab this one! Copy the text into notepad and save as a .scl file).
Pinetone combines Porcupine – arguably the best way to add the 11th harmonic to major and minor harmonies in a seven-note scale – 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 (and the 13th harmonic as well!).
While there aren't as many consonant major and minor triads as we are used to, they are more consonant in Pinetone.
As opposed to in 12edo, each key is distinctly different in Pinetone scales, both a blessing and a curse.
Additionally available in Pinetone are a two sets of octatonic modes with their own Porcupine-like functional harmony that combine Porcupine[8] with the oneirotonic modes that are gaining popularity at the moment. Finally, Pinetone diminished scales combine Porcupine with the familiar diminished scale.
If you have a Lumatone, you can use the standard Bosanquet mapping for 12edo. The white keys are the Pinetone diatonic, a cross between the Meantone diatonic scale and Porcupine[7], and the black keys give the Pinetone 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♯/A♭ 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 Pinetone octatonic. The white keys and the pink key together make a Pinetone octatonic scale - major-harmonic with G♯ and minor-harmonic for A♭. This is what I call the Pinetone System.
If you don't have a Lumatone, no worries, you can set it up just fine on any keyboard!
For lovers of 15-note scales or diminished tetrads, the adjacent Pinetone diminished system comprises the Pinetone diminished heptatonic adding to the Pinetone diminished octatonic as Pinetone-15.
Or, for those who are able to play in 34edo or 41edo, a list of the steps subtended by the scales of Pinetone in degrees of 34edo and 41edo follows.
- Pentatonic major: 65959, 76B6B (subset of diatonic, harmonic diminished)
- Pentatonic minor: 95695, B67B6 (subset of diatonic, harmonic diminished)
- Diatonic: 4546454, 6567656 (subset of major and minor-harmonic octatonics)
- Harmonic minor: 5455475, 6566585 (subset of major and minor-harmonic octatonic)
- Harmonic major: 5545474, 6656585 (subset of major and minor-harmonic octatonic)
- Diminished heptatonic: 4754545 and 4574545, 5865656 and 5685656 (subset of diminished)
- Major-harmonic octatonic: 54524545, 65625656 (subset of chromatic, diminished chromatic)
- Minor-harmonic octatonic: 54254554, 65265665 (subset of chromatic, diminished chromatic)
- Diminished: 45254545, 56265656 (subset of chromatic, diminished chromatic)
- Harmonic diminished: 45272545, 56292656 (subset of diminished chromatic)
- Chromatic: 144142414414, 155152525515 (subset of hyperchromatic)
- Diminished chromatic: 234234232234 and 232243243243, 245245242245 and 242254254254 (subset of Pinetone-15)
- Pinetone-15: (reduces to Hanson[15] in 34edo), 242324232423242
- Hyperchromatic: 1313113123113131131 and 1311313113213113131, 1414114124114141141 and 1411414114214114141
The Pinetone 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 (represented 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 interval 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 in the form of a step signature and step mapping as 5L 2s = (9/8~10/9, 16/15~27/25). Porcupine[7] instead has a step signature and step mapping 1L 6s = (~9/8, 10/9~27/25), hence the difference between 10/9 and 27/25, i.e., 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, which, as mode 2 of Meantone[7] is 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 Pinetone 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 Pinetone diatonic represents both Porcupine[7] and Meantone[7]. To name this mode of the Pinetone diatonic, we simply add the mode names together, prefixing the Porcupine[7] functional mode names introduced in Table 1., with the Meantone diatonic mode names referenced in Table 2., so mode 0 of the Pinetone diatonic is called Dorian symmetric minor. We continue this naming process with the other 6 modes to arrive at the modes shown in Table 3.
Tables 1. and 2. show the modes of Porcupine[7], and Meantone[7], respectively, in the 5-limit. Given that intervals of tempered scales represent more than a single JI interval each, modes are described in their JI pre-image, the simplest JI ratios each interval above the tonic represents. Along with the step pattern and mode number, the modes' UDP are shown. A mode's UDP shows 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). Table 3. shows the modes of the 5-limit Pinetone diatonic, along with the name and step pattern of the corresponding Porcupine[7] and Meantone[7] modes, which can be arrived from their corresponding Pinetone modes by tempering out the Porcupine and Meantone commas respectively.
In Tables 1.-4., Modes marked with '*' have a consonant triad on their root.
| Mode number | Mode as simplest JI pre-image | Step pattern | UDP | Mode |
|---|---|---|---|---|
| 3 | ~ 9/8 5/4 27/20 3/2 5/3 9/5 2/1 | Lssssss | 6|0 | Bright major* |
| 2 | ~ 10/9 5/4 27/20 3/2 5/3 9/5 2/1 | sLsssss | 5|1 | Dark major* |
| 1 | ~ 10/9 6/5 27/20 3/2 5/3 9/5 2/1 | ssLssss | 4|2 | Bright minor* |
| 0 | ~ 10/9 6/5 4/3 3/2 5/3 9/5 2/1 | sssLsss | 3|3 | Symmetric minor* |
| -1 | ~ 10/9 6/5 4/3 36/25 5/3 9/5 2/1 | ssssLss | 2|4 | Bright diminished |
| -2 | ~ 10/9 6/5 4/3 36/25 8/5 9/5 2/1 | sssssLs | 1|5 | Dark diminished |
| -3 | ~ 10/9 6/5 4/3 36/25 8/5 16/9 2/1 | ssssssL | 0|6 | Magical seventh |
| Mode number | Mode as simplest JI pre-image | Step pattern | UDP | Mode |
|---|---|---|---|---|
| 3 | ~ 9/8 5/4 25/18 3/2 5/3 15/8 2/1 | LLLsLLs | 6|0 | Lydian* |
| 2 | ~ 9/8 5/4 4/3 3/2 5/3 15/8 2/1 | LLsLLLs | 5|1 | Ionian* |
| 1 | ~ 9/8 5/4 4/3 3/2 5/3 9/5 2/1 | LLsLLsL | 4|2 | Mixolydian* |
| 0 | ~ 9/8 6/5 4/3 3/2 5/3 9/5 2/1 | LsLLLsL | 3|3 | Dorian* |
| -1 | ~ 9/8 6/5 4/3 3/2 8/5 9/5 2/1 | LsLLsLL | 2|4 | Aeolian* |
| -2 | ~ 16/15 6/5 4/3 3/2 8/5 9/5 2/1 | sLLLsLL | 1|5 | Phrygian* |
| -3 | ~ 16/15 6/5 4/3 36/25 8/5 9/5 2/1 | sLLsLLL | 0|6 | Locrian |
| Mode number | Mode in JI | Step pattern | Meantone[7] | Diatonic mode | Porcupine[7] | Porcupine[7] mode | Pinetone 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 major* |
| 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 Pinetone 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 that the Locrian mode is the mirror inverse of Lydian.
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 Pinetone 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 Pinetone 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 Pinetone 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 Pinetone diatonic, the small step is 27/25 and the medium step is 10/9. We can access our 11-limit harmonies in Pinetone 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 Pinetone diatonic.
The modes of the Ptolemismic Pinetone 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.
| Mode number | Pinetone diatonic mode | Step pattern | Mode as simplest JI pre-image | Mode in cents |
|---|---|---|---|---|
| 3 | 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 |
| 2 | 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 |
| 1 | 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 |
| 0 | 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 |
| -1 | 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 |
| -2 | 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 |
| -3 | 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 |
Tuning options
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 Pinetone 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)
If we also temper out 144/143 (as described below), we get the step signature and mapping
TE 2.3.5.11.13 Ptolemismic: 1L 4m 2s = (209.54162c, 175.89183c, 142.77537c)
Intervals and chords
The table below show the sizes, interval names, ratios approximated, tuning, and occurrence of all 18 intervals of the 2.3.5.11 Ptolemismic Pinetone diatonic scale within an octave, tuned to TE tuning.
| Interval class | size | Meantone name | Porcupine[7] name | Pinetone name | Generic name | JI ratios approximated | size in cents (TE) | Occurence |
|---|---|---|---|---|---|---|---|---|
| 1-step | s | minor 2nd | minor 2nd | minor 2nd | small 2nd | 27/25, 12/11 | 146.635 | 2 |
| m | major 2nd | minor 2nd | major-minor 2nd | medium 2nd | 10/9, 11/10 | 174.055 | 4 | |
| L | major 2nd | major 2nd | major 2nd | large 2nd | 9/8, 25/22 | 209.779 | 1 | |
| 2-step | m + s | minor 3rd | minor 3rd | minor 3rd | small 3rd | 6/5, 40/33 | 320.690 | 4 |
| m + m | major 3rd | minor 3rd | major-minor 3rd | middle 3rd | 100/81, 11/9 | 348.110 | 1 | |
| L + m | major 3rd | major 3rd | major third | large 3rd | 5/4, 99/80 | 383.834 | 2 | |
| 3-step | 2m + s | perfect 4th | minor 4th | minor 4th | small 4th | 4/3, 33/25 | 494.745 | 4 |
| L + m + s | perfect 4th | major 4th | minor-major 4th | middle 4th | 27/20, 15/11 | 530.469 | 2 | |
| L + 2m | augmented 4th | major 4th | major 4th | large 4th | 25/18, 11/8 | 557.888 | 1 | |
| 4-step | 2m + 2s | diminished 5th | minor 5th | minor 5th | small 5th | 36/25, 16/11 | 641.380 | 1 |
| 3m + s | perfect 5th | minor 5th | major-minor 5th | medium 5th | 40/27, 22/15 | 668.800 | 2 | |
| L + 2m + s | perfect 5th | major 5th | major 5th | large 6th | 3/2, 50/33 | 704.524 | 4 | |
| 5-step | 3m + 2s | minor 6th | minor 6th | minor 6th | small 6th | 8/5, 160/99 | 815.435 | 2 |
| L + 2m + 2s | minor 6th | major 6th | minor-major 6th | medium 6th | 81/50, 18/11 | 851.159 | 1 | |
| L + 3m + s | major 6th | major 6th | major 6th | large 6th | 5/3, 33/20 | 878.579 | 4 | |
| 6-step | 4m + 2s | minor 7th | minor 7th | minor 7th | small 7th | 16/9, 44/25 | 989.490 | 1 |
| L + 3m + 2s | minor 7th | major 7th | minor-major 7th | medium 7th | 9/5, 20/11 | 1025.241 | 4 | |
| L + 4m + s | major 7th | major 7th | major 7th | large 7th | 11/6, 50/27 | 1052.633 | 2 |
Note that there are 3 sizes of interval for each interval class. This property is known as trivalence.
| Pinetone diatonic mode | 2nd | 3rd | 4th | 5th | 6th | 7th | 8ve |
|---|---|---|---|---|---|---|---|
| Lydian dark major | Major-minor | Major | Major | Major | Major | Major | Perfect |
| Mixolydian bright minor | Major | Minor-major | Minor-major | ||||
| Ionian bright diminished | Major-minor | Major-minor | Minor | Major-minor | Major | ||
| Dorian symmetric minor | Minor | Major | Minor-major | ||||
| Phrygian bright minor | Minor | Minor-major | Minor-major | ||||
| Aeolian magical seventh | Major-minor | Minor | Major-minor | Minor | Minor | ||
| Locrian dark diminished | Minor | Minor | Minor-major |
| Pinetone diatonic mode | 2nd | 3rd | 4th | 5th | 6th | 7th | 8ve |
|---|---|---|---|---|---|---|---|
| Lydian dark major | Medium | Large | Large | Large | Large | Large | Perfect |
| Mixolydian bright minor | Large | Medium | Medium | ||||
| Ionian bright diminished | Medium | Medium | Small | Medium | Large | ||
| Dorian symmetric minor | Small | Large | Medium | ||||
| Phrygian bright minor | Small | Medium | Medium | ||||
| Aeolian magical seventh | Medium | Small | Medium | Small | Small | ||
| Locrian dark diminished | Small | Small | Medium |
| Root note | Triad notes | Meantone triad | Porcupine triad | Pinetone triad | JI chord approximated by triad |
|---|---|---|---|---|---|
| D | D-F-A | minor | minor | minor | 10:12:15 |
| E | E-G-B | minor | minor | minor | 10:12:15 |
| F | F-A-C | major | major | major | 4:5:6 |
| G | G-B-D | major | major | major | 4:5:6 |
| A | A-C-E | minor | diminished | minor diminished | 15:18:22 |
| B | B-D-F | diminished | diminished | diminished | 25:30:36 |
| C | C-E-G | major | diminished | major diminished | 27:33:40 |
Porcupine tetrads in the table below are named after the third above the tonic and the third above the fifth, apart from tetrads with a diminished fifth.
| Root note | Triad notes | Meantone tetrad | Porcupine tetrad | Pinetone tetrad | JI chord approximated by tetrad |
|---|---|---|---|---|---|
| D | D-F-A-C | minor 7 | minor 7 | minor 7 | 10:12:15:18 |
| E | E-G-B-D | minor 7 | minor 7 | minor 7 | 10:12:15:18 |
| F | F-A-C-E | major 7 | major minor 7 | major major-minor 7 | 12:15:18:22 |
| G | G-B-D-F | major minor 7 | major minor 7 | major minor 7 | 20:25:30:36 |
| A | A-C-E-G | minor 7 | diminished 7 | minor diminished 7 | 45:54:66:80 |
| B | B-D-F-A | half diminished 7 | half diminished 7 | half diminished 7 | 25:30:36:45 |
| C | C-E-G-B | major 7 | half diminished 7 | major half diminished 7 | 27:33:40:50 |
Quartal Chords
Also of interest are the quartal triads of the Pinetone diatonic. We describe these as stacked 3-step intervals (fourths) of the scale, with major and minor designating the large and small 3-step intervals (fourths) respectively. This may seem an odd way to describe quartal chords, but it is consistent with the naming scheme I introduce for the Porcupine[7] 3-step (quartal) triads along side the quartal triads of the Pinetone diatonic.
| Root note | Triad notes | Meantone triad | Porcupine[7] triad | Pinetone triad | JI chord approximated by triad |
|---|---|---|---|---|---|
| D | D-G-C | sus | 3-step minor major | sus minor major | 15:20:27 |
| E | E-A-D | sus | 3-step major minor | sus major minor | 11:15:20 |
| F | F-B-E | sus ♭2 | 3-step major minor | sus ♭2 major minor | 24:33:44 |
| G | G-C-F | sus | 3-step major minor | sus major minor | 11:15:20 |
| A | A-D-G | sus | 3-step minor minor | sus minor minor | 9:12:16 |
| B | B-E-A | sus | 3-step minor major | sus minor major | 15:20:27 |
| C | C-F-B | sus ♯4 | 3-step minor major | sus ♯4 minor major | 6:8:11 |
We can see that compared to tertian (2-step stacked) chords, quartal (3-step stacked) chords are much more evenly spread in the Pinetone diatonic. 3-step pentads ala the quartal voicings of modern jazz harmony may also be of interest. Two octaves are spanned by four 3-step intervals and a 2-step remainder. Given that Pinetone cannot be generated by 4/3 perfect fourths like Meantone can these pentads have much more variety in Pinetone than in Meantone. As in modern jazz, variety is found through utilizing all inversions of the pentad. The root position and 2nd inversion are shown, where the second inversion spans an octave with a symmetric pattern of two 3-step intervals on either side of a 2-step interval. Root position pentads on A and B (2nd inversion on G and A) cover the notes of the Pinetone major and minor pentatonics respectively, introduced below.
| Root note | Notes | Meantone intervals | Porcupine intervals | Pinetone intervals | JI chord approximated by triad | JI Intervals |
|---|---|---|---|---|---|---|
| D | D-G-C-F-B | P4, P4, P4, A4, (m3) | m4, M4, m4, M4, (m3) | m4, mM4, m4, M4, (m3) | 15:20:27:36:50 | 4/3, 27/20, 4/3, 25/18, (6/5) |
| E | E-A-D-G-C | P4, P4, P4, P4, (M3) | M4, m4, m4, M4, (m3) | mM4, m4, m4, mM4, (Mm3) | 20:27:36:48:65 | 27/20, 4/3, 4/3, 65/48, (16/13) |
| F | F-B-E-A-D | A4, P4, P4, P4, (m3) | M4, m4, M4, m4, (m3) | M4, m4, mM4, m4, (m3) | 24:33:44:60:80 | 11/8, 4/3, 15/11, 4/3, (6/5) |
| G | G-C-F-B-E | P4, P4, A4, P4, (m3) | M4, m4, M4, m4, (m3) | mM4, m4, M4, m4, (m3) | 20:27:36:50:66 | 27/20, 4/3, 25/18, 33/25, (40/33) |
| A | A-D-G-C-F | P4, P4, P4, P4, (M3) | m4, m4, M4, m4, (M3) | m4, m4, mM4, m4, (M3) | 25:33:44:60:80 | 33/25, 4/3, 15/11, 4/3, (5/4) |
| B | B-E-A-D-G | P4, P4, P4, P4, (M3) | m4, M4, m4, m4, (M3) | m4, mM4, m4, m4, (M3) | 15:20:27:36:48 | 4/3, 27/20, 4/3, 4/3, (5/4) |
| C | C-F-B-E-A | P4, A4, P4, P4, (m3) | m4, M4, m4, M4, (m3) | m4, M4, m4, mM4, (m3) | 18:24:33:44:60 | 4/3, 11/8, 4/3, 15/11, (6/5) |
| Root note | Notes | Meantone intervals | Porcupine intervals | Pinetone intervals | JI chord approximated by triad | JI Intervals |
|---|---|---|---|---|---|---|
| D | D-G-C-E-A-D | P4, P4, M3, P4, P4 | m4, M4, m3, M4, m4 | m4, mM4, Mm3, mM4, m4 | 15:20:27:33:45:60 | 4/3, 27/20, 11/9, 15/11, 4/3 |
| E | E-A-D-F-B-E | P4, P4, m3, A4, P4 | M4, m4, m3, M4, m4 | mM4, m4, m3, M4, m4 | 11:15:20:24:33:44 | 15/11, 4/3, 6/5, 11/8, 4/3 |
| F | F-B-E-G-C-F | A4, P4, m3, P4, P4 | M4, m4, m3, M4, m4 | M4, m4, m3, mM4, m4 | 18:25:33:40:54:72 | 25/18, 33/25, 40/33, 27/20, 4/3 |
| G | G-C-F-A-D-G | P4, P4, M3, P4, P4 | M4, m4, M3, m4, m4 | mM4, m4, M3, m4, m4 | 11:15:20:25:33:44 | 15/11, 4/3, 5/4, 33/25, 4/3 |
| A | A-D-G-B-E-A | P4, P4, M3, P4, P4 | m4, m4, M3, m4, M4 | m4, m4, M3, m4, mM4 | 27:36:48:60:80:108 | 4/3, 4/3, 5/4, 4/3, 27/20 |
| B | B-E-A-C-F-B | P4, P4, m3, P4, A4 | m4, M4, m3, m4, M4 | m4, mM4, m3, m4, M4 | 33:44:60:72:96:132 | 4/3, 15/11, 6/5, 4/3, 11/8 |
| C | C-F-B-D-G-C | P4, A4, m3, P4, P4 | m4, M4, m3, m4, M4 | m4, M4, m3, m4, mM4 | 27:36:50:60:80:108 | 4/3, 25/18, 6/5, 4/3, 27/20 |
The mode names in the table below link to sequenced examples of the root position 3-step triads, and to root position, 1st inversion, 2nd inversion (with double octave), 3rd inversion, and 4th inversion 3-step pentads, on the tonic of each mode, tuned to TE 2.3.5.11 Ptolemismic.
| Mode | Triad root position | Pentad root position | Pentad 1st inversion | Pentad 2nd inversion | Pentad 3rd intervsion | Pentad fourth inversion |
|---|---|---|---|---|---|---|
| Dorian symmetric minor | D-G-C | D-G-C-F-B | D-G-C-F-A | D-G-C-E-A-D | D-G-B-E-A* | D-F-B-E-A |
| Phrygian bright minor | E-A-D | E-A-D-G-C | E-A-D-G-B* | E-A-D-F-B-E | E-A-C-F-B | E-G-C-F-B |
| Lydian dark major | F-B-E | F-B-E-A-D | F-B-E-A-C | F-B-E-G-C-F | F-B-D-G-C | F-A-D-G-C |
| Mixolydian bright major | G-C-F | G-C-F-B-E | G-C-F-B-D | G-C-F-A-D-G | G-C-E-A-D | G-B-E-A-D* |
| Aeolian magical seventh | A-D-G | A-D-G-C-F | A-D-G-C-E | A-D-G-B-E-A* | A-D-F-B-E | A-C-F-B-E |
| Locrian dark diminished | B-E-A | B-E-A-D-G* | B-E-A-D-F | B-E-A-C-F-B | B-E-G-C-F | B-D-G-C-F |
| Ionian bright diminished | C-F-B | C-F-B-E-A | C-F-B-E-G | C-F-B-D-G-C | C-F-A-D-G | C-E-A-D-G |
The chords marked with '*' are those with the notes of the Pinetone major pentatonic, which sounds particularly consonant as a chord.
The Pinetone pentatonic and the Pinetone chromatic
We know the (meantone) pentatonic scale to be a subset of the (meantone) diatonic scale. Similarly, the Pinetone pentatonic is a subset of the Pinetone 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 Pinetone.
Pinetone pentatonic
Using the familiar Bosanquet 12-note keyboard mapping (the preset for 12edo), we set the Pinetone diatonic scale to the white keys, starting on D. We than add, on F♯/G♭, the Pinetone 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 Pinetone harmonic minor available, and all flats makes the Pinetone 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 Pinetone pentatonic is also a subset of the Pinetone 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 Pinetone pentatonic is chiral (i.e., it is not symmetric, unlike the Pinetone diatonic and procutone chromatic scales, which are achiral). There is a pair of Pinetone pentatonic scales, the right-handed Pinetone pentatonic, 9/8 5/4 3/2 5/3 2/1 in JI, it's mirror inverse the left-handed Pinetone pentatonic, 10/9 5/4 3/2 5/3 2/1 in JI, which tempers to ssLsL in Meantone, but to sLLsL in Father. Since 9/8 5/4 3/2 5/3 2/1 is familiar to us as the 5-limit major pentatonic, we could also call the right-handed Pinetone pentatonic the Pinetone major pentatonic; similarly, 6/5 4/3 3/2 9/5 2/1, a mode of the left-handed Pinetone pentatonic is familiar to us as the 5-limit minor pentatonic, we could also call the left-handed Pinetone pentatonic the Pinetone minor pentatonic. D-F-G-A-C gives the Pinetone minor pentatonic, while G-A-B-D-E, F♯-G♯-A♯-C♯-D♯, and G♭-A♭-B♭-D♭-E♭ are Pinetone major pentatonics.
Pinetone chromatic
Adding the right handed Pinetone pentatonic (on F♯/G♭) to the just Pinetone diatonic, 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 Pinetone 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 Pinetone chromatic is a detempering of Meantone[12], the meantone chromatic scale, just like how the Pinetone diatonic is a detempering of Meantone[7], the meantone diatonic scale.
The Ptolemismic Pinetone 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
Note the more complex intervals: 55/54, 55/36, 72/55, and 108/55. If we temper out an additional comma, we can equate these with simpler intervals, adding prime 13: Tempering out 144/143, these four interval approximate 40/39, 20/13, 13/10, and 39/20 respectively. Tempering out 144/143 also means that the small step of the Pinetone diatonic, equivalently the large step of the Pinetone chromatic approximates 13/12, which, when all three are justly tuned, lies between the other intervals approximated by the step - 27/25, and 12/11.
This leads to a step signature, mapping, and TE tuning of 7L 1m 4s = (27/25~12/11~13/12, 25/24~33/32~27/26, 250/243~55/54~121/120~40/39) = (142.77537c, 66.76626c, 33.11646c).
Mode -3 approximates the JI ratios: 40/39 10/9 6/5 11/9 4/3 11/8 3/2 20/13 5/3 9/5 11/6 2/1, with step pattern sLLsLmLsLLsL.
The TE tuning in cents is: 33.116 175.892 318.667 351.784 494.559 561.325 704.101 737.217 879.992 1022.768 1055.884 1198.660 as D♯ E F F♯ G G♯ A A♯ B C C♯ D
Mode 3 approximates the JI ratios: 12/11 10/9 6/5 13/10 4/3 13/9 3/2 13/8 5/3 9/5 39/20 2/1, with step pattern LsLLsLmLsLLs.
The TE tuning in cents is: 142.775 175.892 318.667 461.443 494.559 637.334 704.101 846.876 879.992 1022.768 1165.543 1198.660 as E♭ E F G♭ G A♭ A B♭ B C D♭ D
I find this tuning to be melodically superior, given the small step is 6 cents large, now a sixth tone rather than an eight tone.
If a full 13-limit tuning is desired, there are two options. The interval approximating 13/10 may either be tempered to approximate 21/16, leading to Supermagic, or 7/6, leading to Thrasher. The Supermagic tuning decreases the size of the small step, and the Starling tuning increases it. The Supermagic tuning reduces to Flattone (where 7/4 is found at a diminished 7th) and Porcupine (where 7/4 is found at a minor seventh), and the Starling tuning reduces to Meanenneadecal and Opossum (both where 7/4 is found at an augmented 6th). If we temper to 13/10 to equate to both 9/7 and 21/16, we get Keema, an extension of Hanson temperament. Keema[7] comprises 4 large steps of 247.695c, and 3 small steps of 69.682c.
The ptolemismic Pinetone chromatic scale is distinctly xenharmonic, and yet is related to the familiar chromatic scale.
Intervals and triads
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 Pinetone chromatic scale:
| Note | Triad class | Triad in meantone | Triad in porcupine | Pinetone triad name | JI triads approximated* | Triads in cents |
|---|---|---|---|---|---|---|
| D | (3, 4) | minor | minor | minor | 10:12:15 | 318.667, 704.101 |
| (4, 3) | major | minor | major-minor | 18:22:27 | 351.784, 704.101 | |
| D♯ | (3, 4) | minor | minor | minor | 10:12:15 | 318.667, 704.101 |
| (4, 3) | dim 4 | min 4 | dim min 4 | 10:13:15 | 461.443, 704.101 | |
| E | (3, 4) | minor | minor | minor | 10:12:15 | 318.667, 704.101 |
| (4, 3) | major | major | major | 4:5:6 | 385.433, 704.101 | |
| F | (3, 4) | aug 2 | maj 2 | aug maj 2 | (14:16:21 or 6:7:9) | 242.658, 704.101 |
| (4, 3) | major | major | major | 4:5:6 | 385.433, 704.101 | |
| F♯ | (3, 4) | minor | major | minor-major | 22:27:33 | 352.317, 704.101 |
| (4, 3) | major | major | major | 4:5:6 | 385.433, 704.101 | |
| G | (3, 4) | aug 2 | maj 2 | aug maj 2 | (14:16:21 or 6:7:9) | 242.658, 704.101 |
| (4, 3) | major | major | major | 4:5:6 | 385.433, 704.101 | |
| G♯ | (3, 4) | minor | diminished | minor diminished | 15:18:22 | 318.667, 670.451 |
| (4, 3) | dim 4 | dim minor 4 | dim min 4 diminished | 30:39:44 | 461.443, 670.451 | |
| A | (3, 4) | minor | diminished | minor diminished | 15:18:22 | 318.667, 670.451 |
| (4, 3) | major | diminished | major diminished | 27:33:40 | 351.784, 670.451 | |
| A♯ | (3, 4) | minor dim 6 | minor (sub) min 6 | minor (sub) dim min 6 | 25:30:39 | 318.667, 780.120 |
| (4, 3) | dim 4 dim 6 | min 4 sub minor 6 | dim min 4 (sub) dim min 6 | 50:65:78 | 461.443, 780.120 | |
| B | (3, 4) | minor | diminished | minor diminished | 15:18:22 | 318.667, 670.451 |
| (4, 3) | major | diminished | major diminished | 27:33:40 | 351.784, 670.451 | |
| C | (3, 4) | aug 2 | dim min 2 | aug min 2 diminished | 243:275:360 | 209.008, 670.451 |
| (4, 3) | major | diminished | major diminished | 27:33:40 | 351.784, 670.451 | |
| C♯ | (3, 4) | minor | minor | minor | 10:12:15 | 318.667, 704.101 |
| (4, 3) | dim 4 | min 4 | dim min 4 | 10:13:15 | 461.443, 704.101 |
* Non-bracketed JI ratios are those approximated in 2.3.5.11.13 Ptolemismic Pinetone; bracketed JI ratios are the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 as in Supermagic temperament.
Note the major minor third approximating 11/9 and the minor major third approximating 27/22 are very similar in size in this tuning (351.784 and 352.317 respectively). If we equate these intervals, we additionally temper out 243/242, leading to an extension of Tetracot (Wollemia or Monkey if we also equate 13/10 with 9/7 or 21/16 respectively).
The step signature, mapping, and size for Tetracot[7] is
6L 1s = (10/9~11/10, 27/25~12/11) = (175.8871c, 144.0106c) in the 2.3.5.11 subgroup, or
6L 1s = (10/9~11/10, 27/25~12/11~13/12) = (176.0044, 142.6653) in the 2.3.5.11.13 subgroup, or
6L 1s = (10/9~11/10, 27/25~12/11~13/12) = (175.6125, 146.2576) as 13-limit Monkey, or
6L 1s = (10/9~11/10~28/25, 27/25~15/14~12/11~13/12) = (176.8600, 136.3262) as Wollemia.
We can see that the large step of Tetracot[7] is the medium step of the Pinetone diatonic, and the small step of Tetracot[7] is the small step of the Pinetone diatonic. The large step of the Pinetone diatonic is the augmented second of tetracot[7].
Tuning options
As with the Pinetone diatonic, tuning the Pinetone 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) (dim min 4 is 9/7 - Starling)
34edo: 7L 1m 4s = (4, 2, 1) = (141.1765c, 70.5882c, 35.2941c) (dim min 4 is 9/7 and 21/16 - Supermagic or Starling)
41edo: 7L 1m 4s = (5, 2, 1) = (146.3415c, 58.5366c, 29.2683c) (dim min 4 is 21/16 - Supermagic)
All three of these edos also temper out 243/242, so the major minor and minor major thirds collapse to a single interval - the neutral third, and the Pinetone diatonic can be considered a MODMOS of Tetracot[7] i.e. Pinetone msmLmsm = Tetracot LsLALsL.
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, mapping, and sizes for the (2.3.5.11.13) Ptolemismic porcupine chromatic is
the TE step signature, mapping, and sizes for the 13-limit Supermagic Pinetone chromatic is
and the TE step signature, mapping, and sizes for the 13-limit Thrasher Pinetone chromatic is
and if optimization just to the 2.3.5.11 subgroup is desired,TE step signature, mapping, and sizes for the (2.3.5.11) Ptolemismic Pinetone chromatic is
7L 1m 4s = (27/25~12/11, 25/24~33/32, 250/243~55/54~121/120) = (146.63528c, 63.14327c, 27.41960c).
Or we might tune to TE 2.3.5.11.13 Tetracot for
7L 1m 4s = (27/25~12/11~13/12, 25/24~33/32~27/26, 81/80~250/243~55/54~121/120~40/39) = (142.6653, 66.6782, 33.3391), which we note is very similar to 2.3.5.11.13 Ptolemismic.
Pinetone octatonic scales
The Porcupine comma is the small step of the Pinetone chromatic, so tempering the Pinetone 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 Pinetone chromatic scale, respectively, are set to D so that this is preserved in Pinetone. This leads to the Pinetone 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 Pinetone diatonic (the Zarlino/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 Pinetone octatonic with G♯ is called the Pinetone major-harmonic octatonic, and the Pinetone octatonic with A♭ is called the Pinetone minor-harmonic octatonic. These names will make sense to the reader after further reading on these scales and the chords they contain, and on the Pintone diminished octatonic introduced below. The mirror inverse of any mode of the Pinetone major-harmonic octatonic is a mode of the Pinetone minor-harmonic octatonic (see chirality). This is true similarly of the familiar harmonic minor and harmonic major scales.
On a keyboard with standard (Bosanquet or 12edo) mapping, the Pinetone octatonic is the C Major bebop scale! On my Lumatone I chose to colour the G♯/A♭ pink, and the rest of the chromatic notes blue, so the Pinetone octatonic is on the white and pink keys, while there's a Pinetone diatonic on the white keys and a Pinetone 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 10/9~27/25. The table below introduces a set of functional mode names for Porcupine[8]. Along with the step pattern and mode number, the modes' UDP are show in the table. A mode's UDP shows 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. The eighth note of Porcupine[8] is typically called 'H', and is equivalent to the note A♭ of Porcupine[7], but we will show the modes for G# as the eighth note as well, since we may use G# in our Pinetone chromatic and octatonic scales.
The step signature and mapping of 5-limit Porcupine[8] is 7L 1s = (10/9~27/25, 25/24~81/80)
| Mode number | Step pattern | UDP | Mode name | Mode as simplest JI pre-image | 3-step stacked triad on root (with G♯) | (with A♭ = H) | Triad name | JI triad approximated |
|---|---|---|---|---|---|---|---|---|
| 4 | LLLLLLLs | 7|0 | Bright quartal | ~ 10/9 6/5 4/3 36/25 8/5 16/9 48/25 2/1 | G♯-C-F | A-D-G | [8] augmented | 9:12:16 |
| 3 | LLLLLLsL | 6|1 | Dark quartal | ~ 10/9 6/5 4/3 36/25 8/5 16/9 9/5 2/1 | A-D-G | B-E-A♭ = B-E-H | [8] augmented | 9:12:16 |
| 2 | LLLLLsLL | 5|2 | Bright major | ~ 10/9 6/5 4/3 36/25 8/5 5/3 9/5 2/1 | B-E-G♯ | C-F-A | [8] major | 3:4:5 |
| 1 | LLLLsLLL | 4|3 | Middle major | ~ 10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1 | C-F-A | D-G-B | [8] major | 3:4:5 |
| -1 | LLLsLLLL | 3|4 | Dark major | ~ 10/9 6/5 4/3 25/18 3/2 5/3 9/5 2/1 | D-G-B | E-A♭-C = E-H-C | [8] major | 3:4:5 |
| -2 | LLsLLLLL | 2|5 | Bright minor | ~ 10/9 6/5 5/4 25/18 3/2 5/3 9/5 2/1 | E-G♯-C | F-A-D | [8] minor | 12:15:20 |
| -3 | LsLLLLLL | 1|6 | Middle minor | ~ 10/9 9/8 5/4 25/18 3/2 5/3 9/5 2/1 | F-A-D | G-B-E | [8] minor | 12:15:20 |
| -4 | sLLLLLLL | 0|7 | Dark minor | ~ 25/24 9/8 5/4 25/18 3/2 5/3 9/5 2/1 | G-B-E | A♭-C-F = H-C-F | [8] minor | 12:15:20 |
All of these triads are pretty consonant, shoutout to Porcupine[8]!
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.
The step signature and mapping of 5-limit Father[8] is 5L 3s = (10/9~25/24~32/27, 27/25~81/80),
| Mode number | Step pattern | UDP | Mode name | 3-step stacked triad on root | JI triads approximated |
|---|---|---|---|---|---|
| 4 | LLsLLsLs | 7|0 | Dylathian (də-LA(H)TH-iən) | [8] perfect | 3:4:5, 9:12:16 |
| 3 | LLsLsLLs | 6|1 | Illarnekian (ill-ar-NEK-iən) | [8] perfect | 3:4:5, 9:12:16 |
| 2 | LsLLsLLs | 5|2 | Celephaïsian (kel-ə-FAY-zhən) | [8] perfect | 3:4:5, 9:12:16 |
| 1 | LsLLsLsL | 4|3 | Ultharian (ul-THA(I)R-iən) | [8] perfect | 3:4:5, 9:12:16 |
| -1 | LsLsLLsL | 3|4 | Mnarian (mə-NA(I)R-iən) | [8] perfect | 3:4:5, 9:12:16 |
| -2 | sLLsLLsL | 2|5 | Kadathian (kə-DA(H)TH-iən) | [8] perfect | 3:4:5, 9:12:16 |
| -3 | sLLsLsLL | 1|6 | Hlanithian (lə-NITH-iən) | [8] major diminished | 160:200:243 |
| -4 | sLsLLsLL | 0|7 | Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn" | [8] minor diminished | 200:243:324 |
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 Pinetone diatonic, the modes are listed in order of brightest, with the brightest mode at the top, and the darkest mode at the bottom.
| Mode in JI (height order) | Step pattern | Porcupine[8]
step pattern and UDP |
Porcupine[8]
mode |
Father[8]
step pattern and UDP |
Oneirotonic
mode |
Pinetone octatonic
mode |
Comments |
|---|---|---|---|---|---|---|---|
| 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*† | Major with 10:12:15 on root |
| 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*† | Minor with 4:5:6 on root |
| 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†† | 4:5:6 and 10:12:15 on root |
| 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*† | Minor with 4:5:6 on root |
| Mode in JI (height order) | Step pattern | Porcupine[8]
step pattern and UDP |
Porcupine[8]
mode |
Father[8]
step pattern and UDP |
Oneirotonic
mode |
Pinetone octatonic
mode |
Comments |
|---|---|---|---|---|---|---|---|
| 10/9 6/5 4/3 40/27 8/5 16/9 48/25 2/1 | LMLLMLMs | LLLLLLLs 7|0 | Bright quartal | LsLLsLsL 4|3 | Ultharian | Ultharian bright quartal* | |
| 10/9 100/81 4/3 40/27 8/5 5/3 50/27 2/1 | LLMLMsLM | LLLLLsLL 5|2 | Bright major | LLsLsLLs 6|1 | Illarnekian | Illarnekian bright major* | |
| 27/25 6/5 4/3 36/25 8/5 216/125 9/5 2/1 | MLLMLMsL | LLLLLLsL 6|1 | Dark quartal | sLLsLsLL 1|6 | Hlanithian | Hlanithian dark quartal | |
| 10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1 | LMLMsLML | LLLLsLLL 4|3 | Middle major | LsLsLLsL 3|4 | Mnarian | Mnarian middle major*† | Major root 10:12:15 |
| 10/9 6/5 5/4 25/18 3/2 5/3 50/27 2/1 | LMsLMLLM | LLsLLLLL 2|5 | Bright minor | LsLLsLLs 5|2 | Celephaïsian | Celephaïsian bright minor*†† | Minor root 4:5:6, 10:12:15 |
| 27/25 6/5 162/125 27/20 3/2 81/50 9/5 2/1 | MLMsLMLL | LLLsLLLL 3|4 | Middle minor | sLsLLsLL 0|7 | Sarnathian | Sarnathian dark major† | 10:12:15 on root |
| 25/24 125/108 5/4 25/18 125/81 5/3 50/27 2/1 | sLMLLMLM | sLLLLLLL 0|7 | Bright minor | LLsLLsLs 7|0 | Dylathian | Dylathian dark minor* | |
| 27/25 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*† | Minor root 4:5:6 |
Note that the darkest mode of the major-harmonic octatonic is the mirror-inverse of the brightest mode of the minor-harmonic 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 Pinetone major and minor-harmonic octatonic scales, along with the scale steps in cents. See TE tuning.
Tempering out 144/143 as well, the large step is tuned to 175.89183c TE, medium step (142.77537c TE) also represents 13/12, and the small step (66.76626c TE) also represents 27/26. See TE tuning.
| Mode (height order) | Step pattern | Mode as simplest JI pre-image | Mode in cents | Comments |
|---|---|---|---|---|
| Celephaïsian dark quartal* | LMLLMLsM | ~ 10/9 6/5 4/3 22/15 8/5 16/9 11/6 2/1 | 175.892 318.667 494.559 670.451 813.226 989.118 1055.884 1198.660 | |
| Sarnathian bright quartal | MLMLLMLs | ~ 12/11 6/5 13/10 13/9 8/5 26/15 48/25 2/1 | 142.775 318.667 461.443 637.334 813.226 956.002 1131.893 1198.660 | |
| Dylathian middle major* | LLMLsMLM | ~ 10/9 11/9 4/3 22/15 20/13 5/3 11/6 2/1 | 175.892 351.784 494.559 670.451 737.217 879.992 1055.884 1198.660 | |
| Kadathian bright major* | MLLMLsML | ~ 12/11 6/5 4/3 13/9 8/5 5/3 9/5 2/1 | 142.775 318.667 494.559 637.334 813.226 879.992 1022.768 1198.660 | |
| Ultharian dark major*† | LMLsMLML | ~ 10/9 6/5 4/3 11/8 3/2 5/3 9/5 2/1 | 175.892 318.667 494.559 561.325 704.101 879.992 1022.768 1198.660 | Major with 10:12:15 on root |
| Illarnekian middle minor*† | LsMLMLLM | ~ 10/9 15/13 5/4 11/8 3/2 5/3 11/6 2/1 | 175.892 242.658 385.433 561.325 704.101 879.992 1055.884 1198.660 | Minor with 4:5:6 on root |
| Hlanithian bright minor(*)†† | MLsMLMLL | ~ 12/11 6/5 5/4 15/11 3/2 13/8 9/5 2/1 | 142.775 318.667 385.433 528.209 704.101 846.876 1022.768 1198.660 | 4:5:6 and 10:12:15 on root |
| Mnarian dark minor*† | sMLMLLML | ~ 25/24 9/8 5/4 15/11 3/2 5/3 9/5 2/1 | 66.766 209.542 385.433 528.209 704.101 879.992 1022.768 1198.660 | Minor with 4:5:6 on root |
| Mode (height order) | Step pattern | Mode as simplest JI pre-image | Mode in cents | Comments |
|---|---|---|---|---|
| Ultharian bright quartal* | LMLLMLMs | ~ 10/9 6/5 4/3 22/15 8/5 16/9 48/25 2/1 | 175.892 318.667 494.559 670.451 813.226 989.118 1131.983 1198.660 | |
| Illarnekian bright major* | LLMLMsLM | ~ 10/9 11/9 4/3 22/15 8/5 5/3 11/6 2/1 | 175.892 351.784 494.559 670.451 813.226 879.992 1055.884 1198.660 | |
| Hlanithian dark quartal | MLLMLMsL | ~ 12/11 6/5 4/3 13/9 8/5 26/15 10/9 2/1 | 142.775 318.667 494.559 637.334 813.226 956.002 1022.768 1198.660 | |
| Mnarian middle major*† | LMLMsLML | ~ 10/9 6/5 4/3 13/9 3/2 5/3 9/5 2/1 | 175.892 318.667 494.559 637.334 704.101 879.992 1022.768 1198.660 | Major with 10:12:15 on root |
| Celephaïsian bright minor*†† | LMsLMLLM | ~ 10/9 6/5 5/4 11/8 3/2 5/3 11/6 2/1 | 175.892 318.667 385.433 561.325 704.101 879.992 1055.884 1198.660 | Minor with 4:5:6 and 10:12:15 on root |
| Sarnathian dark major† | MLMsLMLL | ~ 12/11 6/5 13/10 15/11 3/2 13/8 9/5 2/1 | 142.775 318.667 461.443 528.209 704.101 846.876 1022.768 1198.660 | 10:12:15 on root |
| Dylathian dark minor* | sLMLLMLM | ~ 25/24 15/13 5/4 11/8 20/13 5/3 11/6 2/1 | 66.766 242.658 385.433 561.325 737.217 879.992 1055.884 1198.660 | |
| Kadathian middle minor*† | MsLMLLML | ~ 12/11 9/8 5/4 15/11 3/2 5/3 9/5 2/1 | 142.775 209.542 385.433 528.209 704.101 879.992 1022.768 1198.660 | Minor with 4:5:6 on root |
Intervals and chords
The following table gives all intervals of the Pinetone harmonic octatonics.
| Interval class | sizes | Oneirotonic name | Porcupine[8] name | Pinetone octatonic name | JI ratios approximated* | size in cents (TE) | Occurence |
|---|---|---|---|---|---|---|---|
| 1-step | s | major step | minor step | minor step | 25/24, 33/32, 27/26 | 66.766 | 1 |
| M | minor step | major step | minor-major step | 27/25, 12/11, 13/12 | 142.775 | 3 | |
| L | major step | major step | major step | 10/9, 11/10 | 175.892 | 4 | |
| 2-step | M + s | minor 2-step | minor 2-step | small 2-step | 9/8 | 209.542 | 1 |
| L + s | major 2-step | minor 2-step | minor 2-step | 15/13 (7/6 or 8/7) | 242.658 | 1 | |
| L + M | minor 2-step | major 2-step | major 2-step | 6/5 | 318.667 | 5 | |
| L + L | major 2-step | major 2-step | large 2-step | 11/9, 16/13 | 351.784 | 1 | |
| 3-step | L + M + s | major 3-step | minor 3-step | minor 3-step | 5/4 | 385.433 | 3 |
| L + 2M | minor 3-step | major 3-step | minor-major 3-step | 13/10 (9/7 or 21/16) | 461.433 | 1 | |
| 2L + M | major 3-step | major 3-step | major 3-step | 4/3 | 494.559 | 4 | |
| 4-step | L + 2M + s | minor 4-step | minor 4-step | small 4-step | 27/20, 15/11 | 528.209 | 2 |
| 2L + M + s | major 4-step | minor 4-step | minor 4-step | 25/18, 11/8, 18/13 | 561.325 | 2 | |
| 2L + 2M | minor 4-step | major 4-step | major 4-step | 36/25, 16/11, 13/9 | 637.334 | 2 | |
| 3L + M | major 4-step | major 4-step | large 4-step | 40/27, 22/15 | 670.451 | 2 | |
| 5-step | 2L + 2M + s | minor 5-step | minor 5-step | minor 5-step | 3/2 | 704.101 | 4 |
| 3L + M + s | major 5-step | minor 5-step | major-minor 5-step | 20/13 (14/9 or 32/16) | 737.217 | 1 | |
| 3L + 2M | minor 5-step | major 5-step | major 5-step | 8/5 | 813.227 | 3 | |
| 6-step | 2L + 3M + s | minor 6-step | minor 6-step | small 6-step | 18/11, 13/8 | 846.876 | 1 |
| 3L + 2M + s | major 6-step | minor 6-step | minor 6-step | 5/3 | 879.992 | 5 | |
| 3L + 3M | minor 6-step | major 6-step | major 6-step | 26/15 (12/7 or 7/4) | 956.002 | 1 | |
| 4L + 2M | major 6-step | major 6-step | large 6-step | 16/9 | 989.118 | 1 | |
| 7-step | 3L + 3M + s | minor 7-step | minor 7-step | minor 7-step | 9/5, 20/11 | 1022.768 | 4 |
| 4L + 2M + s | major 7-step | minor 7-step | major-minor 7-step | 50/27, 11/6, 24/13 | 1055.884 | 3 | |
| 4L + 3M | minor 7-step | major 7-step | major 7-step | 48/25, 64/33, 52/27 | 1131.983 | 1 |
* Non-bracketed JI ratios are those approximated in 2.3.5.11.13 Ptolemismic Pinetone; bracketed JI ratios are shows in pairs: the first interval in each pair is the 2.3.7 interval approximated by additionally tempering out 91/90 or 126/125 (Starling); the second is the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 (Supermagic).
The following two tables detail the 3-step stacked triads of the Pinetone harmonic octatonics:
| Mode (rotational order) | Step pattern | 3-step stacked triad on root | Oneirotonic name | Porcupine[8] name | Pinetone octatonic name | JI triad approximated* |
|---|---|---|---|---|---|---|
| Sarnathian bright quartal | MLMLLMLs | G♯-C-F | [8] minor diminished | [8] augmented | [8] minor augmented | 30:39:52 (7:9:12, 16:21:28) |
| Celephaïsian dark quartal | LMLLMLsM | A-D-G | [8] perfect | [8] augmented | [8] augmented | 9:12:16 |
| Kadathian bright major | MLLMLsML | B-E-G♯ | [8] perfect | [8] major | [8] major | 3:4:5 |
| Dylathian middle major | LLMLsMLM | C-F-A | [8] perfect | [8] major | [8] major | 3:4:5 |
| Ultharian dark major | LMLsMLML | D-G-B | [8] perfect | [8] major | [8] major | 3:4:5 |
| Hlanithian bright minor | MLsMLMLL | E-G♯-C | [8] major diminished | [8] minor | [8] diminished minor | 8:10:13 |
| Illarnekian middle minor | LsMLMLLM | F-A-D | [8] perfect | [8] minor | [8] minor | 12:15:20 |
| Mnarian dark minor | sMLMLLML | G-B-E | [8] perfect | [8] minor | [8] minor | 12:15:20 |
* Non-bracketed JI ratios are those approximated in 2.3.5.11.13 Ptolemismic Pinetone; bracketed JI ratios are shows in pairs: the first interval in each pair is the 2.3.7 interval approximated by additionally tempering out 91/90 or 126/125 (Starling); the second is the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 (Supermagic).
| Mode (rotational order) | Step pattern | 3-step stacked triad on root | Oneirotonic name | Porcupine[8] name | Pinetone octatonic name | JI triad approximated* |
|---|---|---|---|---|---|---|
| Ultharian bright quartal | LMLLMLMs | A-D-G | [8] perfect | [8] augmented | [8] augmented | 9:12:16 |
| Hlanithian dark quartal | MLLMLMsL | B-E-A♭ | [8] major diminished | [8] augmented | [8] major augmented | 15:20:26 (12:16:21) |
| Illarnekian bright major | LLMLMsLM | C-F-A | [8] perfect | [8] major | [8] major | 3:4:5 |
| Mnarian middle major | LMLMsLML | D-G-B | [8] perfect | [8] major | [8] major | 3:4:5 |
| Sarnathian dark major | MLMsLMLL | E-A♭-C | [8] minor diminished | [8] major | [8] diminished major | 55:72:90 (16:21:26) |
| Celephaïsian bright minor | LMsLMLLM | F-A-D | [8] perfect | [8] minor | [8] minor | 12:15:20 |
| Dylathian dark minor | sLMLLMLM | A♭-C-F | [8] perfect | [8] minor | [8] minor | 12:15:20 |
| Kadathian middle minor | MsLMLLML | G-B-E | [8] perfect | [8] minor | [8] minor | 12:15:20 |
* Non-bracketed JI ratios are those approximated in 2.3.5.11.13 Ptolemismic Pinetone; bracketed JI ratios are the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 as in Supermagic temperament.
We could alternatively treat either Pinetone harmonic octatonic as a bebop scale, using 2-step stacked tetrads. Since the scale has 8 notes, there are only 2 different 2-step stacked tetrads. In 12edo these are the major add 6 and the fully diminished tetrads. The meantone C major add 6 tunes to 45:55:66:75 in Pinetone. Using the G♯, as in the Pinetone major-harmonic octatonic, the G♯ diminished tetrad tunes to 33:40:48:55 (when B is the bottom note). Using the A♭, as in the Pinetone minor-harmonic octatonic, the B diminished tetrad also tunes to 33:40:48:55 (when D is the bottom note).
Unlike the Pinetone diatonic, and chromatic scales, the Pinetone harmonic octatonics are chiral, and is therefore not step-nested scales. As we can see, they more complex than the Pinetone diatonic. The Pinetone pentatonic and diatonic scales is also wakalix / PWF, and it can be seen that the Pinetone harmonic octatonics are more complex than the Pinetone pentatonic as well. It is left as an exercise for the reader to determine the complexity of the Pinetone chromatic, and compare that to the Pinetone harmonic octatonics.
Pinetone diminished octatonic
Modifying the right or left-handed Pinetone octatonic by switching the order of adjacent pairs of large and medium steps, i.e., by modifying steps of the scale by the L-M chroma - the difference between the large and medium steps - leads to similar Porcupine[8] detempers. Since L and M temper together under Porcupine tempering, any resulting scale tempers to Porcupine[8] just as before, but the scale it tempers to under Diminished and Father temperaments are modified.
Take the Ultharian dark major mode of the Pinetone major-harmonic octatonic, for example: LMLsMLML. Raising the sixth and eighth degrees of the scale by the L-M chroma leads to the mode LMLsLMLM. Similarly, taking the Mnarian middle major mode LMLMsLML of the Pinetone minor-harmonic octatonic and lowering the second and the fourth degree by the L-M chroma leads to the mode MLMLsLML. We can see that LMLsLMLM and MLMLsLML are modes of the same scale. We call these modes the dark major diminished and the middle major diminished respectively.
In the Pinetone chromatic with sharps, which contains the Pinetone major-harmonic octatonic as the naturals plus G♯, the Ultharian dark major mode can be expressed as D E F G G♯ A B C. The Pinetone diminished mode on D, the dark major diminished, is therefore D E F G G♯ A♯ B C♯. In the Pinetone chromatic with flats, which contains the Pinetone minor-harmonic octatonic as the naturals plus A♭, the Mnarian middle major mode can be expressed as D E F G A♭ A B C. The Pinetone diminished mode on D, the middle major diminished, is therefore D E♭ F G♭ A♭ A B C.
We know that this scale tempers to Porcupine[8]; tempering M=s instead leads to LsLsLsLs, i.e., Diminished[8]; and finally tempering L=s leads to LsLLLsLs, a MODMOS of Father[8]. Like the Pinetone chromatic and diatonic scale, this scale is an SN scale, and is therefor achiral. We may name this scale perhaps the Porcupine-Diminished scale, or we may include it in Pinetone as the Pinetone diminished scale, or the Pinetone diminished octatonic
Every other step of any mode of the Pinetone diminished scale gives an inversion of the 5-limit diminished tetrad; therefore every second step of the Pinetone diminished scales only comes in two different sizes, as opposed to the four different sizes of every second step of the Pinetone harmonic octatonics. Although the Pinetone diminished is simpler in this way, the Pinetone major and minor-harmonic octatonic provides more major and minor triads respectively (this is why I have named them the Pinetone major and minor-harmonic octatonics).
| Mode in JI (height order) | Step pattern | Porcupine[8]
step pattern and UDP |
Porcupine[8]
mode |
Diminished[8]
step pattern and UDP |
Pinetone diminished
mode |
Comments |
|---|---|---|---|---|---|---|
| 10/9 6/5 4/3 36/25 8/5 216/125 48/25 2/1 | LMLMLMLs | LLLLLLLs 7|0 | Bright quartal | LsLsLsLs 1|0 (4) | Bright quartal diminished | |
| 10/9 6/5 4/3 36/25 8/5 5/3 50/27 2/1 | LMLMLsLM | LLLLLsLL 5|2 | Bright major | LsLsLsLs 1|0 (4) | Bright major diminished* | |
| 27/25 6/5 162/125 36/25 972/625 216/125 9/5 2/1 | MLMLMLsL | LLLLLLsL 6|1 | Dark quartal | sLsLsLsL 0|1 (4) | Dark quartal diminished | |
| 10/9 6/5 4/3 25/18 125/81 5/3 50/27 2/1 | LMLsLMLM | LLLsLLLL 3|4 | Dark major | LsLsLsLs 1|0 (4) | Dark major diminished* | |
| 27/25 6/5 162/125 36/25 3/2 5/3 9/5 2/1 | MLMLsLML | LLLLsLLL 4|3 | Middle major | sLsLsLsL 0|1 (4) | Middle major diminished† | 10:12:15 on root |
| 10/9 125/108 625/486 25/18 125/81 5/3 50/27 2/1 | LsLMLMLM | LsLLLLLL 1|6 | Middle minor | LsLsLsLs 1|0 (4) | Middle minor diminished | |
| 27/25 6/5 5/4 25/18 3/2 5/3 9/5 2/1 | MLsLMLML | LLsLLLLL 2|5 | Bright minor | sLsLsLsL 0|1 (4) | Bright minor diminished*†† | Minor root 4:5:6, 10:12:15 |
| 25/24 125/108 5/4 25/18 3/2 5/3 9/5 2/1 | sLMLMLML | sLLLLLLL 0|7 | Dark minor | sLsLsLsL 0|1 (4) | Dark minor diminished*† | Minor with 4:5:6 on root |
| Mode (height order) | Step pattern | Mode as simplest JI pre-image | Mode in cents | Comments |
|---|---|---|---|---|
| Bright quartal diminished | LMLMLMLs | ~ 10/9 6/5 4/3 13/9 8/5 26/15 48/25 2/1 | 175.892 318.667 494.559 637.334 813.226 956.002 1131.893 1198.660 | |
| Bright major diminished* | LMLMLsLM | ~ 10/9 6/5 4/3 13/9 8/5 5/3 11/6 2/1 | 175.892 318.667 494.559 637.334 813.226 879.993 1055.884 1198.660 | |
| Dark quartal diminished | MLMLMLsL | ~ 12/11 6/5 13/10 13/9 39/25 26/15 9/5 2/1 | 142.775 318.667 461.443 637.334 780.120 956.002 1022.768 1198.660 | |
| Dark major diminished* | LMLsLMLM | ~ 10/9 6/5 4/3 13/9 20/13 5/3 11/6 2/1 | 175.892 318.667 494.559 561.325 737.218 879.993 1055.884 1198.660 | |
| Middle major diminished† | MLMLsLML | ~ 12/11 6/5 13/10 13/9 3/2 5/3 9/5 2/1 | 142.775 318.667 461.443 637.334 704.101 879.993 1022.768 1198.660 | 10:12:15 on root |
| Middle minor diminished | LsLMLMLM | ~ 10/9 15/13 50/39 11/8 20/13 5/3 11/6 2/1 | 175.892 242.658 418.550 561.325 737.218 879.993 1055.884 1198.660 | |
| Bright minor diminished*†† | MLsLMLML | ~ 12/11 6/5 5/4 11/8 3/2 5/3 9/5 2/1 | 142.775 318.667 385.433 561.325 704.101 879.993 1022.768 1198.660 | Minor root 4:5:6, 10:12:15 |
| Dark minor diminished*† | sLMLMLML | ~ 25/24 15/13 5/4 11/8 3/2 5/3 9/5 2/1 | 66.766 242.658 385.433 561.325 704.101 879.993 1022.768 1198.660 | Minor with 4:5:6 on root |
| Interval class | sizes | Diminished[8] name | Porcupine[8] name | Pinetone octatonic name | JI ratios approximated* | size in cents (TE) | Occurence |
|---|---|---|---|---|---|---|---|
| 1-step | s | minor step | minor step | minor step | 25/24, 33/32, 27/26 | 66.766 | 1 |
| M | minor step | major step | minor-major step | 27/25, 12/11, 13/12 | 142.775 | 3 | |
| L | major step | major step | major step | 10/9, 11/10 | 175.892 | 4 | |
| 2-step | L + s | perfect 2-step | minor 2-step | minor 2-step | 15/13 (7/6 or 8/7) | 242.658 | 2 |
| L + M | perfect 2-step | major 2-step | major 2-step | 6/5, 40/33 | 318.667 | 6 | |
| 3-step | L + M + s | minor 3-step | minor 3-step | minor 3-step | 5/4 | 385.433 | 2 |
| 2L + s | major 3-step | minor 3-step | major-minor 3-step | 33/26, 50/39 | 418.550 | 1 | |
| L + 2M | minor 3-step | major 3-step | minor-major 3-step | 13/10 (9/7 or 21/16) | 461.433 | 2 | |
| 2L + M | major 3-step | major 3-step | major 3-step | 4/3 | 494.559 | 3 | |
| 4-step | 2L + M + s | perfect 4-step | minor 4-step | minor 4-step | 25/18, 11/8, 18/13 | 561.325 | 4 |
| 2L + 2M | perfect 4-step | major 4-step | major 4-step | 36/25, 16/11, 13/9 | 637.334 | 4 | |
| 5-step | 2L + 2M + s | minor 5-step | minor 5-step | minor 5-step | 3/2 | 704.101 | 3 |
| 3L + M + s | major 5-step | minor 5-step | major-minor 5-step | 20/13 (14/9 or 32/16) | 737.217 | 2 | |
| 2L + 3M | minor 5-step | major 5-step | minor-major 5-step | 39/25, 52/33 | 780.120 | 1 | |
| 3L + 2M | major 5-step | major 5-step | major 5-step | 8/5 | 813.227 | 2 | |
| 6-step | 3L + 2M + s | perfect 6-step | minor 6-step | minor 6-step | 5/3, 33/20 | 879.992 | 6 |
| 3L + 3M | perfect 6-step | major 6-step | major 6-step | 26/15 (12/7 or 7/4) | 956.002 | 2 | |
| 7-step | 3L + 3M + s | minor 7-step | minor 7-step | minor 7-step | 9/5, 20/11 | 1022.768 | 4 |
| 4L + 2M + s | major 7-step | minor 7-step | major-minor 7-step | 50/27, 11/6, 24/13 | 1055.884 | 3 | |
| 4L + 3M | major 7-step | major 7-step | major 7-step | 48/25, 64/33, 52/27 | 1131.983 | 1 |
* Non-bracketed JI ratios are those approximated in 2.3.5.11.13 Ptolemismic Pinetone; bracketed JI ratios are shows in pairs: the first interval in each pair is the 2.3.7 interval approximated by additionally tempering out 91/90 or 126/125 (Starling); the second is the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 (Supermagic).
| Mode (rotational order) | Step pattern | 3-step stacked triad on root | Diminished[8] name | Porcupine[8] name | Pinetone octatonic name | JI triad approximated* |
|---|---|---|---|---|---|---|
| Bright quartal diminished | LMLMLMLs | G♯-C♯-F, A-D-G♭ | [8] major | [8] augmented | [8] major augmented | 15:20:26 (12:16:21) |
| Dark quartal diminished | MLMLMLsL | A♯-D-G, B-E♭-A♭ | [8] minor | [8] augmented | [8] minor augmented | 30:39:52 (16:21:28) |
| Bright major diminished | LMLMLsLM | B-E-G♯, C-F-A | [8] major | [8] major | [8] major | 3:4:5 |
| Middle major diminished | MLMLsLML | C♯-F-A♯, D-G♭-B | [8] minor | [8] major | [8] minor major | 30:39:50 |
| Dark major diminished | LMLsLMLM | D-G-B, E♭-A♭-C | [8] major | [8] major | [8] major | 3:4:5 |
| Bright minor diminished | MLsLMLML | E-G♯-C♯, F-A-D | [8] minor | [8] minor | [8] minor | 12:15:20 |
| Middle minor diminished | LsLMLMLM | F-A♯-D, G♭-B-E♭ | [8] major | [8] minor | [8] major minor | 39:50:65 |
| Dark minor diminished | sLMLMLML | G-B-E, A♭-C-F | [8] minor | [8] minor | [8] minor | 12:15:20 |
* Non-bracketed JI ratios are those approximated in 2.3.5.11.13 Ptolemismic Pinetone; bracketed JI ratios are the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 as in Supermagic temperament.
The following 13 notes are used in total for these scales: E♭, G♭, A♭ D, E, F, G, A, B, C, G♯, A♯, C♯
Pinetone diminished chromatic
We can extend the Pinetone diminished into an alternative chromatic scale: Starting with the bright minor diminished scale, MLsLMLML, we add a small step into the bottom or top of every large step, leading to the scales LsMssMLsMLsM and LMssMsLMsLMs respectively, modes of mirror-inversions of one another. In 5-limit just intonation this pair of scales comprises 3 large steps of 27/25, 4 medium steps of 16/15, and 5 small steps of 25/24, i.e., 27/25 9/8 6/5 5/4 125/96 25/18 3/2 25/16 5/3 9/5 15/8 2/1 and 27/25 144/125 6/5 5/4 4/3 25/18 3/2 8/5 5/3 9/5 48/25 2/1 respectively. In other modes, they can be expressed as 25/24 10/9 6/5 5/4 4/3 36/25 3/2 8/5 5/3 125/72 50/27 2/1, and 25/24 10/9 125/108 5/4 4/3 25/18 3/2 8/5 5/3 9/5 48/25 2/1, i.e., sMLsMLsMssML and sMsLMsLMsLMs respectively.
We could call these scales, which temper to Diminished[12], the left and right-handed Pinetone diminished chromatic.
Tempering M=L alternatively leads to sLLsLLsLssLL and sLsLLsLLsLLs, which are MODMOS of Meantone[12], i.e., C C♯ D E♭ E F G♭ G A♭ A A♯ B and G G♯ A A♯ B C C♯ D E♭ E F G♭.
Tempering out 100/99 and 144/143 leads to the following simplest pre-images 25/24 10/9 6/5 5/4 4/3 13/9 3/2 8/5 5/3 45/26 11/6 2/1 and 25/24 10/9 15/13 5/4 4/3 11/8 3/2 8/5 5/3 9/5 48/25 2/1.
Tuned to TE 2.3.5.11.13 Ptolemismic, the Pinetone diminished chromatic scales in those modes in cents are:
66.766 175.892 318.667 385.433 494.559 637.334 704.101 813.227 879.992 956.002 1055.884 1198.660 and 66.766 175.892 242.658 385.433 494.559 561.325 704.101 813.227 879.992 946.759 1131.983 1198.660.
The Pinetone diminished chromatic scales provide the same amount of major and minor triads as the Pinetone chromatic (4 of each), however, they are more evenly spread in the diminished chromatic scales.
From sMsLMsLMsLMs, putting a small step below the top of each large step (replacing L with sm, where m is the small step of the 12-note scale, the medium step of the 15-note scale) leads to
mLm(sm)Lm(sm)Lm(sm)Lm -> mLmsmLmsmLmsmLm, which we later introduce as Pinetone-15.
Pinetone diminished heptatonic
Starting with the 5-limit diminished tetrad 6/5 36/25 5/3 2/1, putting 10/9 above each note takes us to the Pinetone diminished octatonic 10/9 6/5 4/3 36/25 8/5 5/3 50/27 2/1. The 5-limit diminished tetrad has 3 large steps of 6/5, and one small step of 125/108. If we only put 10/9 above the large steps, we get the scale 10/9 6/5 4/3 36/25 5/3 50/27 2/1, with step pattern msmsLms, comprising 3 small steps of 27/25, 3 medium steps of 10/9, and one large step of 125/108.
Tempering the small and medium steps together gives us the scale ssssLss, Porcupine[7]; tempering L=m implies tempering out 25/24, which leads to Dicot[7] as sLsLssL; and tempering s=L gives us sLsLLsL as Sixix[7]. Tempering out 100/99 leads to simplest JI pre-image 10/9 6/5 4/3 16/11 5/3 11/6 2/1, and additionally tempering out 144/143 to 10/9 6/5 4/3 13/9 5/3 11/6 2/1. The Pinetone diminished heptatonic's large step tempers to 55/48 under 2.3.5.11 Ptolemismic, and to 15/13 under 2.3.5.11.13 Ptolemismic. The scale is chiral, with mirror-inverse msmLsms, approximating 12/11 6/5 11/8 3/2 5/3 9/5 2/1, or in other modes, Lmsmsms and smsmsmL, the brightest and darkest modes of the pair of scales, approximating 15/13 33/26 11/8 20/13 5/3 11/6 2/1 and 12/11 6/5 13/10 13/9 39/25 26/15 2/1. The scales, like the Pinetone diatonic, are also trivalent. There's only one major triad and one minor triad available in the scale. In their simplest modes with 3/2 above the root, the pairs of scales in 5-limit JI are 27/25 5/4 25/18 3/2 5/3 9/5 2/1 and 27/25 6/5 25/18 3/2 5/3 2/1, and as Ptolemismic tempered scales their simplest pre-images are 12/11 5/4 11/8 3/2 5/3 9/5 2/1 and 12/11 6/5 11/8 3/2 5/3 9/5 2/1, with step patterns sLmsmsm and smLsmsm respectively.
| Interval class | size | Porcupine name | Meantone name | Pinetone name | Generic name | JI ratios approximated | size in cents (TE) | Occurence |
|---|---|---|---|---|---|---|---|---|
| 1-step | s | minor 2nd | minor 2nd | minor second | small 2nd | 27/25, 12/11, 13/12 | 142.775 | 3 |
| m | minor 2nd | major 2nd | minor-major 2nd | medium 2nd | 10/9, 11/10 | 175.892 | 3 | |
| L | major 2nd | augmented 2nd | augmented 2nd | large 2nd | 125/128, 55/48, 15/13 | 242.658 | 1 | |
| 2-step | m + s | minor 3rd | minor 3rd | minor 3rd | small 3rd | 6/5, 40/33 | 318.667 | 5 |
| L + s | major 3rd | major 3rd | major 3rd | middle 3rd | 5/4 | 385.433 | 1 | |
| L + m | major 3rd | augmented 3rd | augmented 3rd | large 3rd | 33/26, 50/39 | 418.550 | 1 | |
| 3-step | m + 2s | minor 4th | diminished 4th | diminished 4th | small 4th | 13/10 | 461.433 | 2 |
| 2m + s | minor 4th | perfect 4th | minor 4th | middle 4th | 4/3, 33/25 | 494.559 | 2 | |
| L + m + s | major 4th | augmented 4th | major 4th | large 4th | 25/18, 11/8, 18/13 | 561.325 | 3 | |
| 4-step | 2m + 2s | minor 5th | diminished 5th | minor 5th | small 5th | 36/25, 16/11, 13/9 | 637.334 | 3 |
| L + m + 2s | major 5th | perfect 5th | major 5th | medium 5th | 3/2, 50/33 | 704.524 | 2 | |
| L + 2m + s | major 5th | augmented 5th | augmented 5th | large 5th | 20/13 | 737.217 | 2 | |
| 5-step | 3m + 2s | minor 6th | diminished 6th | diminished 6th | small 6th | 39/25, 52/33 | 780.120 | 1 |
| L + 2m + 2s | major 6th | minor 6th | major-minor 6th | medium 6th | 8/5 | 813.227 | 1 | |
| L + 3m + s | major 6th | major 6th | major 6th | large 6th | 5/3, 33/20 | 879.992 | 5 | |
| 6-step | 3m + 3s | minor 7th | diminished 7th | diminished 7th | small 7th | 256/125, 55/24, 26/15 | 956.002 | 1 |
| L + 2m + 3s | major 7th | minor 7th | major-minor 7th | medium 7th | 9/5, 20/11 | 1022.768 | 3 | |
| L + 3m + 2s | major 7th | major 7th | major 7th | large 7th | 11/6, 50/27 | 1055.884 | 3 |
Summary for xen-math nerds
Pinetone scales are built via step nesting from the 5-limit minor seventh tetrad: 6/5 3/2 9/5 2/1. The bounds for its scales are the set of temperings of the rank-3 step-nested children of the 4-note SNS 6/5 3/2 9/5 2/1. Pinetone diminished scales are built via step-nesting from the 5-limit diminished tetrad: 6/5 36/25 5/3 2/1. The bounds for its scales are the set of temperings of the rank-3 step-nested children of the 4-note MOS scale 6/5 36/25 5/3 2/1.
The Pinetone chromatic 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♯ / A♭ 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 Pinetone 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) comprising 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 Pinetone major and minor-harmonic octatonics are the 8-note rank-3 Porcupine[8] x Father[8] Fokker blocks with unison vectors of 250/243, 16/15, and 648/625; comprising 4 large steps of 10/9 (L x L), 3 medium steps of 27/25 (L x s), and one small step of 25/24 (s x L).
The Pinetone diminished scale is a step-nested scale and a Porcupine[8] x Diminished[8] Fokker block with unison vectors of 250/243, 648/625, and 16/15; comprising 4 large steps of 10/9 (L x L), 3 medium steps of 27/25 (L x s), and one small step of 25/24 (s x s).
Pinetone harmonic minor and harmonic major
Additionally, we have another set of Porcupine[7] modes contained in the Pinetone harmonic octatonics: 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.
We note that there are fewer consonant triads available in these scales than in the Pinetone diatonic and octatonic scales, so they may be useful for melody only.
On D we get the scale:
174.055 320.69 557.888 704.524 878.579 1025.214 1199.269 as the notes D E F G♯ A B C D, representing 10/9 6/5 25/18~11/8 3/2 5/3 9/5 2/1
We get the following 7 modes of Pinetone harmonic minor scale:
- Lsmsmms Lydian ♯2 bright major starting on F
- mmsLsms Ionian ♯5 symmetric minor starting on C
- msLsmsm Ukranian dorian bright minor starting on D
- sLsmsmm Phyrgian dominant dark major starting on E
- msmmsLs harmonic minor dark diminished starting on A
- smmsLsm Locrian ♮6 bright diminished starting on B
- smsmmsL altered diminished magical seventh starting on G♯
Replacing the A with an A♭ instead, we get the modes of the Pinetone harmonic major scale. Starting on D we get the mode:
174.055 320.69 494.745 641.38 878.579 1025.214 1199.269 as the notes D E F G A♭ B C D, representing 10/9 6/5 4/3 36/25~13/9 5/3 9/5 2/1.
Which has modes:
- Lsmmsms Lydian Augmented ♯2 bright major starting on A♭
- msLsmms Lydian ♭3 bright minor starting on F
- sLsmmsm Mixolydian ♭2 dark major starting on G
- mmsmsLs harmonic major bright diminished starting on C
- msmsLsm Dorian ♭5 dark diminished starting on D
- smsLsmm Phrygian ♭4 symmetric minor starting on E
- smmsmsL Locrian magical ♭♭7 starting on B
We can see that this scale differs from the Pinetone diminished heptatonic by only a single note - 9/5 instead of 50/27.
The augmented step of the Pinetone harmonic minor and major scales is the same as of the Pinetone diminished heptatonic, representing 15/13 when 325/324 is tempered out (the difference between 100/99 and 144/143).
The Pintone harmonic minor and harmonic minor have step patterns msmmsLs, and mmsmsLs respectively, or, represented as MODMOS of detempered Meantone[7], LsLLsAs and LLsLsAs, the step patterns of the familiar harmonic minor and harmonic major scales.
Pinetone hyperchromatic scales
Maybe you have a Lumatone, and you're wondering, ok so you can either have sharps or flats? Por queno los dos?
Indeed we can have both!
From the Pinetone chromatic with sharps (mode -3), we add another Pinetone diatonic scale, mode 0 starting on D♭, leading to the left-handed Pinetone hyperchromatic scale, with step pattern, sLsLssLsmLssLsLssLs.
Or, from the Pinetone chromatic with flats (mode 3), we add another Pinetone diatonic scale, mode 0 starting on D♯, leading to the right-handed Pinetone hyperchromatic scale, with step pattern, sLssLsLssLmsLssLsLs.
If 81/80 were additionally tempered out (tempering out the difference between the small step and the medium step), these scales would temper to Flattone[19], reflected in their layout on the lumatone. These scale comprises 7 large steps approximating 117/110 (the difference between the large and small steps of the Pinetone chromatic), the medium step of the Pinetone chromatic, approximating 25/24, 33/32, and 27/26, and 11 small steps, the same as the small step of the pinetone chromatic, approximating 250/243, 55/54, 121/120, and 40/39.
We note that sLss, the interval from D to E♯, for example, is very near 9/8, and that sLsL, the interval from D to F♭, for an example, is very near 32/27. If we recognize these approximates, we additionally temper out 243/242, or 352/351, leading to Tetracot temperament, in which case the large step approximates 16/15. This also adds 81/80 to the list of intervals approximated by the small step. Adding an additional small step above G, for the left handed hyperchromatic, or below A, for the right handed hyperchromatic, would give us a MODMOS of Tetracot[20], splitting the one medium step into two small steps (we note also that TE 2.3.5.11.13 ptolemismic tunes the medium step to 66.76626, which is almost exactly twice the size of its small step of 33.11646c).
In 2.3.5.11.13 Tetracot, the left-handed Pinetone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 32/27 6/5 11/9 13/10 4/3 11/8 22/15 3/2 20/13 13/8 5/3 16/9 9/5 11/6 39/20 2/1, and the right-handed Pinetone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 9/8 6/5 11/9 13/10 4/3 15/11 13/9 3/2 20/13 13/8 5/3 27/16 9/5 11/6 39/20 2/1.
Tuned to TE 2.3.5.11.13 Tetracot (with a large step of 109.3262 and a small step of 33.3391c), the left-handed Pinetone hyperchromatic in cents is
33.3391 142.6653 176.0044 285.3306 318.6697 352.0088 461.335 494.6741 561.3532 670.6785 704.0176 737.3567 846.6829 880.022 989.3482 1022.6873 1056.0264 1165.3526 1198.6917,
and the right-handed Pinetone hyperchromatic in cents is
33.3391 142.6653 176.0044 209.3435 318.6697 352.0088 461.335 494.6741 528.0132 637.3394 704.0176 737.3567 846.6829 880.022 913.3611 1022.6873 1056.0264 1165.3526 1198.6917.
The Pinetone hyperchromatic scales may alternatively be tuned to 27edo, 34edo, or 41edo:
27edo: 7L 1m 11s = (2, 2, 1) = (88.8889c, 88.8889c, 44.4444c)
34edo: 7L 1m 11s = (3, 2, 1) = (105.8824c, 70.5882c, 35.2941c)
41edo: 7L 1m 11s = (4, 2, 1) = (117.0732c, 58.5366c, 29.2683c).
Pinetone-15
Alternatively, a 15-note scale can be built from the Pinetone diminished. The resulting scale tempers to Porcupine[15], as well as to Hanson[15].
From the Pinetone diminished scale: MLsLMLML, shown in the bright minor mode as Pinetone bright minor diminished, putting a small step into the bottom of each medium and large step leads to the child SNS of the Pinetone diminished scale: the fifteen note SNS msmLmmLmsmLmsmL, or mLmsmLmsmLmsmLm in it's symmetric mode, comprising 4 large steps of 16/15, 8 medium steps of 25/24 and 3 small steps of 648/625, i.e.,
25/24 10/9 125/108 6/5 5/4 4/3 25/18 36/25 3/2 8/5 5/3 216/125 9/5 48/25 2/1.
Tempering m = s (tempering out 15625/15552, the Hanson comma) results in sLsssLsssLsssLs, which is Hanson[15];
tempering L = m (tempering out 128/125, the Augmented comma) results in LLLsLLLsLLLsLLL, which is a MODMOS of Augmented[15]
tempering L = s (tempering out 250/243, the Porcupine comma) results in sLsLsLsLsLsLsLs, which is Porcupine[15].
tempering out s would lead to sLLsLsL, which is Dicot[7];
tempering out m would lead to ssLsLssLsssL, which is a MODMOS of Diminished[12].
Tempering out 100/99 and 144/143 leads to the simplest pre-image: 25/24 10/9 15/13 6/5 5/4 4/3 11/8 13/9 3/2 8/5 5/3 26/15 9/5 48/25 2/1.
With TE 2.3.5.11.13 Ptolemismic tuning applied, the sizes of the steps shift enough for the size order to change. Pinetone-15 comprises
- 4 large steps of 109.12557c, approximating 16/15;
- 3 medium steps of 76.00911c, approximating 648/625, 128/121, and 26/25; and
- 8 small steps of 66.76626c, approximating 25/24, 33/32, and 27/26.
In cents, TE 2.3.5.11.13 Ptolemismic Pinetone-15, in the symmetric mode, is
66.766 175.892 242.658 318.667 385.433 494.559 561.325 637.334 704.101 813.226 879.993 956.002 1022.768 1131.893 1198.660 as sLsmsLsmsLsmsLs.
Accordingly Pinetone-15 would temper to two step sizes in 19edo (Hanson), 22edo (Porcupine), 34edo (Hanson), and 27edo (Augmented). If we wish to keep the 3-step size structure, we can tune to 26edo or 41edo with (L, m, s) = (3, 2, 1), and (4, 3, 2) respectively.
Tempering out the 325/324, the difference between 100/99 and 144/143 rather than both of 100/99 and 144/143 leads to a more accurate temperament that does not include the whole 2.3.5.11.13 subgroup., rather just the 2.3.5.13 subgroup. The simplest JI pre-image in this temperament would be 25/24 10/9 15/13 6/5 5/4 4/3 18/13 13/9 3/2 8/5 5/3 26/15 9/5 48/25 2/1, which differs only from the simplest pre-image of the scale under 2.3.5.11.13 Ptolemismic tempering by the inclusion of 18/13 rather than 11/8.
2.3.5.13 325/324 may be better tuned to 46edo, with (L, m, s) = (4, 2, 1).
Pinetone-15 may be represented with note names representing the corresponding Meantone notes as: D D♯ E E♯ F F♯ G G♯ A♭ A B♭ B C♭ C D♭ D, though we note that these notes do not necessarily correspond to the notes of the Pinetone chromatic or hyperchromatic of the same name.
Major triads are available on D, E, F, G, B♭, B, and D♭, and minor triads are available on D, E, F, G, G♯, B♭, and B.
Pinetone harmonic diminished octatonic
As a subset of Pinetone-15 we may find modified Pinetone octatonics built on MODMOS of Porcupine[8]. The Porcupine[8]'s 4M (minimally modified MODMOS) is useful given that it still comprises consonant 3-step triads on all notes, but with a more spread-out distribution, so that the triads of each type do not all occur adjacent to each other as in Porcupine[7] and Porcupine[8]. This scale may be found either by lowering G or raising B by a Porcupine[8] chroma, which represents 16/15, the large step of Pinetone-15.
2.3.5.11.13 ptolemismic Pinetone-15 has simplest JI pre-image 25/24 10/9 15/13 6/5 5/4 4/3 11/8 13/9 3/2 8/5 5/3 26/15 9/5 48/25 2/1 as sLsmsLsmsLsmsLs which may be grouped as a detempered Porcupine[8], large step of sL or sm, small step of s in the following ways: 12222222, 21222222, 22122222, 22212222, 22221222, 22222122, 22222212, 22222221. Using
22122222 -> 13122222, 22131222, 22221312 i.e., s(Lsm)s(Ls)(ms)(Ls)(ms)(Ls) or (sL)(sm)(sL)(sm)(sL)s(msL)s, (sL)(sm)s(Lsm)s(Ls)(ms)(Ls) or (sL)(sm)(sL)s(msL)s(ms)(Ls), (sL)(sm)(sL)(sm)s(Lsm)s(Ls) or (sL)s(msL)s(ms)(Ls)(ms)(Ls), 6 modes of 2 scales.
The first pair of modes temper to Diminished[8] (m = 0), and to Father[8] (L=0). The scale has 1 augmented step of 52/45, 3 large steps of 10/9~11/10, 2 medium steps of 12/11~13/12, and 2 small steps of 25/24~33/32~27/26.
| Mode (height order) | Step pattern | Oneirotonic step pattern | Porcupine[8] step pattern | Mode in 5-limit JI | Comments |
|---|---|---|---|---|---|
| Hlanithian diminished | AsLMLMLs | sLLsLsLL (Hlanithian) | AsLLLLLs | 144/125 6/5 4/3 36/25 8/5 216/125 48/25 2/1 | |
| Mnarian diminished* | LMLMLsAs | LsLsLLsL (Mnarian) | LLLLLsAs | 10/9 6/5 4/3 36/25 8/5 5/3 48/25 2/1 | |
| Celephaïsian diminished* | LMLsAsLM | LsLLsLLs (Celephaïsian) | LLLsAsLL | 10/9 6/5 4/3 25/18 8/5 5/3 50/27 2/1 | |
| Sarnathian diminished† | MLMLsAsL | sLsLLsLL (Sarnathian) | LLLLsAsL | 27/25 6/5 162/125 36/25 3/2 216/125 9/5 2/1 | 10:12:15 on the root |
| Dylathian diminished* | LsAsLMLM | LLsLLsLs (Dylathian) | LsAsLLLL | 10/9 125/108 4/3 25/18 125/81 5/3 50/27 2/1 | |
| Kadathian diminished*†† | MLsAsLML | sLLsLLsL (Kadathian) | LLsAsLLL | 27/25 6/5 5/4 36/25 3/2 5/3 9/5 2/1 | root 4:5:6,10:12:15 |
| Ultharian diminished*†† | sAsLMLML | LsLLsLsL (Ultharian) | sAsLLLLL | 25/24 6/5 5/4 25/18 3/2 5/3 9/5 2/1 | root 4:5:6,10:12:15 |
| Illarnekian diminished*† | sLMLMLsA | LLsLsLLs (Illarnekian) | sLLLLLsA | 25/24 125/108 5/4 25/18 3/2 5/3 125/72 2/1 | root 4:5:6 |
| Mode (height order) | Step pattern | Mode as simplest JI pre-image 5-limit JI | Mode in cents | Comments |
|---|---|---|---|---|
| Hlanithian diminished | AsLMLMLs | ~ 52/45 6/5 4/3 13/9 8/5 26/15 48/25 2/1 | 251.901 318.667 494.559 637.334 813.226 956.002 1131.893 1198.660 | |
| Mnarian diminished* | LMLMLsAs | ~ 10/9 6/5 4/3 13/9 8/5 5/3 48/25 2/1 | 175.892 318.667 494.559 637.334 813.226 879.993 1131.893 1198.660 | |
| Celephaïsian diminished* | LMLsAsLM | ~ 10/9 6/5 4/3 11/8 8/5 5/3 11/6 2/1 | 175.892 318.667 494.559 561.325 813.226 879.993 1055.884 1198.660 | |
| Sarnathian diminished† | MLMLsAsL | ~ 12/11 6/5 13/10 13/9 3/2 26/15 9/5 2/1 | 142.775 318.667 461.443 637.334 704.101 956.002 1022.768 1198.660 | 10:12:15 on the root |
| Dylathian diminished* | LsAsLMLM | ~ 10/9 15/13 4/3 11/8 20/13 5/3 11/6 2/1 | 175.892 242.658 494.559 561.325 737.217 879.993 1055.884 1198.660 | |
| Kadathian diminished*†† | MLsAsLML | ~ 12/11 6/5 5/4 13/9 3/2 5/3 9/5 2/1 | 142.775 318.667 385.433 637.334 704.101 879.993 1022.768 1198.660 | root 4:5:6,10:12:15 |
| Ultharian diminished*†† | sAsLMLML | ~ 25/24 6/5 5/4 11/8 3/2 5/3 9/5 2/1 | 66.766 318.667 385.433 561.325 704.101 879.993 1022.768 1198.660 | root 4:5:6,10:12:15 |
| Illarnekian diminished*† | sLMLMLsA | ~ 25/24 15/13 5/4 11/8 3/2 5/3 45/26 2/1 | 66.766 242.658 385.433 561.325 704.101 879.993 946.759 1198.660 | root 4:5:6 |
| Interval class | sizes | Diminished[8] name | Porcupine[8] name | Pinetone octatonic name | JI ratios approximated* | size in cents (TE) | Occurence |
|---|---|---|---|---|---|---|---|
| 1-step | s | minor step | minor step | minor step | 25/24, 33/32, 27/26 | 66.766 | 2 |
| M | minor step | major step | minor-major step | 27/25, 12/11, 13/12 | 142.775 | 2 | |
| L | major step | major step | major step | 10/9, 11/10 | 175.892 | 3 | |
| A | major step | augmented step | augmented step | 144/125, 64/55, 52/45 | 251.901 | 1 | |
| 2-step | L + s | perfect 2-step | minor 2-step | minor 2-step | 15/13 (7/6 or 8/7) | 242.658 | 2 |
| L + M = A + s | perfect 2-step | major 2-step | major 2-step | 6/5 | 318.667 | 6 | |
| 3-step | L + M + s = A + 2s | minor 3-step | minor 3-step | minor 3-step | 5/4 | 385.433 | 3 |
| L + 2M | minor 3-step | major 3-step | minor-major 3-step | 13/10 (9/7 or 21/16) | 461.433 | 1 | |
| 2L + M = A + L + s | major 3-step | major 3-step | major 3-step | 4/3 | 494.559 | 4 | |
| 4-step | 2L + M + s = A + L + 2s | perfect 4-step | minor 4-step | minor 4-step | 25/18, 11/8, 18/13 | 561.325 | 4 |
| 2L + 2M = A + L + M + s | perfect 4-step | major 4-step | major 4-step | 36/25, 16/11, 13/9 | 637.334 | 4 | |
| 5-step | 2L + 2M + s = A + L + M + 2s | minor 5-step | minor 5-step | minor 5-step | 3/2 | 704.101 | 4 |
| A + 2L + 2s | major 5-step | minor 5-step | major-minor 5-step | 20/13 (14/9 or 32/16) | 737.217 | 1 | |
| 3L + 2M = A + 2L + M + s | major 5-step | major 5-step | major 5-step | 8/5 | 813.227 | 3 | |
| 6-step | 3L + 2M + s = A + 2L + M + 2s | perfect 6-step | minor 6-step | minor 6-step | 5/3 | 879.992 | 6 |
| A + 2L + 2M + s | perfect 6-step | major 6-step | major 6-step | 26/15 (12/7 or 7/4) | 956.002 | 2 | |
| 7-step | 3L + 2M + 2s | minor 7-step | diminished 7-step | diminished 7-step | 125/72, 55/32, 45/26 | 946.759 | 1 |
| A + 2L + 2M + 2s | minor 7-step | minor 7-step | minor 7-step | 9/5, 20/11 | 1022.768 | 3 | |
| A + 3L + M + 2s | major 7-step | minor 7-step | major-minor 7-step | 50/27, 11/6, 24/13 | 1055.884 | 2 | |
| 4L + 3M | major 7-step | major 7-step | major 7-step | 48/25, 64/33, 52/27 | 1131.983 | 2 |
* Non-bracketed JI ratios are those approximated in 2.3.5.11.13 Ptolemismic Pinetone; bracketed JI ratios are shows in pairs: the first interval in each pair is the 2.3.7 interval approximated by additionally tempering out 91/90 or 126/125 (Starling); the second is the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 (Supermagic).
The Pinetone harmonic diminished is the only Pinetone octatonic that is improper, given that the augmented 2-step interval is larger than the minor 3-step interval (and, equivalently, the diminished 7-step interval is smaller than the major 6-step interval. A scale is proper if there are no interval larger than any intervals from a larger interval class. The Pinetone and Meantone diatonic scales are also proper (for any reasonable tuning).
| Mode | step | 2-step | 3-step | 4-step | 5-step | 6-step | 7-step | 8-step |
|---|---|---|---|---|---|---|---|---|
| Hlanithian diminished | Augmented | Major | Major | Major | Major | Major | Major | Perfect |
| Mnarian diminished | Major | Minor | ||||||
| Celephaïsian diminished | Minor | Major-minor | ||||||
| Sarnathian diminished | Minor-major | Minor-major | Major | Minor | Major | Minor | ||
| Dylathian diminished | Major | Minor | Major | Minor | Major-minor | Minor | Major-minor | |
| Kadathian diminished | Minor-major | Major | Minor | Major | Minor | Minor | ||
| Ultharian diminished | Minor | Minor | ||||||
| Illarnekian diminished | Minor | Diminished |
| Mode (rotational order) | Step pattern | Diminished[8] name | Porcupine[8] name | Pinetone octatonic name | JI triad approximated* |
|---|---|---|---|---|---|
| Mnarian diminished | LMLMLsAs | [8] major | [8] major | [8] major | 3:4:5 |
| Sarnathian diminished | MLMLsAsL | [8] minor | [8] augmented | [8] minor augmented | 30:39:52 (16:21:28) |
| Celephaïsian diminished | LMLsAsLM | [8] major | [8] major | [8] major | 3:4:5 |
| Kadathian diminished | MLsAsLML | [8] minor | [8] minor | [8] minor | 12:15:20 |
| Dylathian diminished | LsAsLMLM | [8] major | [8] major | [8] major | 3:4:5 |
| Ultharian diminished | sAsLMLML | [8] minor | [8] minor | [8] minor | 12:15:20 |
| Hlanithian diminished | AsLMLMLs | [8] major | [8] augmented | [8] major augmented | 15:20:26 (12:16:21) |
| Illarnekian diminished | sLMLMLsA | [8] minor | [8] minor | [8] minor | 12:15:20 |
* Non-bracketed JI ratios are those approximated in 2.3.5.11.13 Ptolemismic Pinetone; bracketed JI ratios are the 2.3.7 interval approximated by additionally tempering out 105/104 or 245/243 as in Supermagic temperament. Alternatively, if the consonances of the triads are to be maximised, the scale could be tempered to 2.3.5.7 245/243 i.e., Sensamagic temperament.
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-A-C-F -> F-A-C-E -> G-B-D-E -> D-A-C-F
F-A-D -> F-B-E -> (G-B-E ->) G-B-D -> F-A-D
Final thoughts
I find the diatonic and the harmonic diminished to be the most desirable of the albitonic Pinetone scales as the only ones with four 3/2s with with mean variety less than or equal to 3, along with and the major and minor harmonic octatonics as they contain 5 major/minor triads (on 4 roots), the most of any albitonic Pinetone scale (in comparison, the diminished scale contains 8 major/minor triads on 4 roots, the Meantone diatonic scale contains 6 major/minor triads on 6 roots, and Porcupine[8] contains 6 major/minor triads on 4 roots). The harmonic diminished, as well as the harmonic minor and harmonic major are not built directly by step-nesting, but found as subsets of child step-nested scales that can be modified by chromas to give step-nested scales.