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'''Definition: Pinetone is a system of related rank-3 microtonal scales: pentatonic, diatonic, octatonic, chromatic, and hyperchromatic.'''
'''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 [[Pinetone chromatic (sharps)|this one]]! Copy the text into notepad and save as a .scl file).
"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 [[Pinetone chromatic (sharps)|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!).   
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!).   
Line 10: Line 10:
As opposed to in [[12edo]], each key is distinctly different in Pinetone scales, both a blessing and a curse.  
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.  
Additionally available in Pinetone are 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 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.''  
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* [[Pinetone#Pinetone pentatonic|Pentatonic major]]: 65959, 76B6B (subset of diatonic, harmonic diminished)
* [[Pinetone#Pinetone pentatonic|Pentatonic major]]: 65959, 76B6B (subset of diatonic, harmonic diminished)
* [[Pinetone#Pinetone pentatonic|Pentatonic minor]]: 95695, B67B6 (subset of diatonic, harmonic diminished)
* [[Pinetone#Pinetone pentatonic|Pentatonic minor]]: 95695, B67B6 (subset of diatonic, harmonic diminished)
* [[Pinetone#Pinetone diatonic|Diatonic]]: 4546454, 6567656 (subset of major and minor-harmonic octatonics)
* [[Pinetone#Pinetone diatonic|Diatonic]]: 5456545, 6567656 (subset of major and minor-harmonic octatonics)
* [[Pinetone#Pinetone harmonic minor and harmonic major|Harmonic minor]]: 5455475, 6566585 (subset of major and minor-harmonic octatonic)
* [[Pinetone#Pinetone harmonic minor and harmonic major|Harmonic minor]]: 5455475, 6566585 (subset of major and minor-harmonic octatonic)
* [[Pinetone#Pinetone harmonic minor and harmonic major|Harmonic major]]: 5545474, 6656585 (subset of major and minor-harmonic octatonic)
* [[Pinetone#Pinetone harmonic minor and harmonic major|Harmonic major]]: 5545474, 6656585 (subset of major and minor-harmonic octatonic)
* [[Pinetone#Additional heptatonic subsets of the Pinetone chromatic|Azurian bright minor]]: 5465455, 6576566 (subset of chromatic)
* [[Pinetone#Additional heptatonic subsets of the Pinetone chromatic|Duradian dark minor]]: 5456455, 6567566 (subset of chromatic)
* [[Pinetone#Additional heptatonic subsets of the Pinetone chromatic|Karakalian bright minor]]: 5465545, 6576656 (subset of chromatic)
* [[Pinetone#Additional heptatonic subsets of the Pinetone chromatic|Phyradian dark minor]]: 4556455, 5667566 (subset of chromatic)
* [[Pinetone#Pinetone diminished heptatonic|Diminished heptatonic]]: 4754545 and 4574545, 5865656 and 5685656 (subset of diminished)
* [[Pinetone#Pinetone diminished heptatonic|Diminished heptatonic]]: 4754545 and 4574545, 5865656 and 5685656 (subset of diminished)
* [[Pinetone#Pinetone octatonic scales|Major-harmonic octatonic]]: 54524545, 65625656 (subset of chromatic, diminished chromatic)
* [[Pinetone#Pinetone octatonic scales|Major-harmonic octatonic]]: 54524545, 65625656 (subset of chromatic, diminished chromatic)
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|}
|}
{| class="wikitable"
{| class="wikitable"
|+Table 2.3. Modes of the just Pinetone diatonic
|+Table 2.3. Modes of the [[SNS (2/1, 3/2, 6/5)-7|just Pinetone diatonic]]
!Mode number
!Mode number
!Mode in JI
!Mode in JI
!Root note
!Step pattern
!Step pattern
!Meantone[7]
!Meantone[7]
Line 158: Line 163:
|3
|3
|10/9 5/4 25/18 3/2 5/3 50/27 2/1
|10/9 5/4 25/18 3/2 5/3 50/27 2/1
|F
|MLMsMMs
|MLMsMMs
|LLLsLLs
|LLLsLLs
Line 167: Line 173:
|2
|2
|9/8 5/4 27/20 3/2 5/3 9/5 2/1
|9/8 5/4 27/20 3/2 5/3 9/5 2/1
|G
|LMsMMsM
|LMsMMsM
|LLsLLsL
|LLsLLsL
Line 176: Line 183:
|1
|1
|10/9 100/81 4/3 40/27 5/3 50/27 2/1
|10/9 100/81 4/3 40/27 5/3 50/27 2/1
|C
|MMsMLMs
|MMsMLMs
|LLsLLLs
|LLsLLLs
Line 185: Line 193:
|0
|0
|10/9 6/5 4/3 3/2 5/3 9/5 2/1
|10/9 6/5 4/3 3/2 5/3 9/5 2/1
|D
|MsMLMsM
|MsMLMsM
|LsLLLsL
|LsLLLsL
Line 194: Line 203:
|  -1
|  -1
|27/25 6/5 27/20 3/2 81/50 9/5 2/1
|27/25 6/5 27/20 3/2 81/50 9/5 2/1
|E
|sMLMsMM
|sMLMsMM
|sLLLsLL
|sLLLsLL
Line 203: Line 213:
|  -2
|  -2
|10/9 6/5 4/3 40/27 8/5 16/9 2/1
|10/9 6/5 4/3 40/27 8/5 16/9 2/1
|A
|MsMMsML
|MsMMsML
|LsLLsLL
|LsLLsLL
Line 212: Line 223:
|  -3
|  -3
|27/25 6/5 4/3 36/25 8/5 9/5 2/1
|27/25 6/5 4/3 36/25 8/5 9/5 2/1
|B
|sMMsMLM
|sMMsMLM
|sLLsLLL
|sLLsLLL
Line 227: Line 239:
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]].  
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]].  
{| class="wikitable"
{| class="wikitable"
|+Table 2.4. Modes of the Ptolemismic Pinetone diatonic
|+Table 2.4. Modes of the [[SNS (2/1, 3/2, 6/5: 100/99)-7|Ptolemismic Pinetone diatonic]]
!Mode number
!Mode number
!Pinetone diatonic mode
!Pinetone diatonic mode
!Root note
!Step pattern
!Step pattern
!Mode as simplest JI pre-image
!Mode as simplest JI pre-image
Line 236: Line 249:
|3
|3
|Lydian dark major*
|Lydian dark major*
|F
|mLmsmms
|mLmsmms
|~ 10/9 5/4 11/8 3/2 5/3 11/6 2/1
|~ 10/9 5/4 11/8 3/2 5/3 11/6 2/1
Line 241: Line 255:
|-
|-
|2
|2
|Mixolydian bright minor*
|Mixolydian bright major*
|G
|Lmsmmsm
|Lmsmmsm
|~ 9/8 5/4 15/11 3/2 5/3 9/5 2/1
|~ 9/8 5/4 15/11 3/2 5/3 9/5 2/1
Line 248: Line 263:
|1
|1
|Ionian bright diminished
|Ionian bright diminished
|C
|mmsmLms
|mmsmLms
|~ 10/9 11/9 4/3 22/15 5/3 11/6 2/1
|~ 10/9 11/9 4/3 22/15 5/3 11/6 2/1
Line 254: Line 270:
|0
|0
|Dorian symmetric minor*
|Dorian symmetric minor*
|D
|msmLmsm
|msmLmsm
|~ 10/9 6/5 4/3 3/2 5/3 9/5 2/1
|~ 10/9 6/5 4/3 3/2 5/3 9/5 2/1
Line 260: Line 277:
| -1
| -1
|Phrygian bright minor*
|Phrygian bright minor*
|E
|smLmsmm
|smLmsmm
|~ 12/11 6/5 15/11 3/2 18/11 9/5 2/1
|~ 12/11 6/5 15/11 3/2 18/11 9/5 2/1
Line 266: Line 284:
| -2
| -2
|Aeolian magical seventh
|Aeolian magical seventh
|A
|msmmsmL
|msmmsmL
|~ 10/9 6/5 4/3 22/15 8/5 16/9 2/1
|~ 10/9 6/5 4/3 22/15 8/5 16/9 2/1
Line 272: Line 291:
| -3
| -3
|Locrian dark diminished
|Locrian dark diminished
|B
|smmsmLm
|smmsmLm
|~ 12/11 6/5 4/3 16/11 8/5 9/5 2/1
|~ 12/11 6/5 4/3 16/11 8/5 9/5 2/1
Line 599: Line 619:


{| class="wikitable"
{| class="wikitable"
|+Table 2.7. Tertian triads of the Ptolemismic Pinetone diatonic on D (dorian symmetric minor)
|+Table 2.7. Tertian triads of the Ptolemismic Pinetone diatonic in D (dorian symmetric minor)
!Root note
!Root note
!Triad notes
!Triad notes
Line 606: Line 626:
!Pinetone triad
!Pinetone triad
!JI chord approximated by triad
!JI chord approximated by triad
!Mode on root
|-
|-
!D
!D
Line 613: Line 634:
|minor
|minor
|10:12:15
|10:12:15
|Dorian symmetric minor
|-
|-
!E
!E
Line 620: Line 642:
|minor
|minor
|10:12:15
|10:12:15
|Phrygian bright minor
|-
|-
!F
!F
Line 627: Line 650:
|major
|major
|4:5:6
|4:5:6
|Lydian dark major
|-
|-
!G
!G
Line 634: Line 658:
|major
|major
|4:5:6
|4:5:6
|Mixolydian bright major
|-
|-
!A
!A
Line 641: Line 666:
|minor diminished
|minor diminished
| 15:18:22
| 15:18:22
|Aeolian magical seventh
|-
|-
!B
!B
Line 648: Line 674:
|diminished
|diminished
|25:30:36
|25:30:36
|Locrian dark diminished
|-
|-
!C
!C
Line 655: Line 682:
| major diminished
| major diminished
| 27:33:40
| 27:33:40
|Ionian bright diminished
|}
|}


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.
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.
{| class="wikitable"
{| class="wikitable"
|+Table 2.8. Tertian tetrads of the Ptolemismic Pinetone diatonic on D (dorian symmetric minor)
|+Table 2.8. Tertian tetrads of the Ptolemismic Pinetone diatonic in D (dorian symmetric minor)
!Root note
!Root note
!Triad notes
!Triad notes
Line 666: Line 694:
!Pinetone tetrad
!Pinetone tetrad
!JI chord approximated by tetrad
!JI chord approximated by tetrad
!Mode on root
|-
|-
!D
!D
Line 673: Line 702:
| minor 7
| minor 7
| 10:12:15:18
| 10:12:15:18
|Dorian symmetric minor
|-
|-
!E
!E
Line 680: Line 710:
| minor 7
| minor 7
| 10:12:15:18
| 10:12:15:18
|Phrygian bright minor
|-
|-
!F
!F
Line 687: Line 718:
|major major-minor 7
|major major-minor 7
|12:15:18:22
|12:15:18:22
|Lydian dark major
|-
|-
!G
!G
Line 694: Line 726:
|major minor 7
|major minor 7
|20:25:30:36
|20:25:30:36
|Mixolydian bright major
|-
|-
!A
!A
Line 701: Line 734:
|minor diminished 7
|minor diminished 7
|45:54:66:80
|45:54:66:80
|Aeolian magical seventh
|-
|-
!B
!B
Line 708: Line 742:
|half diminished 7
|half diminished 7
|25:30:36:45
|25:30:36:45
|Locrian dark diminished
|-
|-
!C
!C
Line 715: Line 750:
|major half diminished 7
|major half diminished 7
|27:33:40:50
|27:33:40:50
|Ionian bright diminished
|}
|}


Line 721: Line 757:
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.
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.
{| class="wikitable"
{| class="wikitable"
|+Table 2.9. 3-step stacked triads of the Ptolemismic Pinetone diatonic on D (dorian symmetric minor)
|+Table 2.9. 3-step stacked triads of the Ptolemismic Pinetone diatonic in D (dorian symmetric minor)
!Root note
!Root note
!Triad notes
!Triad notes
Line 728: Line 764:
!Pinetone triad
!Pinetone triad
!JI chord approximated by triad
!JI chord approximated by triad
!Mode on root
|-
|-
!D
!D
Line 735: Line 772:
|sus minor major
|sus minor major
|15:20:27
|15:20:27
|Dorian symmetric minor
|-
|-
!E
!E
Line 742: Line 780:
|sus major minor
|sus major minor
|11:15:20
|11:15:20
|Phrygian bright minor
|-
|-
!F
!F
Line 749: Line 788:
|sus ♭2 major minor
|sus ♭2 major minor
|24:33:44
|24:33:44
|Lydian dark major
|-
|-
!G
!G
Line 756: Line 796:
|sus major minor
|sus major minor
|11:15:20
|11:15:20
|Mixolydian bright major
|-
|-
!A
!A
Line 763: Line 804:
|sus minor minor
|sus minor minor
|9:12:16
|9:12:16
|Aeolian magical seventh
|-
|-
!B
!B
Line 770: Line 812:
|sus minor major
|sus minor major
|15:20:27
|15:20:27
|Locrian dark diminished
|-
|-
!C
!C
Line 777: Line 820:
|sus ♯4 minor major
|sus ♯4 minor major
|6:8:11
|6:8:11
|Ionian bright diminished
|}
|}
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 [[The Pinetone System#Pinetone pentatonic|Pinetone major and minor pentatonics]] respectively, introduced below.  
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 [[The Pinetone System#Pinetone pentatonic|Pinetone major and minor pentatonics]] respectively, introduced below.  
{| class="wikitable"
{| class="wikitable"
|+Table 2.10. 3-step stacked pentads of the Ptolemismic Pinetone diatonic on D
|+Table 2.10. 3-step stacked pentads of the Ptolemismic Pinetone diatonic in D
!Root note
!Root note
!Notes
!Notes
Line 788: Line 832:
!JI chord approximated by triad
!JI chord approximated by triad
!JI Intervals
!JI Intervals
!Mode on root
|-
|-
!D
!D
Line 796: Line 841:
|15:20:27:36:50
|15:20:27:36:50
|4/3, 27/20, 4/3, 25/18, (6/5)
|4/3, 27/20, 4/3, 25/18, (6/5)
|Dorian symmetric minor
|-
|-
!E
!E
Line 804: Line 850:
| 20:27:36:48:65
| 20:27:36:48:65
|27/20, 4/3, 4/3, 65/48, (16/13)
|27/20, 4/3, 4/3, 65/48, (16/13)
|Phrygian bright minor
|-
|-
! F
! F
Line 812: Line 859:
|24:33:44:60:80
|24:33:44:60:80
|11/8, 4/3, 15/11, 4/3, (6/5)
|11/8, 4/3, 15/11, 4/3, (6/5)
|Lydian dark major
|-
|-
!G
!G
Line 820: Line 868:
|20:27:36:50:66
|20:27:36:50:66
|27/20, 4/3, 25/18, 33/25, (40/33)
|27/20, 4/3, 25/18, 33/25, (40/33)
|Mixolydian bright major
|-
|-
! A
! A
Line 828: Line 877:
|25:33:44:60:80
|25:33:44:60:80
|33/25, 4/3, 15/11, 4/3, (5/4)
|33/25, 4/3, 15/11, 4/3, (5/4)
|Aeolian magical seventh
|-
|-
!B
!B
Line 836: Line 886:
|15:20:27:36:48
|15:20:27:36:48
|4/3, 27/20, 4/3, 4/3, (5/4)
|4/3, 27/20, 4/3, 4/3, (5/4)
|Locrian dark diminished
|-
|-
!C
!C
Line 844: Line 895:
|18:24:33:44:60
|18:24:33:44:60
|4/3, 11/8, 4/3, 15/11, (6/5)
|4/3, 11/8, 4/3, 15/11, (6/5)
|Ionian bright diminished
|}
|}
{| class="wikitable"
{| class="wikitable"
|+Table 2.11. 2nd inversion (symmetric) 3-step stacked pentads of the Ptolemismic Pinetone diatonic on D
|+Table 2.11. 2nd inversion (symmetric) 3-step stacked pentads of the Ptolemismic Pinetone diatonic in D
!Root note
!Root note
!Notes
!Notes
Line 854: Line 906:
!JI chord approximated by triad
!JI chord approximated by triad
!JI Intervals
!JI Intervals
!Mode on root
|-
|-
!D
!D
Line 862: Line 915:
|15:20:27:33:45:60
|15:20:27:33:45:60
|4/3, 27/20, 11/9, 15/11, 4/3
|4/3, 27/20, 11/9, 15/11, 4/3
|Dorian symmetric minor
|-
|-
!E
!E
Line 870: Line 924:
| 11:15:20:24:33:44
| 11:15:20:24:33:44
|15/11, 4/3, 6/5, 11/8, 4/3
|15/11, 4/3, 6/5, 11/8, 4/3
|Phrygian bright minor
|-
|-
!F
!F
Line 878: Line 933:
|18:25:33:40:54:72
|18:25:33:40:54:72
|25/18, 33/25, 40/33, 27/20, 4/3
|25/18, 33/25, 40/33, 27/20, 4/3
|Lydian dark major
|-
|-
!G
!G
Line 886: Line 942:
| 11:15:20:25:33:44
| 11:15:20:25:33:44
|15/11, 4/3, 5/4, 33/25, 4/3
|15/11, 4/3, 5/4, 33/25, 4/3
|Mixolydian bright major
|-
|-
!A
!A
Line 894: Line 951:
|27:36:48:60:80:108
|27:36:48:60:80:108
|4/3, 4/3, 5/4, 4/3, 27/20
|4/3, 4/3, 5/4, 4/3, 27/20
|Aeolian magical seventh
|-
|-
!B
!B
Line 902: Line 960:
| 33:44:60:72:96:132
| 33:44:60:72:96:132
|4/3, 15/11, 6/5, 4/3, 11/8
|4/3, 15/11, 6/5, 4/3, 11/8
|Locrian dark diminished
|-
|-
!C
!C
Line 910: Line 969:
|27:36:50:60:80:108
|27:36:50:60:80:108
|4/3, 25/18, 6/5, 4/3, 27/20
|4/3, 25/18, 6/5, 4/3, 27/20
|Ionian bright diminished
|}
|}
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 [http://x31eq.com/cgi-bin/rt.cgi?ets=1ce_4p_2ce&limit=2_3_5_11 TE 2.3.5.11 Ptolemismic].
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 [http://x31eq.com/cgi-bin/rt.cgi?ets=1ce_4p_2ce&limit=2_3_5_11 TE 2.3.5.11 Ptolemismic].
Line 1,002: Line 1,062:


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
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
 
{| class="wikitable"
|+(2.3.5.11) [[Ptolemismic Pinetone chromatic|Ptolemismic Pinetone Chromatic]]
!Mode number
!Mode as simplest JI pre-image
!Step pattern
!Meantone[12]
!UDP
|-
| -6
|~ 55/54 10/9 121/108 11/9 4/3 110/81 22/15 55/36 5/3 121/72 11/6 2/1
|sLsLLsLmLsLL
|sLsLLsLsLsLL
|<nowiki>1|10</nowiki>
|-
| -5
|~ 55/54 10/9 55/48 5/4 121/96 11/8 3/2 55/36 5/3 121/72 11/6 2/1
|sLmLsLLsLsLL
|sLsLsLLsLsLL
|<nowiki>0|11</nowiki>
|-
| -4
|~ 55/54 10/9 6/5 11/9 4/3 110/81 22/15 8/5 44/27 16/9 11/6 2/1
|sLLsLsLLsLmL
|sLLsLsLLsLsL
|<nowiki>4|7</nowiki>
|-
| -3
|~ 55/54 10/9 6/5 11/9 4/3 11/8 3/2 55/36 5/3 9/5 11/6 2/1
|sLLsLmLsLLsL
|sLLsLsLsLLsL
|<nowiki>3|8</nowiki>
|-
| -2
|~ 25/24 9/8 55/48 5/4 15/11 11/8 3/2 55/36 5/3 9/5 11/6 2/1
|mLsLLsLsLLsL
|sLsLLsLsLLsL
|<nowiki>2|9</nowiki>
|-
| -1
|~ 12/11 10/9 6/5 11/9 4/3 16/11 22/15 8/5 5/3 9/5 11/6 2/1
|LsLsLLsLmLsL
|LsLsLLsLsLsL
|<nowiki>6|5</nowiki>
|-
|1
|~ 12/11 10/9 6/5 5/4 15/11 11/8 3/2 18/11 5/3 9/5 11/6 2/1
|LsLmLsLLsLsL
|LsLsLsLLsLsL
|<nowiki>5|6</nowiki>
|-
|2
|~ 12/11 10/9 6/5 72/55 4/3 16/11 22/15 8/5 96/55 16/9 48/25 2/1
|LsLLsLsLLsLm
|LsLLsLsLLsLs
|<nowiki>9|2</nowiki>
|-
|3
|~ 12/11 10/9 6/5 72/55 4/3 16/11 3/2 18/11 5/3 9/5 108/55 2/1
|LsLLsLmLsLLs
|LsLLsLsLsLLs
|<nowiki>8|3</nowiki>
|-
|4
|~ 12/11 9/8 27/22 5/4 15/11 81/55 3/2 18/11 5/3 9/5 108/55 2/1
|LmLsLLsLsLLs
|LsLsLLsLsLLs
|<nowiki>7|4</nowiki>
|-
|5
|~ 12/11 144/121 6/5 72/55 4/3 16/11 192/121 8/5 96/55 9/5 108/55 2/1
|LLsLsLLsLmLs
|LLsLsLLsLsLs
|<nowiki>11|0</nowiki>
|-
|6
|~ 12/11 144/121 6/5 72/55 15/11 81/55 3/2 18/11 216/121 9/5 108/55 2/1
|LLsLmLsLLsLs
|LLsLsLsLLsLs
|<nowiki>10|1</nowiki>
|}
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.
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.


Line 1,015: Line 1,154:
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
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.
I find this tuning to be melodically superior, given the small step is 6 cents larger, now a sixth tone rather than an eight tone.
 
{| class="wikitable"
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.
|+(2.3.5.11.13) Ptolemismic Pinetone Chromatic
 
!Mode number
!Mode as simplest JI pre-image
!Step pattern
!Meantone[12]
!UDP
|-
| -6
|~ 40/39 10/9 44/39 11/9 4/3 110/81 22/15 20/13 5/3 22/13 11/6 2/1
|sLsLLsLmLsLL
|sLsLLsLsLsLL
|<nowiki>1|10</nowiki>
|-
| -5
|~ 40/39 10/9 15/13 5/4 33/26 11/8 3/2 20/13 5/3 22/13 11/6 2/1
|sLmLsLLsLsLL
|sLsLsLLsLsLL
|<nowiki>0|11</nowiki>
|-
| -4
|~ 40/39 10/9 6/5 11/9 4/3 110/81 22/15 8/5 44/27 16/9 11/6 2/1
|sLLsLsLLsLmL
|sLLsLsLLsLsL
|<nowiki>4|7</nowiki>
|-
| -3
|~ 40/39 10/9 6/5 11/9 4/3 11/8 3/2 20/13 5/3 9/5 11/6 2/1
|sLLsLmLsLLsL
|sLLsLsLsLLsL
|<nowiki>3|8</nowiki>
|-
| -2
|~ 25/24 9/8 15/13 5/4 15/11 11/8 3/2 20/13 5/3 9/5 11/6 2/1
|mLsLLsLsLLsL
|sLsLLsLsLLsL
|<nowiki>2|9</nowiki>
|-
| -1
|~ 12/11 10/9 6/5 11/9 4/3 13/9 22/15 8/5 5/3 9/5 11/6 2/1
|LsLsLLsLmLsL
|LsLsLLsLsLsL
|<nowiki>6|5</nowiki>
|-
|1
|~ 12/11 10/9 6/5 5/4 15/11 11/8 3/2 13/8 5/3 9/5 11/6 2/1
|LsLmLsLLsLsL
|LsLsLsLLsLsL
|<nowiki>5|6</nowiki>
|-
|2
|~ 12/11 10/9 6/5 13/10 4/3 13/9 22/15 8/5 26/15 16/9 48/25 2/1
|LsLLsLsLLsLm
|LsLLsLsLLsLs
|<nowiki>9|2</nowiki>
|-
|3
|~ 12/11 10/9 6/5 13/10 4/3 13/9 3/2 13/8 5/3 9/5 39/20 2/1
|LsLLsLmLsLLs
|LsLLsLsLsLLs
|<nowiki>8|3</nowiki>
|-
|4
|~ 12/11 9/8 27/22 5/4 15/11 81/55 3/2 13/8 5/3 9/5 39/20 2/1
|LmLsLLsLsLLs
|LsLsLLsLsLLs
|<nowiki>7|4</nowiki>
|-
|5
|~ 12/11 13/11 6/5 13/10 4/3 13/9 52/33 8/5 26/15 9/5 39/20 2/1
|LLsLsLLsLmLs
|LLsLsLLsLsLs
|<nowiki>11|0</nowiki>
|-
|6
|~ 12/11 13/11 6/5 13/10 15/11 81/55 3/2 13/8 39/22 9/5 39/20 2/1
|LLsLmLsLLsLs
|LLsLsLsLLsLs
|<nowiki>10|1</nowiki>
|}
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.
The ptolemismic Pinetone chromatic scale is distinctly xenharmonic, and yet is related to the familiar chromatic scale.


Line 1,289: Line 1,507:
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.  
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==
== Additional heptatonic subsets of the Pinetone chromatic ==
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 ''[[Chirality|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.
As well as our natural keys, diatonic scales are available in the Pinetone chromatic in 5 other keys (not 11, since we do not temper out the [[Pythagorean comma]]), totally 5 scales that differ in their interval content.  
 
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)
{| class="wikitable"
{| class="wikitable"
|+Table 4.1. Porcupine[8] modes (G♯-G gamut)
|+Table 4.1.
!Mode number
! colspan="8" |Scale (mode -3 subset)
! colspan="8" |Scale (mode 3 subset)
!JI ratios approximated
!Step pattern
!Step pattern
!UDP
!Dark minor mode
!Mode name
|-
!Mode as simplest JI pre-image
|D
!3-step stacked triad on root (with G♯)
|E
!(with A♭ = H)
|F
!Triad name
|G
!JI triad approximated
|A
|B
|C
|D
|E♭
|F
|G♭
|A♭
|B♭
|C
|D♭
|E♭
|10/9 6/5 4/3 3/2 5/3 9/5 2/1
|msmLmsm
|Dorian
|-
|D
|E
|F♯
|G
|A
|B
|C
|D
|E♭
|F
|G
|A♭
|B♭
|C
|D♭
|E♭
|10/9 11/9 4/3 3/2 5/3 9/5 2/1
|mmsLmsm
|Mixolydian
|-
|D
|E
|F♯
|G
|A
|B
|C♯
|D
|E♭
|F
|G
|A♭
|B♭
|C
|D
|E♭
|10/9 11/9 4/3 3/2 5/3 11/6 2/1
|mmsLmms
|Ionian
|-
|C♯
|D
|E
|F♯
|G♯
|A
|B
|C♯
|D
|E♭
|F
|G
|A
|B♭
|C
|D
|12/11 6/5 4/3 3/2 18/11 9/5 2/1
|smmLsmm
|Locrian
|-
|C♯
|D♯
|E
|F♯
|G♯
|A
|B
|C♯
|D
|E
|F
|G
|A
|B♭
|C
|D
|10/9 6/5 4/3 3/2 18/11 9/5 2/1
|msmLsmm
|Aeolian
|-
|C♯
|D♯
|E
|F♯
|G♯
|A♯
|B
|C♯
|D
|E
|F
|G
|A
|B
|C
|D
|10/9 6/5 4/3 3/2 5/3 9/5 2/1
|msmLmsm
|Dorian
|}
The scale D E F G A B C has step pattern MsMLMsM, which tempers to sssLsss under Porcupine (M=s) and LsLLLsL under Meantone (M=L).
 
Raising an F to an F♯ replaces ~6/5 with ~11/9, i.e., it raises the F by ~55/54, which is tempered out in [[Porcupine]], so the scale D E F♯ G A B C tempers to sssLsss under Porcupine as before, but to LLsLLsL under Meantone (Mixolydian mode rather than Dorian as before). The modes of this scale are detailed in Table 4.3. Similarly, lowering B to B♭ lowers by ~55/54, leading to the Duradian ♯5 shown in Table 4.4, which I consider to be a really beautiful minor mode.
 
To calculate the mode numbers for Tables 4.3-4.5, the mode numbers of their temperings to Porcupine and to [[Meantone]] were added, ordered, and renumbered. When two modes are tied, small changes in tuning will affect their order. Given that Ptolemismic Porcupine is more accurate than Ptolemismic Meantone, the Porcupine mode number is weighted more heavily to determine the mode order in the event of any ties. Modes are named via the [[Tetracot]][7] [[MODMOS]] they temper to when [[243/242]] is tempered out (i.e., the difference between [[11/9]] and [[27/22]]), as in [[27edo]], [[34edo]], and [[41edo]]. Table 4.2 introduces the modes of Tetracot[7]. The Pinetone diatonic, therefore, is also a detempering of a Tetracot MODMOS, with generator chain equivalent to that of the double harmonic major scale (a MODMOS of the Meantone diatonic scale). Tetracot[7] mode names used the Archeotonic mode names [[6L 1s#Proposed names|here]] as a basis, with the substitution of Azurian, Duradian, and Phyradian as Tetracot specific mode names from [https://www.youtube.com/watch?v=xYZwye9PWSo here].
{| class="wikitable"
|+Table 4.2. Modes of Tetracot[7]
!Mode number
!Mode name
!Step pattern
!Mode as simplest JI pre-image
|-
|3
|Ryonian
|LLLLLLs
|~ 10/9 11/9 15/11 3/2 5/3 11/6 2/1
|-
|2
|Karakalian
|LLLLLsL
|~ 10/9 11/9 15/11 3/2 5/3 9/5 2/1
|-
|1
|Azurian
|LLLLsLL
|~ 10/9 11/9 15/11 3/2 18/11 9/5 2/1
|-
|0
|Hornathian
|LLLsLLL
|~ 10/9 11/9 15/11 22/15 18/11 9/5 2/1
|-
| -1
|Oukranian
|LLsLLLL
|~ 10/9 11/9 4/3 22/15 18/11 9/5 2/1
|-
| -2
|Duradian
|LsLLLLL
|~ 10/9 6/5 4/3 22/15 18/11 9/5 2/1
|-
| -3
|Phyradian
|sLLLLLL
|~ 12/11 6/5 4/3 22/15 18/11 9/5 2/1
|}
 
{| class="wikitable"
|+Table 4.3. Modes of the Ptolemismic Pinetone Azurian bright minor
!Mode number
!Mode name
! colspan="2" |Root note
!Step pattern
!Mode as simplest JI pre-image
|-
|3
|Ryonian ♯2♯3*
|G
|A♭
|Lmsmmms
|~ 9/8 5/4 15/11 3/2 5/3 11/6 2/1
|-
|2
|Ryonian ♭5
|C
|D♭
|mmmsLms
|~ 10/9 11/9 110/81 22/15 5/3 11/6 2/1
|-
|1
|Karakalian ♭4
|D
|E♭
|mmsLmsm
|~ 10/9 11/9 4/3 3/2 5/3 9/5 2/1
|-
|0
|Azurian ♭3*
|E
|F
|msLmsmm
|~ 10/9 6/5 15/11 3/2 18/11 9/5 2/1
|-
| -1
|Horthanian ♭2
|F♯
|G
|sLmsmmm
|~ 12/11 27/22 15/11 81/55 18/11 9/5 2/1
|-
| -2
|Phyradian ♭6
|B
|C
|smmmsLm
|~ 12/11 6/5 4/3 22/15 8/5 9/5 2/1
|-
| -3
|Duradian♭7
|A
|B♭
|msmmmsL
|~ 10/9 6/5 4/3 22/15 44/27 16/9 2/1
|}
 
{| class="wikitable"
|+Table 4.4. Modes of the Ptolemismic Pinetone Duradian dark minor
!Mode number
!Mode name
! colspan="2" |Root note
!Step pattern
!Mode as simplest JI pre-image
|-
|3
|Ryonian ♯3*
|F
|E
|mLsmmms
|~ 10/9 5/4 15/11 3/2 5/3 11/6 2/1
|-
|2
|Karakalian ♯2
|G
|F♯
|Lsmmmsm
|~ 9/8 27/22 15/11 3/2 5/3 9/5 2/1
|-
|1
|Horthanian ♯7
|B♭
|A
|mmmsmLs
|~ 10/9 11/9 110/81 22/15 44/27 11/6 2/1
|-
| 0
|Oukranian ♯6
|C
|B
|mmsmLsm
|~ 10/9 11/9 4/3 22/15 5/3 9/5 2/1
|-
| -1
|Duradian ♯5*
|D
|C♯
|msmLsmm
|~ 10/9 6/5 4/3 3/2 18/11 9/5 2/1
|-
| -2
|Phyradian ♯4
|E
|D♯
|smLsmmm
|~ 12/11 6/5 15/11 81/55 11/8 9/5 2/1
|-
| -3
|Phyradian ♭6♭7
|A
|G♯
|smmmsmL
|~ 12/11 6/5 4/3 22/15 8/5 16/9 2/1
|}
Note that the Pinetone melodic major includes a neutral triad 18:22:27 on the root of the Mixolydian symmetric minor, and the Pinetone melodic minor includes it's inverse on the root of the Dorian bright major, a flavour of triad not available in the Pinetone diatonic.
{| class="wikitable"
|+Table 4.5. Modes of the Ptolemismic Pinetone Karakalian bright minor
!Mode number
!Mode name
! colspan="2" |Root note
!Step pattern
!Mode as simplest JI pre-image
|-
|3
|Ryonian ♯2♯3♯4*
|G
|A♭
|Lmmsmms
|~ 9/8 5/4 11/8 3/2 5/3 11/6 2/1
|-
|2
|Ryonian ♭4
|D
|E♭
|mmsLmms
|~ 10/9 11/9 4/3 3/2 5/3 11/6 2/1
|-
| 1
|Karakalian ♭3*
|E
|F
|msLmmsm
|~ 10/9 6/5 15/11 3/2 5/3 9/5 2/1
|-
| 0
|Azurian ♭2
|F♯
|G
|sLmmsmm
|~ 12/11 27/22 15/11 3/2 18/11 9/5 2/1
|-
| -1
|Oukranian ♭7
|A
|B♭
|mmsmmsL
|~ 10/9 11/9 4/3 22/15 44/27 16/9 2/1
|-
| -2
|Phyradian ♭5
|C♯
|D
|smmsLmm
|~ 12/11 6/5 4/3 16/11 18/11 10/9 2/1
|-
| -3
|Duradian ♭6
|B
|C
|msmmsLm
|~ 10/9 6/5 4/3 22/15 8/5 9/5 2/1
|}
{| class="wikitable"
|+Table 4.6. Modes of the Ptolemismic Pinetone Phyradian dark minor
!Mode number
!Mode names
! colspan="2" |Root note
!Step pattern
!Mode as simplest JI pre-image
|-
|3
|Karakalian ♯3*
|F
|E
|mLsmmsm
|~ 10/9 5/4 15/11 3/2 5/3 9/5 2/1
|-
| 2
|Ryonian ♯4
|E♭
|D
|mmLsmms
|~ 10/9 11/9 11/8 3/2 5/3 9/5 2/1
|-
|1
|Azurian ♯2
|G
|F♯
|Lsmmsmm
|~ 9/8 27/22 15/11 3/2 18/11 9/5 2/1
|-
|0
|Oukranian ♯7
|B♭
|A
|mmsmmLs
|~ 10/9 11/9 4/3 22/15 44/27 11/6 2/1
|-
| -1
|Duradian ♯6
|C
|B
|msmmLsm
|~ 10/9 6/5 4/3 22/15 5/3 9/5 2/1
|-
| -2
|Phyradian ♯5*
|D
|C♯
|smmLsmm
|~ 12/11 6/5 4/3 3/2 18/11 9/5 2/1
|-
| -3
|Phyradian ♭5♭6♭7
|A
|G♯
|smmsmmL
|~ 12/11 6/5 4/3 16/11 8/5 16/9 2/1
|}
Of all Pinetone heptatonic scales, Ryonian ♯2♯3♯4 is the brightest, and Phyradian ♭5♭6♭7 is the darkest. Note also that Ryonian ♯2♯3♯4 begins with the harmonic series segment 8:9:10:11:12. In fact, remarkably, the entire mode can be represented as the harmonic series segment 24:27:30:33:36:40:44:48. Note that the Phyradian dark minor and the Karakalian bright minors are tetrachordal, along with the Pinetone diatonic.
 
None of these other diatonic scales are useful as a basis for tonal harmony, since for all of these scales the only two major or minor chords are part of the same minor 7 chord. Instead they may be employed for melodic reasons, as alternate modes to use with the minor 7 chord, or for the harmonic series segment 8:9:10:11:12, etc.
 
== 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 ''[[Chirality|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).
{| class="wikitable"
|+Table 5.1. Porcupine[8] modes (G♯-G gamut)
!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
|<nowiki>7|0</nowiki>
|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
|<nowiki>6|1</nowiki>
|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
|<nowiki>5|2</nowiki>
|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
|<nowiki>4|3</nowiki>
|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
|<nowiki>3|4</nowiki>
|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
|<nowiki>2|5</nowiki>
|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
|-
|-
|4
| -3
|LLLLLLLs
|LsLLLLLL
|<nowiki>7|0</nowiki>
|<nowiki>1|6</nowiki>
|Bright quartal
|Middle minor
|~ 10/9 6/5 4/3 36/25 8/5 16/9 48/25 2/1
|~ 10/9 9/8 5/4 25/18 3/2 5/3 9/5 2/1
|G♯-C-F
|F-A-D
|A-D-G
| G-B-E
|[8] augmented
|[8] minor
|9:12:16
|12:15:20
|-
|-
|3
| -4
|LLLLLLsL
|sLLLLLLL
|<nowiki>6|1</nowiki>
|<nowiki>0|7</nowiki>
|Dark quartal
|Dark minor
|~ 10/9 6/5 4/3 36/25 8/5 16/9 9/5 2/1
|~ 25/24 9/8 5/4 25/18 3/2 5/3 9/5 2/1
|A-D-G
|G-B-E
| B-E-A♭ = B-E-H
| A♭-C-F = H-C-F
|[8] augmented
|[8] minor
|9:12:16
|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).
{| class="wikitable"
|+Table 5.2. Father[8] oneirotonic modes
!Mode number
!Step pattern
!UDP
!Mode name
!3-step stacked triad on root
!JI triads approximated
|-
|-
|2
|4
|LLLLLsLL
|LLsLLsLs
|<nowiki>5|2</nowiki>
|<nowiki>7|0</nowiki>
|Bright major
|Dylathian (də-LA(H)TH-iən)
|~ 10/9 6/5 4/3 36/25 8/5 5/3 9/5 2/1
|[8] perfect
|B-E-G♯
|3:4:5, 9:12:16
|C-F-A
|[8] major
|3:4:5
|-
|-
|1
| 3
|LLLLsLLL
|LLsLsLLs
|<nowiki>6|1</nowiki>
|Illarnekian (ill-ar-NEK-iən)
|[8] perfect
|3:4:5, 9:12:16
|-
| 2
|LsLLsLLs
|<nowiki>5|2</nowiki>
|Celephaïsian (kel-ə-FAY-zhən)
|[8] perfect
|3:4:5, 9:12:16
|-
| 1
|LsLLsLsL
|<nowiki>4|3</nowiki>
|<nowiki>4|3</nowiki>
|Middle major
|Ultharian (ul-THA(I)R-iən)
|~ 10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1
|[8] perfect
|C-F-A
|3:4:5, 9:12:16
| D-G-B
|[8] major
|3:4:5
|-
|-
| -1
| -1
|LLLsLLLL
|LsLsLLsL
|<nowiki>3|4</nowiki>
|<nowiki>3|4</nowiki>
|Dark major
|Mnarian (mə-NA(I)R-iən)
|~ 10/9 6/5 4/3 25/18 3/2 5/3 9/5 2/1
|[8] perfect
|D-G-B
|3:4:5, 9:12:16
| E-A♭-C = E-H-C
|-
|[8] major
| -2
|3:4:5
|sLLsLLsL
|-
| -2
|LLsLLLLL
|<nowiki>2|5</nowiki>
|<nowiki>2|5</nowiki>
|Bright minor
|Kadathian (kə-DA(H)TH-iən)
|~ 10/9 6/5 5/4 25/18 3/2 5/3 9/5 2/1
|[8] perfect
|E-G♯-C
|3:4:5, 9:12:16
|F-A-D
|[8] minor
|12:15:20
|-
|-
| -3
| -3
|LsLLLLLL
|sLLsLsLL
|<nowiki>1|6</nowiki>
|<nowiki>1|6</nowiki>
|Middle minor
|Hlanithian (lə-NITH-iən)
|~ 10/9 9/8 5/4 25/18 3/2 5/3 9/5 2/1
|[8] major diminished
|F-A-D
| 160:200:243
| G-B-E
|[8] minor
|12:15:20
|-
|-
| -4
| -4
|sLLLLLLL
|sLsLLsLL
|<nowiki>0|7</nowiki>
|<nowiki>0|7</nowiki>
|Dark minor
|Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn"
|~ 25/24 9/8 5/4 25/18 3/2 5/3 9/5 2/1
|[8] minor diminished
|G-B-E
| 200:243:324
| A♭-C-F = H-C-F
|[8] minor
|12:15:20
|}
|}
 
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.
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),
{| class="wikitable"
{| class="wikitable"
|+Table 4.2. Father[8] oneirotonic modes
|+Table 5.3. Modes of the just Pinetone major-harmonic octatonic
!Mode number
!Mode in JI (height order)
!Step pattern
!Step pattern
!UDP
!Porcupine[8]
!Mode name
step pattern and UDP
!3-step stacked triad on root
!Porcupine[8]
!JI triads approximated
mode
!Father[8]
step pattern and UDP
!Oneirotonic
mode
!Pinetone octatonic
mode
!Comments
|-
|-
|4
|10/9 6/5 4/3 40/27 8/5 16/9 50/27 2/1
|LLsLLsLs
|LMLLMLsM
|<nowiki>7|0</nowiki>
|<nowiki>LLLLLLsL 6|1</nowiki>
|Dylathian (də-LA(H)TH-iən)
|Dark quartal
|[8] perfect
|<nowiki>LsLLsLLs 5|2</nowiki>
|3:4:5, 9:12:16
|Celephaïsian
|Celephaïsian dark quartal*
|
|-
|-
| 3
|27/25 6/5 162/125 36/25 8/5 216/125 48/25 2/1
|LLsLsLLs
|MLMLLMLs
|<nowiki>6|1</nowiki>
|<nowiki>LLLLLLLs 7|0</nowiki>
|Illarnekian (ill-ar-NEK-iən)
|Bright quartal
|[8] perfect
|<nowiki>sLsLLsLL 0|7</nowiki>
|3:4:5, 9:12:16
|Sarnathian
|Sarnathian bright quartal
|
|-
|-
| 2
|10/9 100/81 4/3 40/27 125/81 5/3 50/27 2/1
|LsLLsLLs
|LLMLsMLM
|<nowiki>5|2</nowiki>
|<nowiki>LLLLsLLL 4|3</nowiki>
|Celephaïsian (kel-ə-FAY-zhən)
|Middle major
|[8] perfect
|<nowiki>LLsLLsLs 7|0</nowiki>
|3:4:5, 9:12:16
|Dylathian
|Dylathian middle major*
|
|-
|-
| 1
|27/25 6/5 4/3 36/25 8/5 5/3 9/5 2/1
|LsLLsLsL
|MLLMLsML
|<nowiki>4|3</nowiki>
|<nowiki>LLLLLsLL 5|2</nowiki>
|Ultharian (ul-THA(I)R-iən)
|Bright major
|[8] perfect
|<nowiki>sLLsLLsL 2|5</nowiki>
|3:4:5, 9:12:16
|Kadathian
|Kadathian bright major*
|
|-
|-
| -1
|10/9 6/5 4/3 25/18 3/2 5/3 9/5 2/1
|LsLsLLsL
|LMLsMLML
|<nowiki>3|4</nowiki>
|<nowiki>LLLsLLLL 3|4</nowiki>
|Mnarian (mə-NA(I)R-iən)
|Dark major
|[8] perfect
|<nowiki>LsLLsLsL 4|3</nowiki>
|3:4:5, 9:12:16
|Ultharian
|Ultharian dark major*<sup>†</sup>
|Major with 10:12:15 on root
|-
|-
| -2
| 10/9 125/108 5/4 25/18 3/2 5/3 50/27 2/1
|sLLsLLsL
|LsMLMLLM
|<nowiki>2|5</nowiki>
|<nowiki>LsLLLLLL 1|6</nowiki>
|Kadathian (kə-DA(H)TH-iən)
|Middle minor
|[8] perfect
|<nowiki>LLsLsLLs 6|1</nowiki>
|3:4:5, 9:12:16
|Illarnekian
|Illarnekian middle minor*<sup>†</sup>
|Minor with 4:5:6 on root
|-
|-
| -3
| 27/25 6/5 5/4 27/20 3/2 81/50 9/5 2/1
|sLLsLsLL
|MLsMLMLL
|<nowiki>1|6</nowiki>
|<nowiki>LLsLLLLL 2|5</nowiki>
|Hlanithian (lə-NITH-iən)
|Bright minor
|[8] major diminished
|<nowiki>sLLsLsLL 1|6</nowiki>
| 160:200:243
|Hlanithian
|Hlanithian bright minor<sup>†</sup><sup>†</sup>
|4:5:6 and 10:12:15 on root
|-
|-
| -4
| 25/24 9/8 5/4 27/20 3/2 5/3 9/5 2/1
|sLsLLsLL
|sMLMLLML
|<nowiki>0|7</nowiki>
|<nowiki>sLLLLLLL 0|7</nowiki>
|Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn"
|Dark minor
|[8] minor diminished
|<nowiki>LsLsLLsL 3|4</nowiki>
| 200:243:324
|Mnarian
|Mnarian dark minor*<sup>†</sup>
|Minor with 4:5:6 on root
|}
|}
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.
For Tables 5.3.-5.6. modes with major and minor [8] triads (3-step) triads on the root are marked with '*'; modes with major and minor tertian (2-step) triads on the root are marked with '<sup>†</sup>' ('<sup>††</sup>' for both major and minor).
{| class="wikitable"
{| class="wikitable"
|+Table 4.3. Modes of the just Pinetone major-harmonic octatonic
|+Table 5.4. Modes of the just Pinetone minor-harmonic octatonic
!Mode in JI (height order)
!Mode in JI (height order)
!Step pattern
!Step pattern
Line 1,477: Line 2,196:
!Comments
!Comments
|-
|-
|10/9 6/5 4/3 40/27 8/5 16/9 50/27 2/1
|10/9 6/5 4/3 40/27 8/5 16/9 48/25 2/1
|LMLLMLsM
|LMLLMLMs
|<nowiki>LLLLLLsL 6|1</nowiki>
|Dark quartal
|<nowiki>LsLLsLLs 5|2</nowiki>
|Celephaïsian
|Celephaïsian dark quartal*
|
|-
|27/25 6/5 162/125 36/25 8/5 216/125 48/25 2/1
|MLMLLMLs
|<nowiki>LLLLLLLs 7|0</nowiki>
|<nowiki>LLLLLLLs 7|0</nowiki>
|Bright quartal
|Bright quartal
|<nowiki>sLsLLsLL 0|7</nowiki>
|<nowiki>LsLLsLsL 4|3</nowiki>
|Sarnathian
|Ultharian
|Sarnathian bright quartal
|Ultharian bright quartal*
|
|
|-
|-
|10/9 100/81 4/3 40/27 125/81 5/3 50/27 2/1
|10/9 100/81 4/3 40/27 8/5 5/3 50/27 2/1
|LLMLsMLM
|LLMLMsLM
|<nowiki>LLLLsLLL 4|3</nowiki>
|Middle major
|<nowiki>LLsLLsLs 7|0</nowiki>
|Dylathian
|Dylathian middle major*
|
|-
|27/25 6/5 4/3 36/25 8/5 5/3 9/5 2/1
|MLLMLsML
|<nowiki>LLLLLsLL 5|2</nowiki>
|<nowiki>LLLLLsLL 5|2</nowiki>
|Bright major
|Bright major
|<nowiki>sLLsLLsL 2|5</nowiki>
|<nowiki>LLsLsLLs 6|1</nowiki>
|Kadathian
|Illarnekian
|Kadathian bright major*
|Illarnekian bright major*
|
|
|-
|-
|10/9 6/5 4/3 25/18 3/2 5/3 9/5 2/1
|27/25 6/5 4/3 36/25 8/5 216/125 9/5 2/1
|LMLsMLML
|MLLMLMsL
|<nowiki>LLLsLLLL 3|4</nowiki>
|<nowiki>LLLLLLsL 6|1</nowiki>
|Dark major
|Dark quartal
|<nowiki>LsLLsLsL 4|3</nowiki>
|<nowiki>sLLsLsLL 1|6</nowiki>
|Ultharian
|Hlanithian
|Ultharian dark major*<sup>†</sup>
|Hlanithian dark quartal
|Major with 10:12:15 on root
|
|-
|-
| 10/9 125/108 5/4 25/18 3/2 5/3 50/27 2/1
|10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1
|LsMLMLLM
|LMLMsLML
|<nowiki>LsLLLLLL 1|6</nowiki>
|<nowiki>LLLLsLLL 4|3</nowiki>
|Middle minor
|Middle major
|<nowiki>LLsLsLLs 6|1</nowiki>
|<nowiki>LsLsLLsL 3|4</nowiki>
|Illarnekian
|Mnarian
|Illarnekian middle minor*<sup>†</sup>
|Mnarian middle major*<sup>†</sup>
|Minor with 4:5:6 on root
|Major root 10:12:15
|-
|-
| 27/25 6/5 5/4 27/20 3/2 81/50 9/5 2/1
|10/9 6/5 5/4 25/18 3/2 5/3 50/27 2/1
|MLsMLMLL
|LMsLMLLM
|<nowiki>LLsLLLLL 2|5</nowiki>
|<nowiki>LLsLLLLL 2|5</nowiki>
|Bright minor
|Bright minor
|<nowiki>sLLsLsLL 1|6</nowiki>
|<nowiki>LsLLsLLs 5|2</nowiki>
|Hlanithian
|Celephaïsian
|Hlanithian bright minor<sup>†</sup><sup>†</sup>
|Celephaïsian bright minor*<sup>†</sup><sup>†</sup>
|4:5:6 and 10:12:15 on root
|Minor root 4:5:6, 10:12:15
|-
|-
| 25/24 9/8 5/4 27/20 3/2 5/3 9/5 2/1
|27/25 6/5 162/125 27/20 3/2 81/50 9/5 2/1
|sMLMLLML
|MLMsLMLL
|<nowiki>sLLLLLLL 0|7</nowiki>
|<nowiki>LLLsLLLL 3|4</nowiki>
|Middle minor
|<nowiki>sLsLLsLL 0|7</nowiki>
|Sarnathian
|Sarnathian dark major<sup>†</sup>
|10:12:15 on root
|-
| 25/24 125/108 5/4 25/18 125/81 5/3 50/27 2/1
|sLMLLMLM
|<nowiki>sLLLLLLL 0|7</nowiki>
|Bright minor
|<nowiki>LLsLLsLs 7|0</nowiki>
|Dylathian
|Dylathian dark minor*
|
|-
|27/25 9/8 5/4 27/20 3/2 5/3 9/5 2/1
|MsLMLLML
|<nowiki>LsLLLLLL 1|6</nowiki>
|Dark minor
|Dark minor
|<nowiki>LsLsLLsL 3|4</nowiki>
|<nowiki>sLLsLLsL 2|5</nowiki>
|Mnarian
|Kadathian
|Mnarian dark minor*<sup>†</sup>
|Kadathian middle minor*<sup>†</sup>
|Minor with 4:5:6 on root
|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 [http://x31eq.com/cgi-bin/rt.cgi?ets=4p%263p%261ce&limit=2.3.5.11 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 [http://x31eq.com/cgi-bin/rt.cgi?ets=4f%263f%261ce&limit=2.3.5.11.13 TE tuning].
{| class="wikitable"
{| class="wikitable"
|+Table 4.4. Modes of the just Pinetone minor-harmonic octatonic
|+Table 5.5. Modes of the Ptolemismic Pinetone major-harmonic octatonic
!Mode in JI (height order)
!Mode (height order)
!Step pattern
!Step pattern
!Porcupine[8]
!Mode as simplest JI pre-image
step pattern and UDP
!Mode in cents
!Porcupine[8]
!Comments
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
|Celephaïsian dark quartal*
|LMLLMLMs
|LMLLMLsM
|<nowiki>LLLLLLLs 7|0</nowiki>
|~ 10/9 6/5 4/3 22/15 8/5 16/9 11/6 2/1
|Bright quartal
|175.892 318.667 494.559 670.451 813.226 989.118 1055.884 1198.660
|<nowiki>LsLLsLsL 4|3</nowiki>
|Ultharian
|Ultharian bright quartal*
|
|
|-
|-
|10/9 100/81 4/3 40/27 8/5 5/3 50/27 2/1
|Sarnathian bright quartal
|LLMLMsLM
|MLMLLMLs
|<nowiki>LLLLLsLL 5|2</nowiki>
|~ 12/11 6/5 13/10 13/9 8/5 26/15 48/25 2/1
|Bright major
|142.775 318.667 461.443 637.334 813.226 956.002 1131.893 1198.660
|<nowiki>LLsLsLLs 6|1</nowiki>
|Illarnekian
|Illarnekian bright major*
|
|
|-
|-
|27/25 6/5 4/3 36/25 8/5 216/125 9/5 2/1
|Dylathian middle major*
|MLLMLMsL
|LLMLsMLM
|<nowiki>LLLLLLsL 6|1</nowiki>
|~ 10/9 11/9 4/3 22/15 20/13 5/3 11/6 2/1
|Dark quartal
|175.892 351.784 494.559 670.451 737.217 879.992 1055.884 1198.660
|<nowiki>sLLsLsLL 1|6</nowiki>
|Hlanithian
|Hlanithian dark quartal
|
|
|-
|-
|10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1
|Kadathian bright major*
|LMLMsLML
|MLLMLsML
|<nowiki>LLLLsLLL 4|3</nowiki>
|~ 12/11 6/5 4/3 13/9 8/5 5/3 9/5 2/1
|Middle major
|142.775 318.667 494.559 637.334 813.226 879.992 1022.768 1198.660
|<nowiki>LsLsLLsL 3|4</nowiki>
|
|Mnarian
|Mnarian middle major*<sup>†</sup>
|Major root 10:12:15
|-
|-
|10/9 6/5 5/4 25/18 3/2 5/3 50/27 2/1
|Ultharian dark major*<sup>†</sup>
|LMsLMLLM
|LMLsMLML
|<nowiki>LLsLLLLL 2|5</nowiki>
|~ 10/9 6/5 4/3 11/8 3/2 5/3 9/5 2/1
|Bright minor
|175.892 318.667 494.559 561.325 704.101 879.992 1022.768 1198.660
|<nowiki>LsLLsLLs 5|2</nowiki>
|Major with 10:12:15 on root
|Celephaïsian
|Celephaïsian bright minor*<sup>†</sup><sup>†</sup>
|Minor root 4:5:6, 10:12:15
|-
|-
|27/25 6/5 162/125 27/20 3/2 81/50 9/5 2/1
|Illarnekian middle minor*<sup>†</sup>
|MLMsLMLL
|LsMLMLLM
|<nowiki>LLLsLLLL 3|4</nowiki>
|~ 10/9 15/13 5/4 11/8 3/2 5/3 11/6 2/1
|Middle minor
|175.892 242.658 385.433 561.325 704.101 879.992 1055.884 1198.660
|<nowiki>sLsLLsLL 0|7</nowiki>
|Minor with 4:5:6 on root
|Sarnathian
|Sarnathian dark major<sup>†</sup>
|10:12:15 on root
|-
|-
| 25/24 125/108 5/4 25/18 125/81 5/3 50/27 2/1
|Hlanithian bright minor(*)<sup>†</sup><sup>†</sup>
|sLMLLMLM
|MLsMLMLL
|<nowiki>sLLLLLLL 0|7</nowiki>
|~ 12/11 6/5 5/4 15/11 3/2 13/8 9/5 2/1
|Bright minor
|142.775 318.667 385.433 528.209 704.101 846.876 1022.768 1198.660
|<nowiki>LLsLLsLs 7|0</nowiki>
|4:5:6 and 10:12:15 on root
|Dylathian
|Dylathian dark minor*
|
|-
|-
|27/25 9/8 5/4 27/20 3/2 5/3 9/5 2/1
|Mnarian dark minor*<sup>†</sup>
|MsLMLLML
|sMLMLLML
|<nowiki>LsLLLLLL 1|6</nowiki>
|~ 25/24 9/8 5/4 15/11 3/2 5/3 9/5 2/1
|Dark minor
|66.766 209.542 385.433 528.209 704.101 879.992 1022.768 1198.660
|<nowiki>sLLsLLsL 2|5</nowiki>
|Minor with 4:5:6 on root
|Kadathian
|Kadathian middle minor*<sup>†</sup>
|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.
For Tables 5.3.-5.6. modes with major and minor [8] triads (3-step) triads on the root are marked with '*'; modes with major and minor tertian (2-step) triads on the root are marked with '<sup>†</sup>' ('<sup>††</sup>' for both major and minor).
 
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 [http://x31eq.com/cgi-bin/rt.cgi?ets=4p%263p%261ce&limit=2.3.5.11 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 [http://x31eq.com/cgi-bin/rt.cgi?ets=4f%263f%261ce&limit=2.3.5.11.13 TE tuning].
{| class="wikitable"
{| class="wikitable"
|+Table 4.5. Modes of the Ptolemismic Pinetone major-harmonic octatonic
|+Table 5.6. Modes of the Ptolemismic Pinetone minor-harmonic octatonic
!Mode (height order)
!Mode (height order)
!Step pattern
!Step pattern
Line 1,650: Line 2,338:
!Comments
!Comments
|-
|-
|Celephaïsian dark quartal*
|Ultharian bright quartal*
|LMLLMLsM
|LMLLMLMs
|~ 10/9 6/5 4/3 22/15 8/5 16/9 11/6 2/1
|~ 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 1055.884 1198.660
|175.892 318.667 494.559 670.451 813.226 989.118 1131.983 1198.660
|
|
|-
|-
|Sarnathian bright quartal
|Illarnekian bright major*
|MLMLLMLs
|LLMLMsLM
|~ 12/11 6/5 13/10 13/9 8/5 26/15 48/25 2/1
|~ 10/9 11/9 4/3 22/15 8/5 5/3 11/6 2/1
|142.775 318.667 461.443 637.334 813.226 956.002 1131.893 1198.660
|175.892 351.784 494.559 670.451 813.226 879.992 1055.884 1198.660
|
|
|-
|-
|Dylathian middle major*
|Hlanithian dark quartal
|LLMLsMLM
|MLLMLMsL
|~ 10/9 11/9 4/3 22/15 20/13 5/3 11/6 2/1
|~ 12/11 6/5 4/3 13/9 8/5 26/15 10/9 2/1
|175.892 351.784 494.559 670.451 737.217 879.992 1055.884 1198.660
|142.775 318.667 494.559 637.334 813.226 956.002 1022.768 1198.660
|
|
|-
|-
|Kadathian bright major*
|Mnarian middle major*<sup>†</sup>
|MLLMLsML
|LMLMsLML
|~ 12/11 6/5 4/3 13/9 8/5 5/3 9/5 2/1
|~ 10/9 6/5 4/3 13/9 3/2 5/3 9/5 2/1
|142.775 318.667 494.559 637.334 813.226 879.992 1022.768 1198.660
|175.892 318.667 494.559 637.334 704.101 879.992 1022.768 1198.660
|
|-
|Ultharian dark major*<sup>†</sup>
|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
|Major with 10:12:15 on root
|-
|-
|Illarnekian middle minor*<sup>†</sup>
|Celephaïsian bright minor*<sup>†</sup><sup>†</sup>
|LsMLMLLM
|LMsLMLLM
|~ 10/9 15/13 5/4 11/8 3/2 5/3 11/6 2/1
|~ 10/9 6/5 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
|175.892 318.667 385.433 561.325 704.101 879.992 1055.884 1198.660
|Minor with 4:5:6 on root
|Minor with 4:5:6 and 10:12:15 on root
|-
|Sarnathian dark major<sup>†</sup>
|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
|-
|-
|Hlanithian bright minor(*)<sup>†</sup><sup>†</sup>
|Dylathian dark minor*
|MLsMLMLL
|sLMLLMLM
|~ 12/11 6/5 5/4 15/11 3/2 13/8 9/5 2/1
|~ 25/24 15/13 5/4 11/8 20/13 5/3 11/6 2/1
|142.775 318.667 385.433 528.209 704.101 846.876 1022.768 1198.660
|66.766 242.658 385.433 561.325 737.217 879.992 1055.884 1198.660
|4:5:6 and 10:12:15 on root
|
|-
|-
|Mnarian dark minor*<sup>†</sup>
|Kadathian middle minor*<sup>†</sup>
|sMLMLLML
|MsLMLLML
|~ 25/24 9/8 5/4 15/11 3/2 5/3 9/5 2/1
|~ 12/11 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
|142.775 209.542 385.433 528.209 704.101 879.992 1022.768 1198.660
|Minor with 4:5:6 on root
|Minor with 4:5:6 on root
|}
|}
=== Intervals and chords===
The following table gives all intervals of the Pinetone harmonic octatonics.
{| class="wikitable"
{| class="wikitable"
|+Table 4.6. Modes of the Ptolemismic Pinetone minor-harmonic octatonic
|+Table 5.7. Intervals of the Ptolemismic Pinetone harmonic octatonics (variety = 24)
!Mode (height order)
!Interval class
!Step pattern
!sizes
!Mode as simplest JI pre-image
!Oneirotonic name
!Mode in cents
!Porcupine[8] name
!Comments
!Pinetone octatonic name
!JI ratios approximated*
!size in cents (TE)
! Occurence
|-
|-
|Ultharian bright quartal*
! rowspan="3" |1-step
|LMLLMLMs
|s
|~ 10/9 6/5 4/3 22/15 8/5 16/9 48/25 2/1
| major step
|175.892 318.667 494.559 670.451 813.226 989.118 1131.983 1198.660
|minor step
|
| minor step
|25/24, 33/32, 27/26
|66.766
|1
|-
|-
|Illarnekian bright major*
|M
|LLMLMsLM
|minor step
|~ 10/9 11/9 4/3 22/15 8/5 5/3 11/6 2/1
|major step
|175.892 351.784 494.559 670.451 813.226 879.992 1055.884 1198.660
|minor-major step
|
|27/25, 12/11, 13/12
|142.775
|3
|-
|-
|Hlanithian dark quartal
|L
|MLLMLMsL
|major step
|~ 12/11 6/5 4/3 13/9 8/5 26/15 10/9 2/1
|major step
|142.775 318.667 494.559 637.334 813.226 956.002 1022.768 1198.660
|major step
|
|10/9, 11/10
|175.892
|4
|-
|-
|Mnarian middle major*<sup>†</sup>
! rowspan="4" |2-step
|LMLMsLML
|M + s
|~ 10/9 6/5 4/3 13/9 3/2 5/3 9/5 2/1
|minor 2-step
|175.892 318.667 494.559 637.334 704.101 879.992 1022.768 1198.660
|minor 2-step
|Major with 10:12:15 on root
|small 2-step
|9/8
|209.542
|1
|-
|-
|Celephaïsian bright minor*<sup>†</sup><sup>†</sup>
|L + s
|LMsLMLLM
|major 2-step
|~ 10/9 6/5 5/4 11/8 3/2 5/3 11/6 2/1
|minor 2-step
|175.892 318.667 385.433 561.325 704.101 879.992 1055.884 1198.660
|minor 2-step
|Minor with 4:5:6 and 10:12:15 on root
|15/13 (7/6 or 8/7)
|242.658
|1
|-
|-
|Sarnathian dark major<sup>†</sup>
|L + M
|MLMsLMLL
|minor 2-step
|~ 12/11 6/5 13/10 15/11 3/2 13/8 9/5 2/1
|major 2-step
|142.775 318.667 461.443 528.209 704.101 846.876 1022.768 1198.660
|major 2-step
|10:12:15 on root
|6/5
|318.667
|5
|-
|L + L
|major 2-step
|major 2-step
|large 2-step
|11/9, 16/13
|351.784
|1
|-
|-
|Dylathian dark minor*
! rowspan="3" |3-step
|sLMLLMLM
|L + M + s
|~ 25/24 15/13 5/4 11/8 20/13 5/3 11/6 2/1
|major 3-step
|66.766 242.658 385.433 561.325 737.217 879.992 1055.884 1198.660
|minor 3-step
|
|minor 3-step
|5/4
|385.433
|3
|-
|-
|Kadathian middle minor*<sup>†</sup>
|L + 2M
|MsLMLLML
|minor 3-step
|~ 12/11 9/8 5/4 15/11 3/2 5/3 9/5 2/1
|major 3-step
|142.775 209.542 385.433 528.209 704.101 879.992 1022.768 1198.660
|minor-major 3-step
|Minor with 4:5:6 on root
|13/10 (9/7 or 21/16)
|}
|461.433
 
=== Intervals and chords===
The following table gives all intervals of the Pinetone harmonic octatonics.
{| class="wikitable"
|+Table 4.7. Intervals of the Ptolemismic Pinetone harmonic octatonics (variety = 24)
!Interval class
!sizes
!Oneirotonic name
!Porcupine[8] name
!Pinetone octatonic name
!JI ratios approximated*
!size in cents (TE)
! Occurence
|-
! rowspan="3" |1-step
|s
| major step
|minor step
| minor step
|25/24, 33/32, 27/26
|66.766
|1
|1
|-
|-
|M
|2L + M
|minor step
|major 3-step
|major step
|major 3-step
|minor-major step
|major 3-step
|27/25, 12/11, 13/12
|4/3
|142.775
|494.559
|3
|4
|-
|-
|L
! rowspan="4" |4-step
|major step
|L + 2M + s
|major step
|minor 4-step
|major step
|minor 4-step
|10/9, 11/10
|small 4-step
|175.892
|27/20, 15/11
|4
|528.209
|2
|-
|-
! rowspan="4" |2-step
|2L + M + s
|M + s
|major 4-step
|minor 2-step
|minor 4-step
|minor 2-step
|minor 4-step
|small 2-step
|25/18, 11/8, 18/13
|9/8
|561.325
|209.542
|2
|1
|-
|-
|L + s
|2L + 2M
|major 2-step
|minor 4-step
|minor 2-step
|major 4-step
|minor 2-step
|major 4-step
|15/13 (7/6 or 8/7)
|36/25, 16/11, 13/9
|242.658
|637.334
|1
|2
|-
|-
|L + M
|3L + M
|minor 2-step
|major 4-step
|major 2-step
|major 4-step
|major 2-step
|large 4-step
|6/5
|40/27, 22/15
|318.667
|670.451
|5
|2
|-
|-
|L + L
! rowspan="3" |5-step
|major 2-step
|2L + 2M + s
|major 2-step
|minor 5-step
|large 2-step
|minor 5-step
|11/9, 16/13
|minor 5-step
|351.784
|3/2
|1
|704.101
|4
|-
|-
! rowspan="3" |3-step
|3L + M + s
|L + M + s
|major 5-step
|major 3-step
|minor 5-step
|minor 3-step
|major-minor 5-step
|minor 3-step
|20/13 (14/9 or 32/16)
|5/4
|737.217
|385.433
|1
|-
|3L + 2M
|minor 5-step
|major 5-step
|major 5-step
|8/5
|813.227
|3
|3
|-
|-
|L + 2M
! rowspan="4" |6-step
|minor 3-step
|2L + 3M + s
|major 3-step
|minor 6-step
|minor-major 3-step
|minor 6-step
|13/10 (9/7 or 21/16)
|small 6-step
|461.433
|18/11, 13/8
|846.876
|1
|1
|-
|-
|2L + M
|3L + 2M + s
|major 3-step
|major 6-step
|major 3-step
|minor 6-step
|major 3-step
|minor 6-step
|4/3
|5/3
|494.559
|879.992
|4
|5
|-
|-
! rowspan="4" |4-step
|3L + 3M
|L + 2M + s
|minor 6-step
|minor 4-step
|major 6-step
|minor 4-step
|major 6-step
|small 4-step
|26/15 (12/7 or 7/4)
|27/20, 15/11
|956.002
|528.209
|1
|2
|-
|-
|2L + M + s
|4L + 2M
|major 4-step
|major 6-step
|minor 4-step
|major 6-step
|minor 4-step
|large 6-step
|25/18, 11/8, 18/13
|16/9
|561.325
|989.118
|2
|1
|-
|-
|2L + 2M
! rowspan="3" |7-step
|minor 4-step
|3L + 3M + s
|major 4-step
|minor 7-step
|major 4-step
|minor 7-step
|36/25, 16/11, 13/9
|minor 7-step
|637.334
|9/5, 20/11
|2
|1022.768
|4
|-
|-
|3L + M
|4L + 2M + s
|major 4-step
|major 7-step
|major 4-step
|minor 7-step
|large 4-step
|major-minor 7-step
|40/27, 22/15
|50/27, 11/6, 24/13
|670.451
|1055.884
|2
|3
|-
|-
! rowspan="3" |5-step
|4L + 3M
|2L + 2M + s
|minor 7-step
|minor 5-step
|major 7-step
|minor 5-step
|major 7-step
|minor 5-step
|48/25, 64/33, 52/27
|3/2
|1131.983
|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
|1
|}
<nowiki>*</nowiki> 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:
{| class="wikitable"
|+Table 5.8. 3-step stacked triads of the Ptolemismic Pinetone major-harmonic octatonic (G♯-G gamut)
!Mode (rotational order)
!Step pattern
!3-step stacked triad on root
!Oneirotonic name
!Porcupine[8] name
!Pinetone octatonic name
!JI triad approximated*
|-
|-
|3L + 2M
|Sarnathian bright quartal
|minor 5-step
|MLMLLMLs
|major 5-step
|G♯-C-F
|major 5-step
|[8] minor diminished
|8/5
| [8] augmented
|813.227
|[8] minor augmented
|3
|30:39:52 (7:9:12, 16:21:28)
|-
|-
! rowspan="4" |6-step
|Celephaïsian dark quartal
|2L + 3M + s
|LMLLMLsM
|minor 6-step
|A-D-G
|minor 6-step
|[8] perfect
|small 6-step
|[8] augmented
|18/11, 13/8
|[8] augmented
|846.876
|9:12:16
|1
|-
|-
|3L + 2M + s
|Kadathian bright major
|major 6-step
|MLLMLsML
|minor 6-step
|B-E-G♯
|minor 6-step
|[8] perfect
|5/3
|[8] major
|879.992
|[8] major
|5
|3:4:5
|-
|-
|3L + 3M
|Dylathian middle major
|minor 6-step
|LLMLsMLM
|major 6-step
|C-F-A
|major 6-step
|[8] perfect
|26/15 (12/7 or 7/4)
|[8] major
|956.002
|[8] major
|1
|3:4:5
|-
|-
|4L + 2M
|Ultharian dark major
|major 6-step
| LMLsMLML
|major 6-step
|D-G-B
|large 6-step
|[8] perfect
|16/9
|[8] major
|989.118
|[8] major
|1
|3:4:5
|-
|-
! rowspan="3" |7-step
|Hlanithian bright minor
|3L + 3M + s
|MLsMLMLL
|minor 7-step
|E-G♯-C
|minor 7-step
|[8] major diminished
|minor 7-step
| [8] minor
|9/5, 20/11
|[8] diminished minor
|1022.768
|8:10:13
|4
|-
|-
|4L + 2M + s
|Illarnekian middle minor
|major 7-step
|LsMLMLLM
|minor 7-step
|F-A-D
|major-minor 7-step
|[8] perfect
|50/27, 11/6, 24/13
|[8] minor
|1055.884
|[8] minor
|3
|12:15:20
|-
|-
|4L + 3M
|Mnarian dark minor
|minor 7-step
| sMLMLLML
|major 7-step
|G-B-E
|major 7-step
|[8] perfect
|48/25, 64/33, 52/27
|[8] minor
|1131.983
|[8] minor
|1
|12:15:20
|}
|}
<nowiki>*</nowiki> 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).
<nowiki>*</nowiki> 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:
{| class="wikitable"
{| class="wikitable"
|+Table 4.8. 3-step stacked triads of the Ptolemismic Pinetone major-harmonic octatonic (G♯-G gamut)
|+Table 5.9. 3-step stacked triads of the Ptolemismic Pinetone minor-harmonic octatonic (G-A♭ gamut)
!Mode (rotational order)
!Mode (rotational order)
!Step pattern
!Step pattern
Line 1,980: Line 2,687:
!JI triad approximated*
!JI triad approximated*
|-
|-
|Sarnathian bright quartal
|Ultharian bright quartal
|MLMLLMLs
|LMLLMLMs
|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
|A-D-G
|[8] perfect
|[8] perfect
Line 1,996: Line 2,695:
|9:12:16
|9:12:16
|-
|-
|Kadathian bright major
|Hlanithian dark quartal
|MLLMLsML
|MLLMLMsL
|B-E-G♯
|B-E-A♭
|[8] perfect
|[8] major diminished
|[8] major
| [8] augmented
|[8] major
|[8] major augmented
|3:4:5
|15:20:26 (12:16:21)
|-
|-
|Dylathian middle major
|Illarnekian bright major
|LLMLsMLM
|LLMLMsLM
|C-F-A
|C-F-A
|[8] perfect
|[8] perfect
Line 2,012: Line 2,711:
|3:4:5
|3:4:5
|-
|-
|Ultharian dark major
|Mnarian middle major
| LMLsMLML
|LMLMsLML
|D-G-B
|D-G-B
|[8] perfect
|[8] perfect
Line 2,020: Line 2,719:
|3:4:5
|3:4:5
|-
|-
|Hlanithian bright minor
|Sarnathian dark major
|MLsMLMLL
| MLMsLMLL
|E-G♯-C
|E-A♭-C
|[8] major diminished
|[8] minor diminished
| [8] minor
| [8] major
|[8] diminished minor
|[8] diminished major
|8:10:13
|55:72:90 (16:21:26)
|-
|-
|Illarnekian middle minor
|Celephaïsian bright minor
|LsMLMLLM
|LMsLMLLM
|F-A-D
|F-A-D
|[8] perfect
|[8] perfect
Line 2,036: Line 2,735:
|12:15:20
|12:15:20
|-
|-
|Mnarian dark minor
|Dylathian dark minor
| sMLMLLML
| sLMLLMLM
|A♭-C-F
|[8] perfect
|[8] minor
|[8] minor
|12:15:20
|-
|Kadathian middle minor
|MsLMLLML
|G-B-E
|G-B-E
|[8] perfect
|[8] perfect
Line 2,044: Line 2,751:
|12:15:20
|12:15:20
|}
|}
<nowiki>*</nowiki> 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).
 
<nowiki>*</nowiki> 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).
 
{| class="wikitable"
{| class="wikitable"
|+Table 4.9. 3-step stacked triads of the Ptolemismic Pinetone minor-harmonic octatonic (G-A♭ gamut)
|+Table 5.10. Modes of the just Pinetone diminished scale
!Mode (rotational order)
!Mode in JI (height order)
!Step pattern
!Step pattern
!3-step stacked triad on root
!Porcupine[8]
!Oneirotonic name
step pattern and UDP
!Porcupine[8] name
!Porcupine[8]
!Pinetone octatonic name
mode
!JI triad approximated*
!Diminished[8]  
step pattern and UDP
!Pinetone diminished
mode
!Comments
|-
|-
|Ultharian bright quartal
|10/9 6/5 4/3 36/25 8/5 216/125 48/25 2/1
|LMLLMLMs
|LMLMLMLs
|A-D-G
|<nowiki>LLLLLLLs 7|0</nowiki>
|[8] perfect
|Bright quartal
|[8] augmented
|<nowiki>LsLsLsLs 1|0 (4)</nowiki>
|[8] augmented
|Bright quartal diminished
|9:12:16
|
|-
|-
|Hlanithian dark quartal
|10/9 6/5 4/3 36/25 8/5 5/3 50/27 2/1
|MLLMLMsL
|LMLMLsLM
|B-E-A♭
|<nowiki>LLLLLsLL 5|2</nowiki>
|[8] major diminished
|Bright major
| [8] augmented
|<nowiki>LsLsLsLs 1|0 (4)</nowiki>
|[8] major augmented
|Bright major diminished*
|15:20:26 (12:16:21)
|
|-
|-
|Illarnekian bright major
|27/25 6/5 162/125 36/25 972/625 216/125 9/5 2/1
|LLMLMsLM
|MLMLMLsL
|C-F-A
|<nowiki>LLLLLLsL 6|1</nowiki>
|[8] perfect
|Dark quartal
|[8] major
|<nowiki>sLsLsLsL 0|1 (4)</nowiki>
|[8] major
|Dark quartal diminished
|3:4:5
|
|-
|-
|Mnarian middle major
|10/9 6/5 4/3 25/18 125/81 5/3 50/27 2/1
|LMLMsLML
|LMLsLMLM
|D-G-B
|<nowiki>LLLsLLLL 3|4</nowiki>
|[8] perfect
|Dark major
|[8] major
|<nowiki>LsLsLsLs 1|0 (4)</nowiki>
|[8] major
|Dark major diminished*
|3:4:5
|
|-
|-
|Sarnathian dark major
|27/25 6/5 162/125 36/25 3/2 5/3 9/5 2/1
| MLMsLMLL
|MLMLsLML
|E-A♭-C
|<nowiki>LLLLsLLL 4|3</nowiki>
|[8] minor diminished
|Middle major
| [8] major
|<nowiki>sLsLsLsL 0|1 (4)</nowiki>
|[8] diminished major
|Middle major diminished<sup>†</sup>
|55:72:90 (16:21:26)
|10:12:15 on root
|-
|-
|Celephaïsian bright minor
| 10/9 125/108 625/486 25/18 125/81 5/3 50/27 2/1
|LMsLMLLM
|LsLMLMLM
|F-A-D
|<nowiki>LsLLLLLL 1|6</nowiki>
|[8] perfect
|Middle minor
|[8] minor
|<nowiki>LsLsLsLs 1|0 (4)</nowiki>
|[8] minor
|Middle minor diminished
|12:15:20
|
|-
|-
|Dylathian dark minor
|27/25 6/5 5/4 25/18 3/2 5/3 9/5 2/1
| sLMLLMLM
|MLsLMLML
|A♭-C-F
|<nowiki>LLsLLLLL 2|5</nowiki>
|[8] perfect
|Bright minor
|[8] minor
|<nowiki>sLsLsLsL 0|1 (4)</nowiki>
|[8] minor
|Bright minor diminished*<sup>†</sup><sup>†</sup>
|12:15:20
|Minor root 4:5:6, 10:12:15
|-
|-
|Kadathian middle minor
|25/24 125/108 5/4 25/18 3/2 5/3 9/5 2/1
|MsLMLLML
|sLMLMLML
|G-B-E
|<nowiki>sLLLLLLL 0|7</nowiki>
|[8] perfect
|Dark minor
|[8] minor
|<nowiki>sLsLsLsL 0|1 (4)</nowiki>
|[8] minor
|Dark minor diminished*<sup>†</sup>
|12:15:20
|Minor with 4:5:6 on root
|}
|}
 
For Tables 5.10. and 5.11. modes with major and minor [8] triads (3-step) triads on the root are marked with '*'; modes with major and minor tertian (2-step) triads on the root are marked with '<sup>†</sup>' ('<sup>††</sup>' for both major and minor).
<nowiki>*</nowiki> 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).
 
{| class="wikitable"
{| class="wikitable"
|+Tabl3 4.10. Modes of the just Pinetone diminished scale
|+Table 5.11. Modes of the Ptolemismic Pinetone diminished scale
!Mode in JI (height order)
!Mode (height order)
!Step pattern
!Step pattern
!Porcupine[8]
!Mode as simplest JI pre-image
step pattern and UDP
!Mode in cents
!Porcupine[8]
mode
!Diminished[8]
step pattern and UDP
!Pinetone diminished
mode
!Comments
!Comments
|-
|-
|10/9 6/5 4/3 36/25 8/5 216/125 48/25 2/1
|Bright quartal diminished
|LMLMLMLs
|LMLMLMLs
|<nowiki>LLLLLLLs 7|0</nowiki>
|~ 10/9 6/5 4/3 13/9 8/5 26/15 48/25 2/1
|Bright quartal
|175.892 318.667 494.559 637.334 813.226 956.002 1131.893 1198.660
|<nowiki>LsLsLsLs 1|0 (4)</nowiki>
|Bright quartal diminished
|
|
|-
|-
|10/9 6/5 4/3 36/25 8/5 5/3 50/27 2/1
|Bright major diminished*
|LMLMLsLM
|LMLMLsLM
|<nowiki>LLLLLsLL 5|2</nowiki>
|~ 10/9 6/5 4/3 13/9 8/5 5/3 11/6 2/1
|Bright major
|175.892 318.667 494.559 637.334 813.226 879.993 1055.884 1198.660
|<nowiki>LsLsLsLs 1|0 (4)</nowiki>
|Bright major diminished*
|
|
|-
|-
|27/25 6/5 162/125 36/25 972/625 216/125 9/5 2/1
|Dark quartal diminished
|MLMLMLsL
|MLMLMLsL
|<nowiki>LLLLLLsL 6|1</nowiki>
|~ 12/11 6/5 13/10 13/9 39/25 26/15 9/5 2/1
|Dark quartal
|142.775 318.667 461.443 637.334 780.120 956.002 1022.768 1198.660
|<nowiki>sLsLsLsL 0|1 (4)</nowiki>
|Dark quartal diminished
|
|
|-
|-
|10/9 6/5 4/3 25/18 125/81 5/3 50/27 2/1
|Dark major diminished*
|LMLsLMLM
|LMLsLMLM
|<nowiki>LLLsLLLL 3|4</nowiki>
|~ 10/9 6/5 4/3 13/9 20/13 5/3 11/6 2/1
|Dark major
|175.892 318.667 494.559 561.325 737.218 879.993 1055.884 1198.660
|<nowiki>LsLsLsLs 1|0 (4)</nowiki>
|Dark major diminished*
|
|
|-
|-
|27/25 6/5 162/125 36/25 3/2 5/3 9/5 2/1
|Middle major diminished<sup>†</sup>
|MLMLsLML
|MLMLsLML
|<nowiki>LLLLsLLL 4|3</nowiki>
|~ 12/11 6/5 13/10 13/9 3/2 5/3 9/5 2/1
|Middle major
|142.775 318.667 461.443 637.334 704.101 879.993 1022.768 1198.660
|<nowiki>sLsLsLsL 0|1 (4)</nowiki>
|Middle major diminished<sup>†</sup>
|10:12:15 on root
|10:12:15 on root
|-
|-
| 10/9 125/108 625/486 25/18 125/81 5/3 50/27 2/1
|Middle minor diminished
|LsLMLMLM
|LsLMLMLM
|<nowiki>LsLLLLLL 1|6</nowiki>
|~ 10/9 15/13 50/39 11/8 20/13 5/3 11/6 2/1
|Middle minor
|175.892 242.658 418.550 561.325 737.218 879.993 1055.884 1198.660
|<nowiki>LsLsLsLs 1|0 (4)</nowiki>
|Middle minor diminished
|
|
|-
|-
|27/25 6/5 5/4 25/18 3/2 5/3 9/5 2/1
|Bright minor diminished*<sup>†</sup><sup>†</sup>
|MLsLMLML
|MLsLMLML
|<nowiki>LLsLLLLL 2|5</nowiki>
|~ 12/11 6/5 5/4 11/8 3/2 5/3 9/5 2/1
|Bright minor
|142.775 318.667 385.433 561.325 704.101 879.993 1022.768 1198.660
|<nowiki>sLsLsLsL 0|1 (4)</nowiki>
|Bright minor diminished*<sup>†</sup><sup>†</sup>
|Minor root 4:5:6, 10:12:15
|Minor root 4:5:6, 10:12:15
|-
|-
|25/24 125/108 5/4 25/18 3/2 5/3 9/5 2/1
|Dark minor diminished*<sup>†</sup>
|sLMLMLML
|sLMLMLML
|<nowiki>sLLLLLLL 0|7</nowiki>
|~ 25/24 15/13 5/4 11/8 3/2 5/3 9/5 2/1
|Dark minor
|66.766 242.658 385.433 561.325 704.101 879.993 1022.768 1198.660
|<nowiki>sLsLsLsL 0|1 (4)</nowiki>
|Dark minor diminished*<sup>†</sup>
|Minor with 4:5:6 on root
|Minor with 4:5:6 on root
|}
|}
{| class="wikitable"
{| class="wikitable"
|+Table 4.11. Modes of the Ptolemismic Pinetone diminished scale
|+Table 5.12. Intervals of the Ptolemismic Pinetone diminished (variety = 20)
!Mode (height order)
!Interval class
!Step pattern
!sizes
!Mode as simplest JI pre-image
!Diminished[8] name
!Mode in cents
!Porcupine[8] name
!Comments
!Pinetone octatonic name
!JI ratios approximated*
!size in cents (TE)
! Occurence
|-
|-
|Bright quartal diminished
! rowspan="3" |1-step
|LMLMLMLs
|s
|~ 10/9 6/5 4/3 13/9 8/5 26/15 48/25 2/1
| minor step
|175.892 318.667 494.559 637.334 813.226 956.002 1131.893 1198.660
|minor step
|
|minor step
|25/24, 33/32, 27/26
|66.766
|1
|-
|-
|Bright major diminished*
|M
|LMLMLsLM
|minor step
|~ 10/9 6/5 4/3 13/9 8/5 5/3 11/6 2/1
|major step
|175.892 318.667 494.559 637.334 813.226 879.993 1055.884 1198.660
|minor-major step
|
|27/25, 12/11, 13/12
|142.775
|3
|-
|-
|Dark quartal diminished
|L
|MLMLMLsL
|major step
|~ 12/11 6/5 13/10 13/9 39/25 26/15 9/5 2/1
|major step
|142.775 318.667 461.443 637.334 780.120 956.002 1022.768 1198.660
|major step
|
|10/9, 11/10
|175.892
|4
|-
|-
|Dark major diminished*
! rowspan="2" |2-step
|LMLsLMLM
|L + s
|~ 10/9 6/5 4/3 13/9 20/13 5/3 11/6 2/1
|perfect 2-step
|175.892 318.667 494.559 561.325 737.218 879.993 1055.884 1198.660
|minor 2-step
|
|minor 2-step
|15/13 (7/6 or 8/7)
|242.658
|2
|-
|-
|Middle major diminished<sup>†</sup>
|L + M
|MLMLsLML
|perfect 2-step
|~ 12/11 6/5 13/10 13/9 3/2 5/3 9/5 2/1
|major 2-step
|142.775 318.667 461.443 637.334 704.101 879.993 1022.768 1198.660
|major 2-step
|10:12:15 on root
|6/5, 40/33
|318.667
|6
|-
|-
|Middle minor diminished
! rowspan="4" |3-step
|LsLMLMLM
|L + M + s
|~ 10/9 15/13 50/39 11/8 20/13 5/3 11/6 2/1
|minor 3-step
|175.892 242.658 418.550 561.325 737.218 879.993 1055.884 1198.660
|minor 3-step
|
|minor 3-step
|5/4
|385.433
|2
|-
|-
|Bright minor diminished*<sup>†</sup><sup>†</sup>
|2L + s
|MLsLMLML
|major 3-step
|~ 12/11 6/5 5/4 11/8 3/2 5/3 9/5 2/1
|minor 3-step
|142.775 318.667 385.433 561.325 704.101 879.993 1022.768 1198.660
|major-minor 3-step
|Minor root 4:5:6, 10:12:15
|33/26, 50/39
|418.550
|1
|-
|-
|Dark minor diminished*<sup>†</sup>
|L + 2M
|sLMLMLML
|minor 3-step
|~ 25/24 15/13 5/4 11/8 3/2 5/3 9/5 2/1
|major 3-step
|66.766 242.658 385.433 561.325 704.101 879.993 1022.768 1198.660
|minor-major 3-step
|Minor with 4:5:6 on root
|13/10 (9/7 or 21/16)
|}
|461.433
{| class="wikitable"
|2
|+Table 4.12. Intervals of the Ptolemismic Pinetone diminished (variety = 20)
!Interval class
!sizes
!Diminished[8] name
!Porcupine[8] name
!Pinetone octatonic name
!JI ratios approximated*
!size in cents (TE)
! Occurence
|-
|-
! rowspan="3" |1-step
|2L + M
|s
|major 3-step
| minor step
|major 3-step
|minor step
|major 3-step
|minor step
|4/3
|25/24, 33/32, 27/26
|494.559
|66.766
|1
|-
|M
|minor step
|major step
|minor-major step
|27/25, 12/11, 13/12
|142.775
|3
|3
|-
|-
|L
! rowspan="2" |4-step
|major step
|2L + M + s
|major step
|perfect 4-step
|major step
|minor 4-step
|10/9, 11/10
|minor 4-step
|175.892
|25/18, 11/8, 18/13
|561.325
|4
|4
|-
|-
! rowspan="2" |2-step
|2L + 2M
|L + s
|perfect 4-step
|perfect 2-step
|major 4-step
|minor 2-step
|major 4-step
|minor 2-step
|36/25, 16/11, 13/9
|15/13 (7/6 or 8/7)
|637.334
|242.658
|4
|2
|-
|-
|L + M
! rowspan="4" |5-step
|perfect 2-step
|2L + 2M + s
|major 2-step
|minor 5-step
|major 2-step
|minor 5-step
|6/5, 40/33
|minor 5-step
|318.667
|3/2
|6
|704.101
|3
|-
|-
! rowspan="4" |3-step
|3L + M + s
|L + M + s
|major 5-step
|minor 3-step
|minor 5-step
|minor 3-step
|major-minor 5-step
|minor 3-step
|20/13 (14/9 or 32/16)
|5/4
|737.217
|385.433
|2
|2
|-
|-
|2L + s
|2L + 3M
|major 3-step
|minor 5-step
|minor 3-step
|major 5-step
|major-minor 3-step
|minor-major 5-step
|33/26, 50/39
|39/25, 52/33
|418.550
|780.120
|1
|1
|-
|-
|L + 2M
|3L + 2M
|minor 3-step
|major 5-step
|major 3-step
|major 5-step
|minor-major 3-step
|major 5-step
|13/10 (9/7 or 21/16)
|8/5
|461.433
|813.227
|2
|2
|-
|-
|2L + M
! rowspan="2" |6-step
|major 3-step
|3L + 2M + s
|major 3-step
|perfect 6-step
|major 3-step
|minor 6-step
|4/3
|minor 6-step
|494.559
|5/3, 33/20
|3
|879.992
|6
|-
|-
! rowspan="2" |4-step
|3L + 3M
|2L + M + s
|perfect 6-step
|perfect 4-step
|major 6-step
|minor 4-step
|major 6-step
|minor 4-step
|26/15 (12/7 or 7/4)
|25/18, 11/8, 18/13
|956.002
|561.325
|2
|4
|-
|-
|2L + 2M
! rowspan="3" |7-step
|perfect 4-step
|3L + 3M + s
|major 4-step
|minor 7-step
|major 4-step
|minor 7-step
|36/25, 16/11, 13/9
|minor 7-step
|637.334
|9/5, 20/11
|1022.768
|4
|4
|-
|-
! rowspan="4" |5-step
|4L + 2M + s
|2L + 2M + s
|major 7-step
|minor 5-step
|minor 7-step
|minor 5-step
|major-minor 7-step
|minor 5-step
|50/27, 11/6, 24/13
|3/2
|1055.884
|704.101
|3
|3
|-
|-
|3L + M + s
|4L + 3M
|major 5-step
|major 7-step
|minor 5-step
|major 7-step
|major-minor 5-step
|major 7-step
|20/13 (14/9 or 32/16)
|48/25, 64/33, 52/27
|737.217
|1131.983
|2
|-
|2L + 3M
|minor 5-step
|major 5-step
|minor-major 5-step
|39/25, 52/33
|780.120
|1
|1
|}
<nowiki>*</nowiki> 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).
{| class="wikitable"
|+Table 5.13. 3-step stacked triads of the Ptolemismic Pinetone diminished
! Mode (rotational order)
!Step pattern
!3-step stacked triad on root
!Diminished[8] name
!Porcupine[8] name
!Pinetone octatonic name
!JI triad approximated*
|-
|-
|3L + 2M
|Bright quartal diminished
|major 5-step
| LMLMLMLs
|major 5-step
|G♯-C♯-F, A-D-G♭
|major 5-step
|[8] major
|8/5
|[8] augmented
|813.227
|[8] major augmented
|2
|15:20:26 (12:16:21)
|-
|-
! rowspan="2" |6-step
|Dark quartal diminished
|3L + 2M + s
| MLMLMLsL
|perfect 6-step
|A♯-D-G, B-E♭-A♭
|minor 6-step
|[8] minor
|minor 6-step
|[8] augmented
|5/3, 33/20
|[8] minor augmented
|879.992
|30:39:52 (16:21:28)
|6
|-
|-
|3L + 3M
|Bright major diminished
|perfect 6-step
| LMLMLsLM
|major 6-step
|B-E-G♯, C-F-A
|major 6-step
|[8] major
|26/15 (12/7 or 7/4)
|[8] major
|956.002
|[8] major
|2
|3:4:5
|-
|-
! rowspan="3" |7-step
|Middle major diminished
|3L + 3M + s
| MLMLsLML
|minor 7-step
|C♯-F-A♯, D-G♭-B
|minor 7-step
|[8] minor
|minor 7-step
|[8] major
|9/5, 20/11
|[8] minor major
|1022.768
|30:39:50
|4
|-
|-
|4L + 2M + s
|Dark major diminished
|major 7-step
| LMLsLMLM
|minor 7-step
|D-G-B, E♭-A♭-C
|major-minor 7-step
|[8] major
|50/27, 11/6, 24/13
|[8] major
|1055.884
|[8] major
|3
|3:4:5
|-
|-
|4L + 3M
|Bright minor diminished
|major 7-step
| MLsLMLML
|major 7-step
|E-G♯-C♯, F-A-D
|major 7-step
|[8] minor
|48/25, 64/33, 52/27
|[8] minor
|1131.983
|[8] minor
|1
|12:15:20
|}
<nowiki>*</nowiki> 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).
 
{| class="wikitable"
|+Table 4.13. 3-step stacked triads of the Ptolemismic Pinetone diminished
! 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
|Middle minor diminished
| LMLMLMLs
| LsLMLMLM
|G♯-C♯-F, A-D-G♭
|F-A♯-D, G♭-B-E♭
|[8] major
|[8] major
|[8] augmented
|[8] minor
|[8] major augmented
|[8] major minor
|15:20:26 (12:16:21)
|39:50:65
|-
|-
|Dark quartal diminished
|Dark minor diminished
| MLMLMLsL
| sLMLMLML
|A♯-D-G, B-E♭-A♭
|G-B-E, A♭-C-F
|[8] minor
|[8] minor
|[8] augmented
|[8] minor
|[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
|[8] minor
|12:15:20
|12:15:20
Line 2,549: Line 3,182:


====Pinetone diminished heptatonic====
====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.   
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#Pinetone diminished octatonic|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 lower note of each of 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.  
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 [[Chirality|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.  


{| class="wikitable"
{| class="wikitable"
|+Table 4.14. Intervals of the 2.3.5.11.13 Ptolemismic Pinetone diminished heptatonic (variety = 18)
|+Table 5.14. Intervals of the 2.3.5.11.13 Ptolemismic Pinetone diminished heptatonic (variety = 18)
!Interval class
!Interval class
!size
!size
Line 2,737: Line 3,370:
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.  
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-[[Step-nested scale|SNS]] on the white keys, and a pentatonic [[Meantone]][5]-[[Father]][5]-[[Bug]][5] [[wakalix]] on the 'black' keys.  
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-[[Step-nested scale|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).
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 vector|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 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]] (product word) with [[Unison vector|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''). As a product word of 5L 2s (standard diatonic scale form) and 1L 6s (Porcupine[7] scale form) the Pinetone diatonic in it's symmetric mode msmLmsm = LsLLLsL x sssLsss, a product of Dorian mode of the standard diatonic scale, and the symmetric minor mode of the Porcupine[7] scale form.
 
The Azurian bright minor, Duradian dark minor, Karakalian bright minor, Phyradian dark minor are alternative heptatonic subsets of the Pinetone chromatic — Fokker blocks of the same unison vectors and step sizes as the Pinetone diatonic arranged in different patterns, corresponding to product words of different combinations of modes of Porcupine[7] and Meantone[7]. Unlike the Pinetone diatonic, these scales are not also Fokker blocks of Dicot[7] with Porcupine[7] or Meantone[7].
 
Azurian bright minor msLmsmm =  LLLLsLL x ssLssss
 
Duradian dark minor msmLsmm = LsLLLLL x sssLsss
 
Karakalian bright minor msLmmsm = LLLLLsL x ssLssss
 
Phyradian dark minor smmLsmm = sLLLLLL x sssLsss


The Pinetone major and minor-harmonic octatonics are the 8-note rank-3 [[Porcupine]][8] x [[Father]][8] [[Fokker block|Fokker blocks]] with [[Unison vector|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 major and minor-harmonic octatonics are the 8-note rank-3 [[Porcupine]][8] x [[Father]][8] [[Fokker block|Fokker blocks]] with [[Unison vector|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'').
Line 2,757: Line 3,400:


We get the following 7 modes of Pinetone harmonic minor scale:
We get the following 7 modes of Pinetone harmonic minor scale:
*Lsmsmms Lydian ♯2 bright major starting on F
*Lsmsmms Lydian ♯2 bright major starting on F
*mmsLsms Ionian ♯5 symmetric minor starting on C
*mmsLsms Ionian ♯5 symmetric minor starting on C
Line 2,765: Line 3,407:
*smmsLsm Locrian ♮6 bright diminished starting on B
*smmsLsm Locrian ♮6 bright diminished starting on B
*smsmmsL altered diminished magical seventh starting on G♯
*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:
Replacing the A with an A♭ instead, we get the modes of the Pinetone harmonic major scale. Starting on D we get the mode:


Line 2,771: Line 3,412:


Which has modes:
Which has modes:
*Lsmmsms Lydian Augmented ♯2 bright major starting on A♭
*Lsmmsms Lydian Augmented ♯2 bright major starting on A♭
*msLsmms Lydian ♭3 bright minor starting on F
*msLsmms Lydian ♭3 bright minor starting on F
Line 2,829: Line 3,469:
tempering L = s (tempering out 250/243, the Porcupine comma) results in sLsLsLsLsLsLsLs, which is Porcupine[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 s would lead to ssLsLssLsssL, which is a MODMOS of Diminished[12].


tempering out m would lead to ssLsLssLsssL, which is a MODMOS of Diminished[12].
tempering out m would lead to sLLsLsL, which is Dicot[7];


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.
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.
Line 2,849: Line 3,489:
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.
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.  
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).
In this tuning the medium and small steps are within 1c of the same size. Tempering them together results in Cata temperament, an extension of Hanson. 2.3.5.13 325/324 may alternatively be tuned to 46edo, with (L, m, s) = (4, 2, 3), or as Cata[15] in 53edo, 72edo, or 87edo as (5, 3, 3), (7, 4, 4), or (8, 5, 5).  


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.
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.
Line 2,866: Line 3,506:
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.  
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.  
{| class="wikitable"
{| class="wikitable"
|+Table 8.1. Modes of the just Pinetone harmonic diminished
|+Table 9.1. Modes of the just Pinetone harmonic diminished
! Mode (height order)
!Mode number
! Mode name
!Step pattern
!Step pattern
!Oneirotonic step pattern
!Oneirotonic step pattern
Line 2,874: Line 3,515:
! Comments
! Comments
|-
|-
|4
|Hlanithian diminished
|Hlanithian diminished
|AsLMLMLs
|AsLMLMLs
Line 2,881: Line 3,523:
|
|
|-
|-
|3
|Mnarian diminished*
|Mnarian diminished*
|LMLMLsAs
|LMLMLsAs
Line 2,888: Line 3,531:
|
|
|-
|-
|2
|Celephaïsian diminished*
|Celephaïsian diminished*
|LMLsAsLM
|LMLsAsLM
Line 2,895: Line 3,539:
|
|
|-
|-
|1
|Sarnathian diminished<sup>†</sup>
|Sarnathian diminished<sup>†</sup>
|MLMLsAsL
|MLMLsAsL
Line 2,902: Line 3,547:
|10:12:15 on the root
|10:12:15 on the root
|-
|-
| -1
|Dylathian diminished*
|Dylathian diminished*
|LsAsLMLM
|LsAsLMLM
Line 2,909: Line 3,555:
|
|
|-
|-
| -2
|Kadathian diminished*<sup>††</sup>
|Kadathian diminished*<sup>††</sup>
|MLsAsLML
|MLsAsLML
Line 2,916: Line 3,563:
|root 4:5:6,10:12:15
|root 4:5:6,10:12:15
|-
|-
| -3
|Ultharian diminished*<sup>††</sup>
|Ultharian diminished*<sup>††</sup>
|sAsLMLML
|sAsLMLML
Line 2,923: Line 3,571:
|root 4:5:6,10:12:15
|root 4:5:6,10:12:15
|-
|-
| -4
|Illarnekian diminished*<sup>†</sup>
|Illarnekian diminished*<sup>†</sup>
|sLMLMLsA
|sLMLMLsA
Line 2,930: Line 3,579:
| root 4:5:6
| root 4:5:6
|}
|}
For Tables 9.1. and 9.2. modes with major and minor [8] triads (3-step) triads on the root are marked with '*'; modes with major and minor tertian (2-step) triads on the root are marked with '<sup>†</sup>' ('<sup>††</sup>' for both major and minor).
{| class="wikitable"
{| class="wikitable"
|+Table 8.2. Modes of the Ptolemismic Pinetone harmonic diminished
|+Table 9.2. Modes of the Ptolemismic Pinetone harmonic diminished
! Mode (height order)
!Mode number
! Mode name
!Step pattern
!Step pattern
!Mode as simplest JI pre-image 5-limit JI
!Mode as simplest JI pre-image 5-limit JI
Line 2,938: Line 3,589:
!Comments
!Comments
|-
|-
|4
|Hlanithian diminished
|Hlanithian diminished
|AsLMLMLs
|AsLMLMLs
Line 2,944: Line 3,596:
|
|
|-
|-
|3
|Mnarian diminished*
|Mnarian diminished*
|LMLMLsAs
|LMLMLsAs
Line 2,950: Line 3,603:
|
|
|-
|-
|2
|Celephaïsian diminished*
|Celephaïsian diminished*
|LMLsAsLM
|LMLsAsLM
Line 2,956: Line 3,610:
|
|
|-
|-
|1
|Sarnathian diminished<sup>†</sup>
|Sarnathian diminished<sup>†</sup>
|MLMLsAsL
|MLMLsAsL
Line 2,962: Line 3,617:
|10:12:15 on the root
|10:12:15 on the root
|-
|-
| -1
|Dylathian diminished*
|Dylathian diminished*
|LsAsLMLM
|LsAsLMLM
Line 2,968: Line 3,624:
|
|
|-
|-
| -2
|Kadathian diminished*<sup>††</sup>
|Kadathian diminished*<sup>††</sup>
|MLsAsLML
|MLsAsLML
Line 2,974: Line 3,631:
|root 4:5:6,10:12:15
|root 4:5:6,10:12:15
|-
|-
| -3
|Ultharian diminished*<sup>††</sup>
|Ultharian diminished*<sup>††</sup>
|sAsLMLML
|sAsLMLML
Line 2,980: Line 3,638:
|root 4:5:6,10:12:15
|root 4:5:6,10:12:15
|-
|-
| -4
|Illarnekian diminished*<sup>†</sup>
|Illarnekian diminished*<sup>†</sup>
|sLMLMLsA
|sLMLMLsA
Line 2,987: Line 3,646:
|}
|}
{| class="wikitable"
{| class="wikitable"
|+Table 8.3. Intervals of the Ptolemismic Pinetone harmonic diminished (variety = 20)
|+Table 9.3. Intervals of the Ptolemismic Pinetone harmonic diminished (variety = 20)
!Interval class
!Interval class
!sizes
!sizes
Line 3,168: Line 3,827:
{| class="wikitable"
{| class="wikitable"
|+
|+
Table 8.4. Intervals of modes of the Pinetone harmonic diminished
Table 9.4. Intervals of modes of the Pinetone harmonic diminished
!Mode
!Mode (height order)
!step
!step
!2-step
!2-step
Line 3,231: Line 3,890:
|}
|}
{| class="wikitable"
{| class="wikitable"
|+Table 8.6. 3-step stacked triads of the Ptolemismic Pinetone harmonic diminished
|+Table 9.5. 3-step stacked triads of the Ptolemismic Pinetone harmonic diminished
! Mode (rotational order)
! Mode (rotational order)
!Step pattern
!Step pattern
Line 3,295: Line 3,954:
|12:15:20
|12:15:20
|}
|}
<nowiki>*</nowiki> 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.
<nowiki>*</nowiki> 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 as in Supermagic temperament. Alternatively, if the consonances of the triads are to be maximised, the scale could be tempered 7-limit rather than 13-limit Supermagic.


==Comma pump==
==Comma pump==
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