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→Pinetone melodic major and minor: re-wrote this section and added some note names to some tables from the preceding section
Line 149: Line 149:
!Mode number
!Mode number
!Mode in JI
!Mode in JI
!Root note
!Step pattern
!Step pattern
!Meantone[7]
!Meantone[7]
Line 158: Line 159:
|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 169:
|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 179:
|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 189:
|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 199:
|  -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 209:
|  -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 219:
|  -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 230: Line 238:
!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 245:
|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 242: Line 252:
|2
|2
|Mixolydian bright minor*
|Mixolydian bright minor*
|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 259:
|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 266:
|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 273:
| -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 280:
| -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 287:
| -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 1,329: Line 1,345:
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 melodic major and minor ==
== Other diatonic subsets of the Pinetone chromatic scale ==
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]]), i.e., in mode -3 the following diatonic scales are avilable:
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.  
 
D E F G A B C
 
D E F♯ G A B C
 
D E F♯ G A B C♯
 
D E F♯ G♯ A B C♯
 
D♯ E F♯ G♯ A  B C♯
 
D♯ E F♯ G♯ A♯  B C♯
 
in mode 3 the following additional 5 diatonic scales are available:
 
D E F G A B♭ C
 
D E♭ F G A B♭ C
 
D E♭ F G A♭ B♭ C
 
D♭ E♭ F G A♭ B♭ C
 
D♭ E♭ F G♭ A♭ B♭ C
 
Each of these 11 scales are different (in terms of the intervals they contain). We will explore the D E F♯ G A B C and D E F G A B♭ C, which we will call the Pinetone melodic major and minor respectively.
 
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), so will call the mode approximating the ratios 10/9 11/9 4/3 3/2 5/3 9/5 2/1 Mixolydian symmetric minor.
{| class="wikitable"
{| class="wikitable"
|+Table 4.1. Modes of the Ptolemismic Pinetone melodic major
|+Table 4.1.
!Mode number
! colspan="8" |Scale (mode 3 subset)
!Pinetone diatonic mode
! colspan="8" |Scale (mode -3 subset)
!JI ratios approximated
!Step pattern
!Step pattern
!Mode as simplest JI pre-image
!Symmetric minor mode
|-
|-
|3
|D
|Ionian bright major*
|E
|Lmsmmms
|F
|~ 9/8 5/4 15/11 3/2 5/3 11/6 2/1
|G
|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
|-
|-
|2
|D
|Lydian bright diminished
|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
|Phrygian
|-
|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), so will call the mode approximating the ratios 10/9 11/9 4/3 3/2 5/3 9/5 2/1 Mixolydian symmetric minor, and use the mode to name the scale as a whole.
 
To calculate the mode numbers for Tables 4.2-4.5, the mode numbers of their temperings to Porcupine and to Meantone were added, ordered, and renumbered.
{| class="wikitable"
|+Table 4.2. Modes of the Ptolemismic Pinetone Mixolydian symmetric minor
!Mode number
!Pinetone diatonic mode
! colspan="2" |Root note
!Step pattern
!Mode as simplest JI pre-image
|-
|3
|Ionian bright major*
|G
|A♭
|Lmsmmms
|~ 9/8 5/4 15/11 3/2 5/3 11/6 2/1
|-
|2
|Lydian bright diminished
|C
|D♭
|mmmsLms
|mmmsLms
|~ 10/9 11/9 110/81 22/15 5/3 11/6 2/1
|~ 10/9 11/9 110/81 22/15 5/3 11/6 2/1
|-
|-
|1
|1
|Mixolydian symmetric minor
|D
|E♭
|mmsLmsm
|~ 10/9 11/9 4/3 3/2 5/3 9/5 2/1
|-
|0
|Aeolian bright minor*
|Aeolian bright minor*
|E
|F
|msLmsmm
|msLmsmm
|~ 10/9 6/5 15/11 3/2 18/11 9/5 2/1
|~ 10/9 6/5 15/11 3/2 18/11 9/5 2/1
|-
| 0
|Mixolydian symmetric minor
|mmsLmsm
|~ 10/9 11/9 4/3 3/2 5/3 9/5 2/1
|-
|-
| -1
| -1
|Locian dark major
|Locian dark major
|F♯
|G
|sLmsmmm
|sLmsmmm
|~ 12/11 27/22 15/11 81/55 18/11 9/5 2/1
|~ 12/11 27/22 15/11 81/55 18/11 9/5 2/1
|-
|-
| -2
| -2
|Phrygian dark diminished
|B
|C
|smmmsLm
|~ 12/11 6/5 4/3 22/15 8/5 9/5 2/1
|-
| -3
|Dorian magical seventh
|Dorian magical seventh
|A
|B♭
|msmmmsL
|msmmmsL
|~ 10/9 6/5 4/3 22/15 44/27 16/9 2/1
|~ 10/9 6/5 4/3 22/15 44/27 16/9 2/1
|-
| -3
|Phrygian dark diminished
|smmmsLm
|~ 12/11 6/5 4/3 22/15 8/5 9/5 2/1
|}
|}
Similarly, lowering B to B♭ lowers by ~55/54, leading to the Aeolian symmetric minor shown in Table 4.2, which I consider to be a really beautiful minor minor mode.
Similarly, lowering B to B♭ lowers by ~55/54, leading to the Aeolian symmetric minor shown in Table 4.2, which I consider to be a really beautiful minor minor mode.
{| class="wikitable"
{| class="wikitable"
|+Table 4.2. Modes of the Ptolemismic Pinetone melodic minor
|+Table 4.3. Modes of the Ptolemismic Pinetone Aeolian symmetric minor
!Mode number
!Mode number
!Pinetone diatonic mode
!Pinetone diatonic mode
! colspan="2" |Root note
!Step pattern
!Step pattern
!Mode as simplest JI pre-image
!Mode as simplest JI pre-image
Line 1,413: Line 1,548:
|3
|3
|Ionian dark major*
|Ionian dark major*
|F
|E
|mLsmmms
|mLsmmms
|~ 10/9 5/4 15/11 3/2 5/3 11/6 2/1
|~ 10/9 5/4 15/11 3/2 5/3 11/6 2/1
Line 1,418: Line 1,555:
|2
|2
|Dorian bright major
|Dorian bright major
|G
|F♯
|Lsmmmsm
|Lsmmmsm
|~ 9/8 27/22 15/11 3/2 5/3 9/5 2/1
|~ 9/8 27/22 15/11 3/2 5/3 9/5 2/1
Line 1,423: Line 1,562:
|1
|1
|Lydian dark diminished
|Lydian dark diminished
|B♭
|A
|mmmsmLs
|mmmsmLs
|~ 10/9 11/9 110/81 22/15 44/27 11/6 2/1
|~ 10/9 11/9 110/81 22/15 44/27 11/6 2/1
|-
|-
| 0
| 0
|Mixolydian bright diminished
|C
|B
|mmsmLsm
|~ 10/9 11/9 4/3 22/15 5/3 9/5 2/1
|-
| -1
|Aeolian symmetric minor*
|Aeolian symmetric minor*
|D
|C♯
|msmLsmm
|msmLsmm
|~ 10/9 6/5 4/3 3/2 18/11 9/5 2/1
|~ 10/9 6/5 4/3 3/2 18/11 9/5 2/1
|-
| -1
|Mixolydian bright diminished
|mmsmLsm
|~ 10/9 11/9 4/3 22/15 5/3 9/5 2/1
|-
|-
| -2
| -2
|Locrian bright minor
|Locrian bright minor
|E
|D♯
|smLsmmm
|smLsmmm
|~ 12/11 6/5 15/11 81/55 11/8 9/5 2/1
|~ 12/11 6/5 15/11 81/55 11/8 9/5 2/1
Line 1,443: Line 1,590:
| -3
| -3
|Phrygian magical seventh
|Phrygian magical seventh
|A
|G♯
|smmmsmL
|smmmsmL
|~ 12/11 6/5 4/3 22/15 8/5 16/9 2/1
|~ 12/11 6/5 4/3 22/15 8/5 16/9 2/1
Line 1,457: Line 1,606:
The step signature and mapping of 5-limit Porcupine[8] is 7L 1s = (10/9~27/25, 25/24~81/80)
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 5.1. Porcupine[8] modes (G♯-G gamut)
!Mode number
!Mode number
!Step pattern
!Step pattern
Line 1,555: Line 1,704:
The step signature and mapping of 5-limit Father[8] is 5L 3s = (10/9~25/24~32/27, 27/25~81/80),
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 5.2. Father[8] oneirotonic modes
!Mode number
!Mode number
!Step pattern
!Step pattern
Line 1,621: Line 1,770:
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 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.
{| class="wikitable"
{| class="wikitable"
|+Table 4.3. Modes of the just Pinetone major-harmonic octatonic
|+Table 5.3. Modes of the just Pinetone major-harmonic octatonic
!Mode in JI (height order)
!Mode in JI (height order)
!Step pattern
!Step pattern
Line 1,708: Line 1,857:
|Minor with 4:5:6 on root
|Minor with 4:5:6 on root
|}
|}
For Tables 4.3.-4.6. 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).
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 4.4. Modes of the just Pinetone minor-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 1,803: Line 1,952:
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].
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 5.5. Modes of the Ptolemismic Pinetone major-harmonic octatonic
!Mode (height order)
!Mode (height order)
!Step pattern
!Step pattern
Line 1,858: Line 2,007:
|Minor with 4:5:6 on root
|Minor with 4:5:6 on root
|}
|}
For Tables 4.3.-4.6. 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).
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 4.6. Modes of the Ptolemismic Pinetone minor-harmonic octatonic
|+Table 5.6. Modes of the Ptolemismic Pinetone minor-harmonic octatonic
!Mode (height order)
!Mode (height order)
!Step pattern
!Step pattern
Line 1,919: Line 2,068:
The following table gives all intervals of the Pinetone harmonic octatonics.
The following table gives all intervals of the Pinetone harmonic octatonics.
{| class="wikitable"
{| class="wikitable"
|+Table 4.7. Intervals of the Ptolemismic Pinetone harmonic octatonics (variety = 24)
|+Table 5.7. Intervals of the Ptolemismic Pinetone harmonic octatonics (variety = 24)
!Interval class
!Interval class
!sizes
!sizes
Line 2,132: Line 2,281:
The following two tables detail the 3-step stacked triads of the Pinetone harmonic octatonics:
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 5.8. 3-step stacked triads of the Ptolemismic Pinetone major-harmonic octatonic (G♯-G gamut)
!Mode (rotational order)
!Mode (rotational order)
!Step pattern
!Step pattern
Line 2,207: Line 2,356:
<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).
{| class="wikitable"
{| class="wikitable"
|+Table 4.9. 3-step stacked triads of the Ptolemismic Pinetone minor-harmonic octatonic (G-A♭ 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 2,299: Line 2,448:


{| class="wikitable"
{| class="wikitable"
|+Tabl3 4.10. Modes of the just Pinetone diminished scale
|+Table 5.10. Modes of the just Pinetone diminished scale
!Mode in JI (height order)
!Mode in JI (height order)
!Step pattern
!Step pattern
Line 2,376: Line 2,525:
|Minor with 4:5:6 on root
|Minor with 4:5:6 on root
|}
|}
For Tables 4.10. and 4.11. 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).
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).
{| class="wikitable"
{| class="wikitable"
|+Table 4.11. Modes of the Ptolemismic Pinetone diminished scale
|+Table 5.11. Modes of the Ptolemismic Pinetone diminished scale
!Mode (height order)
!Mode (height order)
!Step pattern
!Step pattern
Line 2,434: Line 2,583:
|}
|}
{| class="wikitable"
{| class="wikitable"
|+Table 4.12. Intervals of the Ptolemismic Pinetone diminished (variety = 20)
|+Table 5.12. Intervals of the Ptolemismic Pinetone diminished (variety = 20)
!Interval class
!Interval class
!sizes
!sizes
Line 2,614: Line 2,763:


{| class="wikitable"
{| class="wikitable"
|+Table 4.13. 3-step stacked triads of the Ptolemismic Pinetone diminished
|+Table 5.13. 3-step stacked triads of the Ptolemismic Pinetone diminished
! Mode (rotational order)
! Mode (rotational order)
!Step pattern
!Step pattern
Line 2,716: Line 2,865:


{| 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 3,028: Line 3,177:
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.  
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{| 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 (height order)
!Step pattern
!Step pattern
Line 3,092: Line 3,241:
| root 4:5:6
| root 4:5:6
|}
|}
For Tables 8.1. and 8.2. 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).
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).
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{| 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 (height order)
!Step pattern
!Step pattern
Line 3,150: Line 3,299:
|}
|}
{| 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 3,331: Line 3,480:
{| 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
!step
!step
Line 3,394: Line 3,543:
|}
|}
{| class="wikitable"
{| class="wikitable"
|+Table 8.6. 3-step stacked triads of the Ptolemismic Pinetone harmonic diminished
|+Table 9.6. 3-step stacked triads of the Ptolemismic Pinetone harmonic diminished
! Mode (rotational order)
! Mode (rotational order)
!Step pattern
!Step pattern
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