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→Pinetone harmonic diminished octatonic: added more info and a table, edited some other sections
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!Interval class
!Interval class
!size
!size
!Meantone[7] name
!Meantone name
!Porcupine[7] name
!Porcupine[7] name
!Pinetone diatonic names
!Pinetone name
!Generic name
!JI ratios approximated
!JI ratios approximated
!size in cents (TE)
!size in cents (TE)
Line 318: Line 319:
|minor 2nd
|minor 2nd
|minor 2nd
|minor 2nd
|small 2nd, minor 2nd
|minor 2nd
|small 2nd
|27/25, 12/11
|27/25, 12/11
|146.635
|146.635
Line 326: Line 328:
|major 2nd
|major 2nd
|minor 2nd
|minor 2nd
|medium 2nd, major-minor 2nd
|major-minor 2nd
|medium 2nd
| 10/9, 11/10
| 10/9, 11/10
|174.055
|174.055
Line 334: Line 337:
|major 2nd
|major 2nd
|major 2nd
|major 2nd
|large 2nd, major 2nd
|major 2nd
|large 2nd
|9/8, 25/22
|9/8, 25/22
|209.779
|209.779
Line 343: Line 347:
|minor 3rd
|minor 3rd
|minor 3rd
|minor 3rd
|small 3rd, minor 3rd
|minor 3rd
|small 3rd
|6/5, 40/33
|6/5, 40/33
| 320.690
| 320.690
Line 351: Line 356:
|major 3rd
|major 3rd
|minor 3rd
|minor 3rd
|medium 3rd, major-minor 3rd
|major-minor 3rd
|middle 3rd
| 100/81, 11/9
| 100/81, 11/9
|348.110
|348.110
Line 359: Line 365:
|major 3rd
|major 3rd
|major 3rd
|major 3rd
|large 3rd, major third
|major third
| 5/4, 33/20
|large 3rd
| 5/4, 99/80
|383.834
|383.834
|2
|2
Line 368: Line 375:
|perfect 4th
|perfect 4th
|minor 4th
|minor 4th
|small 4th, minor 4th
|minor 4th
|small 4th
|4/3, 33/25
|4/3, 33/25
|494.745
|494.745
Line 376: Line 384:
|perfect 4th
|perfect 4th
|major 4th
|major 4th
|medium 4th, minor-major 4th
|minor-major 4th
|middle 4th
|27/20, 15/11
|27/20, 15/11
|530.469
|530.469
Line 384: Line 393:
|augmented 4th
|augmented 4th
|major 4th
|major 4th
|large 4th, major 4th
|major 4th
|large 4th
|25/18, 11/8
|25/18, 11/8
| 557.888
| 557.888
Line 393: Line 403:
|diminished 5th
|diminished 5th
|minor 5th
|minor 5th
|small 5th, minor 5th
|minor 5th
|small 5th
|36/25, 16/11
|36/25, 16/11
| 641.380
| 641.380
Line 401: Line 412:
|perfect 5th
|perfect 5th
|minor 5th
|minor 5th
|medium 5th, major-minor 5th
|major-minor 5th
|medium 5th
| 40/27, 22/15
| 40/27, 22/15
|668.800
|668.800
Line 409: Line 421:
|perfect 5th
|perfect 5th
|major 5th
|major 5th
|large 5th, major 5th
|major 5th
|large 6th
|3/2, 50/33
|3/2, 50/33
|704.524
|704.524
Line 418: Line 431:
|minor 6th
|minor 6th
|minor 6th
|minor 6th
|small 6th, minor 6th
|minor 6th
|8/5, 40/33
|small 6th
|8/5, 160/99
| 815.435
| 815.435
|2
|2
Line 426: Line 440:
| minor 6th
| minor 6th
|major 6th
|major 6th
| medium 6th, minor-major 6th
| minor-major 6th
|medium 6th
|81/50, 18/11
|81/50, 18/11
|851.159
|851.159
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|major 6th
|major 6th
|major 6th
|major 6th
|large 6th, major 6th
|major 6th
|large 6th
|5/3, 33/20
|5/3, 33/20
|878.579
|878.579
Line 443: Line 459:
| minor 7th
| minor 7th
|minor 7th
|minor 7th
| small 7th, minor 7th
| minor 7th
|small 7th
|16/9, 44/25
|16/9, 44/25
|989.490
|989.490
Line 451: Line 468:
|minor 7th
|minor 7th
| major 7th
| major 7th
|medium 7th, minor-major 7th
|minor-major 7th
|medium 7th
|9/5, 20/11
|9/5, 20/11
|1025.241
|1025.241
Line 459: Line 477:
| major 7th
| major 7th
|major 7th
|major 7th
|large 7th, major 7th
|major 7th
|large 7th
|11/6, 50/27
|11/6, 50/27
|1052.633
|1052.633
Line 2,286: Line 2,305:
|major 2-step
|major 2-step
|major 2-step
|major 2-step
|6/5
|6/5, 40/33
|318.667
|318.667
|6
|6
Line 2,303: Line 2,322:
|minor 3-step
|minor 3-step
|major-minor 3-step
|major-minor 3-step
|50/39
|33/26, 50/39
|418.550
|418.550
|1
|1
Line 2,361: Line 2,380:
|major 5-step
|major 5-step
|minor-major 5-step
|minor-major 5-step
|39/25
|39/25, 52/33
|780.120
|780.120
|1
|1
Line 2,378: Line 2,397:
|minor 6-step
|minor 6-step
|minor 6-step
|minor 6-step
|5/3
|5/3, 33/20
|879.992
|879.992
|6
|6
Line 2,511: Line 2,530:
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 diminished octatonic 10/9 6/5 4/3 36/25 8/5 5/3 50/27 2/1. The 5-limit diminished tetrad has 3 large steps of 6/5, and one small step of 125/108. If we only put 10/9 above the large steps, we get the scale 10/9 6/5 4/3 36/25 5/3 50/27 2/1, with step pattern msmsLms, comprising 3 small steps of 27/25, 3 medium steps of 10/9, and one large step of 125/108.   


Tempering the small and medium steps together gives us the scale sssssLss, Porcupine[7]. Tempering L=m gives us LsLsLLs, Dicot[7], 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 inverse msmLsms, or in other modes, Lmsmsms and smsmsmL, the brightest and darkest modes of the pair of scales. The scales, like the Pinetone diatonic, are also trivalent.  
Tempering the small and medium steps together gives us the scale sssssLss, Porcupine[7]. Tempering L=m gives us LsLsLLs, Dicot[7], 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 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.  


==Summary for xen-math nerds==
{| class="wikitable"
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.
|+Intervals of the 2.3.5.11.13 Ptolemismic Pinetone diminished heptatonic (variety = 18)
 
!Interval class
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.
!size
 
!Porcupine name
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).
!Meantone name
 
!Pinetone name
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'').
!Generic name
 
!JI ratios approximated
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'').
!size in cents (TE)
 
!Occurence
The Pinetone diminished scale is a [[step-nested scale]] and a [[Porcupine]][8] x Diminished[8] [[Fokker block]] with [[Unison vector|unison vectors]] of 250/243, 648/625, and 16/15; comprising 4 large steps of 10/9 (''L'' x ''L''), 3 medium steps of 27/25 (''L'' x ''s''), and one small step of 25/24 (''s'' x ''s'').
|-
 
! rowspan="3" |1-step
*
|s
 
|minor 2nd
==Pinetone harmonic minor and harmonic major==
|minor 2nd
Additionally, we have another set of [[Porcupine]][7] modes contained in the Pinetone harmonic octatonics: Replacing the G with the G♯ changes the mode of the Porcupine[7] scale represented, and replaces diatonic with harmonic minor modes for the [[Meantone]][7] scale represented, now a MODMOS.
|minor second
 
|small 2nd
We note that there are fewer consonant triads available in these scales than in the Pinetone diatonic and octatonic scales, so they may be useful for melody only.
|27/25, 12/11, 13/12
 
|142.775
On D we get the scale:
|3
 
|-
174.055 320.69 557.888 704.524 878.579 1025.214 1199.269 as the notes D E F G♯ A B C D, representing 10/9 6/5 25/18~11/8 3/2 5/3 9/5 2/1
|m
 
|minor 2nd
We get the following 7 modes of Pinetone harmonic minor scale:
|major 2nd
 
|minor-major 2nd
*Lsmsmms Lydian ♯2 bright major starting on F
|medium 2nd
*mmsLsms Ionian ♯5 symmetric minor starting on C
| 10/9, 11/10
*msLsmsm Ukranian dorian bright minor starting on D
|175.892
*sLsmsmm Phyrgian dominant dark major starting on E
|3
*msmmsLs harmonic minor dark diminished starting on A
|-
*smmsLsm Locrian ♮6 bright diminished starting on B
|L
*smsmmsL altered diminished magical seventh starting on G♯
|major 2nd
 
|augmented 2nd
Replacing the A with an A♭ instead, we get the modes of the Pinetone harmonic major scale. Starting on D we get the mode:
|augmented 2nd
 
|large 2nd
174.055 320.69 494.745 641.38 878.579 1025.214 1199.269 as the notes D E F G A♭ B C D, representing 10/9 6/5 4/3 36/25~13/9 5/3 9/5 2/1.  
|125/128, 55/48, 15/13
 
|242.658
Which has modes:
|1
 
|-
*Lsmmsms Lydian Augmented ♯2 bright major starting on A♭
! rowspan="3" |2-step
*msLsmms Lydian ♭3 bright minor starting on F
|m + s
*sLsmmsm Mixolydian ♭2 dark major starting on G
|minor 3rd
*mmsmsLs harmonic major bright diminished starting on C
|minor 3rd
*msmsLsm Dorian ♭5 dark diminished starting on D
|minor 3rd
*smsLsmm Phrygian ♭4 symmetric minor starting on E
|small 3rd
*smmsmsL Locrian magical ♭♭7 starting on B
|6/5, 40/33
We can see that this scale differs from the Pinetone diminished heptatonic by only a single note -  9/5 instead of 50/27.  
| 318.667
 
|6
The augmented step of the Pinetone harmonic minor and major scales is the same as of the Pinetone diminished heptatonic, representing 15/13 when 325/324 is tempered out (the difference between 100/99 and 144/143).
|-
 
|L + s
The Pintone harmonic minor and harmonic minor have step patterns msmmsLs, and mmsmsLs respectively, or, represented as MODMOS of detempered Meantone[7], LsLLsAs and LLsLsAs, the step patterns of the familiar harmonic minor and harmonic major scales.
|major 3rd
 
|major 3rd
==Pinetone hyperchromatic scales==
|major 3rd
Maybe you have a Lumatone, and you're wondering, ok so you can either have sharps or flats? Por queno los dos?
|middle 3rd
 
| 5/4
Indeed we can have both!
|385.433
 
|1
From the Pinetone chromatic with sharps (mode -3), we add another Pinetone diatonic scale, mode 0 starting on D♭, leading to the left-handed Pinetone hyperchromatic scale, with step pattern, sLsLssLsmLssLsLssLs.  
|-
 
|L + m
Or, from the Pinetone chromatic with flats (mode 3), we add another Pinetone diatonic scale, mode 0 starting on D♯, leading to the right-handed Pinetone hyperchromatic scale, with step pattern, sLssLsLssLmsLssLsLs.
|major 3rd
 
|augmented 3rd
If 81/80 were additionally tempered out (tempering out the difference between the small step and the medium step), these scales would temper to Flattone[19], reflected in their layout on the lumatone. These scale comprises 7 large steps approximating 117/110 (the difference between the large and small steps of the Pinetone chromatic), the medium step of the Pinetone chromatic, approximating 25/24, 33/32, and 27/26, and 11 small steps, the same as the small step of the pinetone chromatic, approximating 250/243, 55/54, 121/120, and 40/39.
|augmented 3rd
 
|large 3rd
We note that sLss, the interval from D to E♯, for example, is very near 9/8, and that sLsL, the interval from D to F♭, for an example, is very near 32/27. If we recognize these approximates, we additionally temper out 243/242, or 352/351, leading to [[Tetracot]] temperament, in which case the large step approximates 16/15. This also adds 81/80 to the list of intervals approximated by the small step. Adding an additional small step above G, for the left handed hyperchromatic, or below A, for the right handed hyperchromatic, would give us a MODMOS of Tetracot[20], splitting the one medium step into two small steps (we note also that TE 2.3.5.11.13 ptolemismic tunes the medium step to 66.76626, which is almost exactly twice the size of its small step of 33.11646c).  
| 33/26, 50/39
 
|418.550
In 2.3.5.11.13 Tetracot, the left-handed Pinetone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 32/27 6/5 11/9 13/10 4/3 11/8 22/15 3/2 20/13 13/8 5/3 16/9 9/5 11/6 39/20 2/1, and the right-handed Pinetone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 9/8 6/5 11/9 13/10 4/3 15/11 13/9 3/2 20/13 13/8 5/3 27/16 9/5 11/6 39/20 2/1.
|1
 
|-
Tuned to [http://x31eq.com/cgi-bin/rt.cgi?ets=7%2613cee&limit=2.3.5.11.13 TE 2.3.5.11.13 Tetracot] (with a large step of 109.3262 and a small step of 33.3391c), the left-handed Pinetone hyperchromatic in cents is
! rowspan="3" |3-step
 
|m + 2s
33.3391 142.6653 176.0044 285.3306 318.6697 352.0088 461.335 494.6741 561.3532 670.6785 704.0176 737.3567 846.6829 880.022 989.3482 1022.6873 1056.0264 1165.3526 1198.6917,
|minor 4th
 
|diminished 4th
and the right-handed Pinetone hyperchromatic in cents is
|diminished 4th
 
|small 4th
33.3391 142.6653 176.0044 209.3435 318.6697 352.0088 461.335 494.6741 528.0132 637.3394 704.0176 737.3567 846.6829 880.022 913.3611 1022.6873 1056.0264 1165.3526 1198.6917.
|13/10
 
|461.433
The Pinetone hyperchromatic scales may alternatively be tuned to 27edo, 34edo, or 41edo:
|2
 
|-
27edo: 7L 1m 11s = (2, 2, 1) = (88.8889c, 88.8889c, 44.4444c)
|2m + s
 
|minor 4th
34edo: 7L 1m 11s = (3, 2, 1) = (105.8824c, 70.5882c, 35.2941c)
|perfect 4th
 
|minor 4th
41edo: 7L 1m 11s = (4, 2, 1) = (117.0732c, 58.5366c, 29.2683c).
|middle 4th
 
|4/3, 33/25
==Pinetone-15==
|494.559
Alternatively, a 15-note scale can be built from the Pinetone diminished. The resulting scale tempers to Porcupine[15], as well as to Hanson[15].
|2
 
|-
From Pinetone diminished scale: MLsLMLML, shown in the bright minor mode as Pinetone bright minor diminished, putting a small step into the bottom of each medium and large step leads to the child SNS of the Pinetone diminished scale: the fifteen note SNS msmLmmLmsmLmsmL, or mLmsmLmsmLmsmLm in it's symmetric mode, comprising 4 large steps of 16/15, 8 medium steps of 25/24 and 3 small steps of 648/625, i.e., 
|L + m + s
 
|major 4th
25/24 10/9 125/108 6/5 5/4 4/3 25/18 36/25 3/2 8/5 5/3 216/125 9/5 48/25 2/1.
|augmented 4th
 
|major 4th
Tempering m = s (tempering out 15625/15552, the Hanson comma) results in sLsssLsssLsssLs, which is Hanson[15];
|large 4th
 
|25/18, 11/8, 18/13
tempering L = m (tempering out 128/125, the Augmented comma) results in LLLsLLLsLLLsLLL, which is a MODMOS of Augmented[15]
| 561.325
 
|3
tempering L = s (tempering out 250/243, the Porcupine comma) results in sLsLsLsLsLsLsLs, which is Porcupine[15].
|-
! rowspan="3" |4-step
|2m + 2s
|minor 5th
|diminished 5th
|minor 5th
|small 5th
|36/25, 16/11, 13/9
| 637.334
|3
|-
|L + m + 2s
|major 5th
|perfect 5th
|major 5th
|medium 5th
| 3/2, 50/33
|704.524
|2
|-
|L + 2m + s
|major 5th
|augmented 5th
|augmented 5th
|large 6th
|20/13
|737.217
|2
|-
! rowspan="3" | 5-step
|3m + 2s
|minor 6th
|diminished 6th
|diminished 6th
|small 6th
|39/25, 52/33
| 780.120
|1
|-
|L + 2m + 2s
|major 6th
|minor 6th
| major-minor 6th
|medium 6th
|8/5
|813.227
|1
|-
|L + 3m + s
|major 6th
|major 6th
|major 6th
|large 6th
|5/3, 33/20
|879.992
|6
|-
! rowspan="3" |6-step
|3m + 3s
|minor 7th
|diminished 7th
| diminished 7th
|small 7th
|256/125, 55/24, 26/15
|956.002
|1
|-
|L + 2m + 3s
|major 7th
|minor 7th
|major-minor 7th
|medium 7th
|9/5, 20/11
|1022.768
|3
|-
|L + 3m + 2s
|major 7th
|major 7th
|major 7th
|large 7th
|11/6, 50/27
|1055.884
|3
|}


tempering out s would lead to sLLsLsL, which is Dicot[7];
==Summary for xen-math nerds==
Pinetone scales are built via step nesting from the 5-limit minor seventh tetrad: 6/5 3/2 9/5 2/1. The bounds for its scales are the set of temperings of the rank-3 step-nested children of the 4-note SNS 6/5 3/2 9/5 2/1. Pinetone diminished scales are built via step-nesting from the 5-limit diminished tetrad: 6/5 36/25 5/3 2/1. The bounds for its scales are the set of temperings of the rank-3 step-nested children of the 4-note MOS scale 6/5 36/25 5/3 2/1.


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


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.
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).


With [http://x31eq.com/cgi-bin/rt.cgi?ets=4f%263f%268&limit=2.3.5.11.13 TE 2.3.5.11.13 Ptolemismic tuning applied], the sizes of the steps shift enough for the size order to change. Pinetone-15 comprises
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'').


4 large steps of 109.12557c, approximating 16/15;  
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 diminished scale is a [[step-nested scale]] and a [[Porcupine]][8] x Diminished[8] [[Fokker block]] with [[Unison vector|unison vectors]] of 250/243, 648/625, and 16/15; comprising 4 large steps of 10/9 (''L'' x ''L''), 3 medium steps of 27/25 (''L'' x ''s''), and one small step of 25/24 (''s'' x ''s'').


3 medium steps of 76.00911c, approximating 648/625, 128/121, and 26/25; and
*


8 small steps of 66.76626c, approximating 25/24, 33/32, and 27/26.
==Pinetone harmonic minor and harmonic major==
Additionally, we have another set of [[Porcupine]][7] modes contained in the Pinetone harmonic octatonics: Replacing the G with the G♯ changes the mode of the Porcupine[7] scale represented, and replaces diatonic with harmonic minor modes for the [[Meantone]][7] scale represented, now a MODMOS.  


In cents, TE 2.3.5.11.13 Ptolemismic Pinetone-15, in the symmetric mode, is
We note that there are fewer consonant triads available in these scales than in the Pinetone diatonic and octatonic scales, so they may be useful for melody only.


66.766 175.892 242.658 318.667 385.433 494.559 561.325 637.334 704.101 813.226 879.993 956.002 1022.768 1131.893 1198.660 as sLsmsLsmsLsmsLs.
On D we get the scale:


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.
174.055 320.69 557.888 704.524 878.579 1025.214 1199.269 as the notes D E F G♯ A B C D, representing 10/9 6/5 25/18~11/8 3/2 5/3 9/5 2/1


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.
We get the following 7 modes of Pinetone harmonic minor scale:


2.3.5.13 325/324 may be better tuned to 46edo, with (L, m, s) = (4, 2, 1).
*Lsmsmms Lydian ♯2 bright major starting on F
*mmsLsms Ionian ♯5 symmetric minor starting on C
*msLsmsm Ukranian dorian bright minor starting on D
*sLsmsmm Phyrgian dominant dark major starting on E
*msmmsLs harmonic minor dark diminished starting on A
*smmsLsm Locrian ♮6 bright diminished starting on B
*smsmmsL altered diminished magical seventh starting on G♯


===Pinetone harmonic diminished octatonic===
Replacing the A with an A♭ instead, we get the modes of the Pinetone harmonic major scale. Starting on D we get the mode:
As a subset of Pinetone-15 we may find modified Pinetone octatonics built on MODMOS of Porcupine[8]. The Porcupine[8]'s [[4M]] (''minimally modified MODMOS'') is useful given that it still comprises consonant 3-step triads on all notes, but with a more spread-out distribution, so that the triads of each type do not all occur adjacent to each other as in Porcupine[7] and Porcupine[8]. This scale may be found either by lowering G or raising B by a Porcupine[8] chroma, which represents 16/15, the large step of Pinetone-15.  


2.3.5.11.13 ptolemismic Pinetone-15 has simplest JI pre-image 25/24 10/9 15/13 6/5 5/4 4/3 11/8 13/9 3/2 8/5 5/3 26/15 9/5 48/25 2/1 as sLsmsLsmsLsmsLs which may be grouped as a detempered Porcupine[8], large step of ''sL'' or ''sm'', small step of ''s'' in the following ways: 12222222, 21222222, 22122222, 22212222, 22221222, 22222122, 22222212, 22222221. Using
174.055 320.69 494.745 641.38 878.579 1025.214 1199.269 as the notes D E F G A♭ B C D, representing 10/9 6/5 4/3 36/25~13/9 5/3 9/5 2/1.  


22122222 -> 13122222, 22131222, 22221312 i.e., s(Lsm)s(Ls)(ms)(Ls)(ms)(Ls) or (sL)(sm)(sL)(sm)(sL)s(msL)s, (sL)(sm)s(Lsm)s(Ls)(ms)(Ls) or (sL)(sm)(sL)s(msL)s(ms)(Ls), (sL)(sm)(sL)(sm)s(Lsm)s(Ls) or (sL)s(msL)s(ms)(Ls)(ms)(Ls), 6 modes of 2 scales.
Which has modes:


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, 2 small steps of 25/24~33/32~27/26.
*Lsmmsms Lydian Augmented ♯2 bright major starting on A♭
{| class="wikitable"
*msLsmms Lydian ♭3 bright minor starting on F
|+Modes of the just Pinetone harmonic diminished
*sLsmmsm Mixolydian ♭2 dark major starting on G
! Mode (height order)
*mmsmsLs harmonic major bright diminished starting on C
!Step pattern
*msmsLsm Dorian ♭5 dark diminished starting on D
!Oneirotonic step pattern
*smsLsmm Phrygian ♭4 symmetric minor starting on E
!Porcupine[8] step pattern
*smmsmsL Locrian magical ♭♭7 starting on B
!Mode in 5-limit JI
We can see that this scale differs from the Pinetone diminished heptatonic by only a single note -  9/5 instead of 50/27.
! Comments
 
|-
The augmented step of the Pinetone harmonic minor and major scales is the same as of the Pinetone diminished heptatonic, representing 15/13 when 325/324 is tempered out (the difference between 100/99 and 144/143).
|[https://xenpaper.com/#%7B1%2F1_144%2F125_6%2F5_4%2F3_36%2F25_8%2F5_216%2F125_48%2F25_2%2F1_288%2F125_12%2F5_8%2F3_72%2F25_16%2F5_432%2F125%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Hlanithian diminished]
 
|AsLMLMLs
The Pintone harmonic minor and harmonic minor have step patterns msmmsLs, and mmsmsLs respectively, or, represented as MODMOS of detempered Meantone[7], LsLLsAs and LLsLsAs, the step patterns of the familiar harmonic minor and harmonic major scales.
|sLLsLsLL (Hlanithian)
 
|AsLLLLLs
==Pinetone hyperchromatic scales==
|144/125 6/5 4/3 36/25 8/5 216/125 48/25 2/1
Maybe you have a Lumatone, and you're wondering, ok so you can either have sharps or flats? Por queno los dos?
|
 
|-
Indeed we can have both!
|[https://xenpaper.com/#%7B1%2F1_10%2F9_6%2F5_4%2F3_36%2F25_8%2F5_5%2F3_48%2F25_2%2F1_20%2F9_12%2F5_8%2F3_72%2F25_16%2F5_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Mnarian diminished]*
 
|LMLMLsAs
From the Pinetone chromatic with sharps (mode -3), we add another Pinetone diatonic scale, mode 0 starting on D♭, leading to the left-handed Pinetone hyperchromatic scale, with step pattern, sLsLssLsmLssLsLssLs.
|LsLsLLsL (Mnarian)
 
|LLLLLsAs
Or, from the Pinetone chromatic with flats (mode 3), we add another Pinetone diatonic scale, mode 0 starting on D♯, leading to the right-handed Pinetone hyperchromatic scale, with step pattern, sLssLsLssLmsLssLsLs.
|10/9 6/5 4/3 36/25 8/5 5/3 48/25 2/1
 
|
If 81/80 were additionally tempered out (tempering out the difference between the small step and the medium step), these scales would temper to Flattone[19], reflected in their layout on the lumatone. These scale comprises 7 large steps approximating 117/110 (the difference between the large and small steps of the Pinetone chromatic), the medium step of the Pinetone chromatic, approximating 25/24, 33/32, and 27/26, and 11 small steps, the same as the small step of the pinetone chromatic, approximating 250/243, 55/54, 121/120, and 40/39.
|-
 
|[https://xenpaper.com/#%7B1%2F1_10%2F9_6%2F5_4%2F3_25%2F18_8%2F5_5%2F3_50%2F27_2%2F1_20%2F9_12%2F5_8%2F3_25%2F9_16%2F5_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Celephaïsian diminished]*
We note that sLss, the interval from D to E♯, for example, is very near 9/8, and that sLsL, the interval from D to F♭, for an example, is very near 32/27. If we recognize these approximates, we additionally temper out 243/242, or 352/351, leading to [[Tetracot]] temperament, in which case the large step approximates 16/15. This also adds 81/80 to the list of intervals approximated by the small step. Adding an additional small step above G, for the left handed hyperchromatic, or below A, for the right handed hyperchromatic, would give us a MODMOS of Tetracot[20], splitting the one medium step into two small steps (we note also that TE 2.3.5.11.13 ptolemismic tunes the medium step to 66.76626, which is almost exactly twice the size of its small step of 33.11646c).
|LMLsAsLM
 
|LsLLsLLs (Celephaïsian)
In 2.3.5.11.13 Tetracot, the left-handed Pinetone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 32/27 6/5 11/9 13/10 4/3 11/8 22/15 3/2 20/13 13/8 5/3 16/9 9/5 11/6 39/20 2/1, and the right-handed Pinetone hyperchromatic approximates the JI ratios 40/39 12/11 10/9 9/8 6/5 11/9 13/10 4/3 15/11 13/9 3/2 20/13 13/8 5/3 27/16 9/5 11/6 39/20 2/1.
|LLLsAsLL
 
|10/9 6/5 4/3 25/18 8/5 5/3 50/27 2/1
Tuned to [http://x31eq.com/cgi-bin/rt.cgi?ets=7%2613cee&limit=2.3.5.11.13 TE 2.3.5.11.13 Tetracot] (with a large step of 109.3262 and a small step of 33.3391c), the left-handed Pinetone hyperchromatic in cents is
|
 
|-
33.3391 142.6653 176.0044 285.3306 318.6697 352.0088 461.335 494.6741 561.3532 670.6785 704.0176 737.3567 846.6829 880.022 989.3482 1022.6873 1056.0264 1165.3526 1198.6917,
|[https://xenpaper.com/#%7B1%2F1_10%2F9_125%2F108_4%2F3_25%2F18_125%2F81_5%2F3_50%2F27_2%2F1_20%2F9_125%2F54_8%2F3_25%2F9_250%2F81_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Dylathian diminished]*
 
|LsAsLMLM
and the right-handed Pinetone hyperchromatic in cents is
|LLsLLsLs (Dylathian)
 
|LsAsLLLL
33.3391 142.6653 176.0044 209.3435 318.6697 352.0088 461.335 494.6741 528.0132 637.3394 704.0176 737.3567 846.6829 880.022 913.3611 1022.6873 1056.0264 1165.3526 1198.6917.
|10/9 125/108 4/3 25/18 125/81 5/3 50/27 2/1
 
|
The Pinetone hyperchromatic scales may alternatively be tuned to 27edo, 34edo, or 41edo:
|-
 
|[https://xenpaper.com/#%7B1%2F1_27%2F25_6%2F5_162%2F125_36%2F25_3%2F2_216%2F125_9%2F5_2%2F1_54%2F25_12%2F5_324%2F125_72%2F25_3%2F1_432%2F125%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Sarnathian diminished]<sup>†</sup>
27edo: 7L 1m 11s = (2, 2, 1) = (88.8889c, 88.8889c, 44.4444c)
|MLMLsAsL
 
|sLsLLsLL (Sarnathian)
34edo: 7L 1m 11s = (3, 2, 1) = (105.8824c, 70.5882c, 35.2941c)
|LLLLsAsL
 
|27/25 6/5 162/125 36/25 3/2 216/125 9/5 2/1
41edo: 7L 1m 11s = (4, 2, 1) = (117.0732c, 58.5366c, 29.2683c).
|10:12:15 on the root
 
|-
==Pinetone-15==
|[https://xenpaper.com/#%7B1%2F1_27%2F25%C2%A06%2F5_5%2F4_36%2F25_3%2F2_5%2F3_9%2F5_2%2F1_54%2F25_12%2F5_5%2F2_72%2F25_3%2F1_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Kadathian diminished]*<sup>††</sup>
Alternatively, a 15-note scale can be built from the Pinetone diminished. The resulting scale tempers to Porcupine[15], as well as to Hanson[15].
|MLsAsLML
 
|sLLsLLsL (Kadathian)
From Pinetone diminished scale: MLsLMLML, shown in the bright minor mode as Pinetone bright minor diminished, putting a small step into the bottom of each medium and large step leads to the child SNS of the Pinetone diminished scale: the fifteen note SNS msmLmmLmsmLmsmL, or mLmsmLmsmLmsmLm in it's symmetric mode, comprising 4 large steps of 16/15, 8 medium steps of 25/24 and 3 small steps of 648/625, i.e., 
|LLsAsLLL
 
|27/25 6/5 5/4 36/25 3/2 5/3 9/5 2/1
25/24 10/9 125/108 6/5 5/4 4/3 25/18 36/25 3/2 8/5 5/3 216/125 9/5 48/25 2/1.
|root 4:5:6,10:12:15
 
|-
Tempering m = s (tempering out 15625/15552, the Hanson comma) results in sLsssLsssLsssLs, which is Hanson[15];
|[https://xenpaper.com/#%7B1%2F1_25%2F24_6%2F5_5%2F4_25%2F18_3%2F2_5%2F3_9%2F5_2%2F1_25%2F12_12%2F5_5%2F2_25%2F9_3%2F1_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Ultharian diminished]*<sup>††</sup>
 
|sAsLMLML
tempering L = m (tempering out 128/125, the Augmented comma) results in LLLsLLLsLLLsLLL, which is a MODMOS of Augmented[15]
|LsLLsLsL (Ultharian)
 
|sAsLLLLL
tempering L = s (tempering out 250/243, the Porcupine comma) results in sLsLsLsLsLsLsLs, which is Porcupine[15].
|25/24 6/5 5/4 25/18 3/2 5/3 9/5 2/1
 
|root 4:5:6,10:12:15
tempering out s would lead to sLLsLsL, which is Dicot[7];
|-
 
|[https://xenpaper.com/#%7B1%2F1_25%2F24_125%2F108_5%2F4_25%2F18_3%2F2_5%2F3_125%2F72_2%2F1_25%2F12_125%2F54_5%2F2_25%2F9_3%2F1_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Illarnekian diminished]*<sup>†</sup>
tempering out m would lead to ssLsLssLsssL, which is a MODMOS of Diminished[12].
|sLMLMLsA
 
|LLsLsLLs (Illarnekian)
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.
|sLLLLLsA
 
|25/24 125/108 5/4 25/18 3/2 5/3 125/72 2/1
With [http://x31eq.com/cgi-bin/rt.cgi?ets=4f%263f%268&limit=2.3.5.11.13 TE 2.3.5.11.13 Ptolemismic tuning applied], the sizes of the steps shift enough for the size order to change. Pinetone-15 comprises
| root 4:5:6
 
|}
4 large steps of 109.12557c, approximating 16/15;
{| class="wikitable"
 
|+Modes of the Ptolemismic Pinetone harmonic diminished
3 medium steps of 76.00911c, approximating 648/625, 128/121, and 26/25; and
! Mode (height order)
 
!Step pattern
8 small steps of 66.76626c, approximating 25/24, 33/32, and 27/26.
!Mode as simplest JI pre-image 5-limit JI
 
! Mode in cents
In cents, TE 2.3.5.11.13 Ptolemismic Pinetone-15, in the symmetric mode, is
!Comments
 
66.766 175.892 242.658 318.667 385.433 494.559 561.325 637.334 704.101 813.226 879.993 956.002 1022.768 1131.893 1198.660 as sLsmsLsmsLsmsLs.
 
Accordingly Pinetone-15 would temper to two step sizes in 19edo (Hanson), 22edo (Porcupine), 34edo (Hanson), and 27edo (Augmented). If we wish to keep the 3-step size structure, we can tune to 26edo or 41edo with (L, m, s) = (3, 2, 1), and (4, 3, 2) respectively.
 
Tempering out the 325/324, the difference between 100/99 and 144/143 rather than both of 100/99 and 144/143 leads to a more accurate temperament that does not include the whole 2.3.5.11.13 subgroup., rather just the 2.3.5.13 subgroup. The simplest JI pre-image in this temperament would be 25/24 10/9 15/13 6/5 5/4 4/3 18/13 13/9 3/2 8/5 5/3 26/15 9/5 48/25 2/1, which differs only from the simplest pre-image of the scale under 2.3.5.11.13 Ptolemismic tempering by the inclusion of 18/13 rather than 11/8.
 
2.3.5.13 325/324 may be better tuned to 46edo, with (L, m, s) = (4, 2, 1).
 
===Pinetone harmonic diminished octatonic===
As a subset of Pinetone-15 we may find modified Pinetone octatonics built on MODMOS of Porcupine[8]. The Porcupine[8]'s [[4M]] (''minimally modified MODMOS'') is useful given that it still comprises consonant 3-step triads on all notes, but with a more spread-out distribution, so that the triads of each type do not all occur adjacent to each other as in Porcupine[7] and Porcupine[8]. This scale may be found either by lowering G or raising B by a Porcupine[8] chroma, which represents 16/15, the large step of Pinetone-15.
 
2.3.5.11.13 ptolemismic Pinetone-15 has simplest JI pre-image 25/24 10/9 15/13 6/5 5/4 4/3 11/8 13/9 3/2 8/5 5/3 26/15 9/5 48/25 2/1 as sLsmsLsmsLsmsLs which may be grouped as a detempered Porcupine[8], large step of ''sL'' or ''sm'', small step of ''s'' in the following ways: 12222222, 21222222, 22122222, 22212222, 22221222, 22222122, 22222212, 22222221. Using
 
22122222 -> 13122222, 22131222, 22221312 i.e., s(Lsm)s(Ls)(ms)(Ls)(ms)(Ls) or (sL)(sm)(sL)(sm)(sL)s(msL)s, (sL)(sm)s(Lsm)s(Ls)(ms)(Ls) or (sL)(sm)(sL)s(msL)s(ms)(Ls), (sL)(sm)(sL)(sm)s(Lsm)s(Ls) or (sL)s(msL)s(ms)(Ls)(ms)(Ls), 6 modes of 2 scales.
 
The first pair of modes temper to Diminished[8] (m = 0), and to Father[8] (L=0). The scale has 1 augmented step of 52/45, 3 large steps of 10/9~11/10, 2 medium steps of 12/11~13/12, 2 small steps of 25/24~33/32~27/26.
{| class="wikitable"
|+Modes of the just Pinetone harmonic diminished
! Mode (height order)
!Step pattern
!Oneirotonic step pattern
!Porcupine[8] step pattern
!Mode in 5-limit JI
! Comments
|-
|-
|[https://xenpaper.com/#%7B0c_251.901c_318.667c_494.559c_637.334c_813.226c_956.002c_1131.893c_1198.660c_1450.561c_1517.327c_1693.219c_1835.994c_2011.886c_2154.661c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Hlanithian diminished]
|[https://xenpaper.com/#%7B1%2F1_144%2F125_6%2F5_4%2F3_36%2F25_8%2F5_216%2F125_48%2F25_2%2F1_288%2F125_12%2F5_8%2F3_72%2F25_16%2F5_432%2F125%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Hlanithian diminished]
|AsLMLMLs
|AsLMLMLs
|~ 52/45 6/5 4/3 13/9 8/5 26/15 48/25 2/1
|sLLsLsLL (Hlanithian)
|251.901 318.667 494.559 637.334 813.226 956.002 1131.893 1198.660
|AsLLLLLs
|144/125 6/5 4/3 36/25 8/5 216/125 48/25 2/1
|
|
|-
|-
|[https://xenpaper.com/#%7B0c_175.892c_318.667c_494.559c_637.334c_813.226c_879.993c_1131.893c_1198.660c_1374.551c_1517.327c_1693.219c_1835.994c_2011.886c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Mnarian diminished]*
|[https://xenpaper.com/#%7B1%2F1_10%2F9_6%2F5_4%2F3_36%2F25_8%2F5_5%2F3_48%2F25_2%2F1_20%2F9_12%2F5_8%2F3_72%2F25_16%2F5_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Mnarian diminished]*
|LMLMLsAs
|LMLMLsAs
|~ 10/9 6/5 4/3 13/9 8/5 5/3 48/25 2/1
|LsLsLLsL (Mnarian)
|175.892 318.667 494.559 637.334 813.226 879.993 1131.893 1198.660
|LLLLLsAs
|10/9 6/5 4/3 36/25 8/5 5/3 48/25 2/1
|
|
|-
|-
|[https://xenpaper.com/#%7B0c_175.892c_318.667c_494.559c_561.325c_813.226c_879.993c_1055.884c_1198.660c_1374.551c_1517.327c_1693.219c_1759.985c_2011.886c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Celephaïsian diminished]*
|[https://xenpaper.com/#%7B1%2F1_10%2F9_6%2F5_4%2F3_25%2F18_8%2F5_5%2F3_50%2F27_2%2F1_20%2F9_12%2F5_8%2F3_25%2F9_16%2F5_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Celephaïsian diminished]*
|LMLsAsLM
|LMLsAsLM
|~ 10/9 6/5 4/3 11/8 8/5 5/3 11/6 2/1
|LsLLsLLs (Celephaïsian)
|175.892 318.667 494.559 561.325 813.226 879.993 1055.884 1198.660
|LLLsAsLL
|10/9 6/5 4/3 25/18 8/5 5/3 50/27 2/1
|
|
|-
|-
|[https://xenpaper.com/#%7B0c_175.892c_242.658c_494.559c_561.325c_737.217c_879.993c_1055.884c_1198.660c_1374.551c_1441.318c_1693.219c_1759.985c_1935.877c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Dylathian diminished]*
|[https://xenpaper.com/#%7B1%2F1_27%2F25_6%2F5_162%2F125_36%2F25_3%2F2_216%2F125_9%2F5_2%2F1_54%2F25_12%2F5_324%2F125_72%2F25_3%2F1_432%2F125%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Sarnathian diminished]<sup>†</sup>
|MLMLsAsL
|sLsLLsLL (Sarnathian)
|LLLLsAsL
|27/25 6/5 162/125 36/25 3/2 216/125 9/5 2/1
|10:12:15 on the root
|-
|[https://xenpaper.com/#%7B1%2F1_10%2F9_125%2F108_4%2F3_25%2F18_125%2F81_5%2F3_50%2F27_2%2F1_20%2F9_125%2F54_8%2F3_25%2F9_250%2F81_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Dylathian diminished]*
|LsAsLMLM
|LsAsLMLM
|~ 10/9 15/13 4/3 11/8 20/13 5/3 11/6 2/1
|LLsLLsLs (Dylathian)
|175.892 242.658 494.559 561.325 737.217 879.993 1055.884 1198.660
|LsAsLLLL
|10/9 125/108 4/3 25/18 125/81 5/3 50/27 2/1
|
|
|-
|-
|[https://xenpaper.com/#%7B0c_142.775c_318.667c_461.443c_637.334c_704.101c_956.002c_1022.768c_1198.660c_1341.435c_1517.327c_1660.322c_1835.994c_1902.760c_2154.661c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Sarnathian diminished]<sup>†</sup>
|[https://xenpaper.com/#%7B1%2F1_27%2F25%C2%A06%2F5_5%2F4_36%2F25_3%2F2_5%2F3_9%2F5_2%2F1_54%2F25_12%2F5_5%2F2_72%2F25_3%2F1_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Kadathian diminished]*<sup>††</sup>
|MLMLsAsL
|~ 12/11 6/5 13/10 13/9 3/2 26/15 9/5 2/1
|142.775 318.667 461.443 637.334 704.101 956.002 1022.768 1198.660
|10:12:15 on the root
|-
|[https://xenpaper.com/#%7B0c_142.775c_318.667c_385.433c_637.334c_704.101c_879.993c_1022.768c_1198.660c_1341.435c_1517.327c_1584.903c_1835.994c_1902.760c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Kadathian diminished]*<sup>††</sup>
|MLsAsLML
|MLsAsLML
|~ 12/11 6/5 5/4 13/9 3/2 5/3 9/5 2/1
|sLLsLLsL (Kadathian)
|142.775 318.667 385.433 637.334 704.101 879.993 1022.768 1198.660
|LLsAsLLL
|27/25 6/5 5/4 36/25 3/2 5/3 9/5 2/1
|root 4:5:6,10:12:15
|root 4:5:6,10:12:15
|-
|-
|[https://xenpaper.com/#%7B0c_66.766c_318.667c_385.433c_561.325c_704.101c_879.993c_1022.768c_1198.660c_1265.426c_1517.327c_1584.903c_1759.985c_1902.760c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Ultharian diminished]*<sup>††</sup>
|[https://xenpaper.com/#%7B1%2F1_25%2F24_6%2F5_5%2F4_25%2F18_3%2F2_5%2F3_9%2F5_2%2F1_25%2F12_12%2F5_5%2F2_25%2F9_3%2F1_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Ultharian diminished]*<sup>††</sup>
|sAsLMLML
|sAsLMLML
|~ 25/24 6/5 5/4 11/8 3/2 5/3 9/5 2/1
|LsLLsLsL (Ultharian)
|66.766 318.667 385.433 561.325 704.101 879.993 1022.768 1198.660
|sAsLLLLL
|25/24 6/5 5/4 25/18 3/2 5/3 9/5 2/1
|root 4:5:6,10:12:15
|root 4:5:6,10:12:15
|-
|-
|[https://xenpaper.com/#%7B0c_66.766c_242.658c_385.433c_561.325c_704.101c_879.993c_946.759c_1198.660c_1265.426c_1441.318c_1584.903c_1759.985c_1902.760c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Illarnekian diminished]*<sup>†</sup>
|[https://xenpaper.com/#%7B1%2F1_25%2F24_125%2F108_5%2F4_25%2F18_3%2F2_5%2F3_125%2F72_2%2F1_25%2F12_125%2F54_5%2F2_25%2F9_3%2F1_10%2F3%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Illarnekian diminished]*<sup>†</sup>
|sLMLMLsA
|sLMLMLsA
|~ 25/24 15/13 5/4 11/8 3/2 5/3 45/26 2/1
|LLsLsLLs (Illarnekian)
|66.766 242.658 385.433 561.325 704.101 879.993 946.759 1198.660
|sLLLLLsA
|root 4:5:6
|25/24 125/108 5/4 25/18 3/2 5/3 125/72 2/1
| root 4:5:6
|}
|}
{| class="wikitable"
{| class="wikitable"
|+Intervals of the Ptolemismic Pinetone harmonic diminished (variety = 20)
|+Modes of the Ptolemismic Pinetone harmonic diminished
!Interval class
! Mode (height order)
!sizes
!Step pattern
!Diminished[8] name
!Mode as simplest JI pre-image 5-limit JI
!Porcupine[8] name
! Mode in cents
!Pinetone octatonic name
!Comments
!JI ratios approximated*
!size in cents (TE)
! Occurence
|-
|-
! rowspan="4" |1-step
|[https://xenpaper.com/#%7B0c_251.901c_318.667c_494.559c_637.334c_813.226c_956.002c_1131.893c_1198.660c_1450.561c_1517.327c_1693.219c_1835.994c_2011.886c_2154.661c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Hlanithian diminished]
|s
|AsLMLMLs
| minor step
|~ 52/45 6/5 4/3 13/9 8/5 26/15 48/25 2/1
|minor step
|251.901 318.667 494.559 637.334 813.226 956.002 1131.893 1198.660
|minor step
|
|25/24, 33/32, 27/26
|66.766
|2
|-
|-
|M
|[https://xenpaper.com/#%7B0c_175.892c_318.667c_494.559c_637.334c_813.226c_879.993c_1131.893c_1198.660c_1374.551c_1517.327c_1693.219c_1835.994c_2011.886c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Mnarian diminished]*
|minor step
|LMLMLsAs
|major step
|~ 10/9 6/5 4/3 13/9 8/5 5/3 48/25 2/1
|minor-major step
|175.892 318.667 494.559 637.334 813.226 879.993 1131.893 1198.660
|27/25, 12/11, 13/12
|
|142.775
|2
|-
|-
|L
|[https://xenpaper.com/#%7B0c_175.892c_318.667c_494.559c_561.325c_813.226c_879.993c_1055.884c_1198.660c_1374.551c_1517.327c_1693.219c_1759.985c_2011.886c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Celephaïsian diminished]*
|major step
|LMLsAsLM
|major step
|~ 10/9 6/5 4/3 11/8 8/5 5/3 11/6 2/1
|major step
|175.892 318.667 494.559 561.325 813.226 879.993 1055.884 1198.660
|10/9, 11/10
|
|175.892
|3
|-
|-
|A
|[https://xenpaper.com/#%7B0c_142.775c_318.667c_461.443c_637.334c_704.101c_956.002c_1022.768c_1198.660c_1341.435c_1517.327c_1660.322c_1835.994c_1902.760c_2154.661c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Sarnathian diminished]<sup>†</sup>
|major step
|MLMLsAsL
|augmented step
|~ 12/11 6/5 13/10 13/9 3/2 26/15 9/5 2/1
|augmented step
|142.775 318.667 461.443 637.334 704.101 956.002 1022.768 1198.660
|144/125, 64/55, 52/45
|10:12:15 on the root
|251.901
|1
|-
|-
! rowspan="2" |2-step
|[https://xenpaper.com/#%7B0c_175.892c_242.658c_494.559c_561.325c_737.217c_879.993c_1055.884c_1198.660c_1374.551c_1441.318c_1693.219c_1759.985c_1935.877c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Dylathian diminished]*
|L + s
|LsAsLMLM
|perfect 2-step
|~ 10/9 15/13 4/3 11/8 20/13 5/3 11/6 2/1
|minor 2-step
|175.892 242.658 494.559 561.325 737.217 879.993 1055.884 1198.660
|minor 2-step
|
|15/13 (7/6 or 8/7)
|242.658
|2
|-
|-
|L + M = A + s
|[https://xenpaper.com/#%7B0c_142.775c_318.667c_385.433c_637.334c_704.101c_879.993c_1022.768c_1198.660c_1341.435c_1517.327c_1584.903c_1835.994c_1902.760c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Kadathian diminished]*<sup>††</sup>
|perfect 2-step
|MLsAsLML
|major 2-step
|~ 12/11 6/5 5/4 13/9 3/2 5/3 9/5 2/1
|major 2-step
|142.775 318.667 385.433 637.334 704.101 879.993 1022.768 1198.660
|6/5
|root 4:5:6,10:12:15
|318.667
|6
|-
|-
! rowspan="3" |3-step
|[https://xenpaper.com/#%7B0c_66.766c_318.667c_385.433c_561.325c_704.101c_879.993c_1022.768c_1198.660c_1265.426c_1517.327c_1584.903c_1759.985c_1902.760c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Ultharian diminished]*<sup>††</sup>
|L + M + s = A + 2s
|sAsLMLML
| minor 3-step
|~ 25/24 6/5 5/4 11/8 3/2 5/3 9/5 2/1
|minor 3-step
|66.766 318.667 385.433 561.325 704.101 879.993 1022.768 1198.660
|minor 3-step
|root 4:5:6,10:12:15
|5/4
|385.433
|3
|-
|-
|L + 2M
|[https://xenpaper.com/#%7B0c_66.766c_242.658c_385.433c_561.325c_704.101c_879.993c_946.759c_1198.660c_1265.426c_1441.318c_1584.903c_1759.985c_1902.760c_2078.652c%7D0_1_2_3_4_5_6_7_8_7_6_5_4_3_2_1_0-.._0-_1-_2-_%5B0_3%5D-_%5B1_4%5D-_%5B2_5%5D-_%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D-_%5B2_5%5D-_%5B1_4%5D-_%5B0_3%5D-_2-_1-_0-.._%5B0_3_6%5D-_%5B1_4_7%5D-_%5B2_5_8%5D-_%5B3_6_9%5D-_%5B4_7_10%5D-_%5B5_8_11%5D-_%5B6_9_12%5D-_%5B7_10_13%5D-_%5B8_11_14%5D-_%5B7_10_13%5D-_%5B6_9_12%5D-_%5B5_8_11%5D-_%5B4_7_10%5D-_%5B3_6_9%5D-_%5B2_5_8%5D-_%5B1_4_7%5D-_%5B0_3_6%5D--- Illarnekian diminished]*<sup>†</sup>
|minor 3-step
|sLMLMLsA
|major 3-step
|~ 25/24 15/13 5/4 11/8 3/2 5/3 45/26 2/1
|minor-major 3-step
|66.766 242.658 385.433 561.325 704.101 879.993 946.759 1198.660
|13/10 (9/7 or 21/16)
|root 4:5:6
|461.433
|}
|1
{| class="wikitable"
|-
|+Intervals of the Ptolemismic Pinetone harmonic diminished (variety = 20)
|2L + M = A + L + s
!Interval class
|major 3-step
!sizes
|major 3-step
!Diminished[8] name
|major 3-step
!Porcupine[8] name
|4/3
!Pinetone octatonic name
|494.559
!JI ratios approximated*
|4
!size in cents (TE)
! Occurence
|-
|-
! rowspan="2" |4-step
! rowspan="4" |1-step
|2L + M + s = A + L + 2s
|s
| perfect 4-step
| minor step
|minor 4-step
|minor step
|minor 4-step
|minor step
|25/18, 11/8, 18/13
|25/24, 33/32, 27/26
|561.325
|66.766
|4
|2
|-
|-
|2L + 2M = A + L + M + s
|M
|perfect 4-step
|minor step
|major 4-step
|major step
|major 4-step
|minor-major step
|36/25, 16/11, 13/9
|27/25, 12/11, 13/12
|637.334
|142.775
|4
|2
|-
|-
! rowspan="3" |5-step
|L
|2L + 2M + s = A + L + M + 2s
|major step
| minor 5-step
|major step
|minor 5-step
|major step
|minor 5-step
|10/9, 11/10
|3/2
|175.892
|704.101
|3
|4
|-
|-
|A + 2L + 2s
|A
|major 5-step
|major step
|minor 5-step
|augmented step
|major-minor 5-step
|augmented step
|20/13 (14/9 or 32/16)
|144/125, 64/55, 52/45
|737.217
|251.901
|1
|1
|-
|-
|3L + 2M = A + 2L + M + s
! rowspan="2" |2-step
| major 5-step
|L + s
|major 5-step
|perfect 2-step
|major 5-step
|minor 2-step
|8/5
|minor 2-step
|813.227
|15/13 (7/6 or 8/7)
|3
|242.658
|2
|-
|-
! rowspan="2" |6-step
|L + M = A + s
|3L + 2M + s = A + 2L + M + 2s
|perfect 2-step
|perfect 6-step
|major 2-step
|minor 6-step
|major 2-step
|minor 6-step
|6/5
|5/3
|318.667
|879.992
|6
|6
|-
|-
|A + 2L + 2M + s
! rowspan="3" |3-step
|perfect 6-step
|L + M + s = A + 2s
|major 6-step
| minor 3-step
|major 6-step
|minor 3-step
|26/15 (12/7 or 7/4)
|minor 3-step
|956.002
|5/4
|2
|385.433
|3
|-
|-
! rowspan="4" |7-step
|L + 2M
|3L + 2M + 2s
|minor 3-step
|minor 7-step
|major 3-step
|diminished 7-step
|minor-major 3-step
|diminished 7-step
|13/10 (9/7 or 21/16)
|125/72, 55/32, 45/26
|461.433
|946.759
|1
|1
|-
|-
|A + 2L + 2M + 2s
|2L + M = A + L + s
|minor 7-step
|major 3-step
|minor 7-step
|major 3-step
|minor 7-step
|major 3-step
|9/5, 20/11
|4/3
|1022.768
|494.559
|3
|4
|-
|-
|A + 3L + M + 2s
! rowspan="2" |4-step
|major 7-step
|2L + M + s = A + L + 2s
|minor 7-step
| perfect 4-step
|major-minor 7-step
|minor 4-step
|50/27, 11/6, 24/13
|minor 4-step
|1055.884
|25/18, 11/8, 18/13
|2
|561.325
|4
|-
|2L + 2M = A + L + M + s
|perfect 4-step
|major 4-step
|major 4-step
|36/25, 16/11, 13/9
|637.334
|4
|-
|-
|4L + 3M
! rowspan="3" |5-step
|major 7-step
|2L + 2M + s = A + L + M + 2s
|major 7-step
| minor 5-step
|major 7-step
|minor 5-step
|48/25, 64/33, 52/27
|minor 5-step
|1131.983
|3/2
|704.101
|4
|-
|A + 2L + 2s
|major 5-step
|minor 5-step
|major-minor 5-step
|20/13 (14/9 or 32/16)
|737.217
|1
|-
|3L + 2M = A + 2L + M + s
| major 5-step
|major 5-step
|major 5-step
|8/5
|813.227
|3
|-
! rowspan="2" |6-step
|3L + 2M + s = A + 2L + M + 2s
|perfect 6-step
|minor 6-step
|minor 6-step
|5/3
|879.992
|6
|-
|A + 2L + 2M + s
|perfect 6-step
|major 6-step
|major 6-step
|26/15 (12/7 or 7/4)
|956.002
|2
|-
! rowspan="4" |7-step
|3L + 2M + 2s
|minor 7-step
|diminished 7-step
|diminished 7-step
|125/72, 55/32, 45/26
|946.759
|1
|-
|A + 2L + 2M + 2s
|minor 7-step
|minor 7-step
|minor 7-step
|9/5, 20/11
|1022.768
|3
|-
|A + 3L + M + 2s
|major 7-step
|minor 7-step
|major-minor 7-step
|50/27, 11/6, 24/13
|1055.884
|2
|-
|4L + 3M
|major 7-step
|major 7-step
|major 7-step
|48/25, 64/33, 52/27
|1131.983
|2
|2
|}<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 Pinetone harmonic diminished is the only Pinetone octatonic that is [[Rothenberg propriety|improper]], given that the augmented 2-step interval is larger than the minor 3-step interval (and, equivalently, the diminished 7-step interval is smaller than the major 6-step interval. A scale is ''proper'' if there are no interval larger than any intervals from a larger interval class. The Pinetone and Meantone diatonic scales are also proper (for any reasonable tuning).
The Pinetone harmonic diminished is the only Pinetone octatonic that is [[Rothenberg propriety|improper]], given that the augmented 2-step interval is larger than the minor 3-step interval (and, equivalently, the diminished 7-step interval is smaller than the major 6-step interval. A scale is ''proper'' if there are no interval larger than any intervals from a larger interval class. The Pinetone and Meantone diatonic scales are also proper (for any reasonable tuning).
{| class="wikitable"
|+
!Mode
!2nd
!3rd
!4th
!5th
!6th
!7th
!8th
!9ve
|-
|Hlanithian diminished
|Augmented
| rowspan="4" |Major
| rowspan="3" |Major
| rowspan="2" |Major
| rowspan="3" |Major
|Major
| rowspan="2" |Major
| rowspan="8" |Perfect
|-
|Mnarian diminished
| rowspan="2" |Major
| rowspan="2" |Minor
|-
|Celephaïsian diminished
|Minor
|Major-minor
|-
|Sarnathian diminished
|Minor-major
|Minor-major
|Major
|Minor
|Major
|Minor
|-
|Dylathian diminished
|Major
|Minor
|Major
|Minor
|Major-minor
| rowspan="4" |Minor
|Major-minor
|-
|Kadathian diminished
|Minor-major
| rowspan="2" |Major
| rowspan="3" |Minor
|Major
| rowspan="3" |Minor
| rowspan="2" |Minor
|-
|Ultharian diminished
| rowspan="2" |Minor
| rowspan="2" |Minor
|-
|Illarnekian diminished
|Minor
|Diminished
|}
{| class="wikitable"
{| class="wikitable"
|+3-step stacked triads of the Ptolemismic Pinetone harmonic diminished
|+3-step stacked triads of the Ptolemismic Pinetone harmonic diminished
Retrieved from "https://en.xen.wiki/w/Pinetone"