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Like Meantone[7] and Porcupine[7], and unlike the Ptolemy/Zarlino just major scale, the porcutone diatonic scale is ''mirror symmetric'', meaning that the mirror inverse of any mode of the scale is also a mode of the scale, i.e., if we trace the steps of the mode from the top instead of from the bottom. This is reflected with the mode numbers. The mirror inverse of mode 3, the brightest mode, is mode -3, the darkest mode, and mode 0 is itself a symmetric mode, hence 'symmetric' in the mode name. We may already know this - that the Dorian mode of the familiar diatonic scale is symmetric, and the mirror inverse of the Lydian mode is the Locrian mode.  
Like Meantone[7] and Porcupine[7], and unlike the Ptolemy/Zarlino just major scale, the porcutone diatonic scale is ''mirror symmetric'', meaning that the mirror inverse of any mode of the scale is also a mode of the scale, i.e., if we trace the steps of the mode from the top instead of from the bottom. This is reflected with the mode numbers. The mirror inverse of mode 3, the brightest mode, is mode -3, the darkest mode, and mode 0 is itself a symmetric mode, hence 'symmetric' in the mode name. We may already know this - that the Dorian mode of the familiar diatonic scale is symmetric, and the mirror inverse of the Lydian mode is the Locrian mode.  


Something to note - the Meantone diatonic scale is ''generated'' by the perfect fifth, 3/2, which means that it can be formed by stacking perfect fifths on top of each other, i.e., F-C-G-D-A-E, and all the notes are connected by perfect fifths. Porcupine[7], on the other hand, is generated by 10/9, so all notes are connected by a chain of 10/9s, i.e., A-B-C-D-E-F-G, where the large step of 9/8 then separates G from A. The Zarlino/Ptolemy just major scale 9/8 5/4 4/3 3/2 5/3 15/8 2/1 can be built of two parallel chains of 3/2, i.e., 4/3-2/1-3/2-9/8, 5/3-5/4-15/8. Accordingly it is a ''[[Generator-offset property|generator-offset]]'' scale. If the scale is on C, then D-A is not a 3/2 perfect fifth, but a wolf fifth of 40/27. The porcutone diatonic is not a generator offset scale. Setting the scale to the naturals, D E F G A B C D, 3/2 perfect fifths are available above D, E, F, and C, so there are 1 fewer 3/2 perfect fifths in the porcutone diatonic scale than in the Zarlino/Ptolemy just major scale, and two fewer than in the typical diatonic scale. Porcupine[7] also has 3/2 fifths only above D, E, F, and C. It is because 3/2 perfect fifths are available above D, E, F, and C in both Meantone[7] and Porcupine[7] that they are available above D, E, F, and C in the Porcutone diatonic.   
Something to note - the Meantone diatonic scale is ''generated'' by the perfect fifth, 3/2, which means that it can be formed by stacking perfect fifths on top of each other, i.e., F-C-G-D-A-E, and all the notes are connected by perfect fifths. Porcupine[7], on the other hand, is generated by 10/9, so all notes are connected by a chain of 10/9s, i.e., A-B-C-D-E-F-G, where the large step of 9/8 then separates G from A. The Zarlino/Ptolemy just major scale 9/8 5/4 4/3 3/2 5/3 15/8 2/1 can be built of two parallel chains of 3/2, i.e., 4/3-2/1-3/2-9/8, 5/3-5/4-15/8. Accordingly it is a ''[[Generator-offset property|generator-offset]]'' scale. If the scale is on C, then D-A is not a 3/2 perfect fifth, but a wolf fifth of 40/27. The porcutone diatonic is not a generator offset scale. Setting the scale to the naturals, D E F G A B C D, 3/2 perfect fifths are available above D, E, F, and C, so there are 1 fewer 3/2 perfect fifths in the porcutone diatonic scale than in the Zarlino/Ptolemy just major scale, and two fewer than in the typical diatonic scale. Porcupine[7] also has 3/2 fifths only above D, E, F, and G. It is because 3/2 perfect fifths are available above D, E, F, and G in both Meantone[7] and Porcupine[7] that they are available above D, E, F, and G in the Porcutone diatonic.   


The minor tone small step of Porcupine[7] can also represent the neutral seconds 11/10 and 12/11, since 10/9*11/10*12/11 = 4/3, and 4/3 is subtended by 3 small steps of Porcupine[7], tempering out both [[100/99]] and [[121/120]]. 11/8 is easily reached in Porcupine[7] as a major 4th, subtended by 2 small steps and 1 large step. The small step of Porcupine[7] represents all of 10/9, 11/10, 12/11 and 27/25, in order of largest to smallest. In the porcutone diatonic, the small step is 27/25 and the medium step is 10/9. We can access our 11-limit harmonies in porcutone by tempering out [[100/99]], which separates 10/9 from 11/10, as well as 27/25 from 12/11. This leads to step signature and step mapping 1L 4M 2s = (9/8~25/22, 10/9~11/10, 27/25~12/11). Since [[100/99]] is called the [[Ptolemisma]], we can call the resulting scale the ptolemismic porcutone diatonic.  
The minor tone small step of Porcupine[7] can also represent the neutral seconds 11/10 and 12/11, since 10/9*11/10*12/11 = 4/3, and 4/3 is subtended by 3 small steps of Porcupine[7], tempering out both [[100/99]] and [[121/120]]. 11/8 is easily reached in Porcupine[7] as a major 4th, subtended by 2 small steps and 1 large step. The small step of Porcupine[7] represents all of 10/9, 11/10, 12/11 and 27/25, in order of largest to smallest. In the porcutone diatonic, the small step is 27/25 and the medium step is 10/9. We can access our 11-limit harmonies in porcutone by tempering out [[100/99]], which separates 10/9 from 11/10, as well as 27/25 from 12/11. This leads to step signature and step mapping 1L 4M 2s = (9/8~25/22, 10/9~11/10, 27/25~12/11). Since [[100/99]] is called the [[Ptolemisma]], we can call the resulting scale the ptolemismic porcutone diatonic.  
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41edo with 1200.2039c octave: 1L 4m 2s = (7, 6, 5) = (204.9129c, 175.6396c, 146.3663c)
41edo with 1200.2039c octave: 1L 4m 2s = (7, 6, 5) = (204.9129c, 175.6396c, 146.3663c)
For comparison, the TE tuning has step signature and mapping
[http://x31eq.com/cgi-bin/rt.cgi?ets=1ce_4p_2ce&limit=2_3_5_11 TE ptolemismic: 1L 4m 2s] = (209.77855c, 174.05488c, 146.63528c)


The table below show the sizes, interval names, ratios approximated, tuning, and occurence of all intervals of the ptolemismic porcutone diatonic scale within an octave, tuned to TE tuning.
The table below show the sizes, interval names, ratios approximated, tuning, and occurence of all intervals of the ptolemismic porcutone diatonic scale within an octave, tuned to TE tuning.
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The ptolemismic porcutone chromatic scale is distinctly xenharmonic, and yet is related to the familiar chromatic scale.
The ptolemismic porcutone chromatic scale is distinctly xenharmonic, and yet is related to the familiar chromatic scale.


Mode -3 has 3/2 perfect fifths available above D, D♯, E, F, F♯, G, and G♯.
Mode 3 has 3/2 perfect fifths available above D, E♭, E, F, G♭, G, and D♭.
Mode -3 has 4:5:6 major triads available above E, F, F♯, G, and G♯
Mode 3 has 4:5:6 major triads available above E♭, E, F, G♭, and G
The following tables show the (3, 4) and (4, 3) triads available of mode 3 and mode -3 of the porcutone chromatic scale:
{| class="wikitable"
|+(3, 4) and (4, 3) triads of the porcutone chromatic mode -3
!Note
!Triad class
!Triad in meantone
!Triad in porcupine
!Porcutone triad name
!JI triads approximated
!Triads in cents
|-
|D
|(3, 4)
(4, 3)
|minor
major
|minor
minor
|minor
major minor
|10:12:15
18:22:27
|
|-
|D♯
|(3, 4)
(4, 3)
|minor
dim 4
|minor
min 4
|minor
dim min 4
|10:12:15
110:144:165
|
|-
|E
|(3, 4)
(4, 3)
|minor
major
|minor
major
|minor
major
|10:12:15
4:5:6
|
|-
|F
|(3, 4)
(4, 3)
|aug 2
major
|maj 2
major
|aug maj 2
major
|48:55:72
4:5:6
|
|-
|F♯
|(3, 4)
(4, 3)
|minor
major
|major
major
|minor major
major
|22:27:33
4:5:6
|
|-
|G
|(3, 4)
(4, 3)
|aug 2
major
|aug 2
major
|aug 2
major
|64:75:128
4:5:6
|
|-
|G♯
|(3, 4)
(4, 3)
|minor
dim 4
|major
maj 4
|minor major
dim maj 4
|22:27:33
20:27:30
|
|-
|A
|(3, 4)
(4, 3)
|minor
major
|diminished
diminished
|minor diminished
major diminished
|15:18:22
27:33:40
|
|-
|A♯
|(3, 4)
(4, 3)
|minor dim 6
dim 4 dim 6
|minor (sub) min 6
min 4 sub minor 6
|minor (sub) dim min 6
dim min 4 (sub) dim min 6
|275:330:432
275:360:432
|
|-
|B
|(3, 4)
(4, 3)
|minor
major
|diminished
diminished
|minor diminished
major diminished
|15:18:22
27:33:40
|
|-
|C
|(3, 4)
(4, 3)
|aug 2
major
|diminished min 2
diminished
|aug min 2 diminished
major diminished
|243:275:360
243:297:360
|
|-
|C♯
|(3, 4)
(4, 3)
|minor
dim 4
|minor
min 4
|minor
dim min 4
|10:12:15
110:144:165
|
|}
As with the porcutone diatonic, tuning the porcutone chromatic to 19edo or 31edo collapses it to the Meantone[12] (Meanenneadecal[12]) chromatic scale. Tuning it to 15edo, 22edo, or 29edo collapses it to Porcupine[8]. Step patterns, mappings and sizes for tunings to 27edo, 34edo, and 41edo are as follows:
As with the porcutone diatonic, tuning the porcutone chromatic to 19edo or 31edo collapses it to the Meantone[12] (Meanenneadecal[12]) chromatic scale. Tuning it to 15edo, 22edo, or 29edo collapses it to Porcupine[8]. Step patterns, mappings and sizes for tunings to 27edo, 34edo, and 41edo are as follows:


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!Father[8]  
!Father[8]  
step pattern and UDP
step pattern and UDP
!Father[8]
!Oneirotonic
mode
mode
!Porcutone octatonic  
!Porcutone octatonic  
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!Father[8]  
!Father[8]  
step pattern and UDP
step pattern and UDP
!Father[8]
!Oneirotonic
mode
mode
!Porcutone octatonic  
!Porcutone octatonic  
Line 646: Line 825:
|Kadathian
|Kadathian
|Kadathian middle minor
|Kadathian middle minor
|}
Note that the darkest mode of the LH octatonic is the brightest mode of the RH 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 porcutone octatonic scales, along with the scale steps in cents.
{| class="wikitable"
|+Modes of the left handed ptolemismic porcutone octatonic
!Porcutone ocatonic mode
!Step pattern
!Mode as simplest JI pre-image
!Mode in cents
|-
|Celephaïsian dark quartal
|LMLLMLsM
|~ 10/9 6/5 4/3 22/15 8/5 16/9 11/6 2/1
|174.055 320.690 494.745 668.800 815.435 989.490 1052.633 1199.269
|-
|Sarnathian bright quartal
|MLMLLMLs
|~ 12/11 6/5 72/55 16/11 8/5 96/55 48/25 2/1
|146.635 383.834 467.325 641.380 815.435 962.070 1136.127 1199.269
|-
|Dylathian middle major
|LLMLsMLM
|~ 10/9 11/9 4/3 22/15 55/36 5/3 11/6 2/1
|174.055 348.110 494.745 668.800 731.943 878.579 1052.633 1199.269
|-
|Kadathian bright major
|MLLMLsML
|~ 12/11 6/5 4/3 16/11 8/5 5/3 9/5 2/1
|146.635 320.690 494.745 641.380 815.435 878.579 1025.214 1199.269
|-
|Ultharian dark major
|LMLsMLML
|~ 10/9 6/5 4/3 11/8 3/2 5/3 9/5 2/1
|174.055 320.690 494.745 557.888 704.524 878.579 1025.214 1199.269
|-
|Illarnekian middle minor
|LsMLMLLM
|~ 10/9 55/48 5/4 11/8 3/2 5/3 11/6 2/1
|174.055 237.198 383.834 557.888 704.524 878.579 1052.633 1199.269
|-
|Hlanithian bright minor
|MLsMLMLL
|~ 12/11 6/5 5/4 15/11 3/2 18/11 9/5 2/1
|146.635 320.690 383.834 530.469 704.524 851.159 1025.214 1199.269
|-
|Mnarian dark minor
|sMLMLLML
|~ 25/24 9/8 5/4 15/11 3/2 5/3 9/5 2/1
|63.143 209.779 383.834 530.469 704.524 878.579 1025.214 1199.269
|}
{| class="wikitable"
|+Modes of the right handed ptolemismic porcutone octatonic
!Porcutone ocatonic mode
!Step pattern
!Mode as simplest JI pre-image
!Mode in cents
|-
|Ultharian bright quartal
|LMLLMLMs
|~ 10/9 6/5 4/3 22/15 8/5 16/9 48/25 2/1
|174.055 320.690 494.745 668.800 815.435 989.490 1136.127 1199.269
|-
|Illarnekian bright major
|LLMLMsLM
|~ 10/9 11/9 4/3 22/15 8/5 5/3 11/6 2/1
|174.055 348.110 494.745 668.800 815.435 878.579 1052.633 1199.269
|-
|Hlanithian dark quartal
|MLLMLMsL
|~ 12/11 6/5 4/3 16/11 8/5 96/55 10/9 2/1
|146.635 320.690 494.745 641.380 815.435 962.070 1025.214 1199.269
|-
|Mnarian middle major
|LMLMsLML
|~ 10/9 6/5 4/3 16/11 3/2 5/3 9/5 2/1
|174.055 320.690 494.745 641.380 704.524 878.579 1025.214 1199.269
|-
|Celephaïsian bright minor
|LMsLMLLM
|~ 10/9 6/5 5/4 11/8 3/2 5/3 11/6 2/1
|174.055 320.690 383.834 557.888 704.524 878.579 1052.633 1199.269
|-
|Sarnathian dark major
|MLMsLMLL
|~ 12/11 6/5 72/55 15/11 3/2 18/11 9/5 2/1
|146.635 320.690 467.325 530.469 704.524 851.159 1025.214 1199.269
|-
|Dylathian dark minor
|sLMLLMLM
|~ 25/24 55/48 5/4 11/8 55/36 5/3 11/6 2/1
|63.143 237.198 383.834 557.888 731.943 878.579 1052.633 1199.269
|-
|Kadathian middle minor
|MsLMLLML
|~ 12/11 9/8 5/4 15/11 3/2 5/3 9/5 2/1
|146.635 209.779 383.834 530.469 704.524 878.579 1025.214 1199.269
|}
|}


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