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→The Porcutone diatonic: added hyperlinks, edited for clarity, added reference tables for modes of 5-limit Meantone[7] and Porcupine[7]
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Consider instead the just scale: 10/9 6/5 4/3 3/2 5/3 9/5 2/1, a just Dorian scale. This scale has 1 large step of [[9/8]], 4 medium steps of [[10/9]], and 2 small steps of [[27/25]], with step pattern MsMLMsM (mode 0). It can be represented with [[Step pattern|step signature]] and step mapping 1L 4M 2s = (9/8, 10/9, 27/25). This is our just Porcutone diatonic.If we temper out the difference between L and M, we get LsLLLsL, [[Meantone]][7] mode 0: Dorian; if we temper out instead the difference between [[10/9]] and [[27/25]], we get sssLsss, [[Porcupine]][7] mode 0, which is referred to as ''symmetric minor''. In this way, the just Porcutone diatonic represents both [[Porcupine]][7] and [[Meantone]][7].
Consider instead the just scale: 10/9 6/5 4/3 3/2 5/3 9/5 2/1, a just Dorian scale. This scale has 1 large step of [[9/8]], 4 medium steps of [[10/9]], and 2 small steps of [[27/25]], with step pattern MsMLMsM (mode 0). It can be represented with [[Step pattern|step signature]] and step mapping 1L 4M 2s = (9/8, 10/9, 27/25). This is our just Porcutone diatonic.If we temper out the difference between L and M, we get LsLLLsL, [[Meantone]][7] mode 0: Dorian; if we temper out instead the difference between [[10/9]] and [[27/25]], we get sssLsss, [[Porcupine]][7] mode 0, which is referred to as ''symmetric minor''. In this way, the just Porcutone diatonic represents both [[Porcupine]][7] and [[Meantone]][7].


To name this mode of the Porcutone diatonic, we simply add the mode names together, prefixing the [[Porcupine]][7] functional mode name (which I am introducing here) with the [[Meantone]] diatonic mode name, so mode 0 of the Porcutone diatonic is called ''Dorian symmetric minor''. We continue this process with the other 6 modes:
To name this mode of the Porcutone diatonic, we simply add the mode names together, prefixing the [[Porcupine]][7] functional mode names introduced in Table 1., with the [[Meantone]] diatonic mode names referenced in Table 2., so mode 0 of the Porcutone diatonic is called ''Dorian symmetric minor''. We continue this process with the other 6 modes to arrive at the modes shown in Table 3.
{| class="wikitable"
{| class="wikitable"
|+Modes of the just Porcutone diatonic
|+Table 1. Modes of 5-limit Porcupine[7]
!Mode number
!Mode number
!Mode in JI
!Mode as simplest JI pre-image
!Step pattern
!Step pattern
!Meantone[7]
!Mode
!Diatonic mode
!Porcupine[7]
!Porcupine[7] mode
!Porcutone diatonic mode
|-
|-
|3
|3
|10/9 5/4 25/18 3/2 5/3 50/27 2/1
|~ 9/8 5/4 27/20 3/2 5/3 9/5 2/1
|MLMsMMs
|Lssssss
|LLLsLLs
|Bright major
|Lydian
|-
|2
|~ 10/9 5/4 27/20 3/2 5/3 9/5 2/1
|sLsssss
|sLsssss
|Dark major
|Dark major
|Lydian dark major
|-
|2
|9/8 5/4 27/20 3/2 5/3 9/5 2/1
|LMsMMsM
|LLsLLsL
|Mixolydian
|Lssssss
|Bright major
|Mixolydian bright major
|-
|-
|1
|1
|10/9 100/81 4/3 40/27 5/3 50/27 2/1
|~ 10/9 6/5 27/20 3/2 5/3 9/5 2/1
|MMsMLMs
|ssLssss
|LLsLLLs
|Bright minor
|Ionian
|ssssLss
|Bright diminished
|Ionian bright diminished
|-
|-
|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
|MsMLMsM
|LsLLLsL
|Dorian
|sssLsss
|sssLsss
|Symmetric minor
|Symmetric minor
|Dorian symmetric minor
|-
|-
| -1
| -1
|27/25 6/5 27/20 3/2 81/50 9/5 2/1
|~ 10/9 6/5 4/3 36/25 5/3 9/5 2/1
|sMLMsMM
|ssssLss
|sLLLsLL
|Bright diminished
|Phrygian
|-
|ssLssss
| -2
|Bright minor
|~ 10/9 6/5 4/3 36/25 8/5 9/5 2/1
|Phrygian bright minor
|sssssLs
|Dark diminished
|-
|-
| -2
| -3
|10/9 6/5 4/3 40/27 8/5 16/9 2/1
|~ 10/9 6/5 4/3 36/25 8/5 16/9 2/1
|MsMMsML
|LsLLsLL
|Aeolian
|ssssssL
|ssssssL
|Magical seventh
|Magical seventh
|Aeolian magical seventh
|-
| -3
|27/25 6/5 4/3 36/25 8/5 9/5 2/1
|sMMsMLM
|sLLsLLL
|Locrian
|sssssLs
|Dark diminished
|Locrian dark diminished
|}
|}
{| class="wikitable"
{| class="wikitable"
|+Modes of 5-limit Meantone[7]
|+Table 2. Modes of 5-limit Meantone[7]
!Mode number
!Mode number
!Mode as simplest JI pre-image
!Mode as simplest JI pre-image
Line 137: Line 105:
|Locrian
|Locrian
|}
|}
{| class="wikitable"
{| class="wikitable"
|+Modes of 5-limit Porcupine[7]
|+Table 3. Modes of the just Porcutone diatonic
!Mode number
!Mode number
!Mode as simplest JI pre-image
!Mode in JI
!Step pattern
!Step pattern
!Mode
!Meantone[7]
!Diatonic mode
!Porcupine[7]
!Porcupine[7] mode
!Porcutone diatonic mode
|-
|-
|3
|3
|~ 9/8 5/4 27/20 3/2 5/3 9/5 2/1
|10/9 5/4 25/18 3/2 5/3 50/27 2/1
|MLMsMMs
|LLLsLLs
|Lydian
|sLsssss
|Dark major
|Lydian dark major
|-
|2
|9/8 5/4 27/20 3/2 5/3 9/5 2/1
|LMsMMsM
|LLsLLsL
|Mixolydian
|Lssssss
|Lssssss
|Bright major
|Bright major
|-
|Mixolydian bright major
|2
|~ 10/9 5/4 27/20 3/2 5/3 9/5 2/1
|sLsssss
|Dark major
|-
|-
|1
|1
|~ 10/9 6/5 27/20 3/2 5/3 9/5 2/1
|10/9 100/81 4/3 40/27 5/3 50/27 2/1
|ssLssss
|MMsMLMs
|Bright minor
|LLsLLLs
|Ionian
|ssssLss
|Bright diminished
|Ionian bright diminished
|-
|-
|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
|MsMLMsM
|LsLLLsL
|Dorian
|sssLsss
|sssLsss
|Symmetric minor
|Symmetric minor
|Dorian symmetric minor
|-
|-
|  -1
|  -1
|~ 10/9 6/5 4/3 36/25 5/3 9/5 2/1
|27/25 6/5 27/20 3/2 81/50 9/5 2/1
|ssssLss
|sMLMsMM
|Bright diminished
|sLLLsLL
|Phrygian
|ssLssss
|Bright minor
|Phrygian bright minor
|-
|-
|  -2
|  -2
|~ 10/9 6/5 4/3 36/25 8/5 9/5 2/1
|10/9 6/5 4/3 40/27 8/5 16/9 2/1
|MsMMsML
|LsLLsLL
|Aeolian
|ssssssL
|Magical seventh
|Aeolian magical seventh
|-
|  -3
|27/25 6/5 4/3 36/25 8/5 9/5 2/1
|sMMsMLM
|sLLsLLL
|Locrian
|sssssLs
|sssssLs
|Dark diminished
|Dark diminished
|-
|Locrian dark diminished
|  -3
|~ 10/9 6/5 4/3 36/25 8/5 16/9 2/1
|ssssssL
|Magical seventh
|}
|}
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 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.   
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.   
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