Pinetone: Difference between revisions

The diatonic scale has a step signature of [[5L 2s]], meaning it has 5 large steps and 2 small step arranged in the step pattern LsLLLsL (represented in mode 0, Dorian mode). In Meantone[7], the large step represents both 9/8 and 10/9, the major and minor tones (''tempering out'' the [[81/80]] interval that separates them) hence the name "Meantone". The small step represents 16/15 and 27/25 (which differ again by [[81/80]]). We write this as [[5L 2s]] = (9/8~10/9, 16/15~27/25). Porcupine[7] instead has step step signature and step mapping [[1L 6s]] = (~9/8, 10/9~27/25), hence the difference between 10/9 and 27/25, [[250/243]], is tempered out. In mode 0 it has step pattern sssLsss. [[81/80]] is called the Meantone comma, and [[250/243]] is called the Porcupine comma.

We are familiar with the Zarlino/Ptolemy just major scale: 9/8 5/4 4/3 3/2 5/3 15/8 2/1. This scale has 3 large steps of 9/8, 2 medium steps of 10/9, and 2 small steps of 16/15, with step pattern LMsLMLs. If we temper out the difference between L and M, we get LLsLLLs, the mode 2 of Meantone[7], the familiar Ionian/major mode.  

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 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:

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.   

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.  

We see 11/8 as the 4th in Lydian dark major. In Meantone[7] this is an augmented fourth. The meantone extension representing 11/8 with an augmented fourth is called Meanenneadecal, referencing the fact that it is most at home in [[19edo]]. Tuning the scale to 19edo (or 12edo) will collapse it into a Meanenneadecal[7] diatonic scale. Similarly, tuning the scale to 15edo, 22edo, or 29edo will collapse it to Porcupine[7] scale. 27edo, 34edo, and 41edo are good tunings for the Porcutone diatonic if tuning to an edo is desired.