Pinetone: Difference between revisions

The porcutone system combines [[Porcupine]] – arguably the best way to add the 11th harmonic to major and minor harmonies in a seven-note scale – with with [[Meantone]] – the system underpinning most common practice music from the last several hundred years, so all the same scales (diatonic, harmonic minor, pentatonic, chromatic, etc.) are still available, just with a new Porcupine spin, and the 11th harmonic (and the 13th harmonic as well!)   

As opposed to in [[12edo]], each key is distinctly different in porcutone, both a blessing and a curse.  

Additionally available in porcutone are a set of octatonic modes with their own Porcupine functional harmony, that combine [[Porcupine]][8] with the [[oneirotonic]] modes that are gaining popularity at the moment.  

If you have a [[Lumatone]], you can use the standard Bosanquet mapping for 12edo. The white keys are the porcutone diatonic, a cross between the meantone diatonic scale and Porcupine[7], and then black keys give the porcutone pentatonic, which approximates the [[just intonation]] pentatonic scale 9/8 5/4 3/2 5/3 2/1. I've chosen to colour the G♯/A♭ key pink, and the other chromatic keys blue, because I'm a proud trans woman and a big nerd. You can use any colours, but I find it helps to colour the G♯/A♭ key a different colour since that's the one chromatic key used along with the diatonic keys to make the porcutone octatonic.  

== The porcutone diatonic ==

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:

|+Modes of the just porcutone diatonic

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.  

The modes of the ptolemismic porcutone diatonic are shown below in their simplest JI pre-image (the simplest JI ratios each interval above the tonic represents), and in cents, in an optimized tuning called [[TE tuning]].  

|+Modes of 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.  

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.

|+Intervals of the porcutone diatonic

|+Tertian tetrads of the porcutone diatonic on D

Also of interest are the quartal triads of the porcutone diatonic. We describe these as stacked 3-step intervals (fourths) of the scale, perhaps an unusual interval to write quartal triads in, but it makes more sense here given the naming scheme I've introduced for the Porcupine[7] 3-step (quartal) triads.

|+3-step stacked triads of the porcutone diatonic on D

== The porcutone pentatonic and the porcutone chromatic ==

We know the (meantone) pentatonic scale to be a subset of the (meantone) diatonic scale. Similarly, the porcutone pentatonic is a subset of the porcutone diatonic. We also know that adding a (meantone) pentatonic to a (meantone) diatonic leads to a (meantone) chromatic, i.e., diatonic on white keys + pentatonic on black keys. We can do this with porcutone.  

Using the familiar Bosanquet 12-note keyboard mapping (the preset for 12edo), we set the porcutone diatonic scale to the white keys, starting on D. We than add, on F♯/G♭, the porcutone penatonic as a set of 5 chromatic keys. There are two options for the chromatic keys, either all sharps or all flats. All sharps makes the porcutone harmonic minor available, and all flats makes the porcutone harmonic major available. These scales will be discussed below. In either case, in the just tuning, the chromatic keys give the scale 9/8 5/4 3/2 5/3 2/1, starting from F♯/G♭, tuned to 100/81 (F♯) or 162/125 (G♭) from D. This scale has step pattern msLsL, with step signature and step mapping 2L 1m 2s = (6/5, 9/8, 10/9). The same scale is also available as G-A-B-D-E.   

We are familiar with this scale as the just pentatonic. If we temper m and s together, we get Meantone[5]: ssLsL. If we temper m and L together instead we get a scale called Father[5], tempering out the diatonic semitone 16/15. This mode of Father[5] has step pattern LsLsL. Keep the connection to Father[5] in the back of your minds for now, we'll come back to it. The porcutone pentatonic is also a subset of the porcutone diatonic, since Meantone[5] is a subset of Meantone[7]. It is available as G-A-B-D-E. The scale F-G-A-C-D is a mode of the inverse of G-A-B-D-E. The porcutone pentatonic is [[Chirality|''chiral'']] (i.e., it is not symmetric, unlike the porcutone diatonic and procutone chromatic scales, which are ''achiral''). There is a pair of porcutone pentatonic scales, the right handed porcutone pentatonic, 9/8 5/4 3/2 5/3 2/1 in JI, it's mirror inverse the left handed porcutone pentatonic, 10/9 5/4 3/2 5/3 2/1 in JI, which tempers to ssLsL in Meantone, but to sLLsL in Father. F-G-A-C-D gives a left handed porcutone pentatonic, while G-A-B-D-E, F♯-G♯-A♯-C♯-D♯, and G♭-A♭-B♭-D♭-E♭ are right handed porcutone pentatonics.  

Adding the right handed porcutone pentatonic (on F♯/G♭) to the just porcutone diatonic, a 12-note mirror-symmetric scale with step signature and step mapping of 7L 1m 4s = (27/25, 25/24, 250/243) = (133.2376c, 70.6724c, 49.1661c), i.e., 7 large steps of what was the small step of the just porcutone diatonic, 1 medium step of the chromatic semitone 25/24, the distance between 6/5 and 5/4, and 4 small steps of 250/243, the porcupine comma, that separates 10/9 from 27/25. For the all sharps scale, we set mode -3 on D (for all flats we set mode 3 on D): 250/243 10/9 6/5 100/81 4/3 25/18 3/2 125/81 5/3 9/5 50/27 2/1, with step pattern sLLsLmLsLLsL.  

The now familiar Meantone comma of 81/80 separates the medium step (25/24) from the small step (250/243), so our porcutone chromatic is a ''detempering'' of Meantone[12], the meantone chromatic scale, just like how the porcutone diatonic is a detempering of Meantone[7], the meantone diatonic scale.  

The ptolemismic porcutone chromatic has a step signature, mapping, and TE tuning of 7L 1m 4s = (27/25~12/11, 25/24~33/32, 250/243~55/54~121/120) = (146.6352c, 63.1434c, 27.4197c).  

Note the more complex intervals: 55/54, 55/36, 72/55, and 108/55. If we temper out an additional comma, we can equate these with simpler intervals, adding prime 13: Tempering out 144/143, these four interval approximate 40/39, 20/13, 13/10, and 39/20 respectively. Tempering out 144/143 also means that the small step of the porcutone diatonic, equivalently the large step of the porcutone chromatic approximates 13/12, which, when all three are justly tuned, lies between the other intervals approximated by the step - 27/25, and 12/11.

The ptolemismic porcutone chromatic scale is distinctly xenharmonic, and yet is related to the familiar chromatic scale.

The following tables show the (3, 4) and (4, 3) triads available of mode 3 and mode -3 of the porcutone chromatic scale:

|+(3, 4) and (4, 3) triads of the porcutone chromatic mode -3

We can see that the large step of Tetracot[7] is the medium step of the porcutone diatonic, and the small step of Tetracot[7] is the small step of the porcutone diatonic. The large step of the porcutone diatonic is the augmented second of tetracot[7].

As with the porcutone diatonic, tuning the porcutone chromatic to 19edo collapses it to the Meantone[12] (Flattone[12]) chromatic scale. Tuning it to 15edo, 22edo, or 29edo collapses it to Porcupine[8]. Step signatures, mappings and sizes for tunings to 27edo, 34edo, and 41edo are as follows:

All three of these edos also temper out 243/242, so the major minor and minor major thirds collapse to a single interval - the neutral third, and the porcutone diatonic can be considered a MODMOS of Tetracot[7] i.e. porcutone msmLmsm = Tetracot LsLALsL.

the TE step signature, mapping, and sizes for the 13-limit Supermagic porcutone chromatic is  

and the TE step signature, mapping, and sizes for the 13-limit Thrasher porcutone chromatic is  

and if optimization just to the 2.3.5.11 subgroup is desired,TE step signature, mapping, and sizes for the (2.3.5.11) ptolemismic porcutone chromatic is  

The porcupine comma is the small step of the scale, so tempering the porcutone chromatic scale to porcupine leads from 7L 1m 4s = (27/25, 25/24, 250/243) to 7L 1s = (10/9~27/25, 25/24~81/80), which is Porcupine[8]! The Porcupine[7] scale has its large step between G and A, so the eighth note of Porcupine[8] is either G♯ or A♭, adding another small step of Porcupine[7] below A (for G♯) or above G (A♭). Mode -3 or mode 3 of the porcutone chromatic scale, respectively, are set to D so that this is preserved in The Porcutone System. This leads to the porcutone octatonic scales: D E F G G♯/A♭ A B C. In Just intonation: 10/9 6/5 4/3 25/18 3/2 5/3 9/5 2/1 with G♯, or 10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1 with A♭. This scale has 4 large steps of 10/9, 3 medium steps of 27/25, and 1 small step of 25/24. It is not mirror-symmetric, or equivalentely, it is ''[[Chirality|chiral]]'' so it cannot be uniquely defined with a step signature like Meantone[7], Porcupine[7], Porcupine[8], Meantone[12], and the porcutone diatonic (the Zarlio/Ptolemy just major scale is also not mirror symmetric). Scales that can be uniquely defined by a step signature are called ''step-nested scales''. More on that later. The mirror inverse of any mode of the porcutone octatonic with G♯ is a mode of the porcutone octatonic with A♭. The porcutone octatonic with G♯ is called the left handed porcupine octatonic, and the porcutone octatonic with A♭ is called the right handed porcupine octatonic (see [[chirality]]).  

On a keyboard with standard (Bosanquet or 12edo) mapping, the porcutone octatonic is the C Major bebop scale! On my [[Lumatone]] I chose to colour the G♯/A♭ pink, and the rest of the chromatic notes blue, so the porcutone octatonic is on the white and pink keys, while there's a porcutone diatonic on the white keys and a porcutone pentatonic on the blue and pink keys.  

If we temper out the difference between the large and medium steps, we reduce the scale to Porcupine[8]. As we discussed above, Porcupine is generated by the interval 10/9~27/25. The table below introduces a set of functional mode names for Porcupine[8]. Along with the step pattern and mode number, the modes' ''[[UDP]]'' is show in the table. The UDP show the number of generators in the direction the brighten the intervals of scale, followed the number of generators in the direction that darkens it, (followed by the number of periods per octave, if it is not one. In this case the scale repeats at the octave, so P = 1, and is not shown). Instead of building chords by stacking thirds (2-step intervals), in octatonic scales we can build major and minor triads by stacking 3-step intervals! Instead of diminished, we get modes with two large fourths making a quartal chord: Accordingly we call these modes 'quartal'. When we stack 3-step intervals of 8-note scales out minor triads come in first inversion, and our major triads come in second inversion, as the 3-step intervals of octatonic scales include 5/4 and 4/3. Hence the brightest modes are quartal, and the darkest are minor. The eighth note of Porcupine[8] is typically called 'H', and is equivalent to the note A♭ of Porcupine[7], but we will show the modes for G# as the eighth note as well, since we may use G# in our porcutone chromatic and octatonic scales.

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 porcutone diatonic, the modes are listed in order of brightest, with the brightest mode at the top, and the darkest mode at the bottom.

|+Modes of the left handed just porcutone octatonic

|+Modes of the right handed just porcutone octatonic

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.

|+Modes of the left handed ptolemismic porcutone octatonic

|175.892 318.667 494.559 670.451 815.435 989.118 1055.884 1198.660

|142.775 318.667 461.443 637.334 815.435 956.002 1131.893 1198.660

|142.775 318.667 494.559 637.334 815.435 879.992 1022.768 1198.660

|+Modes of the right handed ptolemismic porcutone octatonic

|175.892 318.667 494.559 670.451 813.227 989.118 1131.983 1198.660

|175.892 351.784 494.559 670.451 813.227 879.992 1055.884 1198.660

|142.775 318.667 494.559 637.334 813.227 956.002 1022.768 1198.660

The following table gives all intervals of the porcutone octatonic.

|+Intervals of the porcutone octatonic

The following two tables detail the 3-step stacked triads of the left and right handed porcutone octatonics:

|+3-step stacked triads of the left handed porcutone octatonic (G♯-G gamut)

|+3-step stacked triads of the right handed porcutone octatonic (G-A♭ gamut)  

We could alternatively treat the porcutone octatonic as a bebop scale, using 2-step stacked tetrads. Since the scale has 8 notes, there are only 2 different 2-step stacked tetrads. In 12edo these are the major add 6 and the fully diminished tetrads. The meantone C major add 6 tunes to 45:55:66:75 in porcutone. Using the G♯, as in the left-handed porcutone octatonic, the G♯ diminished tetrad tunes to 33:40:48:55 (when B is the bottom note). Using the A♭, as in the right-handed porcutone octatonic, the B diminished tetrad also tunes to 33:40:48:55 (when D is the bottom note).

Unlike the Porcutone diatonic, and chromatic scales, the porcutone octatonic is chiral, and is therefore not a step-nested scale. As we can see, it is more complex than the porcutone diatonic. The Porcutone pentatonic and diatonic scales is also wakalix / PWF, and it can be seen that the porcutone octatonic is more complex than the porcutone pentatonic as well. It is left as an excercise for the reader to determine the complexity of the porcutone chromatic, and compare that to the porcutone octatonic.

We note that there are fewer consonant triads available in these scales than in the porcutone diatonic and octatonic scales, so they may be useful for melody only.

We get the following 7 modes of porcutone harmonic minor scale:

Which has porcutone harmonic major modes:

From the porcutone chromatic with sharps (mode -3), we add another porcutone diatonic scale, mode 0 starting on D♭, leading to the left-handed porcutone hyperchromatic scale, with step pattern, sLsLssLsmLssLsLssLs.  

Or, from the porcutone chromatic with flats (mode 3), we add another porcutone diatonic scale, mode 0 starting on D♯, leading to the right-handed porcutone hyperchromatic scale, with step pattern, sLssLsLssLmsLssLsLs.

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 porcutone chromatic), the medium step of the porcutone chromatic, approximating 25/24, 33/32, and 27/26, and 11 small steps, the same as the small step of the pocutone chromatic, approximating 250/243, 55/54, 121/120, and 40/39.

In 2.3.5.11.13 Tetracot, the left handed porcutone 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 porcutone 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.

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 porcutone hyperchromatic in cents is

and the right handed porcutone hyperchromatic in cents is

The porcutone hyperchromatic scales may alternatively be tuned to 27edo, 34edo, or 41edo: