… a demonstration of how to use Tonal Plexus Editor (TPXE) software (Mac and Windows) to create Region Maps on the keyboard.

… a little demonstration of how to use Tonal Plexus Editor (TPXE) software (Mac and Windows) to import scales and create Areas on the keyboard.

My friend Peter Hesterman asked me today how he would play Harry Partch’s 43-Tone scale on the Tonal Plexus. I’ve been asked this question a number of times by email from people interested in Partch’s scale. Usually I have answered that there are any number of ways to play it, because you can program the keys to whatever pitches you want, but the main idea behind the Tonal Plexus 205ET JND master tuning is that you can play anything you want on the keyboard without ever having to retune it or relearn fingering patterns. So, this morning I decided to make this diagram showing how Partch’s scale can be played in the master tuning. Since Partch always used G as his tonic, the scale is shown here in one octave from G to G.

The harmonic structure the scale is made clearer by applying H-chroma to each pitch according to highest prime harmonics. Partch used primes 2, 3, 5, 7, and 11, and the H-chroma values for those primes are 2 = red, 3 = blue, 5 = yellow, 7 = violet, 11 = green.

For example, the blue dots show the most basic pitches of the 3-Limit in the scale. They are A, B-flat, C, D, E and F. The yellow dots show those pitches based on the 5-Limit, and so on. If you look at the dots grouped by color (which you will do automatically unless you are color-blind), focusing on the C#/Db column as the center, the symmetry of the scale becomes apparent.

The scale is given below as a list of ratios, with H-System pitch and interval names. H-System pitch names follow the form JND inflection, comma shift, letter, accidental. Inflections on the keyboard correspond to the size and shape of the key as shown below, where the large natural key is always 3-Limit:

Partch’s scale maps very easily to the 41-tone circle of fifths with JND inflections, and in only two cases are double inflections used.

H-System interval names follow the form JND intonation quality, comma shift type, quality, ordinal size. The JND intonation quality is measured from the lowest limit interval within each zone. In almost every case, Partch’s intervals have Perfect JND qualities; note that where it is not called Perfect, the intonation is being measured against a lower limit interval which falls in that zone. There are 51 basic 13-Limit intervals in 41 zones which have implied Perfect JND qualities in the H-System.

An example reading from the table below: scale degree 8 has a tone ratio of 10/9, the pitch is called "sharp sub-A" and the interval is called a "Perfect Small Major Second".


degree .. tone ratio .. pitch name .. interval name
1 .......... 1/1 .......... G .......... P1
2 .......... 81/80 .......... b+G .......... P.L1
3 .......... 33/32 .......... b~Ab .......... d.Sm2
4 .......... 21/20 .......... bAb .......... m.m2
5 .......... 16/15 .......... b+Ab .......... P.Lm2
6 .......... 12/11 .......... #≈A .......... P.NM2
7 .......... 11/10 .......... bb~A .......... dd.SM2
8 .......... 10/9 .......... #~A .......... P.SM2
9 .......... 9/8 .......... A .......... P.M2
10 .......... 8/7 .......... b+A .......... P.LM2
11 .......... 7/6 .......... #~Bb .......... P.Sm3
12 .......... 32/27 .......... Bb .......... P.m3
13 .......... 6/5 .......... b+Bb .......... P.Lm3
14 .......... 11/9 .......... b‡Bb .......... P.Wm3
15 .......... 5/4 .......... #~B .......... P.SM3
16 .......... 14/11 .......... #B .......... M.M3
17 .......... 9/7 .......... b+B .......... P.LM3
18 .......... 21/16 .......... ~C .......... P.S4
19 .......... 4/3 .......... C .......... P4
20 .......... 27/20 .......... b+C .......... P.L4
21 .......... 11/8 .......... b~Db .......... P.Sd5
22 .......... 7/5 .......... Db .......... P.d5
23 .......... 10/7 .......... C# .......... P.a4
24 .......... 16/11 .......... #+C# .......... P.La4
25 .......... 40/27 .......... #~D .......... P.S5
26 .......... 3/2 .......... D .......... P5
27 .......... 32/21 .......... +D .......... P.L5
28 .......... 14/9 .......... #~Eb .......... P.Sm6
29 .......... 11/7 .......... bEb .......... m.m6
30 .......... 8/5 .......... b+Eb .......... P.Lm6
31 .......... 18/11 .......... #≈E .......... P.NM6
32 .......... 5/3 .......... #~E .......... P.SM6
33 .......... 27/16 .......... E .......... P.M6
34 .......... 12/7 .......... #+E .......... P.LM6
35 .......... 7/4 .......... #~F .......... P.Sm7
36 .......... 16/9 .......... F .......... P.m7
37 .......... 9/5 .......... b+F .......... P.Lm7
38 .......... 20/11 .......... x+F .......... aa.Lm7
39 .......... 11/6 .......... b‡F .......... P.Wm7
40 .......... 15/8 .......... #~F# .......... P.SM7
41 .......... 40/21 .......... #F# .......... M.M7
42 .......... 64/33 .......... #+F# .......... a.LM7
43 .......... 160/81 .......... #~G .......... P.S8
1 .......... 2/1 .......... G .......... P8


I’m working on a feature for TPXE which will create images like those above, mapping pitches to keys using dots. It would be nice to have a version of this pitches-to-dots mapping to add to the website, but I think it would require Flash or something similar, which I don’t have. The feature in TPXE will allow a library of shapes to be created, which you will be able to move around on the keyboard to see how shapes wrap around at top and bottom, which should be useful for anyone learning to play the instrument.

This excerpt by Lassus presents a puzzle for tuning with pure intervals, where the closing chord should be the same as the opening chord, but if strict theoretical rules are followed the closing chord will fall about a quartertone lower than the opening chord. The passage is played twice here on a 2 octave Tonal Plexus keyboard, showing how this pitch shifting occurs.

In the first performance, only pure intervals are used, which lead to the quartertone lower closing chord. The drop in pitch at 2 junctures should be clearly audible.

The second time the quartertone distance between the opening and closing chord is distributed throughout the passage at almost every change in harmony, so that the final chord ends up being the same as the opening chord instead of lower. This means that common tones and pitches normally held as suspensions (which should not change) actually do change in the second example, shifting up by 1 JND at almost every change of harmony. This is usually called ‘adaptive’ tuning. I limited the adaptive shifts to 1 JND, which is the distance between 1 key vertically and the next in the default tuning of 205ET which is used here. The adapting could be done in slightly different ways to end up with the desired outcome.

I personally like the small lift in pitch which is felt with every change in harmony in the adaptive version. This lifting of pitch gives a feeling that the piece is constantly rising when in fact it is just compensating for the naturally falling pitch. I like the subtle changes. A different example could show the opposite, where small lowing of pitch over time could compensate for rising pitch, and a longer passage could mix both things.

Just a note here - I have changed some registers of pitches in this example because the bass notes would not all fit on a 2 octave keyboard. Also, I admit that I did not consult a score for this; I only listened to a tuning example made by Kalle Aho and figured out the passage by ear at the keyboard. The basic voice leading is correct (compare with this vocal performance - the opening phrase; note that in some other performances a B-flat is used in the first chord of the final elaborated cadence instead of a B, which also sounds nice). Thanks to Kalle for posting the adaptive tuning example which gave me the idea to do this. And, thank you for watching and listening (and reading).

TPXE Tonal Plexus Editor software, free for download (Mac and Windows) at http://www.h-pi.com/downloads.html

In this video, I show some basic things about the interface and demonstrate how to tune a key as well as how to link one key to another to build relative tunings.

This is the first in a series of tutorial videos, which I hope will also get better as we go : )