Private lessons with Joseph Monzo are available online via Discord / WhatsApp / Skype: composition, music-theory, tuning-theory, piano, and all woodwinds (sax, clarinet, flute, bassoon, recorder). Current rates US$ 80 per hour (negotiable). Send an email to: monzojoe (AT) gmail.
I was considering taking the radical step of eschewing the pythagorean sharps and flats in approximating the higher primes, using only the pythagorean nominals as a basis. The intention was to simplify the notation. But I realized that depending on the key, the notation may get complicated anyway. So in the interest of inspiring innovation and experimentation, for most primes I present various ways of using the HEWM symbols to alter the pythagorean base-note nominal, with an extensive 53-odd-limit example below the table.
Some of the suggestions are quite far-fetched and are only included because they are the closest pythagorean approximation. For example: for prime 29 it is very unlikely that someone would want to start with a pythagorean augmented-6th and adjust 10 cents up from there, when it is so much easier to start with the familiar pythagorean minor-7th and adjust ~33.5 cents up from there, especially since the adjustment for both is in the same direction. Similarly, I thought that it would be nice to make the symbols for 5 simpler and cleaner by adjusting up 92 cents from the minor-3rd, but in this case the ~21.5 cent down syntonic-comma adjustment from the major-3rd is so much more familiar. Primes 41 and 43 have only one version each because in both cases there really wasn't any other sensible alternative.
Regarding the symbols for the accidentals: with the exception of plain lower-case v already used for the 11-comma-down, and the period . which I didn't use, these are all of the non-alphabetic characters available on a standard American computer keyboard. I tried to keep them in pairs indicating up on the left and down on the right for each prime-comma. The pairing works excellently for 7, 11, 13, 17, 19, and 23, with each pair having symbols which mirror each other exactly. For primes above 23 the symbols don't resemble each other .. but then neither does b # or + - which are accepted.
monzo
lower raise prime 2 3, 5 7 11, 13 17 19, 23 29 31, 37 41 43, 47 53 ratio ~cents pyth base alteration
b # 3 [-11 7, 0 0 0, 0 0 0, 0 0 0, 0 0 0, 0 0 > 2187:2048 113.6850061 p1 up
- + 5 [ -4 4, -1 0 0, 0 0 0, 0 0 0, 0 0 0, 0 0 > 81:80 21.5062896 maj3 down
[ -7 3, 1 0 0, 0 0 0, 0 0 0, 0 0 0, 0 0 > 135:128 92.1787165 min3 up
< > 7 [ 6 -2, 0 -1 0, 0 0 0, 0 0 0, 0 0 0, 0 0 > 64:63 27.2640918 min7 down
[ 2 -3, 0 1 0, 0 0 0, 0 0 0, 0 0 0, 0 0 > 112:108 62.9609038 maj6 up
[ -5 5, 0 -1 0, 0 0 0, 0 0 0, 0 0 0, 0 0 > 243:224 140.9490978 maj7 down
v ^ 11 [ -5 1, 0 0 1, 0 0 0, 0 0 0, 0 0 0, 0 0 > 33:32 53.2729432 p4 up
[ -6 6, 0 0 -1, 0 0 0, 0 0 0, 0 0 0, 0 0 > 5832:5632 60.4120628 aug4 down
{ } 13 [-10 4, 0 0 0, 1 0 0, 0 0 0, 0 0 0, 0 0 > 1053:1024 48.3476652 min6 up
[ -1 3, 0 0 0, -1 0 0, 0 0 0, 0 0 0, 0 0 > 27:26 65.33734083 maj6 down
\ / 17 [ -7 7, 0 0 0, 0 -1 0, 0 0 0, 0 0 0, 0 0 > 34992:34816 8.7295966 aug1 down
[-12 5, 0 0 0, 0 -1 0, 0 0 0, 0 0 0, 0 0 > 4131:4096 14.7304138 min2 up
[ -4 0, 0 0 0, 0 1 0, 0 0 0, 0 0 0, 0 0 > 17:16 104.9554095 p1 up
( ) 19 [ -9 3, 0 0 0, 0 0 1, 0 0 0, 0 0 0, 0 0 > 513:512 3.3780187 min3 up
[-10 9, 0 0 0, 0 0 -1, 0 0 0, 0 0 0, 0 0 > 314928:311296 20.0819917 aug2 down
[ -1 -2, 0 0 0, 0 0 1, 0 0 0, 0 0 0, 0 0 > 19:18 93.6030144 maj2 up
[ ] 23 [ 5 -6, 0 0 0, 0 0 0, 1 0 0, 0 0 0, 0 0 > 11776:11664 16.5443421 aug4 up
[-14 6, 0 0 0, 0 0 0, 1 0 0, 0 0 0, 0 0 > 16767:16384 40.0043525 dim5 up
[ 3 1, 0 0 0, 0 0 0, -1 0 0, 0 0 0, 0 0 > 24:23 73.6806536 p5 down
; ! 29 [ 11 -10, 0 0 0, 0 0 0, 0 1 0, 0 0 0, 0 0 > 950272:944784 10.0271855 aug6 up
[ -8 2, 0 0 0, 0 0 0, 0 1 0, 0 0 0, 0 0 > 261:256 33.4871959 min7 up
[ 16 -7, 0 0 0, 0 0 0, 0 -1 0, 0 0 0, 0 0 > 65536:63423 56.7377998 dim8 down
? " 31 [ 3 -5, 0 0 0, 0 0 0, 0 0 1, 0 0 0, 0 0 > 3968:3888 35.2605681 maj7 up
[ 5 0, 0 0 0, 0 0 0, 0 0 -1, 0 0 0, 0 0 > 32:31 54.9644275 p8 down
& % 37 [ -2 -2, 0 0 0, 0 0 0, 0 0 0, 1 0 0, 0 0 > 37:36 47.4340370 maj2 up
[ 10 -3, 0 0 0, 0 0 0, 0 0 0, -1 0 0, 0 0 > 1024:999 42.7909586 min3 down
@ $ 41 [ 1 -4, 0 0 0, 0 0 0, 0 0 0, 0 1 0, 0 0 > 82:81 21.2424021 maj3 up
, ' 43 [ -7 1, 0 0 0, 0 0 0, 0 0 0, 0 0 1, 0 0 > 129:128 13.4727065 p4 up
: * 47 [ 4 1, 0 0 0, 0 0 0, 0 0 0, 0 0 0, -1 0 > 48:47 36.4483789 p5 down
[ 4 -6, 0 0 0, 0 0 0, 0 0 0, 0 0 0, 1 0 > 752:729 53.7766168 aug4 up
_ | 53 [ 1 3, 0 0 0, 0 0 0, 0 0 0, 0 0 0, 0 -1 > 54:53 32.3604571 maj6 down
[-12 4, 0 0 0, 0 0 0, 0 0 0, 0 0 0, 0 1 > 4293:4096 81.3245489 min6 up
Below I give a list of the odd harmonics up to the 53-limit, showing all of the alternatives given in the table above. Please note that the adjustment indicated by the symbols vary from one version to the other, just as they are listed in the table above. I am doing this in the hope that it does not cause confusion, but rather that it inspires experimentation.
example, showing odd harmonics above A
harmonic hewm-53-limit
1 A
3 E
5 C#- or C+
7 G< or F#> or G#<
9 B
11 D^ or D#v
13 F} or F#{
15 G#- or G+
17 A#\ or Bb/ or A/
19 C) or B#( or B)
21 D< or C#>
23 D#] or Eb] or E[
25 E#-- or E++
27 F#
29 Fx! or G! or Ab;
31 G#" or A?
33 A^ or A#v
35 B<- or A#>- or Bb<+ or A>+
37 B% or C&
39 C} or C#{
41 C#$
43 D'
45 D#- or D+
47 E: or D#*
49 F<< or E>>
51 E#\ or F/ or E/
53 F#_ or F|
Below I give another similar list, this time showing the harmonics from a reference note of C. This will give readers the opportunity to see what use of the different commas looks like in keys which are fairly different. The idea is to try to provide enough information to decide which of the commas in the main list to assign to the accidentals. In most cases my preference is to reduce the number of symbols as much as possible, i.e., E- instead of Eb+ ... but as we saw in the key of A above, this same harmonic would be notated respectively as C#- instead of C+ ... so it is not easy to make a decision based on that. Ultimately it may be best to go with the solution of Daniel Wolf presented further above this page, using the commas with the simplest ratios even when those comma adjustments are larger.
example, showing odd harmonics above C
harmonic hewm-53-limit
1 C
3 G
5 E- or Eb+
7 Bb< or A> or B<
9 D
11 F^ or F#v
13 Ab} or A{
15 B- or Bb+
17 C#\ or Db/ or C/
19 Eb) or D#( or D)
21 F< or F#>
23 F#] or Gb] or G[
25 G#-- or G++
27 A
29 A#! or Bb! or Cb;
31 B" or C?
33 C^ or C#v
35 D<- or C#>- or Db<+ or C>+
37 D% or Eb&
39 Eb} or E{
41 E$
43 F'
45 F#- or F+
47 G: or F#*
49 Ab<< or A>>
51 G#\ or Ab/ or G/
53 A_ or Ab|
I wanted to avoid using letters, but the following pairs might work as substitutes for some given above, or if there is a reason to expand to 71-limit.
lower raise prime s z 59 y k 61 j f 67 p d 71
Please help keep Tonalsoft online! Select your level of support from the menu. Thank you!