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HEWM notation: Appendix

[Joe Monzo]

Various earlier thoughts leading to final specification on HEWM page

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
. . . . . . . . .

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