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37-edo / 37-et / 37-tone equal-temperament

[Joe Monzo]

37-edo in pitch order

Below is a list of pitches in 37-edo, arranged in descending order of pitch, showing cents, genes, and tinas.

37-edo
deg    ~cents     exact cents     genes   exact genes   tinas    exact tinas
  37   1200.000  1200 +  0 / 37    311   311 +  0 / 37   8539   8539 +  0 / 37  
  36   1167.568  1167 + 21 / 37    303   302 + 22 / 37   8308   8308 +  8 / 37  
  35   1135.135  1135 +  5 / 37    294   294 +  7 / 37   8077   8077 + 16 / 37  
  34   1102.703  1102 + 26 / 37    286   285 + 29 / 37   7847   7846 + 24 / 37  
  33   1070.270  1070 + 10 / 37    277   277 + 14 / 37   7616   7615 + 32 / 37  
  32   1037.838  1037 + 31 / 37    269   268 + 36 / 37   7385   7385 +  3 / 37  
  31   1005.405  1005 + 15 / 37    261   260 + 21 / 37   7154   7154 + 11 / 37  
  30    972.973   972 + 36 / 37    252   252 +  6 / 37   6924   6923 + 19 / 37  
  29    940.541   940 + 20 / 37    244   243 + 28 / 37   6693   6692 + 27 / 37  
  28    908.108   908 +  4 / 37    235   235 + 13 / 37   6462   6461 + 35 / 37  
  27    875.676   875 + 25 / 37    227   226 + 35 / 37   6231   6231 +  6 / 37  
  26    843.243   843 +  9 / 37    219   218 + 20 / 37   6000   6000 + 14 / 37  
  25    810.811   810 + 30 / 37    210   210 +  5 / 37   5770   5769 + 22 / 37  
  24    778.378   778 + 14 / 37    202   201 + 27 / 37   5539   5538 + 30 / 37  
  23    745.946   745 + 35 / 37    193   193 + 12 / 37   5308   5308 +  1 / 37  
  22    713.514   713 + 19 / 37    185   184 + 34 / 37   5077   5077 +  9 / 37  
  21    681.081   681 +  3 / 37    177   176 + 19 / 37   4846   4846 + 17 / 37  
  20    648.649   648 + 24 / 37    168   168 +  4 / 37   4616   4615 + 25 / 37  
  19    616.216   616 +  8 / 37    160   159 + 26 / 37   4385   4384 + 33 / 37  
  18    583.784   583 + 29 / 37    151   151 + 11 / 37   4154   4154 +  4 / 37  
  17    551.351   551 + 13 / 37    143   142 + 33 / 37   3923   3923 + 12 / 37  
  16    518.919   518 + 34 / 37    134   134 + 18 / 37   3693   3692 + 20 / 37  
  15    486.486   486 + 18 / 37    126   126 +  3 / 37   3462   3461 + 28 / 37  
  14    454.054   454 +  2 / 37    118   117 + 25 / 37   3231   3230 + 36 / 37  
  13    421.622   421 + 23 / 37    109   109 + 10 / 37   3000   3000 +  7 / 37  
  12    389.189   389 +  7 / 37    101   100 + 32 / 37   2769   2769 + 15 / 37  
  11    356.757   356 + 28 / 37     92    92 + 17 / 37   2539   2538 + 23 / 37  
  10    324.324   324 + 12 / 37     84    84 +  2 / 37   2308   2307 + 31 / 37  
   9    291.892   291 + 33 / 37     76    75 + 24 / 37   2077   2077 +  2 / 37  
   8    259.459   259 + 17 / 37     67    67 +  9 / 37   1846   1846 + 10 / 37  
   7    227.027   227 +  1 / 37     59    58 + 31 / 37   1615   1615 + 18 / 37  
   6    194.595   194 + 22 / 37     50    50 + 16 / 37   1385   1384 + 26 / 37  
   5    162.162   162 +  6 / 37     42    42 +  1 / 37   1154   1153 + 34 / 37  
   4    129.730   129 + 27 / 37     34    33 + 23 / 37    923    923 +  5 / 37  
   3     97.297    97 + 11 / 37     25    25 +  8 / 37    692    692 + 13 / 37  
   2     64.865    64 + 32 / 37     17    16 + 30 / 37    462    461 + 21 / 37  
   1     32.432    32 + 16 / 37      8     8 + 15 / 37    231    230 + 29 / 37  
   0      0.000     0 +  0 / 37      0     0 +  0 / 37      0      0 +  0 / 37  
			

37-edo is the lowest cardinality of equal-temperament which provides good approximations of JI pitches thru the 13-limit, with the caveat that the approximation of 3 is only mediocre.

. . . . . . . . .

A 13-limit Fokker periodicity-block mapped to 37-edo

On 5 April 2012, Joe Monzo posted an inquiry on the Yahoo! tuning-math group stating that the Tenny-Minkowski reduced lattice basis for a 37-tone Fokker periodicity-block did not produce mappings of the unit generators for prime-factors 7, 11, and 13, and that did include pitches which go too far along the 3-axis, considering that 37-edo's approximation of 3 is not very good. Monzo then asked if anyone could provide a set of unison-vectors which would meet his desired conditions, and Gene Ward Smith responded a few times, the last response [Yahoo! tuning-math group, message 20560] giving a complete Scala file with the appropriate block.


! monzoblock37.scl
!
Symmetrical 13-limit Fokker block containing all of the primes as scale degrees
! Commas: 512/507, 121/120, 91/90, 385/384, 64/63; all generators -18 to 18
37
!
1024/1001
33/32
16/15
13/12
12/11
44/39
8/7
7/6
13/11
77/64
16/13
5/4
33/26
13/10
4/3
192/143
11/8
128/91
91/64
16/11
143/96
3/2
20/13
52/33
8/5
13/8
128/77
22/13
12/7
7/4
39/22
11/6
24/13
15/8
64/33
1001/512
2/1

				

Below is a Tonescape® Lattice illustrating this periodicity-block, using triangular geometry in 3,5,7,11,13-prime-space. This is rendered as a 2-dimensional representation of a 3-dimensional shadow of a 5-dimensional structure. Prime-factor 2 is not shown, since it is being employed as the identity-interval.

Download the Tonescape .tuning file of 37-edo using this periodicity-block for its Lattice structure.

Below is an analysis of the mappings and pitches in this periodicity-block:


unison-vectors:

  ratio        2,  3,  5,  7, 11, 13 - monzo

 512:507  =  [ 9, -1,  0,  0,  0, -2>
 121:120  =  [-3, -1, -1,  0,  2,  0>
  91:90   =  [-1, -2, -1,  1,  0,  1>
 385:384  =  [-7, -1,  1,  1,  1,  0>
  64:63   =  [ 6, -2,  0, -1,  0,  0>
 
 
floating-point mappings:

prime   edo-steps   step-error   edo-map

   2 =  37.000000    +0.00    -->     37
   3 =  58.643613    +0.36    -->     59
   5 =  85.911340    +0.09    -->     86
   7 = 103.872132    +0.13    -->    104
  11 = 127.998970    +0.00    -->    128
  13 = 136.916270    +0.08    -->    137


integer (i.e., true) mappings, compared with cents-value of actual prime

map  2   -->     37 = 1200.000000 cents <-- 1200.000000  + 0.0 cents
map  3   -->     59 = 1913.513514 cents <-- 1901.955001  +11.6 cents
map  5   -->     86 = 2789.189189 cents <-- 2786.313714  + 2.9 cents
map  7   -->    104 = 3372.972973 cents <-- 3368.825906  + 4.1 cents
map 11   -->    128 = 4151.351351 cents <-- 4151.317942  + 0.0 cents
map 13   -->    137 = 4443.243243 cents <-- 4440.527662  + 2.7 cents

--------------

37-tone Fokker periodicity-block, using gws-basis (Gene Ward Smith):

              ratio        -->     37 -edo mapping:

     ratio         cents  error       edo          cents  name

       2:1     =  1200.0  + 0.0  -->  37/37 = 1200.0  (octave)
    1001:512   =  1160.7  + 6.9  -->  36/37 = 1167.6  ()
      64:33    =  1146.7  -11.6  -->  35/37 = 1135.1  ()
      15:8     =  1088.3  +14.4  -->  34/37 = 1102.7  (just major-7th)
      24:13    =  1061.4  + 8.8  -->  33/37 = 1070.3  ()
      11:6     =  1049.4  -11.5  -->  32/37 = 1037.8  ()
      39:22    =   991.2  +14.2  -->  31/37 = 1005.4  ()
       7:4     =   968.8  + 4.1  -->  30/37 =  973.0  (harmonic-7th)
      12:7     =   933.1  + 7.4  -->  29/37 =  940.5  ()
      22:13    =   910.8  - 2.7  -->  28/37 =  908.1  ()
     128:77    =   879.9  - 4.2  -->  27/37 =  875.7  ()
      13:8     =   840.5  + 2.7  -->  26/37 =  843.2  (harmonic-13th)
       8:5     =   813.7  - 2.9  -->  25/37 =  810.8  (just minor-6th)
      52:33    =   787.3  - 8.9  -->  24/37 =  778.4  ()
      20:13    =   745.8  + 0.2  -->  23/37 =  745.9  ()
       3:2     =   702.0  +11.6  -->  22/37 =  713.5  (perfect-5th)
     143:96    =   689.9  - 8.8  -->  21/37 =  681.1  ()
      16:11    =   648.7  - 0.0  -->  20/37 =  648.6  (subharmonic-11th)
      91:64    =   609.4  + 6.9  -->  19/37 =  616.2  ()
     128:91    =   590.6  - 6.9  -->  18/37 =  583.8  ()
      11:8     =   551.3  + 0.0  -->  17/37 =  551.4  (harmonic-11th)
     192:143   =   510.1  + 8.8  -->  16/37 =  518.9  ()
       4:3     =   498.0  -11.6  -->  15/37 =  486.5  (perfect-4th)
      13:10    =   454.2  - 0.2  -->  14/37 =  454.1  ()
      33:26    =   412.7  + 8.9  -->  13/37 =  421.6  ()
       5:4     =   386.3  + 2.9  -->  12/37 =  389.2  (just major-3rd)
      16:13    =   359.5  - 2.7  -->  11/37 =  356.8  (subharmonic-13th)
      77:64    =   320.1  + 4.2  -->  10/37 =  324.3  ()
      13:11    =   289.2  + 2.7  -->   9/37 =  291.9  ()
       7:6     =   266.9  - 7.4  -->   8/37 =  259.5  (subminor-3rd)
       8:7     =   231.2  - 4.1  -->   7/37 =  227.0  (subharmonic-7th, septimal large-2nd)
      44:39    =   208.8  -14.2  -->   6/37 =  194.6  ()
      12:11    =   150.6  +11.5  -->   5/37 =  162.2  ()
      13:12    =   138.6  - 8.8  -->   4/37 =  129.7  ()
      16:15    =   111.7  -14.4  -->   3/37 =   97.3  (just minor-2nd)
      33:32    =    53.3  +11.6  -->   2/37 =   64.9  ()
    1024:1001  =    39.3  - 6.9  -->   1/37 =   32.4  ()
       1:1     =     0.0  + 0.0  -->   0/37 =    0.0  (prime)

				

Below is a pitch-height graph, showing the 37 tones of the Fokker periodicity-block against a grid for the y-axis which represents the degrees of 37-edo:

37-tone 13-limit JI periodicity-block, graphed against 37-edo

37-tone 13-limit periodicity-block graphed against 37-edo
. . . . . . . . .

Some 41-limit JI ratios mapped to 37-edo

Below is a table showing how some of the most common just-intonation ratios in the 41-limit are mapped to 37-edo, using the closest-fit mappings:


37 -edo floating-point mappings:

prime edo-steps   step-error  edo-map

 2 =  37.000000    +0.00 -->     37
 3 =  58.643613    +0.36 -->     59
 5 =  85.911340    +0.09 -->     86
 7 = 103.872132    +0.13 -->    104
11 = 127.998970    +0.00 -->    128
13 = 136.916270    +0.08 -->    137
17 = 151.236125    -0.24 -->    151
19 = 157.173318    -0.17 -->    157
23 = 167.371792    -0.37 -->    167
29 = 179.745297    +0.25 -->    180
31 = 183.305263    -0.31 -->    183
37 = 192.749775    +0.25 -->    193
41 = 198.229424    -0.23 -->    198
43 = 200.771796    +0.23 -->    201

integer (i.e., true) mappings, compared with cents-value of actual prime

map  2 -->     37 = 1200.000000 cents <-- 1200.000000  + 0.0 cents
map  3 -->     59 = 1913.513514 cents <-- 1901.955001  +11.6 cents
map  5 -->     86 = 2789.189189 cents <-- 2786.313714  + 2.9 cents
map  7 -->    104 = 3372.972973 cents <-- 3368.825906  + 4.1 cents
map 11 -->    128 = 4151.351351 cents <-- 4151.317942  + 0.0 cents
map 13 -->    137 = 4443.243243 cents <-- 4440.527662  + 2.7 cents
map 17 -->    151 = 4897.297297 cents <-- 4904.955410  - 7.7 cents
map 19 -->    157 = 5091.891892 cents <-- 5097.513016  - 5.6 cents
map 23 -->    167 = 5416.216216 cents <-- 5428.274347  -12.1 cents
map 29 -->    180 = 5837.837838 cents <-- 5829.577194  + 8.3 cents
map 31 -->    183 = 5935.135135 cents <-- 5945.035572  - 9.9 cents
map 37 -->    193 = 6259.459459 cents <-- 6251.344039  + 8.1 cents
map 41 -->    198 = 6421.621622 cents <-- 6429.062406  - 7.4 cents
map 43 -->    201 = 6518.918919 cents <-- 6511.517706  + 7.4 cents

--------------

examples:

              ratio                      -->     37 -edo mapping:

      ratio            cents   error          edo         cents   name

       2:1           = 1200.0  +  0.0    -->  37/37     = 1200.0  (octave)

     243:128         = 1109.8  + 57.8    -->  36/37     = 1167.6  (pythagorean major-7th)

      31:16          = 1145.0  -  9.9    -->  35/37     = 1135.1  (31st harmonic)
      21:11          = 1119.5  + 15.7    -->  35/37     = 1135.1  (undecimal diminished-8ve)

      15:8           = 1088.3  + 14.4    -->  34/37     = 1102.7  (15th harmonic, just major-7th, 5*3)

      13:7           = 1071.7  -  1.4    -->  33/37     = 1070.3  (tridecimal superminor-7th)
      24:13          = 1061.4  +  8.8    -->  33/37     = 1070.3  (tridecimal major-7th)

      11:6           = 1049.4  - 11.5    -->  32/37     = 1037.8  (undecimal submajor[neutral]-7th)
      20:11          = 1035.0  +  2.8    -->  32/37     = 1037.8  (undecimal superminor[neutral]-7th)
      29:16          = 1029.6  +  8.3    -->  32/37     = 1037.8  (29th harmonic)
       9:5           = 1017.6  + 20.2    -->  32/37     = 1037.8  (just minor-7th)

    4096:2187        = 1086.3  - 80.9    -->  31/37     = 1005.4  (pythagorean diminished-8ve)

      16:9           =  996.1  - 23.1    -->  30/37     =  973.0  (pythagorean minor-7th)
       7:4           =  968.8  +  4.1    -->  30/37     =  973.0  (7th harmonic, septimal subminor-7th)

      19:11          =  946.2  -  5.7    -->  29/37     =  940.5  (nondecimal supermajor-6th)
      12:7           =  933.1  +  7.4    -->  29/37     =  940.5  (septimal supermajor-6th)
      27:16          =  905.9  + 34.7    -->  29/37     =  940.5  (27th harmonic, pythagorean major-6th)

      22:13          =  910.8  -  2.7    -->  28/37     =  908.1  (tridecimal augmented-6th)
    6561:4096        =  815.6  + 92.5    -->  28/37     =  908.1  (pythagorean augmented-5th)

       5:3           =  884.4  -  8.7    -->  27/37     =  875.7  (just major-6th)
      18:11          =  852.6  + 23.1    -->  27/37     =  875.7  (undecimal superminor[neutral]-6th)

      13:8           =  840.5  +  2.7    -->  26/37     =  843.2  (13th harmonic)
      21:13          =  830.3  + 13.0    -->  26/37     =  843.2  (tridecimal ?)

       8:5           =  813.7  -  2.9    -->  25/37     =  810.8  (just minor-6th)

      11:7           =  782.5  -  4.1    -->  24/37     =  778.4  (undecimal augmented-5th)
      25:16          =  772.6  +  5.8    -->  24/37     =  778.4  (25th harmonic, just augmented-5th)

     128:81          =  792.2  - 46.2    -->  23/37     =  745.9  (pythagorean minor-6th)
      14:9           =  764.9  - 19.0    -->  23/37     =  745.9  (septimal subminor-6th)
      17:11          =  753.6  -  7.7    -->  23/37     =  745.9  (septendecimal diminished-6th)
      20:13          =  745.8  +  0.2    -->  23/37     =  745.9  (tridecimal augmented-5th)

       3:2           =  702.0  + 11.6    -->  22/37     =  713.5  (perfect-5th)

     729:512         =  611.7  + 69.4    -->  21/37     =  681.1  (pythagorean augmented-4th)

      19:13          =  657.0  -  8.3    -->  20/37     =  648.6  (nondecimal doubly-augmented-4th)
      16:11          =  648.7  -  0.0    -->  20/37     =  648.6  (11th subharmonic, undecimal diminished-4th)

      13:9           =  636.6  - 20.4    -->  19/37     =  616.2  (tridecimal diminished-5th)
      23:16          =  628.3  - 12.1    -->  19/37     =  616.2  (23rd harmonic)
      10:7           =  617.5  -  1.3    -->  19/37     =  616.2  (septimal large-tritone)
      45:32          =  590.2  + 26.0    -->  19/37     =  616.2  (just augmented-4th)

      64:45          =  609.8  - 26.0    -->  18/37     =  583.8  (just diminished-5th)
       7:5           =  582.5  +  1.3    -->  18/37     =  583.8  (septimal small-tritone)
      18:13          =  563.4  + 20.4    -->  18/37     =  583.8  (tridecimal augmented-4th)

      11:8           =  551.3  +  0.0    -->  17/37     =  551.4  (11th harmonic, undecimal sub-augmented-4th)
      15:11          =  537.0  + 14.4    -->  17/37     =  551.4  (undecimal large-4th)

    1024:729         =  588.3  - 69.4    -->  16/37     =  518.9  (pythagorean diminished-5th)

       4:3           =  498.0  - 11.6    -->  15/37     =  486.5  (perfect-4th)
      21:16          =  470.8  + 15.7    -->  15/37     =  486.5  (21st harmonic, septimal-4th, 7*3)

      17:13          =  464.4  - 10.4    -->  14/37     =  454.1  (septendecimal 4th)
      13:10          =  454.2  -  0.2    -->  14/37     =  454.1  (tridecimal diminished-4th)
       9:7           =  435.1  + 19.0    -->  14/37     =  454.1  (septimal supermajor-3rd)
      81:64          =  407.8  + 46.2    -->  14/37     =  454.1  (pythagorean major-3rd)

      41:32          =  429.1  -  7.4    -->  13/37     =  421.6  (41st harmonic)
      14:11          =  417.5  +  4.1    -->  13/37     =  421.6  (undecimal diminished-4th)
   19683:16384       =  317.6  +104.0    -->  13/37     =  421.6  (pythagorean augmented-2nd)

       5:4           =  386.3  +  2.9    -->  12/37     =  389.2  (5th harmonic, just major-3rd)

      16:13          =  359.5  -  2.7    -->  11/37     =  356.8  (tridecimal major[neutral]-3rd)
      39:32          =  342.5  + 14.3    -->  11/37     =  356.8  (39th harmonic, 13*3)

      11:9           =  347.4  - 23.1    -->  10/37     =  324.3  (undecimal neutral-3rd)
       6:5           =  315.6  +  8.7    -->  10/37     =  324.3  (just minor-3rd)

    8192:6561        =  384.4  - 92.5    -->   9/37     =  291.9  (pythagorean diminished-4th)
      19:16          =  297.5  -  5.6    -->   9/37     =  291.9  (19th harmonic)
      13:11          =  289.2  +  2.7    -->   9/37     =  291.9  (tridecimal diminished-3rd)
      75:64          =  274.6  + 17.3    -->   9/37     =  291.9  (just augmented-2nd)

      32:27          =  294.1  - 34.7    -->   8/37     =  259.5  (pythagorean minor-3rd)
       7:6           =  266.9  -  7.4    -->   8/37     =  259.5  (septimal subminor-3rd)
      37:32          =  251.3  +  8.1    -->   8/37     =  259.5  (37th harmonic)
      15:13          =  247.7  + 11.7    -->   8/37     =  259.5  (tridecimal augmented[neutral]-2nd)

       8:7           =  231.2  -  4.1    -->   7/37     =  227.0  (septimal tone, supermajor-2nd)
       9:8           =  203.9  + 23.1    -->   7/37     =  227.0  (pythagorean major-2nd/tone)

    2187:2048        =  113.7  + 80.9    -->   6/37     =  194.6  (pythagorean augmented-prime/apotome)

      10:9           =  182.4  - 20.2    -->   5/37     =  162.2  (just minor-tone)
      11:10          =  165.0  -  2.8    -->   5/37     =  162.2  (undecimal small-tone/submajor-2nd)
      35:32          =  155.1  +  7.0    -->   5/37     =  162.2  (35th harmonic, 7*5)
      12:11          =  150.6  + 11.5    -->   5/37     =  162.2  (undecimal large-semitone)

      13:12          =  138.6  -  8.8    -->   4/37     =  129.7  (tridecimal minor-2nd)
      14:13          =  128.3  +  1.4    -->   4/37     =  129.7  (tridecimal major-2nd)
      15:14          =  119.4  + 10.3    -->   4/37     =  129.7  (septimal chromatic-semitone)

      16:15          =  111.7  - 14.4    -->   3/37     =   97.3  (just diatonic-semitone)
      17:16          =  105.0  -  7.7    -->   3/37     =   97.3  (17th harmonic, septendecimal semitone)

      25:24          =   70.7  -  5.8    -->   2/37     =   64.9  (just chromatic-semitone)
      33:32          =   53.3  + 11.6    -->   2/37     =   64.9  (33rd harmonic, 11*3)

     256:243         =   90.2  - 57.8    -->   1/37     =   32.4  (pythagorean minor-2nd/limma)

       1:1           =    0.0  +  0.0    -->   0/37     =    0.0  (prime)

some commas:

3-limit
     ratio              cents  error          edo         cents  name

  531441:524288      =   23.5  +138.7    -->   5/37     =  162.2  (pythagorean-comma)

5-limit
     ratio              cents  error          edo         cents  name

     648:625         =   62.6  +  34.7    -->   3/37     =    97.3  (major-diesis)
   16875:16384       =   51.1  +  46.2    -->   3/37     =    97.3  (negri-comma)
     250:243         =   49.2  -  49.2    -->   0/37     =     0.0  (maximal-diesis)
     128:125         =   41.1  -   8.6    -->   1/37     =    32.4  (enharmonic-diesis)
34171875:33554432    =   31.6  +  98.2    -->   4/37     =   129.7  (ampersand-comma)
    3125:3072        =   29.6  +   2.8    -->   1/37     =    32.4  (magic-comma)
   20000:19683       =   27.7  -  92.5    -->  -2/37     =  - 64.9  (tetracot-comma)
      81:80          =   21.5  +  43.4    -->   2/37     =    64.9  (syntonic-comma)
    2048:2025        =   19.6  -  52.0    -->  -1/37     =  - 32.4  (diaschisma)
  393216:390625      =   11.4  -  11.4    -->   0/37     =     0.0  (wuerschmidt-comma)
 2109375:2097152     =   10.1  +  54.8    -->   2/37     =    64.9  (semicomma)
   15625:15552       =    8.1  -  40.5    -->  -1/37     =  - 32.4  (kleisma)
   32805:32768       =    2.0  +  95.3    -->   3/37     =    97.3  (skhisma)
   76294:76256       =    0.9  - 260.3    -->  -8/37     = - 259.5  (ennealimma (~ratio))
  292300:292297      =    0.0  -1005.4    --> -31/37     = -1005.4  (atom (~ratio))

7-limit
     ratio              cents  error          edo         cents  name

      36:35          =   48.8  +16.1    -->   2/37     =   64.9  (septimal-diesis)
      49:48          =   35.7  - 3.3    -->   1/37     =   32.4  (slendro diesis (7/6 : 8/7))
      50:49          =   35.0  - 2.5    -->   1/37     =   32.4  (tritonic diesis, jubilisma)
      64:63          =   27.3  -27.3    -->   0/37     =    0.0  (septimal-comma)
     225:224         =    7.7  +24.7    -->   1/37     =   32.4  (septimal-kleisma)

11-limit
     ratio              cents  error          edo         cents  name

      22:21          =   80.5  -15.7    -->   2/37     =   64.9  ()
      33:32          =   53.3  +11.6    -->   2/37     =   64.9  (undecimal-diesis)
      45:44          =   38.9  +26.0    -->   2/37     =   64.9  ()
    8192:8019        =   37.0  -69.4    -->  -1/37     =  -32.4  (pyth dim-5th: 11/8)
      55:54          =   31.8  -31.8    -->   0/37     =    0.0  ()
      56:55          =   31.2  + 1.2    -->   1/37     =   32.4  ()
      99:98          =   17.6  +14.9    -->   1/37     =   32.4  (mothwellsma)
     100:99          =   17.4  -17.4    -->   0/37     =    0.0  (ptolemisma)
     121:120         =   14.4  -14.4    -->   0/37     =    0.0  (biyatisma (11/10 : 12/11))

13-limit
     ratio              cents  error          edo         cents  name

      40:39          =   43.8  -11.4    -->   1/37     =   32.4  ((5/3 : 13/8))
      65:64          =   26.8  + 5.6    -->   1/37     =   32.4  ((13/8 : 8/5))
    6656:6561        =   24.9  -89.8    -->  -2/37     =  -64.9  (13/8 : pyth aug-5th)
      91:90          =   19.1  -19.1    -->   0/37     =    0.0  (superleap)
     144:143         =   12.1  +20.4    -->   1/37     =   32.4  ((18/11 : 13/8))
     169:168         =   10.3  -10.3    -->   0/37     =    0.0  (dhanvantarisma)
				

Below is a graph showing the degrees of 37-edo as the Y-axis grid, with the pitch of the JI ratios given in the table above plotted against it:

. . . . . . . . .

Below are graphs of the error of 12-edo and 37-edo for approximations to the prime-factors up to 43, given as a percentage of one degree of each tuning. It can easily be seen that compared to 12-edo, 37-edo's approximations of the prime-factors are:

However, Monzo feels that since jazz musical styles generally place less harmonic emphasis on the 3-identity of a chord (often omitting it entirely), 37-edo offers a great avenue of exploration for microtonality in jazz.

[Joe Monzo]