  # gene, 311-edo

[Joe Monzo]

A small unit of interval measurement, noted first by Gene Ward Smith in 1969, and suggested as an interval measurement by Joe Monzo in April 2007. (See Yahoo tuning group message 71213 [Sun Apr 15, 2007 2:45 pm PST].)

A gene divides the octave into 311 equal parts. Its use is valuable because it has very low error for the just-intonation ratios thru the 41-prime-limit, thus obviating the need for decimal places in measuring those intervals logarithmically.

The gene is therefore calculated as the 311th root of 2, or 2(1/311), with a ratio of approximately 1:1.00223125462134. It is an irrational number.

A gene is:

• exactly 3 267/311 (= ~3.858520900 [the denominator is prime, so the decimal part is huge and continues from here] = ~ 3 6/7 ) cents,

The formula for calculating the gene-value of any ratio r is: genes = log10(r) * [ 311 / log10(2) ] or genes = log2(r) * 311

A gene represents one degree of 311-edo tuning.

The 12-edo semitone is exactly 25 11/12 (= 25.91,6...) genes.

. . . . . . . . .

### some intervals mapped to 311-edo

```prime   edo-steps   step-error  edo-map

2 =  311.000000    +0.00 -->    311
3 =  492.923338    +0.08 -->    493
5 =  722.119638    -0.12 -->    722
7 =  873.087381    -0.09 -->    873
11 = 1075.883233    +0.12 -->   1076
13 = 1150.836752    +0.16 -->   1151
17 = 1271.200944    -0.20 -->   1271
19 = 1321.105457    -0.11 -->   1321
23 = 1406.827768    +0.17 -->   1407
29 = 1510.832089    +0.17 -->   1511
31 = 1540.755053    +0.24 -->   1541
37 = 1620.139997    -0.14 -->   1620
41 = 1666.198673    -0.20 -->   1666

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

map  2 -->    311 = 1200.000000 cents <-- 1200.000000  +0.0 cents
map  3 -->    493 = 1902.250804 cents <-- 1901.955001  +0.3 cents
map  5 -->    722 = 2785.852090 cents <-- 2786.313714  -0.5 cents
map  7 -->    873 = 3368.488746 cents <-- 3368.825906  -0.3 cents
map 11 -->   1076 = 4151.768489 cents <-- 4151.317942  +0.5 cents
map 13 -->   1151 = 4441.157556 cents <-- 4440.527662  +0.6 cents
map 17 -->   1271 = 4904.180064 cents <-- 4904.955410  -0.8 cents
map 19 -->   1321 = 5097.106109 cents <-- 5097.513016  -0.4 cents
map 23 -->   1407 = 5428.938907 cents <-- 5428.274347  +0.7 cents
map 29 -->   1511 = 5830.225080 cents <-- 5829.577194  +0.6 cents
map 31 -->   1541 = 5945.980707 cents <-- 5945.035572  +0.9 cents
map 37 -->   1620 = 6250.803859 cents <-- 6251.344039  -0.5 cents
map 41 -->   1666 = 6428.295820 cents <-- 6429.062406  -0.8 cents

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

examples:

ratio                -->      311-edo mapping:

ratio           cents   error       edo     cents    name

2:1         =  1200.0   +0.0 -->   311/311  = 1200.0  (octave)
65536:32805     =  1198.0   -1.9 -->   310/311  = 1196.1  (minimal just dim-2)
2025:1024      =  1180.4   +0.3 -->   306/311  = 1180.7  (small just aug-7th)
1048576:531441    =  1176.5   -3.5 -->   304/311  = 1173.0  (pythagorean dim-2nd)
125:64        =  1158.9   -1.4 -->   300/311  = 1157.6  (minimal just aug-7th)
31:16        =  1145.0   +0.9 -->   297/311  = 1146.0  (31st harmonic)
48:25        =  1129.3   +1.2 -->   293/311  = 1130.5  (small just dim-8ve)
21:11        =  1119.5   -0.5 -->   290/311  = 1119.0  (undecimal diminished-8ve)
243:128       =  1109.8   +1.5 -->   288/311  = 1111.3  (pythagorean major-7th)
256:135       =  1107.8   -0.4 -->   287/311  = 1107.4  (minimal just dim-8ve)
15:8         =  1088.3   -0.2 -->   282/311  = 1088.1  (15th harmonic, large just major-7th, 5*3)
4096:2187      =  1086.3   -2.1 -->   281/311  = 1084.2  (pythagorean diminished-8ve)
13:7         =  1071.7   +1.0 -->   278/311  = 1072.7  (tridecimal superminor-7th)
50:27        =  1066.8   -1.8 -->   276/311  = 1065.0  (small just maj-7th)
24:13        =  1061.4   -0.3 -->   275/311  = 1061.1  (tridecimal major-7th)
11:6         =  1049.4   +0.2 -->   272/311  = 1049.5  (undecimal submajor[neutral]-7th)
20:11        =  1035.0   -0.9 -->   268/311  = 1034.1  (undecimal superminor[neutral]-7th)
29:16        =  1029.6   +0.6 -->   267/311  = 1030.2  (29th harmonic)
59049:32768     =  1019.6   +3.0 -->   265/311  = 1022.5  (pythagorean aug-6th)
9:5         =  1017.6   +1.1 -->   264/311  = 1018.6  (just minor-7th)
3645:2048      =   998.0   +1.3 -->   259/311  =  999.4  (large just aug-6th)
16:9         =   996.1   -0.6 -->   258/311  =  995.5  (pythagorean minor-7th)
225:128       =   976.5   -0.3 -->   253/311  =  976.2  (small just augmented-6th)
7:4         =   968.8   -0.3 -->   251/311  =  968.5  (7th harmonic, septimal subminor-7th)
216:125       =   946.9   +2.3 -->   246/311  =  949.2  (large just dim-7)
19:11        =   946.2   -0.9 -->   245/311  =  945.3  (nondecimal supermajor-6th)
12:7         =   933.1   +0.6 -->   242/311  =  933.8  (septimal supermajor-6th)
128:75        =   925.4   +0.6 -->   240/311  =  926.0  (small just dim-7th)
22:13        =   910.8   -0.2 -->   236/311  =  910.6  (tridecimal augmented-6th)
27:16        =   905.9   +0.9 -->   235/311  =  906.8  (27th harmonic, pythagorean major-6th)
2048:1215      =   903.9   -1.0 -->   234/311  =  902.9  (minimal just dim-7)
5:3         =   884.4   -0.8 -->   229/311  =  883.6  (just major-6th)
32768:19683     =   882.4   -2.7 -->   228/311  =  879.7  (pythagorean dim-7th)
18:11        =   852.6   +0.1 -->   221/311  =  852.7  (undecimal superminor[neutral]-6th)
13:8         =   840.5   +0.6 -->   218/311  =  841.2  (13th harmonic)
21:13        =   830.3   -0.7 -->   215/311  =  829.6  (tridecimal ?)
6561:4096      =   815.6   +2.4 -->   212/311  =  818.0  (pythagorean augmented-5th)
8:5         =   813.7   +0.5 -->   211/311  =  814.1  (just minor-6th)
405:256       =   794.1   +0.7 -->   206/311  =  794.9  (large just aug-5th)
128:81        =   792.2   -1.2 -->   205/311  =  791.0  (pythagorean minor-6th)
11:7         =   782.5   +0.8 -->   203/311  =  783.3  (undecimal augmented-5th)
25:16        =   772.6   -0.9 -->   200/311  =  771.7  (25th harmonic, small just augmented-5th)
14:9         =   764.9   -0.9 -->   198/311  =  764.0  (septimal subminor-6th)
17:11        =   753.6   -1.2 -->   195/311  =  752.4  (septendecimal diminished-6th)
20:13        =   745.8   -1.1 -->   193/311  =  744.7  (tridecimal augmented-5th)
192:125       =   743.0   +1.7 -->   193/311  =  744.7  (large just dim-6)
1024:675       =   721.5   +0.0 -->   187/311  =  721.5  (small just dim-6th)
3:2         =   702.0   +0.3 -->   182/311  =  702.3  (pythagorean perfect-5th)
16384:10935     =   700.0   -1.6 -->   181/311  =  698.4  (minimal just dim-6)
262144:177147    =   678.5   -3.3 -->   175/311  =  675.2  (pythagorean dim-6th)
19:13        =   657.0   -1.0 -->   170/311  =  655.9  (nondecimal doubly-augmented-4th)
16:11        =   648.7   -0.5 -->   168/311  =  648.2  (11th subharmonic, undecimal diminished-4th)
13:9         =   636.6   +0.0 -->   165/311  =  636.7  (tridecimal diminished-5th)
23:16        =   628.3   +0.7 -->   163/311  =  628.9  (23rd harmonic)
10:7         =   617.5   -0.1 -->   160/311  =  617.4  (septimal large-tritone)
729:512       =   611.7   +1.8 -->   159/311  =  613.5  (pythagorean augmented-4th)
64:45        =   609.8   -0.1 -->   158/311  =  609.6  (just diminished-5th)
45:32        =   590.2   +0.1 -->   153/311  =  590.4  (large just augmented-4th)
1024:729       =   588.3   -1.8 -->   152/311  =  586.5  (pythagorean diminished-5th)
7:5         =   582.5   +0.1 -->   151/311  =  582.6  (septimal small-tritone)
25:18        =   568.7   -1.5 -->   147/311  =  567.2  (small just aug-4th)
18:13        =   563.4   -0.0 -->   146/311  =  563.3  (tridecimal augmented-4th)
11:8         =   551.3   +0.5 -->   143/311  =  551.8  (11th harmonic, undecimal sub-augmented-4th)
15:11        =   537.0   -0.6 -->   139/311  =  536.3  (undecimal large-4th)
177147:131072    =   521.5   +3.3 -->   136/311  =  524.8  (pythagorean aug-3rd)
10935:8192      =   500.0   +1.6 -->   130/311  =  501.6  (large just aug-3rd)
4:3         =   498.0   -0.3 -->   129/311  =  497.7  (pythagorean perfect-4th)
675:512       =   478.5   -0.0 -->   124/311  =  478.5  (small just aug-3rd)
21:16        =   470.8   -0.0 -->   122/311  =  470.7  (21st harmonic, septimal-4th, 7*3)
17:13        =   464.4   -1.4 -->   120/311  =  463.0  (septendecimal 4th)
125:96        =   457.0   -1.7 -->   118/311  =  455.3  (minimal just aug-3rd)
13:10        =   454.2   +1.1 -->   118/311  =  455.3  (tridecimal diminished-4th)
9:7         =   435.1   +0.9 -->   113/311  =  436.0  (septimal supermajor-3rd)
41:32        =   429.1   -0.8 -->   111/311  =  428.3  (41st harmonic)
32:25        =   427.4   +0.9 -->   111/311  =  428.3  (small just dim-4th)
14:11        =   417.5   -0.8 -->   108/311  =  416.7  (undecimal diminished-4th)
81:64        =   407.8   +1.2 -->   106/311  =  409.0  (pythagorean major-3rd)
512:405       =   405.9   -0.7 -->   105/311  =  405.1  (minimal just dim-4)
5:4         =   386.3   -0.5 -->   100/311  =  385.9  (5th harmonic, just major-3rd)
8192:6561      =   384.4   -2.4 -->    99/311  =  382.0  (pythagorean diminished-4th)
16:13        =   359.5   -0.6 -->    93/311  =  358.8  (tridecimal major[neutral]-3rd)
11:9         =   347.4   -0.1 -->    90/311  =  347.3  (undecimal neutral-3rd)
39:32        =   342.5   +0.9 -->    89/311  =  343.4  (39th harmonic, 13*3)
19683:16384     =   317.6   +2.7 -->    83/311  =  320.3  (pythagorean augmented-2nd)
6:5         =   315.6   +0.8 -->    82/311  =  316.4  (just minor-3rd)
19:16        =   297.5   -0.4 -->    77/311  =  297.1  (19th harmonic)
1215:1024      =   296.1   +1.0 -->    77/311  =  297.1  (large just aug-2nd)
32:27        =   294.1   -0.9 -->    76/311  =  293.2  (pythagorean minor-3rd)
13:11        =   289.2   +0.2 -->    75/311  =  289.4  (tridecimal diminished-3rd)
75:64        =   274.6   -0.6 -->    71/311  =  274.0  (small just augmented-2nd)
7:6         =   266.9   -0.6 -->    69/311  =  266.2  (septimal subminor-3rd)
37:32        =   251.3   -0.5 -->    65/311  =  250.8  (37th harmonic)
15:13        =   247.7   -0.8 -->    64/311  =  246.9  (tridecimal augmented[neutral]-2nd)
144:125       =   245.0   +2.0 -->    64/311  =  246.9  (large just dim-3)
8:7         =   231.2   +0.3 -->    60/311  =  231.5  (septimal tone, supermajor-2nd)
256:225       =   223.5   +0.3 -->    58/311  =  223.8  (small just dim-3rd)
9:8         =   203.9   +0.6 -->    53/311  =  204.5  (pythagorean major-2nd/tone)
4096:3645      =   202.0   -1.3 -->    52/311  =  200.6  (minimal just dim-3)
10:9         =   182.4   -1.1 -->    47/311  =  181.4  (just minor-tone)
65536:59049     =   180.4   -3.0 -->    46/311  =  177.5  (pythagorean dim-3rd)
11:10        =   165.0   +0.9 -->    43/311  =  165.9  (undecimal small-tone/submajor-2nd)
35:32        =   155.1   -0.8 -->    40/311  =  154.3  (35th harmonic, 7*5)
12:11        =   150.6   -0.2 -->    39/311  =  150.5  (undecimal large-semitone)
13:12        =   138.6   +0.3 -->    36/311  =  138.9  (tridecimal minor-2nd)
14:13        =   128.3   -1.0 -->    33/311  =  127.3  (tridecimal major-2nd)
15:14        =   119.4   +0.2 -->    31/311  =  119.6  (septimal chromatic-semitone)
2187:2048      =   113.7   +2.1 -->    30/311  =  115.8  (pythagorean augmented-prime/apotome)
16:15        =   111.7   +0.2 -->    29/311  =  111.9  (just diatonic-semitone)
17:16        =   105.0   -0.8 -->    27/311  =  104.2  (17th harmonic, septendecimal semitone)
135:128       =    92.2   +0.4 -->    24/311  =   92.6  (large just aug-prime)
256:243       =    90.2   -1.5 -->    23/311  =   88.7  (pythagorean minor-2nd/limma)
25:24        =    70.7   -1.2 -->    18/311  =   69.5  (small just aug-prime, chromatic-semitone)
33:32        =    53.3   +0.7 -->    14/311  =   54.0  (33rd harmonic, 11*3)
128:125       =    41.1   +1.4 -->    11/311  =   42.4  (large just dim-2, diesis)
2048:2025      =    19.6   -0.3 -->     5/311  =   19.3  (small just dim-2nd, diaschisma)
32805:32768     =     2.0   +1.9 -->     1/311  =    3.9  (large just aug-7th, skhisma)
1:1         =     0.0   +0.0 -->     0/311  =    0.0  (prime, unison)

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

some commas:

ratio             cents  error        edo       cents  name

3-limit

531441:524288    =    23.5   +3.5 -->     7/311  =   27.0  (pythagorean-comma)

5-limit

648:625       =    62.6   +3.0 -->    17/311  =   65.6  (major-diesis)
16875:16384     =    51.1   -1.0 -->    13/311  =   50.2  (negri-comma)
250:243       =    49.2   -2.9 -->    12/311  =   46.3  (maximal-diesis)
128:125       =    41.1   +1.4 -->    11/311  =   42.4  (enharmonic-diesis)
34171875:33554432  =    31.6   -0.7 -->     8/311  =   30.9  (ampersand-comma)
3125:3072      =    29.6   -2.6 -->     7/311  =   27.0  (magic-comma)
20000:19683     =    27.7   -4.5 -->     6/311  =   23.2  (tetracot-comma)
81:80        =    21.5   +1.6 -->     6/311  =   23.2  (syntonic-comma)
2048:2025      =    19.6   -0.3 -->     5/311  =   19.3  (diaschisma)
393216:390625    =    11.4   +4.0 -->     4/311  =   15.4  (wuerschmidt-comma)
2109375:2097152   =    10.1   -2.3 -->     2/311  =    7.7  (semicomma)
15625:15552     =     8.1   -4.2 -->     1/311  =    3.9  (kleisma)
32805:32768     =     2.0   +1.9 -->     1/311  =    3.9  (skhisma)
76294:76256     =     0.9  -16.3 -->    -4/311  =  -15.4  (ennealimma (~ratio))
292300:292297    =     0.0  -19.3 -->    -5/311  =  -19.3  (atom (~ratio))

7-limit

36:35        =    48.8   +1.4 -->    13/311  =   50.2  (septimal-diesis)
49:48        =    35.7   -1.0 -->     9/311  =   34.7  (slendro diesis (7/6 : 8/7))
50:49        =    35.0   -0.2 -->     9/311  =   34.7  (tritonic diesis, jubilisma)
64:63        =    27.3   -0.3 -->     7/311  =   27.0  (septimal-comma)
225:224       =     7.7   +0.0 -->     2/311  =    7.7  (septimal-kleisma)

11-limit

22:21        =    80.5   +0.5 -->    21/311  =   81.0  ()
33:32        =    53.3   +0.7 -->    14/311  =   54.0  (undecimal-diesis)
45:44        =    38.9   -0.3 -->    10/311  =   38.6  ()
8192:8019      =    37.0   -2.2 -->     9/311  =   34.7  (pyth dim-5th: 11/8)
55:54        =    31.8   -0.9 -->     8/311  =   30.9  ()
56:55        =    31.2   -0.3 -->     8/311  =   30.9  ()
99:98        =    17.6   +1.7 -->     5/311  =   19.3  (mothwellsma)
100:99        =    17.4   -2.0 -->     4/311  =   15.4  (ptolemisma)
121:120       =    14.4   +1.1 -->     4/311  =   15.4  (biyatisma (11/10 : 12/11))

13-limit

40:39        =    43.8   -1.4 -->    11/311  =   42.4  ((5/3 : 13/8))
65:64        =    26.8   +0.2 -->     7/311  =   27.0  ((13/8 : 8/5))
6656:6561      =    24.9   -1.7 -->     6/311  =   23.2  (13/8 : pyth aug-5th)
91:90        =    19.1   +0.2 -->     5/311  =   19.3  (superleap)
144:143       =    12.1   -0.5 -->     3/311  =   11.6  ((18/11 : 13/8))
169:168       =    10.3   +1.3 -->     3/311  =   11.6  (dhanvantarisma)

```
. . . . . . . . .

### genes calculator

Ratio may be entered as fraction or floating-point decimal number.
(value must be greater than 1)

For EDOs (equal-temperaments), type: "a/b" (without quotes)
where "a" = EDO degree and "b" = EDO cardinality.
(value must be less than 1)

Enter ratio: = genes (= )

. . . . . . . . .

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