# tina

[Joe Monzo]

A very small unit of interval measurement, advocated by Joe Monzo primarly because of its accuracy for use as logarithmic integer values to describe just-intonation intervals thru the 31-limit, first suggested by privately by George Secor in September 2004, then publicly by Gene Ward Smith on 22 April 2007. A few days later, Monzo independently found, by using an error calculation that weighted the percent error for prime-factors as 3:*10, 5:*6, 7:*3, 11:*2, and all the others up to 41 *1, that 8539-edo had a lower "score" than any other EDO with cardinality smaller than 19932-edo, giving excellent approximations to all primes in the 41-limit except 37. Dave Keenan and George Secor referred to it as "tina", a name which Monzo adopted.

A tina divides the octave into 8,539 equal parts. Because that number is prime, it does not offer any other divisibility, which is its chief disadvantage.

The tina is therefore calculated as the 8539th root of 2, or 2(1/8539), with a ratio of approximately 1:1.000081178. It is an irrational number.

A tina is:

• exactly 12/8539 (= ~0.001405 = ~ 1/712 ) of a 12-edo semitone,
• exactly 19/8539 (= ~0.002225 = ~ 1/449 ) of a 19-edo "third-tone",
• exactly 31/8539 (= ~0.003630 = ~ 1/275 ) of a 31-edo diesis,
• exactly 43/8539 (= ~0.005036 = ~ 1/199 ) of a 43-edo méride,
• exactly 53/8539 (= ~0.006207 = ~ 1/161 ) of a 53-edo mercator,
• exactly 72/8539 (= ~0.008432 = ~ 1/119 ) of a 72-edo morion,
• exactly 144/8539 (= ~0.016864 = ~ 1/59 ) of a 144-edo farab,
• exactly 300/8539 (= ~0.035133 = ~ 1/28 ) of a 300-edo savart,
• exactly 301/8539 (= ~0.035250 = ~ 1/28 ) of a 301-edo heptaméride,
• exactly 612/8539 (= ~0.071671 = ~ 1/14 ) of a 612-edo sk,
• approximately 0.071883 (= ~ 1/14 ) of a grad,
• exactly 730/8539 (= ~0.085490 = ~ 1/12 ) of a 730-edo woolhouse-unit,
• exactly 768/8539 (= ~0.089940 = ~ 1/11 ) of a 768-edo 6mu,
• exactly 1000/8539 (= ~0.117110 = ~ 1/9 ) of a 1000-edo millioctave,
• exactly 1024/8539 (= ~0.119920 = ~ 1/8 ) of a 1024-edo yamaha-unit,
• exactly 1200/8539 (= ~0.140532 = ~ 1/7 ) of a 1200-edo cent,
• exactly 2460/8539 (= ~0.288090 = ~ 2/7 ) of a 2460-edo mina,
• exactly 1 2061/8539 (= ~1.241363 = ~ 1 1/4 ) 10600-edo türk-sents,
• exactly 3 4486/8539 (= ~3.525354 = ~ 3 1/2 ) 30103-edo jots,
• approximately 4.312991 (= ~ 4 1/3 ) tuning-units,
• exactly 5 3337/8539 (= ~5.390795 = ~ 5 2/5 ) 46032-edo flus,
• exactly 5 6457/8539 (= ~5.756178 = ~ 5 3/4 ) 49152-edo 12mus,
• exactly 6 7739/8539 (= ~6.906312 = ~ 6 10/11 ) 58973-edo units,
• exactly 23 211/8539 (= ~23.024710 = ~ 23 1/40 ) 196608-edo 14mus,

The formula for calculating the tina-value of any ratio r is: tinas = log10(r) * [ 8539 / log10(2) ] or tinas = log2(r) * 8539

A tina represents one degree of 8539-edo tuning.

The 12-edo semitone is exactly 711 7/12 (= 711.58,3...) tinas, a value which is nearly midway between two adjacent tinas. The unfortunate result of this is that there is rounding error when using tinas to represent 12-edo. In actual practice it is not really a problem because one tina is so small, but it does cause problems with the mathematics. Below is a table (arranged in descending order of pitch) showing the tina values for 12-edo based on the chain-of-5ths from -5...+6 generators:

```12edo

s-t .. tinas

12 .. 8539
11 .. 7827
10 .. 7116
9 .. 6404
8 .. 5693
7 .. 4981
6 .. 4269
5 .. 3558
4 .. 2846
3 .. 2135
2 .. 1423
1 ... 712
0 ..... 0
```

Below is a table of tina values for common 31-limit JI interval sizes; tina values are given both in integer and floating-point form, expressly to point out how unnecessary it is to use the decimal places for most of the cases:

```                                   integer  floating-pt  ------------- monzo ---------------
interval name                        tinas   tinas       2   3,  5  7 11, 13 17 19, 23 29 31     ratio

octave .............................. 8539  8539.00  [  1> ...................................... 1/1
8386  8385.96  [  5  -4,  1> ............................ 160/81
8345  8344.99  [ -5   2,  0  1> .......................... 63/32
8247  8246.83  [ -6   0,  3> ............................ 125/64
8160  8159.92  [  6  -1,  0  0 -1> ....................... 64/33
31st harmonic ....................... 8148  8147.88  [ -4   0,  0  0  0,  0  0  0,  0  0  1> ... 31/16
septimal major-7th .................. 8091  8090.98  [ -1   3,  0 -1> .......................... 27/14
8068  8068.15  [ -3   0, -1  1  1> ....................... 77/40
8036  8036.11  [  4   1, -2> ............................. 48/25
7938  7937.95  [  3  -1,  1 -1> .......................... 40/21
pyth major-7th ...................... 7897  7896.97  [ -7   5> ................................ 243/128
7883  7883.07  [  8  -3, -1> ............................ 256/135
7814  7814.13  [  1   1, -1 -1  1>........................ 66/35
17th subharmonic .................... 7792  7792.15  [  5   0,  0  0  0,  0 -1> ................ 32/17
just major-7th / 15th harmonic .....  7744  7743.94  [ -3   1,  1> ............................. 15/8
7730  7730.04  [ 12  -7> ............................... 4096/2187
7689  7689.06  [  2  -1, -1  1> .......................... 28/15
undecimal neutral-7th ............... 7467  7467.09  [ -1  -1,  0  0  1> ....................... 11/6
7435  7435.05  [  6   0, -1 -1> .......................... 64/35
7365  7364.86  [  2   0,  1  0 -1 ........................ 20/11
29th harmonic ....................... 7326  7326.30  [ -4   0,  0  0  0,  0  0  0,  0  1> ...... 29/16
7273  7273.09  [ -7   1,  0  1  1> ...................... 231/128
pyth aug-6th .......................  7255  7254.95  [-15  10> .............................. 59049/32768
just minor-7th ...................... 7241  7241.05  [  0   2, -1> .............................. 9/5
7102  7101.91  [-11   6,  1> ........................... 3645/2048
pyth minor-7th / 9th subharmonic .... 7088  7088.01  [  4  -2> ................................. 16/9
just aug-6th ........................ 6949  6948.88  [ -7   2,  2> ............................ 225/128
7th harmonic ........................ 6894  6894.00  [ -2   0,  0  1> ........................... 7/4
septimal minor-7th .................. 6894  6894.00  [ -2   0,  0  1> ........................... 7/4
6862  6861.96  [  5   1, -1  0 -1 ........................ 96/55
6672  6672.03  [ -5   0,  1  0  1 ........................ 55/32
septimal major-6th .................. 6640  6639.99  [  2   1,  0 -1> .......................... 12/7
6599  6599.02  [ -8   7, -1> ........................... 2187/1280
6585  6585.12  [  7  -1, -2> ............................ 128/75
pyth major-6th  / 27th harmonic ..... 6446  6445.98  [ -4   3> ................................. 27/16
19th subharmonic .................... 6422  6421.95  [  5   0,  0  0  0,  0  0 -1> ............. 32/19
6363  6363.14  [  4  -1, -1 -1  1> ...................... 176/105
just major-6th ...................... 6293  6292.95  [  0  -1,  1> .............................. 5/3
6169  6169.14  [ -2   1, -1  0  1> ....................... 33/20
6099  6098.94  [ -6   1,  1  1> ......................... 105/64
6067  6066.90  [  1   2,  0  0 -1> ....................... 18/11
13th harmonic ....................... 5981  5981.05  [ -3   0,  0  0  0,  1> ................... 13/8
5822  5822.10  [ -4  -1,  0  1  1> ....................... 77/48
pyth aug-5th .......................  5804  5803.96  [-12   8> ............................... 6561/4096
just minor-6th / 5th subharmonic .... 5790  5790.06  [  3   0, -1> .............................. 8/5
5651  5650.92  [ -8   4,  1> ............................ 405/256
pyth minor-6th ...................... 5637  5637.02  [  7  -4> ................................ 128/81
5568  5568.08  [  0   0,  0 -1  1> ....................... 11/7
just aug-5th ........................ 5498  5497.89  [ -4   0,  2> ............................. 25/16
5443  5443.01  [  1  -2,  0  1> .......................... 14/9
5411  5410.97  [  8  -1, -1  0 -1> ...................... 256/165
5374  5374.08  [ -6   2,  0  0  1> ....................... 99/64
septimal 5th / 21st subharmonic ..... 5189  5189.00  [  5  -1,  0 -1> .......................... 32/21
5148  5148.03  [ -5   5, -1> ............................ 243/160
perfect-5th / 3rd harmonic .......... 4995  4994.99  [ -1   1> .................................. 3/2
just wolf-5th ....................... 4842  4841.96  [  3  -3,  1> ............................. 40/27
4801  4800.99  [ -7   3,  0  1> ......................... 189/128
4718  4718.15  [  1  -1, -1  0  1> ....................... 22/15
4703  4702.83  [ -8   1,  3> ............................ 375/256
4648  4647.95  [ -3  -1,  1  1> .......................... 35/24
11th subharmonic .................... 4616  4615.91  [  4   0,  0  0 -1> ....................... 16/11
4524  4524.14  [ -5   1, -1  1  1> ...................... 231/160
large just dim-5th .................. 4492  4492.10  [  2   2,  2> ............................. 36/25
23rd harmonic ....................... 4471  4470.70  [ -4   0,  0  0  0,  0  0  0,  1> ......... 23/16
septimal aug-4th .................... 4394  4393.94  [  1   0,  1 -1> .......................... 10/7
pyth aug-4th / tritone .............. 4353  4352.97  [ -9   6> ................................ 729/512
small just dim-5th .................. 4339  4339.07  [  6  -2, -1> ............................. 64/45
just aug-4th / tritone .............. 4200  4199.93  [ -5   2,  1> ............................. 45/32
pyth dim-5th ........................ 4186  4186.03  [ 10  -6> ............................... 1024/729
septimal dim-5th .................... 4145  4145.06  [  0   0, -1  1> ........................... 7/5
23rd subharmonic .................... 4068  4068.30  [  5   0,  0  0  0,  0  0  0, -1> ......... 32/23
4047  4046.90  [ -1  -2,  2> ............................. 25/18
11th harmonic ....................... 3923  3923.09  [ -3   0,  0  0  1> ....................... 11/8
3891  3891.05  [  4   1, -1 -1> .......................... 48/35
3836  3836.17  [  9  -1, -3> ............................ 512/375
3821  3820.85  [  0   1,  1  0 -1> ....................... 15/11
3711  3710.94  [-17  11> ............................. 177147/131072
3697  3697.04  [ -2   3, -1> ............................. 27/20
3558  3557.91  [-13   7> .............................. 10935/8192
perfect-4th / 3rd subharmonic ....... 3544  3544.01  [  2  -1> .................................. 4/3
3405  3404.87  [ -9   3> ................................ 675/512
septimal-4th / 21st harmonic ........ 3350  3350.00  [ -4   1,  0  1> .......................... 21/16
3252  3251.84  [ -5  -1,  3> ............................ 125/96
3165  3164.92  [  7  -2,  0  0 -1> ...................... 128/99
3128  3128.03  [ -7   1,  1  0  1> ...................... 165/128
septimal major-3rd .................. 3096  3095.99  [  0   2,  0 -1> ........................... 9/7
3073  3073.15  [ -2  -1, -1  1  1> ....................... 77/60
3041  3041.11  [  5   0, -2> ............................. 32/25
pyth major-3rd / ditone ............. 2902  2901.98  [ -6   4> ................................. 81/64
2888  2888.08  [  9  -4, -1> ............................ 512/405
2819  2819.14  [  2   0, -1 -1  1> ....................... 44/35
just major-3rd  / 5th harmonic .....  2749  2748.94  [ -2   0,  1> .............................. 5/4
13th subharmonic .................... 2558  2557.95  [  4   0,  0  0  0, -1> ................... 16/13
undecimal neutral-3rd ............... 2472  2472.10  [  0  -2,  0  0  1> ....................... 11/9
2440  2440.06  [  7  -1, -1 -1> ......................... 128/105
2370  2369.86  [  3  -1,  1  0 -1> ....................... 40/33
2278  2278.09  [ -6   0,  0  1  1> ....................... 77/64
pyth aug-2nd ........................ 2260  2259.95  [-14   9> .............................. 19683/16384
just minor-3rd ...................... 2246  2246.05  [  1   1, -1> .............................. 6/5
19th harmonic ....................... 2117  2117.05  [ -4   0,  0  0  0,  0  0  1> ............. 19/16
2107  2106.92  [-10   5,  1> ........................... 1215/1024
pyth minor-3rd / trihemitone ........ 2093  2093.02  [  5  -3> ................................. 32/27
2024  2024.08  [ -2   1,  0 -1  1> ....................... 33/28
1954  1953.88  [ -6   1,  2> ............................. 75/64
septimal minor-3rd .................. 1899  1899.01  [ -1  -1,  0  1> ........................... 7/6
1867  1866.97  [  6   0, -1  0 -1> ....................... 64/55
1677  1677.04  [ -4  -1,  1  0  1> ....................... 55/48
septimal major-2nd / 7th subharmonic  1645  1645.00  [  3   0,  0 -1> ........................... 8/7
1604  1604.02  [ -7   6, -1> ............................ 729/640
pyth major-2nd / tone / 9th harmonic  1451  1450.99  [ -3   2> .................................. 9/8
just minor-2nd / small tone ......... 1298  1297.95  [  1  -2,  1> ............................. 10/9
29th subharmonic .................... 1213  1212.70  [  5   0,  0  0  0,  0  0  0,  0 -1> ...... 32/29
1174  1174.14  [ -1   0, -1  0  1> ....................... 11/10
1104  1103.95  [ -5   0,  1  1> .......................... 35/32
1072  1071.91  [  2   1,  0  0 -1> ....................... 12/11
948   948.10  [  0   3, -2> ............................. 27/25
850   849.94  [ -1   1,  1 -1> .......................... 15/14
pyth aug-prime / apotome ............  809   808.96  [-11   7> ............................... 2187/2048
just minor-2nd / 15th subharmonic ...  795   795.06  [  4  -1, -1> ............................. 16/15
17th harmonic .......................  747   746.85  [ -4   0,  0  0  0,  0  1> ................ 17/16
large just aug-prime ................  656   655.93  [ -7   3,  1> ............................ 135/128
pyth minor-2nd / limma ..............  642   642.03  [  8  -5> ................................ 256/243
601   601.05  [ -2   1, -1  1> .......................... 21/20
573   573.09  [  1  -1,  0 -1  1> ....................... 22/21
small just aug-prime ................  503   502.89  [ -3  -1,  2> ............................. 25/24
pyth tricomma ......................   501   500.81  [-57  36> ........................... 1.50E+17/1.44E+17
31st subharmonic ....................  391   391.12  [  5   0,  0  0  0,  0  0  0,  0  0  -1> .. 32/31
undecimal-diesis / 33rd harmonic ....  379   379.08  [ -5   1,  0  0  1> ....................... 33/32
maximal-diesis ......................  350   349.86  [  1  -5,  3> ............................ 250/243
septimal-diesis .....................  347   347.04  [  2   2, -1 -1> .......................... 36/35
enharmonic diesis ...................  292   292.17  [  7   0, -3> ............................ 128/125
large biseptimal-comma ..............  254   254.01  [ -4  -1,  0  2> .......................... 49/48
small biseptimal-comma ..............  249   248.88  [  1   0,  2 -2> .......................... 50/49
magic-comma ........................   211   210.73  [-10  -1,  5> ........................... 3125/3072
septimal comma ......................  194   194.01  [  6  -2,  0 -1> .......................... 64/63
pythagorean-comma ..................   167   166.94  [-19  12> ............................. 531441/524288
syntonic-comma ......................  153   153.04  [ -4   4, -1> ............................. 81/80
diaschisma ..........................  139   139.13  [ 11  -4, -2> ........................... 2048/2025
semicomma ..........................    72    71.59  [-21   3,  7> ........................ 2109375/2097152
kleisma .............................   58    57.69  [ -6  -5,  6> .......................... 15625/15552
septimal-kleisma ....................   55    54.87  [ -5   2,  2 -1> ......................... 225/224
septimal-schisma ....................   27    27.07  [ 25 -14,  0 -1> .................... 33554432/33480783
mercator-comma .....................    26    25.72  [-84  53> ........................... 1.94E+25/1.93E+25
nondecimal-schisma ..................   24    24.04  [ -9   3,  0  0  0,  0  0  1> ............ 513/512
skhisma ............................    14    13.90  [-15   8,  1> .......................... 32805/32768
monzisma ............................    2     2.08  [ 54 -37,  2> ....................... 4.50E+17/4.50E+17
nanisma ............................     1     1.35  [109 -67,  0 -1> .................... 6.49E+32/6.49E+32
origin / prime / unison .............    0     0.00  [  0> ...................................... 1/1
```

Out of the 148 intervals listed in this table, only 9 have an error of around 30 percent or more (listed here in order of decreasing error):

```interval ......... tinas .. percent error

semicomma ........... 72 .. 41
nanisma .............. 1 .. 35
kleisma ............. 58 .. 31
23rd subharmonic .. 4068 .. 30
29th harmonic ..... 7326 .. 30
23rd harmonic ..... 4471 .. 30
29th subharmonic .. 1213 .. 30
mercator-comma ...... 26 .. 28
magic-comma ........ 211 .. 27
```

Of those, the only really important ones for most people will be the semicomma, kleisma, and magic-comma. All the rest of the JI intervals in the table (except one) have an error of 17 percent or less. This a remarkably low level of error for an integer logarithmic measurement unit.

Below is a table of the tina values for all of the commonly used intervals, in all of the standard keys, in some of the most important EDO meantones. The intervals are listed as a chain-of-5ths, in decreasing generator order, with the tonic of each key as the zeroth generator. 53-edo is also shown for comparison, as a representation of pythagorean tuning. The percentage errors for 12-, 55-, and 31-edo are quite low, those for 43-, 50-, and 19-edo not as good, and the error for 53-edo almost as bad as it can get, at 49% (i.e., the 5th of 53-edo is almost midway between two tina values, at ~4994.5094) -- the values shown in the 53-edo column are actually quite accurate for real pythagorean JI tuning.

```                                                                                                 EDOs ---------------------------------
..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  generator 53    12    55    43    31    50    19

..  ..  ..  ..  ..  .. ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  .. Ax . 24 .. 334  8537  8225  8153  7985  7865  7649
..  ..  ..  ..  ..  .. ..  ..  ..  ..  ..  ..  ..  ..  ..  .. Ax .Dx . 23 . 3878  3556  3257  3188  3027  2912  2705
..  ..  ..  ..  ..  .. ..  ..  ..  ..  ..  ..  ..  ..  .. Ax .Dx .Gx . 22 . 7422  7114  6828  6762  6608  6498  6300
..  ..  ..  ..  ..  .. ..  ..  ..  ..  ..  ..  ..  .. Ax .Dx .Gx .Cx . 21 . 2427  2133  1860  1797  1650  1545  1356
..  ..  ..  ..  ..  .. ..  ..  ..  ..  ..  ..  .. Ax .Dx .Gx .Cx .Fx . 20 . 5971  5691  5431  5371  5231  5131  4951
dbl aug-7th .....  ..  ..  ..  ..  ..  .. ..  ..  ..  ..  ..  .. Ax .Dx .Gx .Cx .Fx .B# . 19 .  976   710   463   406   273   178     7
dbl aug-3rd .....  ..  ..  ..  ..  ..  .. ..  ..  ..  ..  .. Ax .Dx .Gx .Cx .Fx .B# .E# . 18 . 4520  4268  4034  3980  3854  3764  3602
dbl aug-6th .....  ..  ..  ..  ..  ..  .. ..  ..  ..  .. Ax .Dx .Gx .Cx .Fx .B# .E# .A# . 17 . 8064  7826  7605  7554  7435  7350  7197
dbl aug-2nd .....  ..  ..  ..  ..  ..  .. ..  ..  .. Ax .Dx .Gx .Cx .Fx .B# .E# .A# .D# . 16 . 3069  2845  2637  2589  2477  2397  2253
dbl aug-5th .....  ..  ..  ..  ..  ..  .. ..  .. Ax .Dx .Gx .Cx .Fx .B# .E# .A# .D# .G# . 15 . 6613  6403  6208  6163  6058  5983  5848
dbl aug-prime      ..  ..  ..  ..  ..  .. .. Ax .Dx .Gx .Cx .Fx .B# .E# .A# .D# .G# .C# . 14 . 1618  1422  1240  1198  1100  1030   904
dbl aug-4th       ..  ..  ..  ..  ..  .. Ax .Dx .Gx .Cx .Fx .B# .E# .A# .D# .G# .C# .F# . 13 . 5162  4980  4811  4772  4681  4616  4499
aug-7th ........ ..  ..  ..  ..  ..  Ax .Dx .Gx .Cx .Fx .B# .E# .A# .D# .G# .C# .F# .B .. 12 .. 167  8538  8382  8346  8262  8202  8094
aug-3rd ........ ..  ..  ..  ..  Ax .Dx .Gx .Cx .Fx .B# .E# .A# .D# .G# .C# .F# .B ..E .. 11 . 3711  3557  3414  3381  3304  3249  3150
aug-6th ........ ..  ..  ..  Ax .Dx .Gx .Cx .Fx .B# .E# .A# .D# .G# .C# .F# .B ..E ..A .. 10 . 7255  7115  6985  6955  6885  6835  6745
aug-2nd ........ ..  ..  Ax .Dx .Gx .Cx .Fx .B# .E# .A# .D# .G# .C# .F# .B ..E ..A ..D ..  9 . 2260  2134  2017  1990  1927  1882  1801
aug-5th ........     Ax. Dx .Gx .Cx .Fx .B# .E# .A# .D# .G# .C# .F# .B ..E ..A ..D ..G ..  8 . 5804  5692  5588  5564  5508  5468  5396
Chromatic s-t ...Ax  Dx .Gx .Cx .Fx .B# .E# .A# .D# .G# .C# .F# .B ..E ..A ..D ..G ..C ..  7 .  809   711   620   599   550   515   452
aug-4th ........ Dx  Gx .Cx .Fx .B# .E# .A# .D# .G# .C# .F# .B ..E ..A ..D ..G ..C ..F ..  6 . 4353  4269  4191  4173  4131  4101  4047

major-7th ...... Gx  Cx .Fx .B# .E# .A# .D# .G# .C# .F# .B ..E ..A ..D ..G ..C ..F ..Bb .  5 . 7897  7827  7762  7747  7712  7687  7642
major-3rd ...... Cx  Fx .B# .E# .A# .D# .G# .C# .F# .B ..E ..A ..D ..G ..C ..F ..Bb .Eb .  4 . 2902  2846  2794  2782  2754  2734  2698
major-6th ...... Fx  B# .E# .A# .D# .G# .C# .F# .B ..E ..A ..D ..G ..C ..F ..Bb .Eb .Ab .  3 . 6446  6404  6365  6356  6335  6320  6293
major-2nd ...... B#  E# .A# .D# .G# .C# .F# .B ..E ..A ..D ..G ..C ..F ..Bb .Eb .Ab .Db .  2 . 1451  1423  1397  1391  1377  1367  1349
perfect-5th .....E#  A# .D# .G# .C# .F# .B ..E ..A ..D ..G ..C ..F ..Bb .Eb .Ab .Db .Gb .  1 . 4995  4981  4968  4965  4958  4953  4944

Prime .......... A#  D# .G# .C# .F# .B ..E ..A ..D ..G ..C ..F ..Bb .Eb .Ab .Db .Gb .Cb .  0 ..   0     0     0     0     0     0     0

perfect-4th .... D#  G# .C# .F# .B ..E ..A ..D ..G ..C ..F ..Bb .Eb .Ab .Db .Gb .Cb .Fb . -1 . 3544  3558  3571  3574  3581  3586  3595
minor-7th .......G#  C# .F# .B ..E ..A ..D ..G ..C ..F ..Bb .Eb .Ab .Db .Gb .Cb .Fb .Bbb  -2 . 7088  7116  7142  7148  7162  7172  7190
minor-3rd ...... C#  F# .B ..E ..A ..D ..G ..C ..F ..Bb .Eb .Ab .Db .Gb .Cb .Fb .Bbb Ebb  -3 . 2093  2135  2174  2183  2204  2219  2246
minor-6th .......F#  B ..E ..A ..D ..G ..C ..F ..Bb .Eb .Ab .Db .Gb .Cb .Fb .Bbb Ebb Abb  -4 . 5637  5693  5745  5757  5785  5805  5841

Diatonic s-t ..  B   E ..A ..D ..G ..C ..F ..Bb .Eb .Ab .Db .Gb .Cb .Fb .Bbb Ebb Abb Dbb  -5 .. 642   712   777   792   827   852   897
dim-5th ........ E   A ..D ..G ..C ..F ..Bb .Eb .Ab .Db .Gb .Cb .Fb .Bbb Ebb Abb Dbb Gbb  -6 . 4186  4270  4348  4366  4408  4438  4492
dim-8ve .......  A   D ..G ..C ..F ..Bb .Eb .Ab .Db .Gb .Cb .Fb .Bbb Ebb Abb Dbb Gbb ..   -7 . 7730  7828  7919  7940  7989  8024  8087
dim-4th ........ D   G ..C ..F ..Bb .Eb .Ab .Db .Gb .Cb .Fb .Bbb Ebb Abb Dbb Gbb ..  ..   -8 . 2735  2847  2951  2975  3031  3071  3143
dim-7th ........ G   C ..F ..Bb .Eb .Ab .Db .Gb .Cb .Fb .Bbb Ebb Abb Dbb Gbb ..  ..  ..   -9 . 6279  6405  6522  6549  6612  6657  6738
dim-3rd .......  C   F ..Bb .Eb .Ab .Db .Gb .Cb .Fb .Bbb Ebb Abb Dbb Gbb ..  ..  ..  ..  -10   1284  1424  1554  1584  1654  1704  1794
dim-6th .......  F   Bb .Eb .Ab .Db .Gb .Cb .Fb .Bbb Ebb Abb Dbb Gbb ..  ..  ..  ..  ..  -11   4828  4982  5125  5158  5235  5290  5389
dim-2nd .......  Bb  Eb .Ab .Db .Gb .Cb .Fb .Bbb Ebb Abb Dbb Gbb ..  ..  ..  ..  ..  ..  -12   8372     1   157   193   277   337   445
dbl dim-5th ...  Eb  Ab .Db .Gb .Cb .Fb .Bbb Ebb Abb Dbb Gbb ..  ..  ..  ..  ..  ..  ..  -13   3377  3559  3728  3767  3858  3923  4040
dbl dim-8ve ...  Ab  Db .Gb .Cb .Fb .Bbb Ebb Abb Dbb Gbb ..  ..  ..  ..  ..  ..  ..  ..  -14   6921  7117  7299  7341  7439  7509  7635
dbl dim-4th ...  Db  Gb .Cb .Fb .Bbb Ebb Abb Dbb Gbb ..  ..  ..  ..  ..  ..  ..  ..  ..  -15 . 1926  2136  2331  2376  2481  2556  2691
dbl dim-7th ...  Gb  Cb .Fb .Bbb Ebb Abb Dbb Gbb ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  -16 . 5470  5694  5902  5950  6062  6142  6286
dbl dim-3rd ...  Cb  Fb .Bbb Ebb Abb Dbb Gbb ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  -17    475   713   934   985  1104  1189  1342
dbl dim-6th ...  Fb  Bbb Ebb Abb Dbb Gbb ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  -18   4019  4271  4505  4559  4685  4775  4937
dbl dim-2nd ...  Bbb Ebb Abb Dbb Gbb ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  -19   7563  7829  8076  8133  8266  8361  8532
Ebb Abb Dbb Gbb ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  -20   2568  2848  3108  3168  3308  3408  3588
Abb Dbb Gbb ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  -21   6112  6406  6679  6742  6889  6994  7183
Dbb Gbb ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  -22   1117  1425  1711  1777  1931  2041  2239
Gbb ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  -23   4661  4983  5282  5351  5512  5627  5834
```

Below are graphs of the tina-values for some of the EDO chain-of-5ths tunings in the above table:

. . . . . . . . .

### tinas 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: = tinas

. . . . . . . . .

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