# tritone

[Joe Monzo]

A musical interval composed of three successive 'whole tones', which is what the word "tritone" literally means. This interval is technically the "augmented 4th", but the word "tritone" is also used generally to refer to its complement, the diminished 5th.

The tritone in the 12-tone equal tempered scale divides the 'octave' precisely in half, and thus has a size of exactly 600 cents or 6.00 Semitones. Mathematically, since the "octave" is the ratio 2:1, this tritone is the square-root of 2, also written 2(1/2).

Any equal-temperament with an even number of degrees has this 'perfect tritone'.

Scales tuned to just intonation, meantone, and other tuning systems based on just intonation tend to have two different-sized 'tritones' which fall on either side of this exact division.

The tritone, as both "augmented 4th" and "diminished 5th", originally arose in pythagorean tuning. In this system, the "whole-tone" has the ratio 9:8. Three of them are mathematically (9/8)3 = ratio 729/512, expressed in decimal form as exactly 1.423828125, with a logarithmic interval size of ~611.7300052 cents, which is the augmented 4th:

```    note       F    G      A       B
ratio (C=n0)  4/3  3/2   27/16  243/128
prime-factor  3-1   31     33      35
ratio (F=n0)  1/1  9/8   81/64  729/512
prime-factor   n0   32     34      36
~cents      0   204    408     612
\/     \/      \/
9/8    9/8     9/8
32     32      32
204    204     204
tone   tone    tone
```

The diminished 5th is the result of ascending two tones and two diatonic semitones. In pythagorean tuning this is mathematically (9/8)2(256/243)2 = ratio 1024/729, expressed in decimal form as exactly 1.404663923182441700960219478737997256515775034293552812071330589849108367626886145... (the decimal part repeats after the 81st place), with a logarithmic interval size of 5.88 Semitones = ~588.269994807675 cents:

```    note        B      C        D      E        F
ratio (C=n0) 243/128  1/1      9/8   81/64     4/3
prime-factor    35     n0       32     34      3-1
ratio (B=n0)   1/1  256/243   32/27   4/3   1024/729
prime-factor    n0    3-5      3-3     3-1      3-6
~cents       0     90       294    498      588
\/     \/       \/      \/
256/243   9/8      9/8   256/243
3-5      32       32      3-5
90      204      204     90
semitone  tone     tone   semitone
```

The "standard" 5-limit just intonation tritone (augmented-4th) is the ratio 45:32, expressed in decimal form as exactly 1.40625, with a logarithmic interval size of 5.90 Semitones = ~590.22371559561 cents:

```    note       F     G      A       B
ratio (C=n0)  4/3   3/2    5/3    15/8
prime-factor  3-1    31    3-151   3151
ratio (F=n0)  1/1   9/8    5/4    45/32
prime-factor   n0    32     51     3251
~cents      0    204    386     590
\/     \/     \/
9/8   10/9    9/8
32   3-251    32
204    182    204
tone   tone   tone
```

Also frequently encountered is its complement 64:45 (=~ 610 cents):

```    note        B      C        D      E        F
ratio (C=n0)  15/8    1/1      9/8    5/4      4/3
prime-factor   3151    n0       32     51      3-1
ratio (B=n0)   1/1   16/15     6/5    4/3     64/45
prime-factor    n0   3-15-1    315-1   3-1    3-25-1
~cents       0     112      316    498      610
\/     \/       \/      \/
16/15    9/8      10/9   16/15
3-5      32       32      3-5
112     204      182     112
semitone  tone     tone   semitone
```

Typical 7-limit tritones are 7:5 (= ~583 cents) and its complement 10:7 (= ~617 cents), whose fractions have much smaller integer terms than the 3- and 5-limit tritones.

Below is a table of many 13-limit ratios which may represent the tritone. These are all the tritones (i.e., between 580 and 620 cents) which are elements of the set of just intonation pitch-classes bounded by 3(-10...10) * 5(-2...2) * 7(-1...1) * 11(-1...1) * 13(-1...1). Interval names are from Manuel Op de Coul's Scala file "intnam.par".

```		      prime-factor
2   3   5   7  11  13       ratio     numerator denominator   cents      interval name

9  -9   1   0   1   0    1.430676218     28160 / 19683     620.0366484
25  -8  -2   0  -1  -1    1.430552523  33554432 / 23455575  619.8869612
-3  -7   2   1   1   1    1.430326932     25025 / 17496     619.6139323
13  -6  -1   1  -1   0    1.430203267     57344 / 40095     619.464245
-14   8   2  -1   0   0    1.430184501    164025 / 114688    619.4415282
-2   0  -2   0   1   1    1.43             143  / 100       619.2181764
1   0   1  -1   0   0    1.428571429        10 / 7         617.4878074  Euler's tritone
-11   2   2   0   0   1    1.428222656      2925 / 2048      617.0650912
-10   9  -2  -1   0   1    1.427896205    255879 / 179200    616.6693354
-4   3   0   0   1  -1    1.427884615       297 / 208       616.6552832
-16   5   1   1   1   0    1.427536011     93555 / 65536     616.232567
0   6  -2   1  -1  -1    1.427412587      5103 / 3575      616.0828798
16  -8   0  -1   0   0    1.426960176     65536 / 45927     615.5340866
4  -6   1   0   0   1    1.426611797      1040 / 729       615.1113704
11  -5  -1   0   1  -1    1.426274137     22528 / 15795     614.7015624
-1  -3   0   1   1   0    1.425925926        77 / 54        614.2788462
-4   3   1  -1  -1   1    1.424512987      1755 / 1232      612.5625294
3   4  -1  -1   0  -1    1.424175824       648 / 455       612.1527214
-9   6   0   0   0   0    1.423828125       729 / 512       611.7300052  Pythagorean tritone
-21   8   1   1   0   1    1.423480511   2985255 / 2097152   611.307289
-14   9  -1   1   1  -1    1.423143592   1515591 / 1064960   610.897481
11  -5   0  -1  -1   1    1.422906312     26624 / 18711     610.6088086
18  -4  -2  -1   0  -1    1.422569529    262144 / 184275    610.1990006
6  -2  -1   0   0   0    1.422222222        64 / 45        609.7762844  2nd tritone
-6   0   0   1   0   1    1.421875           91 / 64        609.3535682
1   1  -2   1   1  -1    1.42153842        462 / 325       608.9437602
9  -2   2  -1  -1  -1    1.42080142      12800 / 9009      608.0459154
21 -10  -2   0   0   0    1.420618131   2097152 / 1476225   607.8225636
9  -8  -1   1   0   1    1.4202713       46592 / 32805     607.3998475
-18   6   2  -1   1   1    1.42025266    2606175 / 1835008   607.3771306
-2   7  -1  -1  -1   0    1.42012987       2187 / 1540      607.2274434
-14   9   0   0  -1   1    1.419783159    255879 / 180224    606.8047272
-8   3   2   1   0  -1    1.419771635      4725 / 3328      606.790675
-7  10  -2   0   0  -1    1.419447115     59049 / 41600     606.3949192
24 -10   1  -1  -1  -1    1.419198932  83886080 / 59108049  606.0921946
12  -8   2   0  -1   0    1.418852448    102400 / 72171     605.6694784
-3  -2   1  -1   1   1    1.418650794       715 / 504       605.4234098
13  -1  -2  -1  -1   0    1.418528139      8192 / 5775      605.2737226
1   1  -1   0  -1   1    1.418181818        78 / 55        604.8510064
7  -5   1   1   0  -1    1.418170307      4480 / 3159      604.8369542
-4   4  -2   1   0   0    1.4175            567 / 400       604.0184822
12 -10   0  -1   1   1    1.41705073     585728 / 413343    603.469689
16  -7  -2   0  -1   1    1.416582284    851968 / 601425    602.8972856
-1   4   1   0  -1  -1    1.416083916       405 / 286       602.2881132
-16  10   0  -1   1   0    1.415882656    649539 / 458752    602.0420445
-13   6   2   1  -1   0    1.415738192    127575 / 90112     601.865397
10  -7   2  -1   1  -1    1.414954501    281600 / 199017    600.9067958
14  -4   0   0  -1  -1    1.414486748     16384 / 11583     600.3343924
-1   2  -1  -1   1   0    1.414285714        99 / 70        600.0883238  2nd quasi-equal tritone
2  -2   1   1  -1   0    1.414141414       140 / 99        599.9116762  quasi-equal tritone
-13   4   0   0   1   1    1.41394043      11583 / 8192      599.6656076
-9   7  -2   1  -1   1    1.413473011    199017 / 140800    599.0932042
14  -6  -2  -1   1   0    1.412690574    180224 / 127575    598.134603
17 -10   0   1  -1   0    1.412546437    917504 / 649539    597.9579555
2  -4  -1   0   1   1    1.412345679       572 / 405       597.7118868
-15   7   2   0   1  -1    1.411848802    601425 / 425984    597.1027144
-11  10   0   1  -1  -1    1.411382075    413343 / 292864    596.530311
5  -4   2  -1   0   0    1.410934744       800 / 567       595.9815178
-6   5  -1  -1   0   1    1.410267857      3159 / 2240      595.1630458
0  -1   1   0   1  -1    1.41025641         55 / 39        595.1489936
-12   1   2   1   1   0    1.409912109      5775 / 4096      594.7262774
4   2  -1   1  -1  -1    1.40979021       1008 / 715       594.5765902
-11   8  -2   0   1   0    1.409589844     72171 / 51200     594.3305216
-23  10  -1   1   1   1    1.409245706  59108049 / 41943040  593.9078054
8 -10   2   0   0   1    1.408999306     83200 / 59049     593.6050808
9  -3  -2  -1   0   1    1.408677249      6656 / 4725      593.209325
15  -9   0   0   1  -1    1.408665815    360448 / 255879    593.1952728
3  -7   1   1   1   0    1.40832190       3080 / 2187      592.7725566
19  -6  -2   1  -1  -1    1.40820014    3670016 / 2606175   592.6228694
-8   8   1  -1   0  -1    1.408181662     32805 / 23296     592.6001525
-20  10   2   0   0   0    1.407837868   1476225 / 1048576   592.1774364
-8   2  -2   1   1   1    1.40765625       9009 / 6400      591.9540846
0  -1   2  -1  -1   1    1.406926407       325 / 231       591.0562398
7   0   0  -1   0  -1    1.406593407       128 / 91        590.6464318
-5   2   1   0   0   0    1.40625            45 / 32        590.2237156  tritone
-17   4   2   1   0   1    1.40590677     184275 / 131072    589.8009994
-10   5   0   1   1  -1    1.40557398      18711 / 13312     589.3911914
15  -9   1  -1  -1   1    1.40533956    2129920 / 1515591   589.102519
22  -8  -1  -1   0  -1    1.405006942   4194304 / 2985255   588.692711
10  -6   0   0   0   0    1.404663923      1024 / 729       588.2699948  Pythagorean diminished fifth
-2  -4   1   1   0   1    1.404320988       455 / 324       587.8472786
5  -3  -1   1   1  -1    1.403988604      2464 / 1755      587.4374706
2   3   0  -1  -1   0    1.402597403       108 / 77        585.7211538
-10   5   1   0  -1   1    1.402254972     15795 / 11264     585.2984376
-3   6  -1   0   0  -1    1.401923077       729 / 520       584.8886296
-15   8   0   1   0   0    1.401580811     45927 / 32768     584.4659134
1  -6   2  -1   1   1    1.401136586      7150 / 5103      583.9171202
17  -5  -1  -1  -1   0    1.401015445    131072 / 93555     583.767433
5  -3   0   0  -1   1    1.400673401       416 / 297       583.3447168
11  -9   2   1   0  -1    1.400662032    358400 / 255879    583.3306646
12  -2  -2   0   0  -1    1.40034188       4096 / 2925      582.9349088
0   0  -1   1   0   0    1.4                 7 / 5         582.5121926  septimal or Huygens' tritone, BP fourth
3   0   2   0  -1  -1    1.39860139        200 / 143       580.7818236
15  -8  -2   1   0   0    1.39842097     229376 / 164025    580.5584718
-12   6   1  -1   1   0    1.398402623     40095 / 28672     580.535755
4   7  -2  -1  -1  -1    1.398281718     34992 / 25025     580.3860677
-24   8   2   0   1   1    1.398061216  23455575 / 16777216  580.1130388
-8   9  -1   0  -1   0    1.397940341     19683 / 14080     579.9633
```

Below is a graph of these tritones's cents values.

24:17 (= ~597 cents) and 17:12 (= ~603 cents) are two 17-limit tritones.

Below is a table and graph of the tritones for some of the most common fraction-of-a-comma meantones. They show both the "augmented 4th" (+6 generator) and "diminished 5th" (-6 generator) for each meantone. The numerator and denominator of the fraction-of-a-comma tempering of each "5th" (generator) is shown in blue on the left. Values are given in cents.

Note that for all intents and purposes, 1/11-comma meantone is identical to the standard 12-edo scale, and that 1/3-comma is nearly identical to 19-edo, 3/10-comma is essentially the same as LucyTuning, both 4/15-comma and 7/26-comma can represent golden meantone, 1/4-comma resembles 31-edo, and 1/6-comma resembles 55-edo.

. . . . . . . . .

The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. Thank you.

 support level donor: \$5 USD friend: \$25 USD patron: \$50 USD savior: \$100 USD angel of tuning: \$500 USD microtonal god: \$1000 USD