# tredek, 270-edo

[Joe Monzo]

A small unit of interval measurement, noted first by Erv Wilson and John Chalmers in the 1970s, and suggested as an interval measurement by Joe Monzo in November 2013.

A tredek divides the octave into 270 equal parts. Its use is valuable because it has extremely low relative error for the just-intonation ratios thru the 13-prime-limit, thus obviating the need for decimal places in measuring those intervals logarithmically.

The tredek is therefore calculated as the 270th root of 2, or 2^(1/270), with a ratio of approximately 1:1.00257050989. It is an irrational number.

A tredek is:

• exactly 2/45 (= 0.0,4... = ~ 1/23 ) semitone.
• exactly 4/45 (= 0.0,8... = ~ 1/11 ) quarter-tone.
• exactly 31/270 (= 0.1,148... = ~ 1/9 ) diesis.
• exactly 43/270 (= 0.1,592... = ~ 1/6 ) méride.
• exactly 4/15 (= 0.2,6... = ~ 1/4 ) morion.
• exactly 2 + 38/45 (= 2.8,4... = ~ 2 + 5/6 ) 6mus.
• exactly 3 + 19/27 (= 3.703... = ~ 3 + 2/3 ) millioctaves.
• exactly 3 + 107/135 (= 3.7,925... = ~ 3 + 4/5 ) yamaha-units.
• exactly 4 + 4/9 (= ~4.4... = ~ 4 + 1/2 ) cents.
• exactly 9 + 1/9 (= 9.1... ) minas.
• exactly 31 + 169/270 (= 31.625,925... = ~ 31 +2/3 ) tinas.
• exactly 39 + 7/27 (= 39.259... = ~ 39 + 1/4 ) türk-sents.
• exactly 182 + 2/45 (= 182.0,4... = ~ 182 + 1/22 ) 12mus.
• exactly 728 + 8/45 (= 728.1,7... = ~ 728 + 1/6 ) 14mus.

`		`

The formula for calculating the floating-point tredek-value of any ratio r is: tredeks = log_10(r) * [ 270 / log_10(2) ] or tredeks = log_2(r) * 270

However, the primary benefit in using tredeks as a logarithmic interval measurement for 13-limit JI is that there is hardly any significance to any of the values after the decimal point, so that one may memorize the integer tredek values of 13-limit ratios and compute with them, with little concern for rounding errors.

A tredek represents one degree of 270-edo tuning.

The 12-edo semitone is exactly 22 + 1/2 (= 22.5) tredeks.

. . . . . . . . .

### some intervals mapped to 270-edo

(Note that while 270-edo is extremely accurate for prime-factors 2, 3, 5, 7, 13, 19, and 43, ratios listed here which include other prime-factors above 13 are not represented well in this tuning, and are given only for comparison with their mappings in other tunings. This can be seen easily in the "error" column.)

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

2 =  270.000000    +0.00 -->    270
3 =  427.939875    +0.06 -->    428
5 =  626.920586    +0.08 -->    627
7 =  757.985829    +0.01 -->    758
11 =  934.046537    -0.05 -->    934
13 =  999.118724    -0.12 -->    999
17 = 1103.614967    +0.39 -->   1104
19 = 1146.940429    +0.06 -->   1147
23 = 1221.361728    -0.36 -->   1221
29 = 1311.654869    +0.35 -->   1312
31 = 1337.633004    +0.37 -->   1338
37 = 1406.552409    +0.45 -->   1407
41 = 1446.539041    +0.46 -->   1447
43 = 1465.091484    -0.09 -->   1465

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

map  2 -->    270 = 1200.000000 cents <-- 1200.000000  +0.0 cents
map  3 -->    428 = 1902.222222 cents <-- 1901.955001  +0.3 cents
map  5 -->    627 = 2786.666667 cents <-- 2786.313714  +0.4 cents
map  7 -->    758 = 3368.888889 cents <-- 3368.825906  +0.1 cents
map 11 -->    934 = 4151.111111 cents <-- 4151.317942  -0.2 cents
map 13 -->    999 = 4440.000000 cents <-- 4440.527662  -0.5 cents
map 17 -->   1104 = 4906.666667 cents <-- 4904.955410  +1.7 cents
map 19 -->   1147 = 5097.777778 cents <-- 5097.513016  +0.3 cents
map 23 -->   1221 = 5426.666667 cents <-- 5428.274347  -1.6 cents
map 29 -->   1312 = 5831.111111 cents <-- 5829.577194  +1.5 cents
map 31 -->   1338 = 5946.666667 cents <-- 5945.035572  +1.6 cents
map 37 -->   1407 = 6253.333333 cents <-- 6251.344039  +2.0 cents
map 41 -->   1447 = 6431.111111 cents <-- 6429.062406  +2.0 cents
map 43 -->   1465 = 6511.111111 cents <-- 6511.517706  -0.4 cents

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

examples:

ratio                -->      270-edo mapping:

ratio           cents   error       edo     cents    name

2:1       	= 1200.0  +0.0	--> 270/270	= 1200.0  (octave)
65536:32805   	= 1198.0  -2.5	--> 269/270	= 1195.6  (minimal just dim-2)
2025:1024    	= 1180.4  +1.8	--> 266/270	= 1182.2  (small just aug-7th)
1048576:531441  	= 1176.5  -3.2	--> 264/270	= 1173.3  (pythagorean dim-2nd)
125:64      	= 1158.9  +1.1	--> 261/270	= 1160.0  (minimal just aug-7th)
31:16      	= 1145.0  +1.6	--> 258/270	= 1146.7  (31st harmonic)
48:25      	= 1129.3  -0.4	--> 254/270	= 1128.9  (small just dim-8ve)
21:11      	= 1119.5  +0.5	--> 252/270	= 1120.0  (undecimal diminished-8ve)
243:128     	= 1109.8  +1.3	--> 250/270	= 1111.1  (pythagorean major-7th)
256:135     	= 1107.8  -1.2	--> 249/270	= 1106.7  (minimal just dim-8ve)
15:8       	= 1088.3  +0.6	--> 245/270	= 1088.9  (15th harmonic, just major-7th, 5*3)
4096:2187    	= 1086.3  -1.9	--> 244/270	= 1084.4  (pythagorean diminished-8ve)
13:7       	= 1071.7  -0.6	--> 241/270	= 1071.1  (tridecimal superminor-7th)
50:27      	= 1066.8  -0.1	--> 240/270	= 1066.7  (small just maj-7th)
24:13      	= 1061.4  +0.8	--> 239/270	= 1062.2  (tridecimal major-7th)
11:6       	= 1049.4  -0.5	--> 236/270	= 1048.9  (undecimal submajor[neutral]-7th)
20:11      	= 1035.0  +0.6	--> 233/270	= 1035.6  (undecimal superminor[neutral]-7th)
29:16      	= 1029.6  +1.5	--> 232/270	= 1031.1  (29th harmonic)
59049:32768   	= 1019.6  +2.7	--> 230/270	= 1022.2  (pythagorean aug-6th)
9:5       	= 1017.6  +0.2	--> 229/270	= 1017.8  (just minor-7th)
3645:2048    	=  998.0  +2.0	--> 225/270	= 1000.0  (large just aug-6th)
16:9       	=  996.1  -0.5	--> 224/270	=  995.6  (pythagorean minor-7th)
225:128     	=  976.5  +1.2	--> 220/270	=  977.8  (small just augmented-6th)
7:4       	=  968.8  +0.1	--> 218/270	=  968.9  (7th harmonic, septimal subminor-7th)
216:125     	=  946.9  -0.3	--> 213/270	=  946.7  (large just dim-7)
19:11      	=  946.2  +0.5	--> 213/270	=  946.7  (nondecimal supermajor-6th)
12:7       	=  933.1  +0.2	--> 210/270	=  933.3  (septimal supermajor-6th)
128:75      	=  925.4  -1.0	--> 208/270	=  924.4  (small just dim-7th)
22:13      	=  910.8  +0.3	--> 205/270	=  911.1  (tridecimal augmented-6th)
27:16      	=  905.9  +0.8	--> 204/270	=  906.7  (27th harmonic, pythagorean major-6th)
2048:1215    	=  903.9  -1.7	--> 203/270	=  902.2  (minimal just dim-7)
5:3       	=  884.4  +0.1	--> 199/270	=  884.4  (just major-6th)
32768:19683   	=  882.4  -2.4	--> 198/270	=  880.0  (pythagorean dim-7th)
18:11      	=  852.6  +0.7	--> 192/270	=  853.3  (undecimal superminor[neutral]-6th)
13:8       	=  840.5  -0.5	--> 189/270	=  840.0  (13th harmonic)
21:13      	=  830.3  +0.9	--> 187/270	=  831.1  (tridecimal ?)
6561:4096    	=  815.6  +2.1	--> 184/270	=  817.8  (pythagorean augmented-5th)
8:5       	=  813.7  -0.4	--> 183/270	=  813.3  (just minor-6th)
405:256     	=  794.1  +1.4	--> 179/270	=  795.6  (large just aug-5th)
128:81      	=  792.2  -1.1	--> 178/270	=  791.1  (pythagorean minor-6th)
11:7       	=  782.5  -0.3	--> 176/270	=  782.2  (undecimal augmented-5th)
25:16      	=  772.6  +0.7	--> 174/270	=  773.3  (25th harmonic, just augmented-5th)
14:9       	=  764.9  -0.5	--> 172/270	=  764.4  (septimal subminor-6th)
17:11      	=  753.6  +1.9	--> 170/270	=  755.6  (septendecimal diminished-6th)
20:13      	=  745.8  +0.9	--> 168/270	=  746.7  (tridecimal augmented-5th)
192:125     	=  743.0  -0.8	--> 167/270	=  742.2  (large just dim-6)
1024:675     	=  721.5  -1.5	--> 162/270	=  720.0  (small just dim-6th)
3:2       	=  702.0  +0.3	--> 158/270	=  702.2  (perfect-5th)
16384:10935   	=  700.0  -2.2	--> 157/270	=  697.8  (minimal just dim-6)
262144:177147  	=  678.5  -2.9	--> 152/270	=  675.6  (pythagorean dim-6th)
19:13      	=  657.0  +0.8	--> 148/270	=  657.8  (nondecimal doubly-augmented-4th)
16:11      	=  648.7  +0.2	--> 146/270	=  648.9  (11th subharmonic, undecimal diminished-4th)
13:9       	=  636.6  -1.1	--> 143/270	=  635.6  (tridecimal diminished-5th)
23:16      	=  628.3  -1.6	--> 141/270	=  626.7  (23rd harmonic)
10:7       	=  617.5  +0.3	--> 139/270	=  617.8  (septimal large-tritone)
729:512     	=  611.7  +1.6	--> 138/270	=  613.3  (pythagorean augmented-4th)
64:45      	=  609.8  -0.9	--> 137/270	=  608.9  (just diminished-5th)
45:32      	=  590.2  +0.9	--> 133/270	=  591.1  (just augmented-4th)
1024:729     	=  588.3  -1.6	--> 132/270	=  586.7  (pythagorean diminished-5th)
7:5       	=  582.5  -0.3	--> 131/270	=  582.2  (septimal small-tritone)
25:18      	=  568.7  +0.2	--> 128/270	=  568.9  (small just aug-4th)
18:13      	=  563.4  +1.1	--> 127/270	=  564.4  (tridecimal augmented-4th)
11:8       	=  551.3  -0.2	--> 124/270	=  551.1  (11th harmonic, undecimal sub-augmented-4th)
15:11      	=  537.0  +0.8	--> 121/270	=  537.8  (undecimal large-4th)
177147:131072  	=  521.5  +2.9	--> 118/270	=  524.4  (pythagorean aug-3rd)
43:32      	=  511.5  -0.4	--> 115/270	=  511.1  (43rd harmonic)
10935:8192    	=  500.0  +2.2	--> 113/270	=  502.2  (large just aug-3rd)
4:3       	=  498.0  -0.3	--> 112/270	=  497.8  (perfect-4th)
675:512     	=  478.5  +1.5	--> 108/270	=  480.0  (small just aug-3rd)
21:16      	=  470.8  +0.3	--> 106/270	=  471.1  (21st harmonic, septimal-4th, 7*3)
17:13      	=  464.4  +2.2	--> 105/270	=  466.7  (septendecimal 4th)
125:96      	=  457.0  +0.8	--> 103/270	=  457.8  (minimal just aug-3rd)
13:10      	=  454.2  -0.9	--> 102/270	=  453.3  (tridecimal diminished-4th)
9:7       	=  435.1  +0.5	-->  98/270	=  435.6  (septimal supermajor-3rd)
41:32      	=  429.1  +2.0	-->  97/270	=  431.1  (41st harmonic)
32:25      	=  427.4  -0.7	-->  96/270	=  426.7  (small just dim-4th)
14:11      	=  417.5  +0.3	-->  94/270	=  417.8  (undecimal diminished-4th)
81:64      	=  407.8  +1.1	-->  92/270	=  408.9  (pythagorean major-3rd)
512:405     	=  405.9  -1.4	-->  91/270	=  404.4  (minimal just dim-4)
5:4       	=  386.3  +0.4	-->  87/270	=  386.7  (5th harmonic, just major-3rd)
8192:6561    	=  384.4  -2.1	-->  86/270	=  382.2  (pythagorean diminished-4th)
16:13      	=  359.5  +0.5	-->  81/270	=  360.0  (tridecimal major[neutral]-3rd)
11:9       	=  347.4  -0.7	-->  78/270	=  346.7  (undecimal neutral-3rd)
39:32      	=  342.5  -0.3	-->  77/270	=  342.2  (39th harmonic, 13*3)
19683:16384   	=  317.6  +2.4	-->  72/270	=  320.0  (pythagorean augmented-2nd)
6:5       	=  315.6  -0.1	-->  71/270	=  315.6  (just minor-3rd)
19:16      	=  297.5  +0.3	-->  67/270	=  297.8  (19th harmonic)
1215:1024    	=  296.1  +1.7	-->  67/270	=  297.8  (large just aug-2nd)
32:27      	=  294.1  -0.8	-->  66/270	=  293.3  (pythagorean minor-3rd)
13:11      	=  289.2  -0.3	-->  65/270	=  288.9  (tridecimal diminished-3rd)
75:64      	=  274.6  +1.0	-->  62/270	=  275.6  (just augmented-2nd)
7:6       	=  266.9  -0.2	-->  60/270	=  266.7  (septimal subminor-3rd)
37:32      	=  251.3  +2.0	-->  57/270	=  253.3  (37th harmonic)
15:13      	=  247.7  +1.1	-->  56/270	=  248.9  (tridecimal augmented[neutral]-2nd)
144:125     	=  245.0  -0.5	-->  55/270	=  244.4  (large just dim-3)
8:7       	=  231.2  -0.1	-->  52/270	=  231.1  (septimal tone, supermajor-2nd)
256:225     	=  223.5  -1.2	-->  50/270	=  222.2  (small just dim-3rd)
9:8       	=  203.9  +0.5	-->  46/270	=  204.4  (pythagorean major-2nd/tone)
4096:3645    	=  202.0  -2.0	-->  45/270	=  200.0  (minimal just dim-3)
10:9       	=  182.4  -0.2	-->  41/270	=  182.2  (just minor-tone)
65536:59049   	=  180.4  -2.7	-->  40/270	=  177.8  (pythagorean dim-3rd)
11:10      	=  165.0  -0.6	-->  37/270	=  164.4  (undecimal small-tone/submajor-2nd)
35:32      	=  155.1  +0.4	-->  35/270	=  155.6  (35th harmonic, 7*5)
12:11      	=  150.6  +0.5	-->  34/270	=  151.1  (undecimal large-semitone)
13:12      	=  138.6  -0.8	-->  31/270	=  137.8  (tridecimal minor-2nd)
14:13      	=  128.3  +0.6	-->  29/270	=  128.9  (tridecimal major-2nd)
15:14      	=  119.4  +0.6	-->  27/270	=  120.0  (septimal chromatic-semitone)
2187:2048    	=  113.7  +1.9	-->  26/270	=  115.6  (pythagorean augmented-prime/apotome)
16:15      	=  111.7  -0.6	-->  25/270	=  111.1  (just diatonic-semitone)
17:16      	=  105.0  +1.7	-->  24/270	=  106.7  (17th harmonic, septendecimal semitone)
135:128     	=   92.2  +1.2	-->  21/270	=   93.3  (large just aug-prime)
256:243     	=   90.2  -1.3	-->  20/270	=   88.9  (pythagorean minor-2nd/limma)
25:24      	=   70.7  +0.4	-->  16/270	=   71.1  (just chromatic-semitone)
33:32      	=   53.3  +0.1	-->  12/270	=   53.3  (33rd harmonic, 11*3)
128:125     	=   41.1  -1.1	-->   9/270	=   40.0  (large just dim-2, diesis)
2048:2025    	=   19.6  -1.8	-->   4/270	=   17.8  (small just dim-2nd, diaschisma)
32805:32768   	=    2.0  +2.5	-->   1/270	=    4.4  (large just aug-7th, skhisma)
1:1       	=    0.0  +0.0	-->   0/270	=    0.0  (prime)

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

some commas:

ratio            cents   error       edo     cents   name

3-limit

531441:524288  	=   23.5  +3.2	-->   6/270	=   26.7  (pythagorean-comma)
~1938:1934   	=    3.6  +14.2	-->   4/270	=   17.8  (mercator-comma (~ratio: actual = 3^53 : 2^84)

5-limit

648:625     	=   62.6  -0.3	-->  14/270	=   62.2  (major-diesis)
16875:16384   	=   51.1  +2.2	-->  12/270	=   53.3  (negri-comma)
250:243     	=   49.2  -0.3	-->  11/270	=   48.9  (maximal-diesis)
128:125     	=   41.1  -1.1	-->   9/270	=   40.0  (enharmonic-diesis)
34171875:33554432	=   31.6  +4.0	-->   8/270	=   35.6  (ampersand-comma)
3125:3072    	=   29.6  +1.5	-->   7/270	=   31.1  (magic-comma)
20000:19683   	=   27.7  -1.0	-->   6/270	=   26.7  (tetracot-comma)
81:80      	=   21.5  +0.7	-->   5/270	=   22.2  (syntonic-comma)
2048:2025    	=   19.6  -1.8	-->   4/270	=   17.8  (diaschisma)
393216:390625  	=   11.4  -2.6	-->   2/270	=    8.9  (wuerschmidt-comma)
2109375:2097152 	=   10.1  +3.3	-->   3/270	=   13.3  (semicomma)
15625:15552   	=    8.1  +0.8	-->   2/270	=    8.9  (kleisma)
32805:32768   	=    2.0  +2.5	-->   1/270	=    4.4  (skhisma)
~76294:76256   	=    0.9  -0.9	-->   0/270	=    0.0  (ennealimma (~ratio: actual = 2 * 5^18 : 3^27))
~292300:292298  	=    0.0  -26.7	-->  -6/270	=  -26.7  (atom (~ratio: actual = 2^161 : 3^84 * 5^12))

7-limit

36:35      	=   48.8  +0.1	-->  11/270	=   48.9  (septimal-diesis)
49:48      	=   35.7  -0.1	-->   8/270	=   35.6  (slendro diesis (7/6 : 8/7))
50:49      	=   35.0  +0.6	-->   8/270	=   35.6  (tritonic diesis, jubilisma)
64:63      	=   27.3  -0.6	-->   6/270	=   26.7  (septimal-comma)
225:224     	=    7.7  +1.2	-->   2/270	=    8.9  (septimal-kleisma)

11-limit

22:21      	=   80.5  -0.5	-->  18/270	=   80.0  ()
33:32      	=   53.3  +0.1	-->  12/270	=   53.3  (undecimal-diesis)
45:44      	=   38.9  +1.1	-->   9/270	=   40.0  ()
8192:8019    	=   37.0  -1.4	-->   8/270	=   35.6  (pyth dim-5th: 11/8)
55:54      	=   31.8  -0.7	-->   7/270	=   31.1  ()
56:55      	=   31.2  -0.1	-->   7/270	=   31.1  ()
99:98      	=   17.6  +0.2	-->   4/270	=   17.8  (mothwellsma)
100:99      	=   17.4  +0.4	-->   4/270	=   17.8  (ptolemisma)
121:120     	=   14.4  -1.0	-->   3/270	=   13.3  (biyatisma (11/10 : 12/11))

13-limit

40:39      	=   43.8  +0.6	-->  10/270	=   44.4  ((5/3 : 13/8))
65:64      	=   26.8  -0.2	-->   6/270	=   26.7  ((13/8 : 8/5))
6656:6561    	=   24.9  -2.7	-->   5/270	=   22.2  (13/8 : pyth aug-5th)
91:90      	=   19.1  -1.4	-->   4/270	=   17.8  (superleap)
144:143     	=   12.1  +1.3	-->   3/270	=   13.3  ((18/11 : 13/8))
169:168     	=   10.3  -1.4	-->   2/270	=    8.9  (dhanvantarisma)

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

### tredeks 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: = tredeks (= )

. . . . . . . . .

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