# spook, 581-ed2

[Joe Monzo, Tonalsoft Encyclopedia of Microtonal Music Theory]

A small unit of interval measurement.

A spook is one degree of 581-edo, the logarithmic division of the octave into 581 equal parts. It is calculated as the 581st root of 2, expressed mathematically as 581√2, or 2(1/581), with a ratio of approximately 1:1.0011937363438692. It is an irrational number. The formula for calculating the absolute floating-point spook value of any ratio is: spooks = log10r * [581 / log102] or spooks = log2r * 581 , where r is the ratio.

Some tuning-theorists advocate the use of the spook as a logarithmic measurement used to compare interval sizes because it approximates 23-limit JI to high degree of accuracy.

. . . . . . . . .

### some intervals mapped to 581-edo

(Note that while 581-edo is very accurate for prime-factors 5, 7, 11, 13, and 19, and also quite accurate for factors 3, 17, and 23, ratios listed here which include other prime-factors above 23 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.)

```581 -edo floating-point mappings:

prime    edo-steps   step-error  edo-map
2  =   581.000000    +0.00 -->    581
3  =   920.863213    +0.14 -->    921
5  =  1349.040223    -0.04 -->   1349
7  =  1631.073210    -0.07 -->   1631
11  =  2009.929770    +0.07 -->   2010
13  =  2149.955476    +0.04 -->   2150
17  =  2374.815911    +0.18 -->   2375
19  =  2468.045885    -0.05 -->   2468
23  =  2628.189496    -0.19 -->   2628
29  =  2822.486958    -0.49 -->   2822
31  =  2878.388056    -0.39 -->   2878
37  =  3026.692405    +0.31 -->   3027
41  =  3112.737715    +0.26 -->   3113
43  =  3152.659822    +0.34 -->   3153

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

map  2 -->    581 = 1200.000000 cents <-- 1200.000000  +0.0 cents
map  3 -->    921 = 1902.237522 cents <-- 1901.955001  +0.3 cents
map  5 -->   1349 = 2786.230637 cents <-- 2786.313714  -0.1 cents
map  7 -->   1631 = 3368.674699 cents <-- 3368.825906  -0.2 cents
map 11 -->   2010 = 4151.462995 cents <-- 4151.317942  +0.1 cents
map 13 -->   2150 = 4440.619621 cents <-- 4440.527662  +0.1 cents
map 17 -->   2375 = 4905.335628 cents <-- 4904.955410  +0.4 cents
map 19 -->   2468 = 5097.418244 cents <-- 5097.513016  -0.1 cents
map 23 -->   2628 = 5427.882960 cents <-- 5428.274347  -0.4 cents
map 29 -->   2822 = 5828.571429 cents <-- 5829.577194  -1.0 cents
map 31 -->   2878 = 5944.234079 cents <-- 5945.035572  -0.8 cents
map 37 -->   3027 = 6251.979346 cents <-- 6251.344039  +0.6 cents
map 41 -->   3113 = 6429.604131 cents <-- 6429.062406  +0.5 cents
map 43 -->   3153 = 6512.220310 cents <-- 6511.517706  +0.7 cents

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

examples:

ratio               -->      581 -edo mapping:

ratio         cents  error       edo          cents  name

2:1       	= 1200.0  +0.0	--> 581/581	= 1200.0  (octave)
65536:32805   	= 1198.0  -2.2	--> 579/581	= 1195.9  (minimal just dim-2)
2025:1024    	= 1180.4  +1.0	--> 572/581	= 1181.4  (small just aug-7th)
1048576:531441  	= 1176.5  -3.4	--> 568/581	= 1173.1  (pythagorean dim-2nd)
125:64      	= 1158.9  -0.2	--> 561/581	= 1158.7  (minimal just aug-7th)
31:16      	= 1145.0  -0.8	--> 554/581	= 1144.2  (31st harmonic)
48:25      	= 1129.3  +0.4	--> 547/581	= 1129.8  (small just dim-8ve)
21:11      	= 1119.5  -0.0	--> 542/581	= 1119.4  (undecimal diminished-8ve)
243:128     	= 1109.8  +1.4	--> 538/581	= 1111.2  (pythagorean major-7th)
256:135     	= 1107.8  -0.8	--> 536/581	= 1107.1  (minimal just dim-8ve)
15:8       	= 1088.3  +0.2	--> 527/581	= 1088.5  (15th harmonic, just major-7th, 5*3)
4096:2187    	= 1086.3  -2.0	--> 525/581	= 1084.3  (pythagorean diminished-8ve)
13:7       	= 1071.7  +0.2	--> 519/581	= 1071.9  (tridecimal superminor-7th)
50:27      	= 1066.8  -1.0	--> 516/581	= 1065.7  (small just maj-7th)
24:13      	= 1061.4  +0.2	--> 514/581	= 1061.6  (tridecimal major-7th)
11:6       	= 1049.4  -0.1	--> 508/581	= 1049.2  (undecimal submajor[neutral]-7th)
20:11      	= 1035.0  -0.2	--> 501/581	= 1034.8  (undecimal superminor[neutral]-7th)
29:16      	= 1029.6  -1.0	--> 498/581	= 1028.6  (29th harmonic)
59049:32768   	= 1019.6  +2.8	--> 495/581	= 1022.4  (pythagorean aug-6th)
9:5       	= 1017.6  +0.6	--> 493/581	= 1018.2  (just minor-7th)
3645:2048    	=  998.0  +1.6	--> 484/581	=  999.7  (large just aug-6th)
16:9       	=  996.1  -0.6	--> 482/581	=  995.5  (pythagorean minor-7th)
225:128     	=  976.5  +0.4	--> 473/581	=  976.9  (small just augmented-6th)
7:4       	=  968.8  -0.2	--> 469/581	=  968.7  (7th harmonic, septimal subminor-7th)
216:125     	=  946.9  +1.1	--> 459/581	=  948.0  (large just dim-7)
19:11      	=  946.2  -0.2	--> 458/581	=  946.0  (nondecimal supermajor-6th)
12:7       	=  933.1  +0.4	--> 452/581	=  933.6  (septimal supermajor-6th)
128:75      	=  925.4  -0.1	--> 448/581	=  925.3  (small just dim-7th)
22:13      	=  910.8  +0.1	--> 441/581	=  910.8  (tridecimal augmented-6th)
27:16      	=  905.9  +0.8	--> 439/581	=  906.7  (27th harmonic, pythagorean major-6th)
2048:1215    	=  903.9  -1.3	--> 437/581	=  902.6  (minimal just dim-7)
5:3       	=  884.4  -0.4	--> 428/581	=  884.0  (just major-6th)
32768:19683   	=  882.4  -2.5	--> 426/581	=  879.9  (pythagorean dim-7th)
18:11      	=  852.6  +0.4	--> 413/581	=  853.0  (undecimal superminor[neutral]-6th)
13:8       	=  840.5  +0.1	--> 407/581	=  840.6  (13th harmonic)
21:13      	=  830.3  +0.0	--> 402/581	=  830.3  (tridecimal ?)
6561:4096    	=  815.6  +2.3	--> 396/581	=  817.9  (pythagorean augmented-5th)
8:5       	=  813.7  +0.1	--> 394/581	=  813.8  (just minor-6th)
405:256     	=  794.1  +1.0	--> 385/581	=  795.2  (large just aug-5th)
128:81      	=  792.2  -1.1	--> 383/581	=  791.0  (pythagorean minor-6th)
11:7       	=  782.5  +0.3	--> 379/581	=  782.8  (undecimal augmented-5th)
25:16      	=  772.6  -0.2	--> 374/581	=  772.5  (25th harmonic, just augmented-5th)
14:9       	=  764.9  -0.7	--> 370/581	=  764.2  (septimal subminor-6th)
17:11      	=  753.6  +0.2	--> 365/581	=  753.9  (septendecimal diminished-6th)
20:13      	=  745.8  -0.2	--> 361/581	=  745.6  (tridecimal augmented-5th)
192:125     	=  743.0  +0.5	--> 360/581	=  743.5  (large just dim-6)
1024:675     	=  721.5  -0.7	--> 349/581	=  720.8  (small just dim-6th)
3:2       	=  702.0  +0.3	--> 340/581	=  702.2  (perfect-5th)
16384:10935   	=  700.0  -1.9	--> 338/581	=  698.1  (minimal just dim-6)
262144:177147  	=  678.5  -3.1	--> 327/581	=  675.4  (pythagorean dim-6th)
19:13      	=  657.0  -0.2	--> 318/581	=  656.8  (nondecimal doubly-augmented-4th)
16:11      	=  648.7  -0.1	--> 314/581	=  648.5  (11th subharmonic, undecimal diminished-4th)
13:9       	=  636.6  -0.5	--> 308/581	=  636.1  (tridecimal diminished-5th)
23:16      	=  628.3  -0.4	--> 304/581	=  627.9  (23rd harmonic)
10:7       	=  617.5  +0.1	--> 299/581	=  617.6  (septimal large-tritone)
729:512     	=  611.7  +1.7	--> 297/581	=  613.4  (pythagorean augmented-4th)
64:45      	=  609.8  -0.5	--> 295/581	=  609.3  (just diminished-5th)
45:32      	=  590.2  +0.5	--> 286/581	=  590.7  (just augmented-4th)
1024:729     	=  588.3  -1.7	--> 284/581	=  586.6  (pythagorean diminished-5th)
7:5       	=  582.5  -0.1	--> 282/581	=  582.4  (septimal small-tritone)
25:18      	=  568.7  -0.7	--> 275/581	=  568.0  (small just aug-4th)
18:13      	=  563.4  +0.5	--> 273/581	=  563.9  (tridecimal augmented-4th)
11:8       	=  551.3  +0.1	--> 267/581	=  551.5  (11th harmonic, undecimal sub-augmented-4th)
15:11      	=  537.0  +0.1	--> 260/581	=  537.0  (undecimal large-4th)
177147:131072  	=  521.5  +3.1	--> 254/581	=  524.6  (pythagorean aug-3rd)
43:32      	=  511.5  +0.7	--> 248/581	=  512.2  (43rd harmonic)
10935:8192    	=  500.0  +1.9	--> 243/581	=  501.9  (large just aug-3rd)
4:3       	=  498.0  -0.3	--> 241/581	=  497.8  (perfect-4th)
675:512     	=  478.5  +0.7	--> 232/581	=  479.2  (small just aug-3rd)
21:16      	=  470.8  +0.1	--> 228/581	=  470.9  (21st harmonic, septimal-4th, 7*3)
17:13      	=  464.4  +0.3	--> 225/581	=  464.7  (septendecimal 4th)
125:96      	=  457.0  -0.5	--> 221/581	=  456.5  (minimal just aug-3rd)
13:10      	=  454.2  +0.2	--> 220/581	=  454.4  (tridecimal diminished-4th)
9:7       	=  435.1  +0.7	--> 211/581	=  435.8  (septimal supermajor-3rd)
41:32      	=  429.1  +0.5	--> 208/581	=  429.6  (41st harmonic)
32:25      	=  427.4  +0.2	--> 207/581	=  427.5  (small just dim-4th)
14:11      	=  417.5  -0.3	--> 202/581	=  417.2  (undecimal diminished-4th)
81:64      	=  407.8  +1.1	--> 198/581	=  409.0  (pythagorean major-3rd)
512:405     	=  405.9  -1.0	--> 196/581	=  404.8  (minimal just dim-4)
5:4       	=  386.3  -0.1	--> 187/581	=  386.2  (5th harmonic, just major-3rd)
8192:6561    	=  384.4  -2.3	--> 185/581	=  382.1  (pythagorean diminished-4th)
16:13      	=  359.5  -0.1	--> 174/581	=  359.4  (tridecimal major[neutral]-3rd)
11:9       	=  347.4  -0.4	--> 168/581	=  347.0  (undecimal neutral-3rd)
39:32      	=  342.5  +0.4	--> 166/581	=  342.9  (39th harmonic, 13*3)
19683:16384   	=  317.6  +2.5	--> 155/581	=  320.1  (pythagorean augmented-2nd)
6:5       	=  315.6  +0.4	--> 153/581	=  316.0  (just minor-3rd)
19:16      	=  297.5  -0.1	--> 144/581	=  297.4  (19th harmonic)
1215:1024    	=  296.1  +1.3	--> 144/581	=  297.4  (large just aug-2nd)
32:27      	=  294.1  -0.8	--> 142/581	=  293.3  (pythagorean minor-3rd)
13:11      	=  289.2  -0.1	--> 140/581	=  289.2  (tridecimal diminished-3rd)
75:64      	=  274.6  +0.1	--> 133/581	=  274.7  (just augmented-2nd)
7:6       	=  266.9  -0.4	--> 129/581	=  266.4  (septimal subminor-3rd)
37:32      	=  251.3  +0.6	--> 122/581	=  252.0  (37th harmonic)
15:13      	=  247.7  +0.1	--> 120/581	=  247.8  (tridecimal augmented[neutral]-2nd)
144:125     	=  245.0  +0.8	--> 119/581	=  245.8  (large just dim-3)
8:7       	=  231.2  +0.2	--> 112/581	=  231.3  (septimal tone, supermajor-2nd)
256:225     	=  223.5  -0.4	--> 108/581	=  223.1  (small just dim-3rd)
9:8       	=  203.9  +0.6	-->  99/581	=  204.5  (pythagorean major-2nd/tone)
4096:3645    	=  202.0  -1.6	-->  97/581	=  200.3  (minimal just dim-3)
10:9       	=  182.4  -0.6	-->  88/581	=  181.8  (just minor-tone)
65536:59049   	=  180.4  -2.8	-->  86/581	=  177.6  (pythagorean dim-3rd)
11:10      	=  165.0  +0.2	-->  80/581	=  165.2  (undecimal small-tone/submajor-2nd)
35:32      	=  155.1  -0.2	-->  75/581	=  154.9  (35th harmonic, 7*5)
12:11      	=  150.6  +0.1	-->  73/581	=  150.8  (undecimal large-semitone)
13:12      	=  138.6  -0.2	-->  67/581	=  138.4  (tridecimal minor-2nd)
14:13      	=  128.3  -0.2	-->  62/581	=  128.1  (tridecimal major-2nd)
15:14      	=  119.4  +0.4	-->  58/581	=  119.8  (septimal chromatic-semitone)
2187:2048    	=  113.7  +2.0	-->  56/581	=  115.7  (pythagorean augmented-prime/apotome)
16:15      	=  111.7  -0.2	-->  54/581	=  111.5  (just diatonic-semitone)
17:16      	=  105.0  +0.4	-->  51/581	=  105.3  (17th harmonic, septendecimal semitone)
135:128     	=   92.2  +0.8	-->  45/581	=   92.9  (large just aug-prime)
256:243     	=   90.2  -1.4	-->  43/581	=   88.8  (pythagorean minor-2nd/limma)
25:24      	=   70.7  -0.4	-->  34/581	=   70.2  (just chromatic-semitone)
33:32      	=   53.3  +0.4	-->  26/581	=   53.7  (33rd harmonic, 11*3)
128:125     	=   41.1  +0.2	-->  20/581	=   41.3  (large just dim-2, diesis)
2048:2025    	=   19.6  -1.0	-->   9/581	=   18.6  (small just dim-2nd, diaschisma)
32805:32768   	=    2.0  +2.2	-->   2/581	=    4.1  (large just aug-7th, skhisma)
1:1       	=    0.0  +0.0	-->   0/581	=    0.0  (prime)

some commas:

3-limit
ratio         cents  error       edo          cents  name

531441:524288  	=   23.5  +3.4	-->  13/581	=   26.9  (pythagorean-comma)
19383245667680019896796723:19342813113834066795298816	=    3.6  +15.0	-->   9/581	=   18.6  (mercator-comma)

5-limit
ratio         cents  error       edo          cents  name

648:625     	=   62.6  +1.5	-->  31/581	=   64.0  (major-diesis)
16875:16384   	=   51.1  +0.5	-->  25/581	=   51.6  (negri-comma)
250:243     	=   49.2  -1.7	-->  23/581	=   47.5  (maximal-diesis)
128:125     	=   41.1  +0.2	-->  20/581	=   41.3  (enharmonic-diesis)
34171875:33554432	=   31.6  +1.5	-->  16/581	=   33.0  (ampersand-comma)
3125:3072    	=   29.6  -0.7	-->  14/581	=   28.9  (magic-comma)
20000:19683   	=   27.7  -2.9	-->  12/581	=   24.8  (tetracot-comma)
81:80      	=   21.5  +1.2	-->  11/581	=   22.7  (syntonic-comma)
2048:2025    	=   19.6  -1.0	-->   9/581	=   18.6  (diaschisma)
393216:390625  	=   11.4  +0.9	-->   6/581	=   12.4  (wuerschmidt-comma)
2109375:2097152 	=   10.1  +0.3	-->   5/581	=   10.3  (semicomma)
15625:15552   	=    8.1  -1.9	-->   3/581	=    6.2  (kleisma)
32805:32768   	=    2.0  +2.2	-->   2/581	=    4.1  (skhisma)
76294:76256   	=    0.9  -9.1	-->  -4/581	=   -8.3  (ennealimma (~ratio))
292300:292297  	=    0.0  -22.7	--> -11/581	=  -22.7  (atom (~ratio))

7-limit
ratio         cents  error       edo          cents  name

36:35      	=   48.8  +0.8	-->  24/581	=   49.6  (septimal-diesis)
49:48      	=   35.7  -0.6	-->  17/581	=   35.1  (slendro diesis (7/6 : 8/7))
50:49      	=   35.0  +0.1	-->  17/581	=   35.1  (tritonic diesis, jubilisma)
64:63      	=   27.3  -0.4	-->  13/581	=   26.9  (septimal-comma)
225:224     	=    7.7  +0.6	-->   4/581	=    8.3  (septimal-kleisma)

11-limit
ratio         cents  error       edo          cents  name

22:21      	=   80.5  +0.0	-->  39/581	=   80.6  ()
33:32      	=   53.3  +0.4	-->  26/581	=   53.7  (undecimal-diesis)
45:44      	=   38.9  +0.3	-->  19/581	=   39.2  ()
8192:8019    	=   37.0  -1.8	-->  17/581	=   35.1  (pyth dim-5th: 11/8)
55:54      	=   31.8  -0.8	-->  15/581	=   31.0  ()
56:55      	=   31.2  -0.2	-->  15/581	=   31.0  ()
99:98      	=   17.6  +1.0	-->   9/581	=   18.6  (mothwellsma)
100:99      	=   17.4  -0.9	-->   8/581	=   16.5  (ptolemisma)
121:120     	=   14.4  +0.1	-->   7/581	=   14.5  (biyatisma (11/10 : 12/11))

13-limit
ratio         cents  error       edo          cents  name

40:39      	=   43.8  -0.5	-->  21/581	=   43.4  ((5/3 : 13/8))
65:64      	=   26.8  +0.0	-->  13/581	=   26.9  ((13/8 : 8/5))
6656:6561    	=   24.9  -2.2	-->  11/581	=   22.7  (13/8 : pyth aug-5th)
91:90      	=   19.1  -0.5	-->   9/581	=   18.6  (superleap)
144:143     	=   12.1  +0.3	-->   6/581	=   12.4  ((18/11 : 13/8))
169:168     	=   10.3  +0.1	-->   5/581	=   10.3  (dhanvantarisma)
```
. . . . . . . . .

### spooks 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: = spooks

This calculator's result is an absolute spook value of a ratio. However, the true value in employing the spook is that it maps the prime-factors 3, 5, 7, 11, 13, 17, 19, and 23 so accurately that once the mapping of those primes is known, the ratio simply needs to be factored and then the spook-values may be added, the integer result providing a very accurate measurement of a ratio, obviating the need for decimal-places. (See the comparison chart at the bottom of edo prime-error)

. . . . . . . . .

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