Small comma

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Revision as of 05:49, 13 January 2019 by TallKite (talk | contribs) (Moved all commas (except 3-limit ones) with extremely high odd limits to the end of their tables, because they are harder to pump, and less useful. Admins, feel free to revert this edit if you disagree!)
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Definition

A comma is a small interval that occurs in just intonation, between two intervals which are close in pitch. Commas are always greater than 1 (0 cents) and not a power such as a square or cube of any other interval, and generally in a low prime limit. The word is often used in reference to regular tempering when the comma is tempered out, which is to say reduced to a unison, by the temperament, thereby equating the intervals separated by that comma. For example the Syntonic or Didymus comma of 81/80, which occurs between 10/9 and 9/8, as well as 81/64 and 5/4, is tempered out by meantone temperament. Commas are usually written as ratios, but they can also be written as monzos. The following lists mostly give both forms, as well as the size in cents. Except for the 3-limit commas, the color name refers to both the comma and the temperament created when it is tempered out.

For commas over 100 cents in size, see Large commas; and for commas under 3.5 cents in size, see Unnoticeable commas

Due to the wide range of sizes, cents values here and on the other commas pages are listed to 5 significant digits, instead of to a fixed number of decimal places as is the convention elsewhere on the wiki.

List of commas, by prime limit

3-limit commas

Name(s) Color Name Ratio Monzo Cents
limma, Pythagorean minor second small wa 2nd sw2 256 / 243 [8 -5 90.225
Pythagorean comma wa comma LLw-2 531441 / 524288 [-19 12 23.460
17-comma wa-17 comma ssw3 134217728 / 129140163 [27 -17 66.765
29-comma wa-29 comma s4w4 70368744177664 / 68630377364883 [46 -29 43.305
41-comma wa-41 comma s6w5 36893488147419103232 /
36472996377170786403
[65 -41 19.845
Mercator's comma, 53-comma wa-53 comma L8w-6 19383245667680019896796723 /
19342813113834066795298816
[-84 53 3.6150
70-comma wa-70 comma s10w8 [111 -70 63.150
77-comma wa-77 comma L11w-8 [-122 77 50.535
200-comma, Pythagorean integer cents comma wa-200 comma w-200 [317 -200 8.9998

5-limit commas

Name(s) Color Name Ratio Monzo Cents
Double large yoyo negative 2nd lala-yoyo LLyy-2 4428675 / 4194304 [-22 11 2 94.132
major chroma, major limma, pelogic comma layo-unison Ly1 135 / 128 [-7 3 1 92.179
double small gu 3rd sasagu ssg3 8388608 / 7971615 [23 -13 -1 88.271
ripple quingu g52 6561 / 6250 [-1 8 -5 84.071
doublewide quadbiyo y8-2 390625 / 373248 [-9 -6 8 78.780
sevond sasepyo sy71 5000000 / 4782969 [6 -14 7 76.826
nusecond legu g113 51018336 / 48828125 [5 13 -11 75.964
sentinel trila-quadyo L3y4-3 8968066875 / 8589934592 [-33 15 4 74.580
large triyo negative 2nd latriyo Ly3-2 273375 / 262144 [-18 7 3 72.626
chromatic semitone yoyo yy1 25 / 24 [-3 -1 2 70.672
superpyth, grave minor second sayo sy2 20480 / 19683 [12 -9 1 68.719
major diesis, diminished comma quadgu g42 648 / 625 [3 4 -4 62.565
passion saquingu sg53 262144 / 253125 [18 -4 -5 60.611
shibboleth tritriyo y9-2 1953125 / 1889568 [-5 -10 9 57.273
mynic quinbigu g103 10077696 / 9765625 [9 9 -10 54.458
tritonic lalaquinyo LLy5-3 553584375 / 536870912 [-29 11 5 53.074
negri comma, double augmentation diesis laquadyo Ly4-2 16875 / 16384 [-14 3 4 51.120
maximal diesis, porcupine triyo y31 250 / 243 [1 -5 3 49.166
immunity sasayoyo ssyy2 1638400 / 1594323 [16 -13 2 47.212
double large green comma lalagu LLg-2 43046721 / 41943040 [-23 16 -1 44.966
diesis, augmented comma trigu g32 128 / 125 [7 0 -3 41.059
wizard lala-quadtriyo LLy12-4 2197265625 / 2147483648 [-31 2 12 39.675
sycamore laleyo Ly11-3 48828125 / 47775744 [-16 -6 11 37.721
fifive saquinbiyo sy10-2 9765625 / 9565938 [-1 -14 10 35.767
unicorn laquadbigu Lg82 1594323 / 1562500 [-2 13 -8 34.905
valentine comma satritrigu sg93 1990656 / 1953125 [13 5 -9 32.952
ampersand, Ampersand's comma lala-tribiyo LLy6-3 34171875 / 33554432 [-25 7 6 31.567
mabila comma sasa-quinbigu ssg104 268435456 / 263671875 [28 -3 -10 30.998
small diesis, magic comma quinyo Ly5-2 3125 / 3072 [-10 -1 5 29.614
minimal diesis, tetracot comma saquadyo sy41 20000 / 19683 [5 -9 4 27.660
quanic comma sepsa-quinyo s7y55 [74 -54 5 25.999
roda sasa-triyo ssy32 131072000 / 129140163 [20 -17 3 25.706
trisedodge comma saquintrigu sg154 30958682112 / 30517578125 [19 10 -15 24.844
maja saseyo sy17-3 762939453125 / 753145430616 [-3 -23 17 22.368
syntonic / Didymos / meantone comma gu g1 81 / 80 [-4 4 -1 21.506
maquila comma trisa-segu s3g176 562949953421312 / 556182861328125 [49 -6 -17 20.937
diaschisma sagugu sgg2 2048 / 2025 [11 -4 -2 19.553
ditonma lala-theyo LLy13-4 1220703125 / 1207959552 [-27 -2 13 18.168
misty comma sasa-trigu ssg33 67108864 / 66430125 [26 -12 -3 17.599
pental comma trila-quingu L3g5-2 847288609443 / 838860800000 [-28 25 -5 17.306
undim comma trisa-quadgu s3g44 2199023255552 / 2179240250625 [41 -20 -4 15.645
graviton, gravity comma lala-tribigu LLg61 129140163 / 128000000 [-13 17 -6 15.353
majvam sasa-lebigu ssg226 [40 7 -22 14.783
quartonic saleyo sy11-2 390625000 / 387420489 [3 -18 11 14.261
medium semicomma, sensipent comma sepgu g72 78732 / 78125 [2 9 -7 13.399
Würschmidt comma saquabigu sg83 393216 / 390625 [17 1 -8 11.445
semicomma, Fokker comma lasepyo Ly7-3 2109375 / 2097152 [-21 3 7 10.061
escapade comma sasa-tritrigu ssg94 4294967296 / 4271484375 [32 -7 -9 9.4916
kleisma, semicomma majeur tribiyo y6-2 15625 / 15552 [-6 -5 6 8.1073
qintosec comma quadsa-quinbigu s4g105 140737488355328 / 140126044921875 [47 -15 -10 7.5378
unidecma laquadtrigu Lg122 31381059609 / 31250000000 [-7 22 -12 7.2455
mutt comma trila-septriyo L3y21-6 476837158203125 / 474989023199232 [-44 -3 21 6.7230
amity comma saquinyo sy51 1600000 / 1594323 [9 -13 5 6.1536
parakleisma thegu g133 1224440064 / 1220703125 [8 14 -13 5.2917
gammic comma laquinquadyo Ly20-5 95367431640625 / 95105071448064 [-29 -11 20 4.7693
vulture comma sasa-quadyo ssy42 10485760000 / 10460353203 [24 -21 4 4.1998
lafa comma tribisa-quatrigu s6g127 [77 -31 -12 3.6304
10-15-comma trila-quinbiyo L3y10-4 576650390625 / 549755813888 [-39 10 10 82.687
11-15-comma trisa-legu s3g115 8796093022208 / 8649755859375 [43 -11 -11 29.044
21-15-comma [-82 21 21 53.643
Twenty-two thirds limma trila-lebiyo L3y22-6 2384185791015625 / 2251799813685248 [-51 0 22 98.902
23-3/5-comma twenty-thegu g235 12339534735212544 / 11920928955078125 [17 23 -23 59.750
sibbolet quadla-quinquinyo L4y25-7 298023223876953125 / 288230376151711744 [-58 0 25 57.843
Hunt 5-cycle comma quadla-sepquadyo L4y28-8 37252902984619140625 / 36893488147419103232 [-65 0 28 16.784
acute thirds comma s4g319 [72 0 -31 24.275
32-15-comma [-125 32 32 24.599
34-5-comma s5g3410 [79 0 -34 65.334
34-3/5-comma sebiyo y343 [-25 -34 34 68.196
Huntian 15-cycle comma [168 -43 -43 4.4453
Schismic thirds chromatic semitone L8y53-14 [-123 0 53 74.627
53-15-comma [-207 53 53 78.242
53-3/5-comma fifty-theyo y537 [-39 -53 53 71.012
54-15-comma [211 -54 -54 33.489
56-syntonic comma [-225 224 -56 4.3522
59-5-comma s9g5916 [137 0 -59 7.4909
76-15-comma [297 -76 -76 91.578

7-limit commas

Name(s) Color Name Ratio Monzo Cents
minor semitone zogu zg2 21 / 20 [-2 1 -1 1 84.467
Laruru negative 2nd laruru Lrr-2 6561 / 6272 [-7 8 0 -2 77.988
bapbo rurugu rrg1 256 / 245 [8 0 -1 -2 76.034
Archytas' 1/3-tone zo z2 28 / 27 [2 -3 0 1 62.961
greenwoodma ruruyo rry-2 405 / 392 [-3 4 1 -2 56.482
Harrison's comma laru Lr-2 59049 / 57344 [-13 10 0 -1 50.724
keega trizogu z3g33 1029 / 1000 [-3 1 -3 3 49.492
septimal quarter tone rugu rg1 36 / 35 [2 2 -1 -1 48.770
5-21-comma saquinru sr5-2 4194304 / 4084101 [22 -5 0 -5 46.095
cloudy laquinzo Lz53 16807 / 16384 [-14 0 0 5 44.130
Avicenna, Avicenna's enharmonic diesis lazoyoyo Lzyy1 525 / 512 [-9 1 2 1 43.408
slendro diesis zozo zz2 49 / 48 [-4 -1 0 2 35.697
tritonic diesis, jubilisma biruyo rryy-2 50 / 49 [1 0 2 -2 34.976
eric sepru r7-4 839808 / 823543 [7 8 0 -7 33.859
senga trizoagugu z3gg3 686 / 675 [1 -3 -2 3 27.985
septimal / Archytas' comma, Leipziger Komma ru r1 64 / 63 [6 -2 0 -1 27.264
blackjackisma lasepruaquabiyo Lr7y8-6 854296875 / 843308032 [-10 7 8 -7 22.413
squalentine laquadzoatrigu Lz4g33 64827 / 64000 [-9 3 -3 4 22.227
keema zotriyo zy31 875 / 864 [-5 -3 3 1 21.902
gariboh triruaquinyo r3y5-3 3125 / 3087 [0 -2 5 -3 21.181
nuwell quadruayo r4y-3 2430 / 2401 [1 5 1 -4 20.785
tolerma sazoyoyo szyy2 179200 / 177147 [10 -11 2 1 19.948
mirwomo labizoyo Lzzyy1 33075 / 32768 [-15 3 2 2 16.144
trimyna quinzogu z5g54 50421 / 50000 [-4 1 -5 5 14.516
sensamagic zozoyo zzy2 245 / 243 [0 -5 1 2 14.191
septimal semicomma, starling comma zotrigu zg32 126 / 125 [1 2 -3 1 13.795
octagar rurutriyo rry3-2 4000 / 3969 [5 -4 3 -2 13.469
orwellisma, orwell comma triruagu r3g-2 1728 / 1715 [6 3 -1 -3 13.074
quince lasepzoagugu Lz7gg4 823543 / 819200 [-15 0 -2 7 9.1539
gamelisma latrizo Lz32 1029 / 1024 [-10 1 0 3 8.4327
varunisma labizogugu Lzzg42 321489 / 320000 [-9 8 -4 2 8.0370
septimal kleisma, marvel comma ruyoyo ryy-2 225 / 224 [-5 2 2 -1 7.7115
dimcomp quadruyoyo r4y8-4 390625 / 388962 [-1 -4 8 -4 7.3861
cataharry labirugu Lrrgg-2 19683 / 19600 [-4 9 -2 -2 7.3158
mirkwai quinruaquadyo r5y4-4 16875 / 16807 [0 3 4 -5 6.9903
stearnsma latribiru Lr6-4 118098 / 117649 [1 10 0 -6 6.5946
hemimage satrizoagu sz3g3 10976 / 10935 [5 -7 -1 3 6.4790
hemimean zozoquingu zzg53 3136 / 3125 [6 0 -5 2 6.0832
hemifamity saruyo sry1 5120 / 5103 [10 -6 1 -1 5.7578
linus saquinbizogu sz10g108 578509309952 / 576650390625 [11 -10 -10 10 5.5719
porwell sarurutrigu srrg31 6144 / 6125 [11 1 -3 -2 5.3620
garischisma sasaru ssr2 33554432 / 33480783 [25 -14 0 -1 3.8041
wadisma latritrizoayo Lz9y4 201768035 / 201326592 [-26 -1 1 9 3.7919
quasiorwellisma sazoquinbigu szg104 29360128 / 29296875 [22 -1 -10 1 3.7338
218edo (stacked 7/4 and 10/9) comma trisa-triru-quadquadyo s3r3y16-3 640000000000000000 / 635585924776181463 [22 -32 16 -3 11.982
8-35-comma lala-quabizoyo LLz8y82 2251875390625 / 2199023255552 [-41 0 8 8 41.117
15-35-comma [77 0 -15 -15 72.906
hexadecatone comma, 16-7-comma sasa-quaquadru ssr16-6 35184372088832 / 33232930569601 [45 0 0 -16 98.785
18-21-comma [-79 18 0 18 74.056
21-7-comma 576460752303423488 / 558545864083284007 [59 0 0 -21 54.656
23-21-comma [-101 23 0 23 27.961
23-35-comma [118 0 -23 -23 31.789
quarter tone comma quinquinrugu [49 50 -25 -25 19.260
Hunt 7-cycle comma [73 0 0 -26 10.526
Huntian 21-cycle comma [123 -28 0 -28 18.135
29-7/5-comma [-14 0 -29 29 92.854
German sixth comma [-87 0 0 31 33.603
31-7/5-comma [-15 0 -31 31 57.878
31-35-comma [-159 0 31 31 9.3282
33-7/5-comma [-16 0 -33 33 22.902
33-21-comma [145 -33 0 -33 64.230
35-7/5-comma [17 0 35 -35 12.073
36-7-comma [-101 0 0 36 77.733
37-7/5-comma [18 0 37 -37 47.049
39-7/5-comma [19 0 39 -39 82.024
39-35-comma [-200 0 39 39 50.445
47-7-comma [132 0 0 -47 65.182
47-35-comma [-241 0 47 47 91.562
Huntian 35-cycle comma [277 0 -54 -54 22.461
slendro 34-comma [-137 -34 0 68 13.692
34-jubilismic comma [-33 0 -68 68 10.829
slendro integer cents comma [1123 0 0 -400 69.637

11-limit commas

Name(s) Color Name Ratio Monzo Cents
undecimal minor semitone loru 1or1 22 / 21 [1 -1 0 -1 1 80.537
Lalu unison lalu L1u1 729 / 704 [-6 6 0 0 -1 60.412
undecimal comma, al-Farabi's 1/4-tone ilo 1o1 33 / 32 [-5 1 0 0 1 53.273
Hunt flat 5 comma salu s1u2 8192 / 8019 [13 -6 0 0 -1 36.952
sterno lotriru 1or3-2 352 / 343 [5 0 0 -3 1 44.840
1/5-tone luyo 1uy1 45 / 44 [-2 2 1 0 -1 38.906
undecimal diesis luzogu 1uzg2 56 / 55 [3 0 -1 1 -1 31.194
telepathma loyo 1oy1 55 / 54 [-1 -3 1 0 1 31.767
thuja comma saquinluayo s1u5y2 163840 / 161051 [15 0 1 0 -5 29.724
sensmus salozogu s1ozg2 1232 / 1215 [4 -5 -1 1 1 24.055
sevnothrush loquingu 1og52 3168 / 3125 [5 2 -5 0 1 23.659
cassacot biluzoayo 1uuzzy2 245 / 242 [-1 0 1 2 -2 21.330
mothwellsma loruru 1orr-2 99 / 98 [-1 2 0 -2 1 17.576
ptolemisma luyoyo 1uyy1 100 / 99 [2 -2 2 0 -1 17.399
hemimin triluazo 1u3z2 1344 / 1331 [6 1 0 1 -3 16.827
biyatisma lologu 1oog1 121 / 120 [-3 -1 -1 0 2 14.367
aphrowe triloaruru 1o3rr-2 1331 / 1323 [0 -3 0 -2 3 10.437
valinorsma lorugugu 1orgg1 176 / 175 [4 0 -2 -1 1 9.8646
pentacircle saluzo s1uz2 896 / 891 [7 -4 0 1 -1 9.6880
orgonisma satriluaruru s1u3rr1 65536 / 65219 [16 0 0 -2 -3 8.3944
rastma lulu 1uu1 243 / 242 [-1 5 0 0 -2 7.1391
myhemiwell lolozotrigu 1oozg32 3388 / 3375 [2 -3 -3 1 2 6.6556
keenanisma lozoyo 1ozy1 385 / 384 [-7 -1 1 1 1 4.5026
werckisma luzozogu 1uzzg2 441 / 440 [-3 2 -1 2 -1 3.9302
moctdel lotriruyo 1or3y3-3 1375 / 1372 [-2 0 3 -3 1 3.7814
11-cycle chroma, 11-11-comma lalalelo LL1o11-3 285311670611 / 274877906944 [-38 0 0 0 11 64.497
the Igs boson, 13-11-comma sasathelu ss1u134 35184372088832 / 34522712143931 [45 0 0 0 -13 32.867
Hunt 11-cycle comma [-83 0 0 0 24 31.631
11-cycle major seventh prote, 35-11-comma [-121 0 0 0 35 96.128

13-limit commas

Name(s) Color Name Ratio Monzo Cents
tridecimal comma thu 3u1 27 / 26 [-1 3 0 0 0 -1 65.337
tridecimal 1/3-tone thogu 3og2 26 / 25 [1 0 -2 0 0 1 67.900
Hunt minor submediant comma latho L3o1 1053 / 1024 [-10 4 0 0 0 1 48.348
tridecimal minor diesis thuyo 3uy1 40 / 39 [3 -1 1 0 0 -1 43.831
astburyp latribitho L3o62 4826809 / 4718592 [-19 -2 0 0 0 6 39.256
wilsorma thoyo 3oy1 65 / 64 [-6 0 1 0 0 1 26.841
winmeanma thulogu 3u1og1 66 / 65 [1 1 -1 0 1 -1 26.432
negustma tholuru 3o1ur1 78 / 77 [1 1 0 -1 -1 1 22.339
superleap thozogu 3ozg2 91 / 90 [-1 -2 -1 1 0 1 19.130
animist thuzoyo 3uzy1 105 / 104 [-3 1 1 1 0 -1 16.567
secorian sathuzo s3uz2 28672 / 28431 [12 -7 0 1 0 -1 14.613
mosaic lathoruru L3orr-2 3159 / 3136 [-6 5 0 -2 0 1 12.651
gassorma thuloruyoyo 3u1oryy-2 275 / 273 [0 -1 2 -1 1 -1 12.637
grossma thulu 3u1u1 144 / 143 [4 2 0 0 -1 -1 12.064
buzurgisma, dhanvantarisma thothoru 3oor1 169 / 168 [-3 -1 0 -1 0 2 10.274
catadictma thologu 3o1og1 1287 / 1280 [-8 2 -1 0 1 1 9.4419
mynucuma thuzozogu 3uzzg2 196 / 195 [2 -1 -1 2 0 -1 8.8554
huntma, nelindic comma thururuyo 3urry-2 640 / 637 [7 0 1 -2 0 -1 8.1342
threedie satritho s3o32 2197 / 2187 [0 -7 0 0 0 3 7.8980
kestrel comma thuthuloru 3uu1or-2 1188 / 1183 [2 3 0 -1 1 -2 7.3017
marveltwin thoyoyo 3oyy1 325 / 324 [-2 -4 2 0 0 1 5.3351
ratwolfsma thorugugu 3orgg1 351 / 350 [-1 3 -2 -1 0 1 4.9393
minthma thulo 3u1o1 352 / 351 [5 -3 0 0 1 -1 4.9253
gentle comma tholuluzo 3o1uuz2 364 / 363 [2 -1 0 1 -2 1 4.7627
cuthbert bithuloazogu 3uu1oozg1 847 / 845 [0 0 -1 1 2 -2 4.0928
Hunt 13-cycle comma laquinbitho L3o102 137858491849 / 137438953472 [-37 0 0 0 0 10 5.2766
silver half cents comma [-8881 0 0 0 0 2400 66.388

17-limit commas

Name(s) Color Name Ratio Monzo Cents
septendecimal 1/4-tone suzoyo 17uzy1 35 / 34 [-1 0 1 1 0 0 -1 50.184
solu 17o1u2 34 / 33 [1 -1 0 0 -1 0 1 51.682
Arabic lute index finger su 17u1 18 / 17 [1 2 0 0 0 0 -1 98.955
17th-partial chroma sogugu 17ogg2 51 / 50 [-1 1 -2 0 0 0 1 34.283
sutho 17u3o1 52 / 51 [2 -1 0 0 0 1 -1 33.617
soruyo 17ory1 85 / 84 [-2 -1 1 -1 0 0 1 20.488
Hunt flat 2 comma laso L17o1 4131 / 4096 [-12 5 0 0 0 0 1 14.730
suruyo 17ury-2 120 / 119 [3 1 1 -1 0 0 -1 14.487
Blume comma sololo 17o1oo1 2057 / 2048 [-11 0 0 0 2 0 1 7.5913
septendecimal kleisma sugu 17ug1 256 / 255 [8 -1 -1 0 0 0 -1 6.7759
tannisma suthozo 17u3oz1 273 / 272 [-4 1 0 1 0 1 -1 6.3532
septendecimal integer cents comma soso 17oo2 289 / 288 [-5 -2 0 0 0 0 2 6.0008
11 semitone comma 35184372088832 / 34271896307633 [45 0 0 0 0 0 -11 45.490
12 semitone comma 582622237229761 / 562949953421312 [-49 0 0 0 0 0 12 59.465
23 semitone comma [-94 0 0 0 0 0 23 13.974
Hunt 17-cycle comma [139 0 0 0 0 0 -34 31.516
17-ripple integer cents comma [327 0 0 0 0 0 -80 3.5672

19-limit commas

Name(s) Color Name Ratio Monzo Cents
undevicesimal semitone ino 19o2 19 / 18 [-1 -2 0 0 0 0 0 1 93.603
small undevicesimal semitone nuyo 19uy1 20 / 19 [2 0 1 0 0 0 0 -1 88.801
nutho 19u3o1 39 / 38 [-1 1 0 0 0 1 0 -1 44.970
noru 19or1 57 / 56 [-3 1 0 -1 0 0 0 1 30.642
nuruyo 19ury-2 135 / 133 [0 3 1 -1 0 0 0 -1 25.840
nogugu 19ogg2 76 / 75 [2 -1 -2 0 0 0 0 1 22.931
approximation to Pyth 53-comma nulozo 19u1oz1 77 / 76 [-2 0 0 1 1 0 0 -1 22.631
19th-partial chroma nugu 19ug1 96 / 95 [5 1 -1 0 0 0 0 -1 18.128
noluzo 19o1uz2 133 / 132 [-2 -1 0 1 -1 0 0 1 13.066
Ganassi's comma nuso 19u17o1 153 / 152 [-3 2 0 0 0 0 1 -1 11.352
nosugu 19o17ug1 171 / 170 [-1 2 -1 0 0 0 -1 1 10.154
Hunt 19-cycle comma saquadnu s19u4-2 131072 / 130321 [17 0 0 0 0 0 0 -4 9.9479
spleen comma nuluzoyo 19u1uzy1 210 / 209 [1 1 1 1 -1 0 0 -1 8.2637
nusu 19u17u-2 324 / 323 [2 4 0 0 0 0 -1 -1 5.3516
go comma nonogu 19oog2 361 / 360 [-3 -2 -1 0 0 0 0 2 4.8023