Table of 3-limit commas: Difference between revisions

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A [[3-limit]] comma is the amount which separates m [[3/2|fifth]]<nowiki/>s from n [[octave]]<nowiki/>s. It is the simplest kind of comma a tuning may [[Tempering out|temper out]]. All commas not equal to the perfect fourth listed here generate interlaced pentatonic scales when tempered out.
A [[3-limit]] comma is the amount which separates ''m'' [[3/2|fifths]] from ''n'' [[octave]]s. It is the simplest kind of comma a tuning may [[Tempering out|temper out]]. All commas not equal to the perfect fourth listed here generate interlaced pentatonic scales and fifths tempered by up to one step of ''n''-[[edo]] when tempered out.
 
{| class="wikitable"
{| class="wikitable"
|-
|-
!Name(s)
! colspan="2" | Name(s)
! colspan="2" |[[Color notation|Color Name]]
! colspan="2" | [[Color notation|Color Name]]
!Ratio
! [[Monzo]]
!Monzo
! [[Cent]]s
!Cents
|-
| whole tone
| 2-comma
| wa 2nd
| w2
| {{Monzo| -3 2 }}
| 203.91
|-
|-
|24-comma
|wa-24 comma
|w-24
|
|
|{{monzo| -37 24}}
| 193-comma
|1246.92
| wa-193 comma
| w-193
| {{Monzo| 306 -193 }}
| 202.46
|-
|-
|48-comma
|wa-48 comma
|w-48
|
|
|{{monzo| 77 -48 }}
| 255-comma
|1106.16
| wa-255 comma
| w-255
| {{Monzo| -404 255 }}
| 198.525
|-
|-
|41b-comma
|wa-41b comma
|w-41b
|
|
|{{monzo| -64 41 }}
| 104-comma
|1080.155
| wa-104 comma
| w-104
| {{Monzo| 165 -104 }}
| 196.68
|-
|-
|29b-comma
|wa-29b comma
|w-29b
|
|
|{{monzo| -45 29 }}
| 157-comma
|1156.695
| wa-157 comma
| w-157
| {{Monzo| 249 -157 }}
| 193.065
|-
|-
|Pythagorean supermajor seventh, 17b-comma
|wa-17b comma
|w-17b
|
|
|{{monzo| -26 17 }}
| 149-comma
|1133.235
| wa-149 comma
| w-149
| {{Monzo| -236 149 }}
| 191.295
|-
|-
|51-comma
|wa-51 comma
|w-51
|
|
|{{monzo| -80 51 }}
| 263-comma
|999.705
| wa-263 comma
| w-263
| {{Monzo| 417 -263 }}
| 185.835
|-
|-
|44-comma
|wa-44 comma
|w-44
|
|
|{{monzo| 69 -44 }}
| 43-comma
|886.02
| wa-43 comma
| w-43
| {{Monzo| -68 43 }}
| 184.065
|-
|-
|34-comma
|wa-34 comma
|w-34
|
|
|{{monzo| -53 34 }}
| 296-comma
|835.485
| wa-296 comma
| w-296
| {{Monzo| -469 296 }}
| 178.68
|-
|-
|19b-comma
|wa-19b comma
|w-19b
|
|
|{{monzo|31 -19}}
| 63-comma
|1062.855
| wa-63 comma
| w-63
| {{Monzo| 100 -63 }}
| 176.835
|-
|-
|31b-comma
|wa-31b comma
|w-31b
|
|
|{{monzo| 50 -31 }}
| 243-comma
|1039.395
| wa-243 comma
| w-243
| {{Monzo| -385 243 }}
| 175.605
|-
|-
|43b-comma
|wa-43b comma
|w-43b
|
|
|{{monzo| 67 -43 }}
| 190-comma
|1015.395
| wa-190 comma
| w-190
| {{Monzo| -301 190 }}
| 171.45
|-
|-
|39b-comma
|wa-39b comma
|w-39b
|
|
|{{monzo| -61 39 }}
| 169-comma
|976.25
| wa-169 comma
| w-169
| {{Monzo| 268 -169}}
| 169.605
|-
|-
|Pythagorean harmonic seventh, 14-comma
|wa-14 comma
|w-14
|
|
|{{Monzo|23 -14}}
| 137-comma
|972.63
| wa-137 comma
| w-137
| {{Monzo| -217 137 }}
| 167.835
|-
|-
|27b-comma
|wa-27b comma
|w-27b
|
|
|{{monzo| -44 27}}
| 275-comma
|952.785
| wa-275 comma
| w-275
| {{Monzo| 436 -275 }}
| 162.375
|-
|-
|Pythagorean supermajor sixth, 15-comma
|wa-15 comma
|w-15
|
|
|{{Monzo|-23 15}}
| [[31-comma_temperaments|31-comma]]
|929.325
| wa-31 comma
| w-31
| {{Monzo| -49 31 }}
| 160.605
|-
|-
|38-comma
|wa-38 comma
|w-38
|
|
|{{monzo| -61 38 }}
| 22-comma
|925.71
| wa-22 comma
| w-22
| {{Monzo| 35 -22 }}
| 156.99
|-
|-
|21-comma
|wa-21 comma
|w-21
|
|
|{{Monzo| 34 -21}}
| 75-comma
|858.945
| wa-75 comma
| w-75
| {{Monzo|119 -75}}
| 153.375
|-
|-
|20-comma
|wa-20 comma
|w-20
|
|
|{{Monzo| -31 20}}
| 128-comma
|839.1
| wa-128 comma
| w-128
| {{Monzo| 203 -128}}
| 149.76
|-
|-
|33b-comma
|wa-33b comma
|w-33b
|
|
|{{monzo| 53 -33 }}
| 181-comma
|835.485
| wa-181 comma
| w-181
| {{Monzo| 287 -181 }}
| 146.145
|-
|-
|37b-comma
|wa-37b comma
|w-37b
|
|
|{{monzo| -58 37 }}
| 125-comma
|772.335
| wa-125 comma
| w-125
| {{Monzo| -198 125}}
| 144.375
|-
|-
|25-comma
|wa-25 comma
|w-25
|
|
|{{Monzo| -39 25}}
| 234-comma
|748.875
| wa-234 comma
| w-234
| {{Monzo| 371 -234 }}
| 142.53
|-
|-
|28-comma
|wa-28 comma
|w-28
|
|
|{{monzo| 45 -28}}
| 19-comma
|745.26
| wa-19 comma
| w-19
| {{Monzo| -30 19 }}
| 137.145
|-
|-
|13-comma
|wa-13 comma
|w-13
|
|
|{{Monzo| -20 13}}
| 272-comma
|725.415
| wa-272 comma
| w-272
| {{Monzo| -431 272 }}
| 131.75
|-
|-
|54-comma
|wa-54 comma
|w-54
|
|
|{{monzo| -85 54 }}
| 219-comma
|705.57
| wa-219 comma
| w-219
| {{Monzo| -347 219 }}
| 128.145
|-
|-
|52-comma
|wa-52 comma
|w-52
|
|
|{{monzo| 83 -52 }}
| 166-comma
|698.33
| wa-166 comma
| w-166
| {{Monzo| -263 166}}
| 124.53
|-
|-
|Pythagorean wolf fifth, 11b-comma
|
|small wa 5th
| 113-comma
|sw5
| wa-113 comma
|262144 / 177147
| w-113
|{{monzo| 18 -11}}
| {{Monzo| 179 -113 }}
|678.495
| 120.915
|-
|-
|47b-comma
| [[2187/2048|Apotome]]
|wa-47b comma
| 7-comma
|w-47b
| lawa unison
|
| Lw1
|{{monzo| 75 -47 }}
| {{Monzo| -11 7 }}
|610.115
| 113.685
|-
|-
|47-comma
| Pythagorean-septendecimal semitone
|wa-47 comma
| 207-comma
|w-47
| wa-207 comma
|
| w-207
|{{monzo| -74 47 }}
| {{Monzo| -328 207 }}
|591.885
| 104.685
|-
|-
|42-comma
|wa-42 comma
|w-42
|
|
|{{monzo| 67 -42 }}
| 46-comma
|517.89
| wa-46 comma
| w-46
| {{Monzo| 73 -46 }}
| 110.07
|-
|-
|23b-comma
|wa-23b comma
|w-23b
|
|
|{{monzo| 37 -23}}
| 99-comma
|655.055
| wa-99 comma
| w-99
| {{Monzo| 157 -99 }}
| 106.455
|-
|-
|30-comma
|wa-30 comma
|w-30
|
|
|{{monzo| -47 30 }}
| 152-comma
|658.65
| wa-152 comma
| w-152
| {{Monzo| 241 -152 }}
| 102.84
|-
|-
|18-comma
|wa-18 comma
|w-18
|
|
|{{monzo|29 -18}}
| 101-comma
|564.81
| wa-101 comma
| w-101
| {{Monzo| -160 101 }}
| 97.455
|-
|-
|23-comma
|wa-23 comma
|w-23
|
|
|{{monzo| -36 23}}
| 258-comma
|544.965
| wa-258 comma
| w-258
| {{Monzo| 409 -258 }}
| 95.61
|-
|-
|Pythagorean wolf fourth, 11-comma
| Limma, Pythagorean minor second
|Large wa 4th
| 5-comma
|Lw4
| sawa 2nd
|177147 / 131072
| sw2
|{{monzo| -17 11}}
| {{Monzo| 8 -5 }}
|521.505
| 90.225
|-
|-
|40-comma
|wa-40 comma
|w-40
|
|
|{{monzo| 63 -40 }}
| 301-comma
|478.2
| wa-301 comma
| w-301
| {{Monzo| -477 301 }}
| 88.455
|-
|-
|13b-comma
|wa-13b comma
|w-13b
|
|
|{{Monzo| 21 -13}}
| 248-comma
|474.585
| wa-248 comma
| w-248
| {{Monzo| -393 248 }}
| 84.840
|-
|-
|perfect fourth
|
|wa 4th
| 111-comma
|w4
| wa-111 comma
|[[4/3|4 / 3]]
| w-111
|{{monzo| 2 -1}}
| {{Monzo| 176 -111 }}
|498.045
| 82.995
|-
|-
|Pythagorean supermajor third, 16-comma
|wa-16 comma
|w-16
|
|
|{{Monzo|-25 16}}
| 195-comma
|431.28
| wa-195 comma
| w-195
| {{Monzo| -309 195 }}
| 81.225
|-
|-
|49-comma
|wa-49 comma
|w-49
|
|
|{{monzo| 78 -49 }}
| 142-comma
|404.205
| wa-142 comma
| w-142
| {{Monzo| -225 142 }}
| 77.610
|-
|-
|37-comma
|wa-37 comma
|w-37
|
|
|{{monzo| 59 -37 }}
| 217-comma
|427.665
| wa-217 comma
| w-217
| {{Monzo| 344 -217 }}
| 75.765
|-
|-
|45-comma
|wa-45 comma
|w-45
|
|
|{{monzo| -71 45 }}
| 89-comma
|387.975
| wa-89 comma
| w-89
| {{Monzo| -141 89 }}
| 73.995
|-
|-
|Pythagorean diminished fourth, 8-comma
|
|small wa 4th
| [[17-comma]]
|sw4
| wa-17 comma
|[[8192/6561|8192 / 6561]]
| w-17
|{{Monzo| 13 -8 }}
| {{Monzo| 27 -17 }}
|384.360
| 66.765
|-
|-
|33-comma
|wa-33 comma
|w-33
|
|
|{{monzo| -52 33 }}
| 289-comma
|364.515
| wa-289 comma
| w-289
| {{Monzo| -458 289 }}
| 64.995
|-
|-
|32-comma
|wa-32 comma
|w-32
|
|
|{{monzo| 51 -32 }}
| 70-comma
|337.44
| wa-70 comma
| w-70
| {{Monzo| 111 -70 }}
| 63.150
|-
|-
|Pythagorean augmented second, 9-comma
|
|large wa 2nd
| 183-comma
|Lw2
| wa-183 comma
|[[19683/16384]]
| w-183
|{{Monzo| -14 9 }}
| {{Monzo| -290 183 }}
|317.595
| 57.765
|-
|-
|50-comma
|wa-50 comma
|w-50
|
|
|{{monzo| -79 50 }}
| 176-comma
|404.205
| wa-176 comma
| w-176
| {{Monzo| 279 -176 }}
| 55.920
|-
|-
|Pythagorean minor third, 3-comma
|
|wa 3rd
| 229-comma
|w3
| wa-229 comma
|[[32/27|32 / 27]]
| w-229
|{{Monzo| 5 -3 }}
| {{Monzo| 363 -229 }}
|294.135
| 52.305
|-
|-
|26-comma
|wa-26 comma
|w-26
|
|
|{{monzo| -41 26 }}
| 77-comma
|250.83
| wa-77 comma
| w-77
| {{Monzo| -122 77 }}
| 50.535
|-
|-
|27-comma
|wa-27 comma
|w-27
|
|
|{{monzo| 43 -27 }}
| 282-comma
|247.215
| wa-282 comma
| w-282
| {{Monzo| 447 -282 }}
| 48.69
|-
|-
|39-comma
|wa-39 comma
|w-39
|
|
|{{monzo| 62 -39 }}
| [[29-comma]]
|223.75
| wa-29 comma
| w-29
| {{Monzo| 46 -29 }}
| 43.305
|-
|-
|55-comma
|wa-55 comma
|w-55
|
|
|{{monzo| -87 55 }}
| 277-comma
|207.525
| wa-277 comma
| w-277
| {{Monzo| -439 277 }}
| 41.535
|-
|-
|whole tone
|
|wa 2nd
| 224-comma
|w2
| wa-224 comma
|[[9/8|9 / 8]]
| w-224
|{{monzo| -3 2 }}
| {{Monzo| -355 224 }}
|203.91
| 37.920
|-
|-
|43-comma
|wa-43 comma
|w-43
|
|
|{{monzo| -68 43 }}
| 135-comma
|184.065
| wa-135 comma
| w-135
| {{Monzo| 214 -135 }}
| 36.075
|-
|-
|63-comma
|wa-63 comma
|w-63
|
|
|{{monzo| 100 -63 }}
| 171-comma
|176.835
| wa-171 comma
| w-171
| {{Monzo| -271 171 }}
| 34.305
|-
|-
|[[31-comma_temperaments|31-comma]]
|
|wa-31 comma
| 118-comma
|w-31
| wa-118 comma
|617673396283947 / 562949953421312
| w-118
|{{monzo| -49 31 }}
| {{Monzo| -187 118 }}
|160.605
| 30.690
|-
|-
|22-comma
|wa-22 comma
|w-22
|34359738368 / 31381059609
|{{monzo| 35 -22 }}
|156.99
|-
|19-comma
|wa-19 comma
|w-19
|1162261467 / 1073741824
|{{monzo| -30 19 }}
|137.145
|-
|apotome
|large wa unison
|Lw1
|[[2187/2048|2187 / 2048]]
|{{monzo| -11 7 }}
|113.685
|-
|46-comma
|wa-46 comma
|w-46
|
|
|{{monzo| 73 -46 }}
| 241-comma
|110.07
| wa-241 comma
| w-241
| {{Monzo| 382 -241 }}
| 28.845
|-
|-
|limma, Pythagorean minor second
| Pythagorean septimal comma
|small wa 2nd
| 65-comma
|sw2
| wa-65 comma
|[[256/243|256 / 243]]
| w-65
|{{monzo| 8 -5 }}
| {{Monzo| -103 65 }}
|90.225
| 27.075
|-
|-
|[[17-comma]]
| [[Pythagorean comma]]
|wa-17 comma
| 12-comma
|ssw3
| lalawa comma
|134217728 / 129140163
| LLw-2
|{{monzo| 27 -17 }}
| {{Monzo| -19 12 }}
|66.765
| 23.460
|-
|-
|70-comma
| Pythagorean countercomma, countercomp comma
|wa-70 comma
| [[41-comma]]
|w-70
| wa-41 comma
|
| w-41
|{{monzo| 111 -70 }}
| {{monzo| 65 -41 }}
|63.15
| 19.845
|-
|77-comma
|wa-77 comma
|w-77
|
|{{monzo| -122 77 }}
|50.535
|-
|-
|[[29-comma]]
| [[Stearnsmic clan #Garistearn|Garistearn comma]]
|wa-29 comma
| 94-comma
|w-29
| wa-94 comma
|70368744177664 / 68630377364883
| w-94
|{{monzo| 46 -29 }}
| {{Monzo| 149 -94 }}
|43.305
| 16.230
|-
|-
|65-comma
|wa-65 comma
|w-65
|
|
|{{monzo| -103 65}}
| 147-comma
|27.075
| wa-147 comma
|-
| w-147
|[[Pythagorean comma]]
| {{Monzo| 233 -147 }}
|lalawa comma
| 12.615
or wa comma
|LLw-2
|531441 / 524288
|{{monzo| -19 12 }}
|23.460
|-
|-
|[[41-comma]]
| Pythagorean integer-cent ET comma
|wa-41 comma
| 200-comma
|w-41
| wa-200 comma
|36893488147419103232 /
| w-200
36472996377170786403
| {{Monzo| 317 -200 }}
|{{monzo| 65 -41 }}
| 8.9998
|19.845
|-
|-
|94-comma
|wa-94 comma
|w-94
|
|
|{{monzo| 149 -94 }}
| 253-comma
|16.230
| wa-253 comma
| w-253
| {{Monzo| 401 -253 }}
| 5.3848
|-
|-
|200-comma, Pythagorean integer cents comma<!--the meaning of "integer cents" is ambiguous, because this comma is both near an integer in cents and involves integer cent fifths (702 cents), causing that term to be inconsistently used for one or the other-->
| [[Mercator's comma]]
|wa-200 comma
| 53-comma
|w-200
| wa-53 comma
|
| w-53
|{{monzo| 317 -200 }}
| {{Monzo| -84 53 }}
|8.9998
| 3.6150
|-
|-
|[[Mercator's comma]], 53-comma
|[[Qian's large comma]]
|wa-53 comma
| 359-comma
|w-53
| wa-359 comma
|19383245667680019896796723 /
| w-359
19342813113834066795298816
| {{Monzo| -569 359 }}
|{{monzo| -84 53 }}
| 1.8453
|3.6150
|-
|-
|359-comma
| [[Qian's small comma]]
|wa-359 comma
| 306-comma
|w-359
| wa-306 comma
|
| w-306
|{{monzo| -569 359 }}
| {{Monzo| 485 -306 }}
|1.8453
| 1.7697
|-
|-
|[[Sasktel comma]]
| [[Satanic comma]]
|wa-306 comma
| 665-comma
|w-306
| wa-665 comma
|
| w-665
|{{monzo| 485 -306 }}
| {{Monzo| -1054 665 }}
|1.7697
| 0.075575
|-
|-
|[[Satanic comma]]
|wa-665 comma
|w-665
|
|
|{{monzo| -1054 665 }}
| 15601-comma
|0.075575
| wa-15601 comma
| w-15601
| {{Monzo| 24727 -15601 }}
| 0.031499
|-
|-
|15601-comma
|wa-15601 comma
|w-15601
|
|
|{{monzo| 24727 -15601 }}
| 31867-comma
|0.031499
| wa-31867 comma
| w-31867
| {{Monzo| -50508 31867 }}
| 0.012577
|-
|-
|31867-comma
| [[Archangelic comma]]
|wa-31867 comma
| 190537-comma
|w-31867
| wa-190537 comma
|
| w-190537
|{{monzo| -50508 31867 }}
| {{Monzo| 301994 -190537 }}
|0.012577
| 0.000112
|}
|}
[[Category:3-limit]]
[[Category:Lists of commas]]
{{Todo| unify precision }}

Latest revision as of 23:07, 3 May 2025

A 3-limit comma is the amount which separates m fifths from n octaves. It is the simplest kind of comma a tuning may temper out. All commas not equal to the perfect fourth listed here generate interlaced pentatonic scales and fifths tempered by up to one step of n-edo when tempered out.

Name(s) Color Name Monzo Cents
whole tone 2-comma wa 2nd w2 [-3 2 203.91
193-comma wa-193 comma w-193 [306 -193 202.46
255-comma wa-255 comma w-255 [-404 255 198.525
104-comma wa-104 comma w-104 [165 -104 196.68
157-comma wa-157 comma w-157 [249 -157 193.065
149-comma wa-149 comma w-149 [-236 149 191.295
263-comma wa-263 comma w-263 [417 -263 185.835
43-comma wa-43 comma w-43 [-68 43 184.065
296-comma wa-296 comma w-296 [-469 296 178.68
63-comma wa-63 comma w-63 [100 -63 176.835
243-comma wa-243 comma w-243 [-385 243 175.605
190-comma wa-190 comma w-190 [-301 190 171.45
169-comma wa-169 comma w-169 [268 -169 169.605
137-comma wa-137 comma w-137 [-217 137 167.835
275-comma wa-275 comma w-275 [436 -275 162.375
31-comma wa-31 comma w-31 [-49 31 160.605
22-comma wa-22 comma w-22 [35 -22 156.99
75-comma wa-75 comma w-75 [119 -75 153.375
128-comma wa-128 comma w-128 [203 -128 149.76
181-comma wa-181 comma w-181 [287 -181 146.145
125-comma wa-125 comma w-125 [-198 125 144.375
234-comma wa-234 comma w-234 [371 -234 142.53
19-comma wa-19 comma w-19 [-30 19 137.145
272-comma wa-272 comma w-272 [-431 272 131.75
219-comma wa-219 comma w-219 [-347 219 128.145
166-comma wa-166 comma w-166 [-263 166 124.53
113-comma wa-113 comma w-113 [179 -113 120.915
Apotome 7-comma lawa unison Lw1 [-11 7 113.685
Pythagorean-septendecimal semitone 207-comma wa-207 comma w-207 [-328 207 104.685
46-comma wa-46 comma w-46 [73 -46 110.07
99-comma wa-99 comma w-99 [157 -99 106.455
152-comma wa-152 comma w-152 [241 -152 102.84
101-comma wa-101 comma w-101 [-160 101 97.455
258-comma wa-258 comma w-258 [409 -258 95.61
Limma, Pythagorean minor second 5-comma sawa 2nd sw2 [8 -5 90.225
301-comma wa-301 comma w-301 [-477 301 88.455
248-comma wa-248 comma w-248 [-393 248 84.840
111-comma wa-111 comma w-111 [176 -111 82.995
195-comma wa-195 comma w-195 [-309 195 81.225
142-comma wa-142 comma w-142 [-225 142 77.610
217-comma wa-217 comma w-217 [344 -217 75.765
89-comma wa-89 comma w-89 [-141 89 73.995
17-comma wa-17 comma w-17 [27 -17 66.765
289-comma wa-289 comma w-289 [-458 289 64.995
70-comma wa-70 comma w-70 [111 -70 63.150
183-comma wa-183 comma w-183 [-290 183 57.765
176-comma wa-176 comma w-176 [279 -176 55.920
229-comma wa-229 comma w-229 [363 -229 52.305
77-comma wa-77 comma w-77 [-122 77 50.535
282-comma wa-282 comma w-282 [447 -282 48.69
29-comma wa-29 comma w-29 [46 -29 43.305
277-comma wa-277 comma w-277 [-439 277 41.535
224-comma wa-224 comma w-224 [-355 224 37.920
135-comma wa-135 comma w-135 [214 -135 36.075
171-comma wa-171 comma w-171 [-271 171 34.305
118-comma wa-118 comma w-118 [-187 118 30.690
241-comma wa-241 comma w-241 [382 -241 28.845
Pythagorean septimal comma 65-comma wa-65 comma w-65 [-103 65 27.075
Pythagorean comma 12-comma lalawa comma LLw-2 [-19 12 23.460
Pythagorean countercomma, countercomp comma 41-comma wa-41 comma w-41 [65 -41 19.845
Garistearn comma 94-comma wa-94 comma w-94 [149 -94 16.230
147-comma wa-147 comma w-147 [233 -147 12.615
Pythagorean integer-cent ET comma 200-comma wa-200 comma w-200 [317 -200 8.9998
253-comma wa-253 comma w-253 [401 -253 5.3848
Mercator's comma 53-comma wa-53 comma w-53 [-84 53 3.6150
Qian's large comma 359-comma wa-359 comma w-359 [-569 359 1.8453
Qian's small comma 306-comma wa-306 comma w-306 [485 -306 1.7697
Satanic comma 665-comma wa-665 comma w-665 [-1054 665 0.075575
15601-comma wa-15601 comma w-15601 [24727 -15601 0.031499
31867-comma wa-31867 comma w-31867 [-50508 31867 0.012577
Archangelic comma 190537-comma wa-190537 comma w-190537 [301994 -190537 0.000112