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46edo: Difference between revisions

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{| class="wikitable center-all right-2 left-3 left-4 left-7"
{| class="wikitable center-all right-2 left-3 left-4 left-7"
|-
|-
! #
! #
! Cents
! Cents
! Approximate Ratios<ref name="interval ratios" group="note">Based on treating 46edo as a 2.3.5.7.11.13.17.23 subgroup, without ratios of 15 (except the superparticulars). 46edo has intervals involving the 15th harmonic poorly approximated, except for 15/8 and 16/15 themselves, because, while the 3rd and 5th harmonics are sharp and their deviations from just intonation add up, 7, 11, and 13 are all tuned flat, making the difference even larger. This prevents it from being [[consistent]] in the [[15-odd-limit]], as there is a discrepancy approximating [[15/13]] and [[26/15]]&mdash;9\46 is closer to 15/13 by a hair, but 10\46 represents the difference between 46edo's 15/8 and 13/8 and is more likely to appear in chords actually functioning as 15/13.</ref>
! Approximate Ratios<ref name="interval ratios" group="note">Based on treating 46edo as a 2.3.5.7.11.13.17.23 subgroup, without ratios of 15 (except the superparticulars). 46edo has intervals involving the 15th harmonic poorly approximated, except for 15/8 and 16/15 themselves, because, while the 3rd and 5th harmonics are sharp and their deviations from just intonation add up, 7, 11, and 13 are all tuned flat, making the difference even larger. This prevents it from being [[consistent]] in the [[15-odd-limit]], as there is a discrepancy approximating [[15/13]] and [[26/15]]&mdash;9\46 is closer to 15/13 by a hair, but 10\46 represents the difference between 46edo's 15/8 and 13/8 and is more likely to appear in chords actually functioning as 15/13.</ref>
Line 750: Line 750:
|+ style="font-size: 105%;" | 46-note circles by gen. with related temperaments organized by period
|+ style="font-size: 105%;" | 46-note circles by gen. with related temperaments organized by period
|-
|-
! [[Interval]] !! [[Closing error|Closing<br>Error]] !! [[Circle #Definitions|Consistency]] !! 1\1 !! 1\2
! [[Interval]] !! [[Closing error|Closing<br />Error]] !! [[Circle #Definitions|Consistency]] !! 1\1 !! 1\2
|-
|-
| [[68/65]] || 25.9% || Normal || [[Valentine]] || [[Semivalentine]]
| [[68/65]] || 25.9% || Normal || [[Valentine]] || [[Semivalentine]]
Line 762: Line 762:
|+ style="font-size: 105%;" | 23-note circles by gen. with related half-octave temperaments
|+ style="font-size: 105%;" | 23-note circles by gen. with related half-octave temperaments
|-
|-
! [[Interval]] !! [[Closing error|Closing<br>Error]] !! [[Circle #Definitions|Consistency]] !! Temperaments
! [[Interval]] !! [[Closing error|Closing<br />Error]] !! [[Circle #Definitions|Consistency]] !! Temperaments
|-
|-
| [[17/16]] || 53.5% || Weak || [[Diaschismic]]
| [[17/16]] || 53.5% || Weak || [[Diaschismic]]
Line 784: Line 784:
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br />8ve Stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning Error
|-
|-
Line 845: Line 845:
{| class="commatable wikitable center-1 center-2 right-4 center-5"
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
|-
! [[Harmonic limit|Prime<br>Limit]]
! [[Harmonic limit|Prime<br />Limit]]
! [[Ratio]]<ref group=note>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cent]]s
! [[Cent]]s
Line 1,019: Line 1,019:
{| class="wikitable center-1 center-2 center-3 center-4"
{| class="wikitable center-1 center-2 center-3 center-4"
|-
|-
! Periods <br>per 8ve
! Periods <br />per 8ve
! Generator
! Generator
! Cents
! Cents
Line 1,036: Line 1,036:
| 78.261
| 78.261
| [[Valentine]]
| [[Valentine]]
| 1L 14s (15-tone)<br>15L 1s (16-tone)<br>16L 15s (31-tone)
| 1L 14s (15-tone)<br />15L 1s (16-tone)<br />16L 15s (31-tone)
| 4:3 ~ [[Maximal evenness|quasi-equal]]<br>3:1<br>2:1 ~ QE
| 4:3 ~ [[Maximal evenness|quasi-equal]]<br />3:1<br />2:1 ~ QE
|-
|-
| 5\46
| 5\46
| 130.435
| 130.435
| [[Twothirdtonic]]
| [[Twothirdtonic]]
| [[1L 8s]] (9-tone)<br>[[9L 1s]] (10-tone)<br>9L 10s (19-tone)<br>9L 19s (28-tone)<br>9L 28s (37-tone)
| [[1L 8s]] (9-tone)<br />[[9L 1s]] (10-tone)<br />9L 10s (19-tone)<br />9L 19s (28-tone)<br />9L 28s (37-tone)
| 6:5 ~ QE<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE
| 6:5 ~ QE<br />5:1<br />4:1<br />3:1<br />2:1 ~ QE
|-
|-
| 7\46
| 7\46
| 182.609
| 182.609
| [[Minortone]] / [[mitonic]]
| [[Minortone]] / [[mitonic]]
| [[1L 5s]] (6-tone)<br>[[6L 1s]] (7-tone)<br>7L 6s (13-tone)<br>13L 7s (20-tone)<br>13L 20s (33-tone)
| [[1L 5s]] (6-tone)<br />[[6L 1s]] (7-tone)<br />7L 6s (13-tone)<br />13L 7s (20-tone)<br />13L 20s (33-tone)
| 11:7<br>7:4<br>4:3 ~ QE<br>3:1<br>2:1 ~ QE
| 11:7<br />7:4<br />4:3 ~ QE<br />3:1<br />2:1 ~ QE
|-
|-
| 9\46
| 9\46
| 234.783
| 234.783
| [[Rodan]]
| [[Rodan]]
| [[1L 4s]] (5-tone)<br>[[1L 5s]] (6-tone)<br>[[5L 6s]] (11-tone)<br>5L 11s (16-tone)<br>5L 16s (21-tone)<br>5L 21s (26-tone)<br>5L 26s (31-tone)<br>5L 31s (36-tone)<br>5L 36s (41-tone)
| [[1L 4s]] (5-tone)<br />[[1L 5s]] (6-tone)<br />[[5L 6s]] (11-tone)<br />5L 11s (16-tone)<br />5L 16s (21-tone)<br />5L 21s (26-tone)<br />5L 26s (31-tone)<br />5L 31s (36-tone)<br />5L 36s (41-tone)
| 10:9 ~QE<br>9:1<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE, Pathological
| 10:9 ~QE<br />9:1<br />8:1<br />7:1<br />6:1<br />5:1<br />4:1<br />3:1<br />2:1 ~ QE, Pathological
|-
|-
| 11\46
| 11\46
| 286.957
| 286.957
| [[Amity_family#Gamity|Gamity]]
| [[Amity_family#Gamity|Gamity]]
| [[4L 1s]] (5-tone)<br>[[4L 5s]] (9-tone)<br>4L 9s (13-tone)<br>4L 13s (17-tone)<br>4L 17s (21-tone)<br>21L 4s (25-tone)
| [[4L 1s]] (5-tone)<br />[[4L 5s]] (9-tone)<br />4L 9s (13-tone)<br />4L 13s (17-tone)<br />4L 17s (21-tone)<br />21L 4s (25-tone)
| 11:2<br>9:2<br>7:2<br>5:2<br>3:2 ~ QE, Golden<br>2:1 ~ QE
| 11:2<br />9:2<br />7:2<br />5:2<br />3:2 ~ QE, Golden<br />2:1 ~ QE
|-
|-
| 13\46
| 13\46
| 339.130
| 339.130
| [[Amity]] / [[hitchcock]]
| [[Amity]] / [[hitchcock]]
| [[4L 3s]] (7-tone)<br>[[7L 4s]] (11-tone)<br>7L 11s (18-tone)<br>7L 18s (25-tone)<br>7L 25s (32-tone)<br>7L 32s (39-tone)
| [[4L 3s]] (7-tone)<br />[[7L 4s]] (11-tone)<br />7L 11s (18-tone)<br />7L 18s (25-tone)<br />7L 25s (32-tone)<br />7L 32s (39-tone)
| 7:6 ~ QE<br>6:1<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE! Pathological
| 7:6 ~ QE<br />6:1<br />5:1<br />4:1<br />3:1<br />2:1 ~ QE! Pathological
|-
|-
| 15\46
| 15\46
| 391.304
| 391.304
| [[Magus]] / [[amigo]]
| [[Magus]] / [[amigo]]
| [[1L 2s]] (3-tone)<br>[[3L 1s]] (4-tone)<br>[[3L 4s]] (7-tone)<br>[[3L 7s]] (10-tone)<br>3L 10s (13-tone)<br>3L 13s (16-tone)<br>3L 16s (19-tone)<br>3L 19s (21-tone)<br>3L 21s (24-tone)<br>3L 24s (27-tone)<br>3L 27s (30-tone)<br>3L 30s (33-tone)<br>3L 33s (36-tone)<br>3L 36s (39-tone)<br>3L 39s (42-tone)
| [[1L 2s]] (3-tone)<br />[[3L 1s]] (4-tone)<br />[[3L 4s]] (7-tone)<br />[[3L 7s]] (10-tone)<br />3L 10s (13-tone)<br />3L 13s (16-tone)<br />3L 16s (19-tone)<br />3L 19s (21-tone)<br />3L 21s (24-tone)<br />3L 24s (27-tone)<br />3L 27s (30-tone)<br />3L 30s (33-tone)<br />3L 33s (36-tone)<br />3L 36s (39-tone)<br />3L 39s (42-tone)
| 16:15 ~ QE<br>15:1<br>14:1<br>13:1<br>12:1<br>11:1<br>10:1<br>9:1<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1 ~ Pathological<br>2:1 ~ QE, Pathological
| 16:15 ~ QE<br />15:1<br />14:1<br />13:1<br />12:1<br />11:1<br />10:1<br />9:1<br />8:1<br />7:1<br />6:1<br />5:1<br />4:1<br />3:1 ~ Pathological<br />2:1 ~ QE, Pathological
|-
|-
| 17\46
| 17\46
| 443.478
| 443.478
| [[Sensi]]
| [[Sensi]]
| [[3L 2s]] (5-tone)<br>[[3L 5s]] (8-tone)<br>[[8L 3s]] (11-tone)<br>8L 11s (19-tone)<br>19L 8s (27-tone)
| [[3L 2s]] (5-tone)<br />[[3L 5s]] (8-tone)<br />[[8L 3s]] (11-tone)<br />8L 11s (19-tone)<br />19L 8s (27-tone)
| 12:5<br>7:5<br>5:2<br>3:2 ~ QE, Golden<br>2:1
| 12:5<br />7:5<br />5:2<br />3:2 ~ QE, Golden<br />2:1
|-
|-
| 19\46
| 19\46
| 495.652
| 495.652
| [[Leapday]]
| [[Leapday]]
| [[2L 3s]] (5-tone)<br>[[5L 2s]] (7-tone)<br>[[5L 7s]] (12-tone)<br>12L 5s (17-tone)<br>17L 12s (29-tone)
| [[2L 3s]] (5-tone)<br />[[5L 2s]] (7-tone)<br />[[5L 7s]] (12-tone)<br />12L 5s (17-tone)<br />17L 12s (29-tone)
| 11:8<br>8:3<br>5:3 ~ Golden<br>3:2 ~ QE, Golden<br>2:1 ~ QE
| 11:8<br />8:3<br />5:3 ~ Golden<br />3:2 ~ QE, Golden<br />2:1 ~ QE
|-
|-
| 21\46
| 21\46
| 547.826
| 547.826
| [[Heinz]]
| [[Heinz]]
| [[2L 3s]] (5-tone)<br>[[2L 5s]] (7-tone)<br>[[2L 7s]] (9-tone)<br>[[2L 9s]] (11-tone)<br>11L 2s (13-tone)<br>11L 13s (24-tone)<br>11L 24s (35-tone)
| [[2L 3s]] (5-tone)<br />[[2L 5s]] (7-tone)<br />[[2L 7s]] (9-tone)<br />[[2L 9s]] (11-tone)<br />11L 2s (13-tone)<br />11L 13s (24-tone)<br />11L 24s (35-tone)
| 17:4<br>13:4<br>9:4<br>5:4 ~ QE<br>4:1<br>3:1<br>2:1 ~ QE
| 17:4<br />13:4<br />9:4<br />5:4 ~ QE<br />4:1<br />3:1<br />2:1 ~ QE
|-
|-
| rowspan="11" | 2
| rowspan="11" | 2
Line 1,103: Line 1,103:
| 52.174
| 52.174
| [[Shrutar]]
| [[Shrutar]]
| 2L 2s (4-tone)<br>[[2L 4s]] (6-tone)<br>[[2L 6s]] (8-tone)<br>[[2L 8s]] (10-tone)<br>[[2L 10s]] (12-tone)<br>2L 12s (14-tone)<br>2L 14s (16-tone)<br>2L 16s (18-tone)<br>2L 18s (20-tone)<br>2L 20s (22-tone)<br>22L 2s (24-tone)
| 2L 2s (4-tone)<br />[[2L 4s]] (6-tone)<br />[[2L 6s]] (8-tone)<br />[[2L 8s]] (10-tone)<br />[[2L 10s]] (12-tone)<br />2L 12s (14-tone)<br />2L 14s (16-tone)<br />2L 16s (18-tone)<br />2L 18s (20-tone)<br />2L 20s (22-tone)<br />22L 2s (24-tone)
| 21:2<br>19:2<br>17:2<br>15:2<br>13:2<br>11:2<br>9:2<br>7:2<br>5:2<br>3:2 ~ QE, Golden<br>2:1 ~ QE
| 21:2<br />19:2<br />17:2<br />15:2<br />13:2<br />11:2<br />9:2<br />7:2<br />5:2<br />3:2 ~ QE, Golden<br />2:1 ~ QE
|-
|-
| 3\46
| 3\46
| 78.261
| 78.261
| [[Semivalentine]]
| [[Semivalentine]]
| 2L 2s (4-tone)<br>[[2L 4s]] (6-tone)<br>[[2L 6s]] (8-tone)<br>[[2L 8s]] (10-tone)<br>[[2L 10s]] (12-tone)<br>2L 12s (14-tone)<br>14L 2s (16-tone)<br>16L 14s (30-tone)
| 2L 2s (4-tone)<br />[[2L 4s]] (6-tone)<br />[[2L 6s]] (8-tone)<br />[[2L 8s]] (10-tone)<br />[[2L 10s]] (12-tone)<br />2L 12s (14-tone)<br />14L 2s (16-tone)<br />16L 14s (30-tone)
| 20:3<br>17:3<br>14:3<br>11:3<br>8:3<br>5:3 ~ Golden<br>3:2 ~ QE, Golden<br>2:1 ~ QE
| 20:3<br />17:3<br />14:3<br />11:3<br />8:3<br />5:3 ~ Golden<br />3:2 ~ QE, Golden<br />2:1 ~ QE
|-
|-
| 4\46
| 4\46
| 104.348
| 104.348
| [[Srutal]] / [[diaschismic]]
| [[Srutal]] / [[diaschismic]]
| 2L 2s (4-tone)<br>[[2L 4s]] (6-tone)<br>[[2L 6s]] (8-tone)<br>[[2L 8s]] (10-tone)<br>[[10L 2s]] (12-tone)<br>12L 10s (22-tone)<br>12L 22s (34-tone)
| 2L 2s (4-tone)<br />[[2L 4s]] (6-tone)<br />[[2L 6s]] (8-tone)<br />[[2L 8s]] (10-tone)<br />[[10L 2s]] (12-tone)<br />12L 10s (22-tone)<br />12L 22s (34-tone)
| 19:4<br>15:4<br>11:4<br>7:4<br>4:3 ~ QE<br>3:1<br>2:1 ~ QE
| 19:4<br />15:4<br />11:4<br />7:4<br />4:3 ~ QE<br />3:1<br />2:1 ~ QE
|-
|-
| 5\46
| 5\46
| 130.435
| 130.435
|  
|  
| 2L 2s (4-tone)<br>[[2L 4s]] (6-tone)<br>[[2L 6s]] (8-tone)<br>[[8L 2s]] (10-tone)<br>8L 10s (18-tone)<br>18L 10s (28-tone)
| 2L 2s (4-tone)<br />[[2L 4s]] (6-tone)<br />[[2L 6s]] (8-tone)<br />[[8L 2s]] (10-tone)<br />8L 10s (18-tone)<br />18L 10s (28-tone)
| 18:5<br>13:5<br>8:5 ~ Golden<br>5:3 ~ Golden<br>3:2 ~ QE, Golden<br>2:1 ~ QE
| 18:5<br />13:5<br />8:5 ~ Golden<br />5:3 ~ Golden<br />3:2 ~ QE, Golden<br />2:1 ~ QE
|-
|-
| 6\46
| 6\46
| 156.522
| 156.522
| [[Bison]]
| [[Bison]]
| 2L 2s (4-tone)<br>[[2L 4s]] (6-tone)<br>[[6L 2s]] (8-tone)<br>8L 6s (14-tone)<br>8L 14s (22-tone)<br>8L 22s (30-tone)<br>8L 30s (38-tone
| 2L 2s (4-tone)<br />[[2L 4s]] (6-tone)<br />[[6L 2s]] (8-tone)<br />8L 6s (14-tone)<br />8L 14s (22-tone)<br />8L 22s (30-tone)<br />8L 30s (38-tone
| 17:6<br>11:6<br>6:5 ~ QE<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE, Pathological
| 17:6<br />11:6<br />6:5 ~ QE<br />5:1<br />4:1<br />3:1<br />2:1 ~ QE, Pathological
|-
|-
| 7\46
| 7\46
| 182.609
| 182.609
| [[Unidec]] / [[hendec]]
| [[Unidec]] / [[hendec]]
| 2L 2s (4-tone)<br>[[2L 4s]] (6-tone)<br>[[6L 2s]] (8-tone)<br>6L 8s (14-tone)<br>6L 14s (20-tone)<br>20L 6s (26-tone)
| 2L 2s (4-tone)<br />[[2L 4s]] (6-tone)<br />[[6L 2s]] (8-tone)<br />6L 8s (14-tone)<br />6L 14s (20-tone)<br />20L 6s (26-tone)
| 16:7<br>9:7<br>7:2<br>5:2<br>3:2 ~ QE, Golden<br>2:1 ~ QE
| 16:7<br />9:7<br />7:2<br />5:2<br />3:2 ~ QE, Golden<br />2:1 ~ QE
|-
|-
| 8\46
| 8\46
| 208.696
| 208.696
| [[Abigail]]
| [[Abigail]]
| 2L 2s (4-tone)<br>[[4L 2s]] (6-tone)<br>[[6L 2s]] (8-tone)<br>6L 8s (14-tone)<br>6L 14s (20-tone)<br>6L 20s (26-tone)<br>6L 26s (32-tone)<br>6L 32s (38-tone)<br>6L 38s (44-tone)
| 2L 2s (4-tone)<br />[[4L 2s]] (6-tone)<br />[[6L 2s]] (8-tone)<br />6L 8s (14-tone)<br />6L 14s (20-tone)<br />6L 20s (26-tone)<br />6L 26s (32-tone)<br />6L 32s (38-tone)<br />6L 38s (44-tone)
| 15:8<br>8:7 ~ QE<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1 ~ Pathological<br>2:1 ~ QE, Pathological
| 15:8<br />8:7 ~ QE<br />8:1<br />7:1<br />6:1<br />5:1<br />4:1<br />3:1 ~ Pathological<br />2:1 ~ QE, Pathological
|-
|-
| 9\46
| 9\46
| 234.783
| 234.783
| [[Echidnic]]
| [[Echidnic]]
| 2L 2s (4-tone)<br>[[4L 2s]] (6-tone)<br>[[6L 4s]] (10-tone)<br>10L 6s (16-tone)<br>10L 16s (26-tone)<br>10L 26s (36-tone)
| 2L 2s (4-tone)<br />[[4L 2s]] (6-tone)<br />[[6L 4s]] (10-tone)<br />10L 6s (16-tone)<br />10L 16s (26-tone)<br />10L 26s (36-tone)
| 14:9<br>9:5<br>5:4 ~ QE<br>4:1<br>3:1<br>2:1 ~ QE
| 14:9<br />9:5<br />5:4 ~ QE<br />4:1<br />3:1<br />2:1 ~ QE
|-
|-
| 10\46
| 10\46
| 260.87
| 260.87
| [[Bamity]]
| [[Bamity]]
| 2L 2s (4-tone)<br>[[4L 2s]] (6-tone)<br>[[4L 6s]] (10-tone)<br>4L 10s (14-tone)<br>14L 4s (18-tone)<br>14L 18s (32-tone)
| 2L 2s (4-tone)<br />[[4L 2s]] (6-tone)<br />[[4L 6s]] (10-tone)<br />4L 10s (14-tone)<br />14L 4s (18-tone)<br />14L 18s (32-tone)
| 13:10<br>10:3<br>7:3<br>4:3 ~ QE<br>3:1<br>2:1 ~ QE
| 13:10<br />10:3<br />7:3<br />4:3 ~ QE<br />3:1<br />2:1 ~ QE
|-
|-
| 11\46
| 11\46
| 286.957
| 286.957
| [[Vines]]
| [[Vines]]
| 2L 2s (4-tone)<br>[[4L 2s]] (6-tone)<br>[[4L 6s]] (10-tone)<br>4L 10s (14-tone)<br>4L 14s (18-tone)<br>4L 18s (22-tone)<br>4L 22s (26-tone)<br>4L 26s (30-tone)<br>4L 30s (34-tone)<br>4L 34s (38-tone)<br>4L 38s (42-tone)
| 2L 2s (4-tone)<br />[[4L 2s]] (6-tone)<br />[[4L 6s]] (10-tone)<br />4L 10s (14-tone)<br />4L 14s (18-tone)<br />4L 18s (22-tone)<br />4L 22s (26-tone)<br />4L 26s (30-tone)<br />4L 30s (34-tone)<br />4L 34s (38-tone)<br />4L 38s (42-tone)
| 12:11 ~ QE<br>11:1<br>10:1<br>9:1<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1 ~ Pathological<br>2:1 ~ QE, Pathological
| 12:11 ~ QE<br />11:1<br />10:1<br />9:1<br />8:1<br />7:1<br />6:1<br />5:1<br />4:1<br />3:1 ~ Pathological<br />2:1 ~ QE, Pathological
|-
|-
| 23
| 23
Retrieved from "https://en.xen.wiki/w/46edo"