21edo: Difference between revisions
m →Theory: grammar |
→Intervals: link ratios + some small changes |
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! Steps | ! Steps | ||
! Cents | ! Cents | ||
! | ! Approximate atios* | ||
! Additional ratios<br>of 17, 19, and 27** | ! Additional ratios<br>of 17, 19, and 27** | ||
!Other ratios† | !Other ratios† | ||
| Line 37: | Line 37: | ||
| 0 | | 0 | ||
| 0.00 | | 0.00 | ||
| colspan="3" | 1/1 | | colspan="3" | [[1/1]] | ||
|- | |- | ||
| 1 | | 1 | ||
| 57.14 | | 57.14 | ||
| 29/28, 30/29, 31/30, 32/31, 33/32 | | [[29/28]], [[30/29]], [[31/30]], [[32/31]], [[33/32]] | ||
| 28/27, 34/33, 39/38 | | [[28/27]], [[34/33]], [[39/38]] | ||
|21/20 | | [[21/20]] | ||
|- | |- | ||
| 2 | | 2 | ||
| 114.29 | | 114.29 | ||
| 15/14, 16/15, 31/29, 33/31, 46/43 | | [[15/14]], [[16/15]], [[31/29]], [[33/31]], [[46/43]] | ||
| 17/16, 29/27 | | [[17/16]], [[29/27]] | ||
|14/13 | | [[14/13]] | ||
|- | |- | ||
| 3 | | 3 | ||
| 171.43 | | 171.43 | ||
| 11/10, 32/29, 31/28, 43/39 | | [[11/10]], [[32/29]], [[31/28]], [[43/39]] | ||
| 10/9, 19/17, 34/31 | | [[10/9]], [[19/17]], [[34/31]] | ||
|''9/8, 13/12'' | | ''[[9/8]], [[13/12]]'' | ||
|- | |- | ||
| 4 | | 4 | ||
| 228.57 | | 228.57 | ||
| 8/7, 33/29 | | [[8/7]], [[33/29]] | ||
| 17/15, 31/27, 38/33, 43/38 | | [[17/15]], [[31/27]], [[38/33]], [[43/38]] | ||
| | | | ||
|- | |- | ||
| 5 | | 5 | ||
| 285.71 | | 285.71 | ||
| 13/11, 33/28, 46/39 | | [[13/11]], [[33/28]], [[46/39]] | ||
| 19/16, 27/23, 32/27, 34/29 | | [[19/16]], [[27/23]], [[32/27]], [[34/29]] | ||
|''6/5,'' 7/6 | |''[[6/5]],'' [[7/6]] | ||
|- | |- | ||
| 6 | | 6 | ||
| 342.86 | | 342.86 | ||
| 28/23, 39/32 | | [[28/23]], [[39/32]] | ||
| 11/9, 17/14, 23/19, 38/31 | | [[11/9]], [[17/14]], [[23/19]], [[38/31]] | ||
|16/13 | | [[16/13]] | ||
|- | |- | ||
| 7 | | 7 | ||
| 400.00 | | 400.00 | ||
| 29/23, 39/31 | | [[29/23]], [[39/31]] | ||
| 19/15, 34/27, 43/34, 54/43 | | [[19/15]], [[34/27]], [[43/34]], [[54/43]] | ||
|5/4, ''9/7'' | | [[5/4]], ''[[9/7]]'' | ||
|- | |- | ||
| 8 | | 8 | ||
| 457.14 | | 457.14 | ||
| 13/10, 30/23, 39/30, 43/33, 56/43 | | [[13/10]], [[30/23]], [[39/30]], [[43/33]], [[56/43]] | ||
| 38/29 | | [[38/29]] | ||
|21/16 | | [[21/16]] | ||
|- | |- | ||
| 9 | | 9 | ||
| 514.29 | | 514.29 | ||
| 31/23, 39/29, 43/32, 58/43 | | [[31/23]], [[39/29]], [[43/32]], [[58/43]] | ||
| 19/14, 23/17 | | [[19/14]], [[23/17]] | ||
|4/3 | | [[4/3]] | ||
|- | |- | ||
| 10 | | 10 | ||
| 571.43 | | 571.43 | ||
| 32/23, 39/28, 46/33, 43/31, 60/43 | | [[32/23]], [[39/28]], [[46/33]], [[43/31]], [[60/43]] | ||
| 18/13, 38/27 | | [[18/13]], [[38/27]] | ||
|7/5 | | [[7/5]] | ||
|- | |- | ||
| 11 | | 11 | ||
| 628.57 | | 628.57 | ||
| 23/16, 56/39, 33/23, 43/30, 62/43 | | [[23/16]], [[56/39]], [[33/23]], [[43/30]], [[62/43]] | ||
| 13/9, 27/19 | | [[13/9]], [[27/19]] | ||
|10/7 | | [[10/7]] | ||
|- | |- | ||
| 12 | | 12 | ||
| 685.71 | | 685.71 | ||
| 46/31, 58/39, 43/29, 64/43 | | [[46/31]], [[58/39]], [[43/29]], [[64/43]] | ||
| 28/19, 34/23 | | [[28/19]], [[34/23]] | ||
|3/2 | | [[3/2]] | ||
|- | |- | ||
| 13 | | 13 | ||
| 742.86 | | 742.86 | ||
| 20/13, 23/15, 60/39, 43/28, 66/43 | | [[20/13]], [[23/15]], [[60/39]], [[43/28]], [[66/43]] | ||
| 29/19 | | [[29/19]] | ||
|32/21 | | [[32/21]] | ||
|- | |- | ||
| 14 | | 14 | ||
| 800.00 | | 800.00 | ||
| 46/29, 62/39 | | [[46/29]], [[62/39]] | ||
| 30/19, 27/17, 43/27, 68/43 | | [[30/19]], [[27/17]], [[43/27]], [[68/43]] | ||
|8/5, ''14/9'' | | [[8/5]], ''[[14/9]]'' | ||
|- | |- | ||
| 15 | | 15 | ||
| 857.14 | | 857.14 | ||
| 23/14, 64/39 | | [[23/14]], [[64/39]] | ||
| 18/11, 28/17, 38/23, 31/19 | | [[18/11]], [[28/17]], [[38/23]], [[31/19]] | ||
|13/8 | | [[13/8]] | ||
|- | |- | ||
| 16 | | 16 | ||
| 914.29 | | 914.29 | ||
| 22/13, 56/33, 39/23 | | [[22/13]], [[56/33]], [[39/23]] | ||
| 32/19, 27/16, 46/27, 29/17 | | [[32/19]], [[27/16]], [[46/27]], [[29/17]] | ||
|''5/3'', 12/7 | |''[[5/3]]'', [[12/7]] | ||
|- | |- | ||
| 17 | | 17 | ||
| 971.43 | | 971.43 | ||
| 7/4, 58/33 | | [[7/4]], [[58/33]] | ||
| 30/17, 54/31, 33/19, 76/43 | | [[30/17]], [[54/31]], [[33/19]], [[76/43]] | ||
| | | | ||
|- | |- | ||
| 18 | | 18 | ||
| 1028.57 | | 1028.57 | ||
| 20/11, 29/16, 56/31, 78/43 | | [[20/11]], [[29/16]], [[56/31]], [[78/43]] | ||
| 9/5, 34/19, 31/17 | | [[9/5]], [[34/19]], [[31/17]] | ||
|''16/9, 24/13'' | |''[[16/9]], [[24/13]]'' | ||
|- | |- | ||
| 19 | | 19 | ||
| 1085.71 | | 1085.71 | ||
| 15/8, 28/15, 58/31, 62/33, 43/23 | | [[15/8]], [[28/15]], [[58/31]], [[62/33]], [[43/23]] | ||
| 32/17, 54/29 | | [[32/17]], [[54/29]] | ||
|13/7 | | [[13/7]] | ||
|- | |- | ||
| 20 | | 20 | ||
| 1142.86 | | 1142.86 | ||
| 29/15, 56/29, 31/16, 60/31, 64/33 | | [[29/15]], [[56/29]], [[31/16]], [[60/31]], [[64/33]] | ||
| 27/14, 33/17, 76/39 | | [[27/14]], [[33/17]], [[76/39]] | ||
|40/21 | | [[40/21]] | ||
|- | |- | ||
| 21 | | 21 | ||
| 1200.00 | | 1200.00 | ||
| colspan="3" | 2/1 | | colspan="3" | [[2/1]] | ||
|} | |} | ||
<nowiki/>* | <nowiki/>*43-odd-limit ratios of the 2.15.7.33.39.23.29.31.43 subgroup | ||
<nowiki>** | <nowiki/>**Odd 27 by direct approximation | ||
†All by patent val | †All by patent val | ||
Note: In the second column, the ratios 9/5, 11/9, 13/9, and their octave complements are all included here, being 27/15, 33/27, and 39/27 respectively. These ratios are mapped inconsistently to their second-best approximations in the patent val, so despite 9/8 and 10/9 being listed in the same row, 21edo is ''not'' a [[meantone]] system. | Note: In the second column, the ratios 9/5, 11/9, 13/9, and their octave complements are all included here, being 27/15, 33/27, and 39/27 respectively. These ratios are mapped inconsistently to their second-best approximations in the patent val, so despite 9/8 and 10/9 being listed in the same row in the table above, 21edo is ''not'' a [[meantone]] system. | ||
== Notation == | == Notation == | ||