57edo: Difference between revisions
→Intervals: don't add the 19edo interpretation of 9 if you don't intend to add the corresponding 3 |
→Notation: SZG notation |
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=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|57}} | {{Harmonics in equal|57}} | ||
{{Harmonics in equal|57|intervals=odd|columns=11|prec=2|start=12|collapsed=true|title=Approximation of odd harmonics in 57edo (continued)}} | |||
=== Subsets and supersets === | === Subsets and supersets === | ||
| Line 24: | Line 25: | ||
| 0 | | 0 | ||
| 0.00 | | 0.00 | ||
| [[1/1]] | |[[1/1]] | ||
| {{UDnote|step=0}} | | {{UDnote|step=0}} | ||
| {{UDnote|fifth=34|step=0}} | | {{UDnote|fifth=34|step=0}} | ||
| Line 42: | Line 43: | ||
| 3 | | 3 | ||
| 63.16 | | 63.16 | ||
| [[29/28]] | |[[29/28]] | ||
| {{UDnote|step=3}} | | {{UDnote|step=3}} | ||
| {{UDnote|fifth=34|step=3}} | | {{UDnote|fifth=34|step=3}} | ||
| Line 48: | Line 49: | ||
| 4 | | 4 | ||
| 84.21 | | 84.21 | ||
| [[20/19]], [[21/20]], [[22/21]] | |[[20/19]], [[21/20]], [[22/21]] | ||
| {{UDnote|step=4}} | | {{UDnote|step=4}} | ||
| {{UDnote|fifth=34|step=4}} | | {{UDnote|fifth=34|step=4}} | ||
| Line 54: | Line 55: | ||
| 5 | | 5 | ||
| 105.26 | | 105.26 | ||
| [[17/16]], [[33/31]] | |[[17/16]], [[33/31]] | ||
| {{UDnote|step=5}} | | {{UDnote|step=5}} | ||
| {{UDnote|fifth=34|step=5}} | | {{UDnote|fifth=34|step=5}} | ||
| Line 60: | Line 61: | ||
| 6 | | 6 | ||
| 126.32 | | 126.32 | ||
| [[14/13]] | |[[14/13]], ''[[16/15]]'' | ||
| {{UDnote|step=6}} | | {{UDnote|step=6}} | ||
| {{UDnote|fifth=34|step=6}} | | {{UDnote|fifth=34|step=6}} | ||
| Line 66: | Line 67: | ||
| 7 | | 7 | ||
| 147.37 | | 147.37 | ||
| [[12/11]] | |[[12/11]] | ||
| {{UDnote|step=7}} | | {{UDnote|step=7}} | ||
| {{UDnote|fifth=34|step=7}} | | {{UDnote|fifth=34|step=7}} | ||
| Line 72: | Line 73: | ||
| 8 | | 8 | ||
| 168.42 | | 168.42 | ||
| [[11/10]], [[32/29]] | |[[11/10]], [[32/29]] | ||
| {{UDnote|step=8}} | | {{UDnote|step=8}} | ||
| {{UDnote|fifth=34|step=8}} | | {{UDnote|fifth=34|step=8}} | ||
| Line 78: | Line 79: | ||
| 9 | | 9 | ||
| 189.47 | | 189.47 | ||
| [[19/17]], [[29/26]] | |[[19/17]], [[29/26]],<br>[[10/9]], ''[[9/8]]'' | ||
| {{UDnote|step=9}} | | {{UDnote|step=9}} | ||
| {{UDnote|fifth=34|step=9}} | | {{UDnote|fifth=34|step=9}} | ||
| Line 84: | Line 85: | ||
| 10 | | 10 | ||
| 210.53 | | 210.53 | ||
| [[26/23]] | |[[26/23]] | ||
| {{UDnote|step=10}} | | {{UDnote|step=10}} | ||
| {{UDnote|fifth=34|step=10}} | | {{UDnote|fifth=34|step=10}} | ||
| Line 90: | Line 91: | ||
| 11 | | 11 | ||
| 231.58 | | 231.58 | ||
| [[8/7]] | |[[8/7]] | ||
| {{UDnote|step=11}} | | {{UDnote|step=11}} | ||
| {{UDnote|fifth=34|step=11}} | | {{UDnote|fifth=34|step=11}} | ||
| Line 96: | Line 97: | ||
| 12 | | 12 | ||
| 252.63 | | 252.63 | ||
| [[22/19]] | |[[22/19]] | ||
| {{UDnote|step=12}} | | {{UDnote|step=12}} | ||
| {{UDnote|fifth=34|step=12}} | | {{UDnote|fifth=34|step=12}} | ||
| Line 102: | Line 103: | ||
| 13 | | 13 | ||
| 273.68 | | 273.68 | ||
| [[7/6]], [[34/29]] | |[[7/6]], [[34/29]] | ||
| {{UDnote|step=13}} | | {{UDnote|step=13}} | ||
| {{UDnote|fifth=34|step=13}} | | {{UDnote|fifth=34|step=13}} | ||
| Line 108: | Line 109: | ||
| 14 | | 14 | ||
| 294.74 | | 294.74 | ||
| [[19/16]] | |[[19/16]] | ||
| {{UDnote|step=14}} | | {{UDnote|step=14}} | ||
| {{UDnote|fifth=34|step=14}} | | {{UDnote|fifth=34|step=14}} | ||
| Line 114: | Line 115: | ||
| 15 | | 15 | ||
| 315.79 | | 315.79 | ||
| [[6/5]] | |[[6/5]] | ||
| {{UDnote|step=15}} | | {{UDnote|step=15}} | ||
| {{UDnote|fifth=34|step=15}} | | {{UDnote|fifth=34|step=15}} | ||
| Line 120: | Line 121: | ||
| 16 | | 16 | ||
| 336.84 | | 336.84 | ||
| [[17/14]], [[28/23]] | |[[17/14]], [[28/23]] | ||
| {{UDnote|step=16}} | | {{UDnote|step=16}} | ||
| {{UDnote|fifth=34|step=16}} | | {{UDnote|fifth=34|step=16}} | ||
| Line 126: | Line 127: | ||
| 17 | | 17 | ||
| 357.89 | | 357.89 | ||
| [[16/13]] | |[[16/13]] | ||
| {{UDnote|step=17}} | | {{UDnote|step=17}} | ||
| {{UDnote|fifth=34|step=17}} | | {{UDnote|fifth=34|step=17}} | ||
| Line 132: | Line 133: | ||
| 18 | | 18 | ||
| 378.95 | | 378.95 | ||
| [[5/4]] | |[[5/4]] | ||
| {{UDnote|step=18}} | | {{UDnote|step=18}} | ||
| {{UDnote|fifth=34|step=18}} | | {{UDnote|fifth=34|step=18}} | ||
| Line 138: | Line 139: | ||
| 19 | | 19 | ||
| 400.00 | | 400.00 | ||
| [[24/19]], [[29/23]] | |[[24/19]], [[29/23]] | ||
| {{UDnote|step=19}} | | {{UDnote|step=19}} | ||
| {{UDnote|fifth=34|step=19}} | | {{UDnote|fifth=34|step=19}} | ||
| Line 144: | Line 145: | ||
| 20 | | 20 | ||
| 421.05 | | 421.05 | ||
| | |[[14/11]], ''[[9/7]]'' | ||
| {{UDnote|step=20}} | | {{UDnote|step=20}} | ||
| {{UDnote|fifth=34|step=20}} | | {{UDnote|fifth=34|step=20}} | ||
| Line 150: | Line 151: | ||
| 21 | | 21 | ||
| 442.11 | | 442.11 | ||
| [[22/17]], [[31/24]] | |[[22/17]], [[31/24]] | ||
| {{UDnote|step=21}} | | {{UDnote|step=21}} | ||
| {{UDnote|fifth=34|step=21}} | | {{UDnote|fifth=34|step=21}} | ||
| Line 156: | Line 157: | ||
| 22 | | 22 | ||
| 463.16 | | 463.16 | ||
| [[17/13]], [[38/29]] | |[[17/13]], [[21/16]], [[38/29]] | ||
| {{UDnote|step=22}} | | {{UDnote|step=22}} | ||
| {{UDnote|fifth=34|step=22}} | | {{UDnote|fifth=34|step=22}} | ||
| Line 162: | Line 163: | ||
| 23 | | 23 | ||
| 484.21 | | 484.21 | ||
| [[33/25]] | |[[33/25]] | ||
| {{UDnote|step=23}} | | {{UDnote|step=23}} | ||
| {{UDnote|fifth=34|step=23}} | | {{UDnote|fifth=34|step=23}} | ||
| Line 168: | Line 169: | ||
| 24 | | 24 | ||
| 505.26 | | 505.26 | ||
| | |[[4/3]] | ||
| {{UDnote|step=24}} | | {{UDnote|step=24}} | ||
| {{UDnote|fifth=34|step=24}} | | {{UDnote|fifth=34|step=24}} | ||
| Line 174: | Line 175: | ||
| 25 | | 25 | ||
| 526.32 | | 526.32 | ||
| [[19/14]], [[23/17]] | |[[19/14]], [[42/31]], [[23/17]] | ||
| {{UDnote|step=25}} | | {{UDnote|step=25}} | ||
| {{UDnote|fifth=34|step=25}} | | {{UDnote|fifth=34|step=25}} | ||
| Line 180: | Line 181: | ||
| 26 | | 26 | ||
| 547.37 | | 547.37 | ||
| [[11/8]], [[26/19]] | |[[11/8]], [[26/19]] | ||
| {{UDnote|step=26}} | | {{UDnote|step=26}} | ||
| {{UDnote|fifth=34|step=26}} | | {{UDnote|fifth=34|step=26}} | ||
| Line 186: | Line 187: | ||
| 27 | | 27 | ||
| 568.42 | | 568.42 | ||
| [[25/18]], [[32/23]] | |[[25/18]], [[32/23]] | ||
| {{UDnote|step=27}} | | {{UDnote|step=27}} | ||
| {{UDnote|fifth=34|step=27}} | | {{UDnote|fifth=34|step=27}} | ||
| Line 192: | Line 193: | ||
| 28 | | 28 | ||
| 589.47 | | 589.47 | ||
| [[31/22]] | |[[31/22]] | ||
| {{UDnote|step=28}} | | {{UDnote|step=28}} | ||
| {{UDnote|fifth=34|step=28}} | | {{UDnote|fifth=34|step=28}} | ||
| Line 198: | Line 199: | ||
| 29 | | 29 | ||
| 610.53 | | 610.53 | ||
| | |[[44/31]] | ||
| {{UDnote|step=29}} | | {{UDnote|step=29}} | ||
| {{UDnote|fifth=34|step=29}} | | {{UDnote|fifth=34|step=29}} | ||
| Line 204: | Line 205: | ||
| 30 | | 30 | ||
| 631.58 | | 631.58 | ||
| [[23/16]] | |[[23/16]] | ||
| {{UDnote|step=30}} | | {{UDnote|step=30}} | ||
| {{UDnote|fifth=34|step=30}} | | {{UDnote|fifth=34|step=30}} | ||
| Line 210: | Line 211: | ||
| 31 | | 31 | ||
| 652.63 | | 652.63 | ||
| [[16/11]], [[19/13]] | |[[16/11]], [[19/13]] | ||
| {{UDnote|step=31}} | | {{UDnote|step=31}} | ||
| {{UDnote|fifth=34|step=31}} | | {{UDnote|fifth=34|step=31}} | ||
| Line 216: | Line 217: | ||
| 32 | | 32 | ||
| 673.68 | | 673.68 | ||
| [[28/19]], [[31/21]], [[34/23]] | |[[28/19]], [[31/21]], [[34/23]] | ||
| {{UDnote|step=32}} | | {{UDnote|step=32}} | ||
| {{UDnote|fifth=34|step=32}} | | {{UDnote|fifth=34|step=32}} | ||
| Line 222: | Line 223: | ||
| 33 | | 33 | ||
| 694.74 | | 694.74 | ||
| | |[[3/2]] | ||
| {{UDnote|step=33}} | | {{UDnote|step=33}} | ||
| {{UDnote|fifth=34|step=33}} | | {{UDnote|fifth=34|step=33}} | ||
| Line 228: | Line 229: | ||
| 34 | | 34 | ||
| 715.79 | | 715.79 | ||
| | |[[50/33]] | ||
| {{UDnote|step=34}} | | {{UDnote|step=34}} | ||
| {{UDnote|fifth=34|step=34}} | | {{UDnote|fifth=34|step=34}} | ||
| Line 234: | Line 235: | ||
| 35 | | 35 | ||
| 736.84 | | 736.84 | ||
| [[26/17]], [[29/19]] | |[[26/17]], [[32/21]], [[29/19]] | ||
| {{UDnote|step=35}} | | {{UDnote|step=35}} | ||
| {{UDnote|fifth=34|step=35}} | | {{UDnote|fifth=34|step=35}} | ||
| Line 240: | Line 241: | ||
| 36 | | 36 | ||
| 757.89 | | 757.89 | ||
| [[17/11]], [[31/20]] | |[[17/11]], [[31/20]] | ||
| {{UDnote|step=36}} | | {{UDnote|step=36}} | ||
| {{UDnote|fifth=34|step=36}} | | {{UDnote|fifth=34|step=36}} | ||
| Line 246: | Line 247: | ||
| 37 | | 37 | ||
| 778.95 | | 778.95 | ||
| [[11/7]], ''[[14/9]]'' | |[[11/7]], ''[[14/9]]'' | ||
| {{UDnote|step=37}} | | {{UDnote|step=37}} | ||
| {{UDnote|fifth=34|step=37}} | | {{UDnote|fifth=34|step=37}} | ||
| Line 252: | Line 253: | ||
| 38 | | 38 | ||
| 800.00 | | 800.00 | ||
| [[19/12]] | |[[19/12]] | ||
| {{UDnote|step=38}} | | {{UDnote|step=38}} | ||
| {{UDnote|fifth=34|step=38}} | | {{UDnote|fifth=34|step=38}} | ||
| Line 258: | Line 259: | ||
| 39 | | 39 | ||
| 821.05 | | 821.05 | ||
| [[8/5]] | |[[8/5]] | ||
| {{UDnote|step=39}} | | {{UDnote|step=39}} | ||
| {{UDnote|fifth=34|step=39}} | | {{UDnote|fifth=34|step=39}} | ||
| Line 264: | Line 265: | ||
| 40 | | 40 | ||
| 842.11 | | 842.11 | ||
| [[13/8]] | |[[13/8]] | ||
| {{UDnote|step=40}} | | {{UDnote|step=40}} | ||
| {{UDnote|fifth=34|step=40}} | | {{UDnote|fifth=34|step=40}} | ||
| Line 270: | Line 271: | ||
| 41 | | 41 | ||
| 863.16 | | 863.16 | ||
| [[23/14]], [[28/17]], [[33/20]] | |[[23/14]], [[28/17]], [[33/20]] | ||
| {{UDnote|step=41}} | | {{UDnote|step=41}} | ||
| {{UDnote|fifth=34|step=41}} | | {{UDnote|fifth=34|step=41}} | ||
| Line 276: | Line 277: | ||
| 42 | | 42 | ||
| 884.21 | | 884.21 | ||
| [[5/3]] | |[[5/3]] | ||
| {{UDnote|step=42}} | | {{UDnote|step=42}} | ||
| {{UDnote|fifth=34|step=42}} | | {{UDnote|fifth=34|step=42}} | ||
| Line 282: | Line 283: | ||
| 43 | | 43 | ||
| 905.26 | | 905.26 | ||
| [[32/19]] | |[[32/19]] | ||
| {{UDnote|step=43}} | | {{UDnote|step=43}} | ||
| {{UDnote|fifth=34|step=43}} | | {{UDnote|fifth=34|step=43}} | ||
| Line 288: | Line 289: | ||
| 44 | | 44 | ||
| 926.32 | | 926.32 | ||
| [[12/7]], [[29/17]] | |[[12/7]], [[29/17]] | ||
| {{UDnote|step=44}} | | {{UDnote|step=44}} | ||
| {{UDnote|fifth=34|step=44}} | | {{UDnote|fifth=34|step=44}} | ||
| Line 294: | Line 295: | ||
| 45 | | 45 | ||
| 947.37 | | 947.37 | ||
| [[19/11]] | |[[19/11]] | ||
| {{UDnote|step=45}} | | {{UDnote|step=45}} | ||
| {{UDnote|fifth=34|step=45}} | | {{UDnote|fifth=34|step=45}} | ||
| Line 300: | Line 301: | ||
| 46 | | 46 | ||
| 968.42 | | 968.42 | ||
| [[7/4]] | |[[7/4]] | ||
| {{UDnote|step=46}} | | {{UDnote|step=46}} | ||
| {{UDnote|fifth=34|step=46}} | | {{UDnote|fifth=34|step=46}} | ||
| Line 306: | Line 307: | ||
| 47 | | 47 | ||
| 989.47 | | 989.47 | ||
| [[23/13]] | |[[23/13]] | ||
| {{UDnote|step=47}} | | {{UDnote|step=47}} | ||
| {{UDnote|fifth=34|step=47}} | | {{UDnote|fifth=34|step=47}} | ||
| Line 312: | Line 313: | ||
| 48 | | 48 | ||
| 1010.53 | | 1010.53 | ||
| [[34/19]], [[52/29]] | |[[34/19]], [[52/29]],<br>[[9/5]], ''[[16/9]]'' | ||
| {{UDnote|step=48}} | | {{UDnote|step=48}} | ||
| {{UDnote|fifth=34|step=48}} | | {{UDnote|fifth=34|step=48}} | ||
| Line 318: | Line 319: | ||
| 49 | | 49 | ||
| 1031.58 | | 1031.58 | ||
| [[20/11]], [[29/16]] | |[[20/11]], [[29/16]] | ||
| {{UDnote|step=49}} | | {{UDnote|step=49}} | ||
| {{UDnote|fifth=34|step=49}} | | {{UDnote|fifth=34|step=49}} | ||
| Line 324: | Line 325: | ||
| 50 | | 50 | ||
| 1052.63 | | 1052.63 | ||
| [[11/6]] | |[[11/6]] | ||
| {{UDnote|step=50}} | | {{UDnote|step=50}} | ||
| {{UDnote|fifth=34|step=50}} | | {{UDnote|fifth=34|step=50}} | ||
| Line 330: | Line 331: | ||
| 51 | | 51 | ||
| 1073.68 | | 1073.68 | ||
| [[13/7]] | |[[13/7]], ''[[15/8]]'' | ||
| {{UDnote|step=51}} | | {{UDnote|step=51}} | ||
| {{UDnote|fifth=34|step=51}} | | {{UDnote|fifth=34|step=51}} | ||
| Line 336: | Line 337: | ||
| 52 | | 52 | ||
| 1094.74 | | 1094.74 | ||
| [[32/17]] | |[[32/17]] | ||
| {{UDnote|step=52}} | | {{UDnote|step=52}} | ||
| {{UDnote|fifth=34|step=52}} | | {{UDnote|fifth=34|step=52}} | ||
| Line 342: | Line 343: | ||
| 53 | | 53 | ||
| 1115.79 | | 1115.79 | ||
| [[19/10]], [[21/11]] | |[[19/10]], [[21/11]] | ||
| {{UDnote|step=53}} | | {{UDnote|step=53}} | ||
| {{UDnote|fifth=34|step=53}} | | {{UDnote|fifth=34|step=53}} | ||
| Line 348: | Line 349: | ||
| 54 | | 54 | ||
| 1136.84 | | 1136.84 | ||
| | |[[56/29]] | ||
| {{UDnote|step=54}} | | {{UDnote|step=54}} | ||
| {{UDnote|fifth=34|step=54}} | | {{UDnote|fifth=34|step=54}} | ||
| Line 366: | Line 367: | ||
| 57 | | 57 | ||
| 1200.00 | | 1200.00 | ||
| [[2/1]] | |[[2/1]] | ||
| {{UDnote|step=57}} | | {{UDnote|step=57}} | ||
| {{UDnote|fifth=34|step=57}} | | {{UDnote|fifth=34|step=57}} | ||
|} | |} | ||
<nowiki>*</nowiki> As a 2.5 | <nowiki>*</nowiki> As a 2.3.5.7.11.13.17.19.23.29.31-subgroup temperament, in ''italics'' if inconsistent | ||
== Notation == | == Notation == | ||
=== | === Stein–Zimmermann–Gould notation === | ||
57edo can be notated using [[Stein–Zimmermann–Gould notation]]: | |||
{{Sharpness-sharp3-szg}} | |||
Here, a sharp raises by three steps, and a flat lowers by three steps, so arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with double arrows. | |||
=== Kite's ups and downs notation === | |||
Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud. | Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud. | ||
{{Ups and downs sharpness}} | {{Ups and downs sharpness}} | ||
=== Sagittal notation === | === Sagittal notation === | ||
This notation uses the same sagittal sequence as | This notation uses the same sagittal sequence as edos [[50edo #Sagittal notation|50]], [[64edo #Sagittal notation|64]], and [[71edo #Sagittal notation|71b]], and is a superset of the notation for [[19edo #Sagittal notation|19edo]]. | ||
==== Evo flavor ==== | ==== Evo flavor ==== | ||
| Line 404: | Line 407: | ||
</imagemap> | </imagemap> | ||
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this | In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo. | ||
== Scales == | == Scales == | ||