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58edo: Difference between revisions

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→Intervals: added 58hi intervals of 19 and 23 (23 incomplete)
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While the [[17/1|17th harmonic]] is a cent and a half flat, the harmonics below it are all a little sharp, giving it the sound of a sharp system. Since {{nowrap|58 {{=}} 2 × 29}}, 58edo shares the same excellent perfect fifth with [[29edo]]. It is the last edo to have exactly one [[5L 2s|diatonic]] perfect fifth and no [[5edo]] or [[7edo]] fifths.  
While the [[17/1|17th harmonic]] is a cent and a half flat, the harmonics below it are all a little sharp, giving it the sound of a sharp system. Since {{nowrap|58 {{=}} 2 × 29}}, 58edo shares the same excellent perfect fifth with [[29edo]]. It is the last edo to have exactly one [[5L 2s|diatonic]] perfect fifth and no [[5edo]] or [[7edo]] fifths.  
The [[19/1|19th]] and [[23/1|23rd]] harmonics are very flat, so it makes sense to use their second-best approximations in the 58hi val. This val is, in fact, one of the first to be [[diamond monotone]] in the 23-odd-limit, past the idiosyncratic 53e val.


=== Prime harmonics ===
=== Prime harmonics ===
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! Cents
! Cents
! Approximate ratios*
! Approximate ratios*
!Additional ratios of 19 and 23
using the 58hi val
! [[Ups and downs notation]]
! [[Ups and downs notation]]
|-
|-
Line 30: Line 34:
| 0.0
| 0.0
| [[1/1]]
| [[1/1]]
|
| {{UDnote|step=0}}
| {{UDnote|step=0}}
|-
|-
Line 35: Line 40:
| 20.7
| 20.7
| [[56/55]], [[64/63]], [[81/80]], [[91/90]], [[105/104]]
| [[56/55]], [[64/63]], [[81/80]], [[91/90]], [[105/104]]
|
| {{UDnote|step=1}}
| {{UDnote|step=1}}
|-
|-
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| 41.4
| 41.4
| [[36/35]], [[40/39]], [[45/44]], [[49/48]], [[50/49]], [[55/54]]
| [[36/35]], [[40/39]], [[45/44]], [[49/48]], [[50/49]], [[55/54]]
|
| {{UDnote|step=2}}
| {{UDnote|step=2}}
|-
|-
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| 62.1
| 62.1
| [[26/25]], [[27/26]], [[28/27]], [[33/32]]
| [[26/25]], [[27/26]], [[28/27]], [[33/32]]
|
| {{UDnote|step=3}}
| {{UDnote|step=3}}
|-
|-
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| 82.8
| 82.8
| [[21/20]], [[22/21]], ''[[25/24]]''
| [[21/20]], [[22/21]], ''[[25/24]]''
|20/19
| {{UDnote|step=4}}
| {{UDnote|step=4}}
|-
|-
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| 103.4
| 103.4
| [[16/15]], [[17/16]], [[18/17]]
| [[16/15]], [[17/16]], [[18/17]]
|19/18
| {{UDnote|step=5}}
| {{UDnote|step=5}}
|-
|-
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| 124.1
| 124.1
| [[14/13]], [[15/14]]
| [[14/13]], [[15/14]]
|
| {{UDnote|step=6}}
| {{UDnote|step=6}}
|-
|-
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| 144.8
| 144.8
| [[12/11]], [[13/12]]
| [[12/11]], [[13/12]]
|
| {{UDnote|step=7}}
| {{UDnote|step=7}}
|-
|-
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| 165.5
| 165.5
| [[11/10]]
| [[11/10]]
|
| {{UDnote|step=8}}
| {{UDnote|step=8}}
|-
|-
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| 186.2
| 186.2
| [[10/9]]
| [[10/9]]
|
| {{UDnote|step=9}}
| {{UDnote|step=9}}
|-
|-
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| 206.9
| 206.9
| [[9/8]], [[17/15]]
| [[9/8]], [[17/15]]
|''19/17''
| {{UDnote|step=10}}
| {{UDnote|step=10}}
|-
|-
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| 227.6
| 227.6
| [[8/7]]
| [[8/7]]
|
| {{UDnote|step=11}}
| {{UDnote|step=11}}
|-
|-
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| 248.3
| 248.3
| [[15/13]]
| [[15/13]]
|22/19
| {{UDnote|step=12}}
| {{UDnote|step=12}}
|-
|-
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| 269.0
| 269.0
| [[7/6]]
| [[7/6]]
|
| {{UDnote|step=13}}
| {{UDnote|step=13}}
|-
|-
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| 289.7
| 289.7
| [[13/11]], [[20/17]]
| [[13/11]], [[20/17]]
|
| {{UDnote|step=14}}
| {{UDnote|step=14}}
|-
|-
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| 310.3
| 310.3
| [[6/5]]
| [[6/5]]
|''19/16''
| {{UDnote|step=15}}
| {{UDnote|step=15}}
|-
|-
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| 331.0
| 331.0
| [[17/14]], [[40/33]]
| [[17/14]], [[40/33]]
|
| {{UDnote|step=16}}
| {{UDnote|step=16}}
|-
|-
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| 351.7
| 351.7
| [[11/9]], [[16/13]]
| [[11/9]], [[16/13]]
|
| {{UDnote|step=17}}
| {{UDnote|step=17}}
|-
|-
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| 372.4
| 372.4
| [[21/17]], [[26/21]]
| [[21/17]], [[26/21]]
|
| {{UDnote|step=18}}
| {{UDnote|step=18}}
|-
|-
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| 393.1
| 393.1
| [[5/4]]
| [[5/4]]
|''24/19''
| {{UDnote|step=19}}
| {{UDnote|step=19}}
|-
|-
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| 413.8
| 413.8
| [[14/11]]
| [[14/11]]
|19/15
| {{UDnote|step=20}}
| {{UDnote|step=20}}
|-
|-
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| 434.5
| 434.5
| [[9/7]]
| [[9/7]]
|
| {{UDnote|step=21}}
| {{UDnote|step=21}}
|-
|-
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| 455.2
| 455.2
| [[13/10]], [[17/13]], [[22/17]]
| [[13/10]], [[17/13]], [[22/17]]
|
| {{UDnote|step=22}}
| {{UDnote|step=22}}
|-
|-
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| 475.9
| 475.9
| [[21/16]]
| [[21/16]]
|
| {{UDnote|step=23}}
| {{UDnote|step=23}}
|-
|-
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| 496.6
| 496.6
| [[4/3]]
| [[4/3]]
|
| {{UDnote|step=24}}
| {{UDnote|step=24}}
|-
|-
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| 517.2
| 517.2
| [[27/20]]
| [[27/20]]
|
| {{UDnote|step=25}}
| {{UDnote|step=25}}
|-
|-
Line 160: Line 190:
| 537.9
| 537.9
| [[15/11]]
| [[15/11]]
|19/14, 26/19
| {{UDnote|step=26}}
| {{UDnote|step=26}}
|-
|-
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| 558.6
| 558.6
| [[11/8]], [[18/13]]
| [[11/8]], [[18/13]]
|
| {{UDnote|step=27}}
| {{UDnote|step=27}}
|-
|-
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| 579.3
| 579.3
| [[7/5]]
| [[7/5]]
|
| {{UDnote|step=28}}
| {{UDnote|step=28}}
|-
|-
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| 600.0
| 600.0
| [[17/12]], [[24/17]]
| [[17/12]], [[24/17]]
|
| {{UDnote|step=29}}
| {{UDnote|step=29}}
|-
|-
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| 620.7
| 620.7
| [[10/7]]
| [[10/7]]
|
| {{UDnote|step=30}}
| {{UDnote|step=30}}
|-
|-
Line 185: Line 220:
| 641.4
| 641.4
| [[13/9]], [[16/11]]
| [[13/9]], [[16/11]]
|''23/16''
| {{UDnote|step=31}}
| {{UDnote|step=31}}
|-
|-
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| 662.1
| 662.1
| [[22/15]]
| [[22/15]]
|19/13, 28/19
| {{UDnote|step=32}}
| {{UDnote|step=32}}
|-
|-
Line 195: Line 232:
| 682.8
| 682.8
| [[40/27]]
| [[40/27]]
|
| {{UDnote|step=33}}
| {{UDnote|step=33}}
|-
|-
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| 703.4
| 703.4
| [[3/2]]
| [[3/2]]
|
| {{UDnote|step=34}}
| {{UDnote|step=34}}
|-
|-
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| 724.1
| 724.1
| [[32/21]]
| [[32/21]]
|
| {{UDnote|step=35}}
| {{UDnote|step=35}}
|-
|-
Line 210: Line 250:
| 744.8
| 744.8
| [[17/11]], [[20/13]], [[26/17]]
| [[17/11]], [[20/13]], [[26/17]]
|
| {{UDnote|step=36}}
| {{UDnote|step=36}}
|-
|-
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| 765.5
| 765.5
| [[14/9]]
| [[14/9]]
|
| {{UDnote|step=37}}
| {{UDnote|step=37}}
|-
|-
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| 786.2
| 786.2
| [[11/7]]
| [[11/7]]
|30/19
| {{UDnote|step=38}}
| {{UDnote|step=38}}
|-
|-
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| 806.9
| 806.9
| [[8/5]]
| [[8/5]]
|''19/12''
| {{UDnote|step=39}}
| {{UDnote|step=39}}
|-
|-
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| 827.6
| 827.6
| [[21/13]], [[34/21]]
| [[21/13]], [[34/21]]
|
| {{UDnote|step=40}}
| {{UDnote|step=40}}
|-
|-
Line 235: Line 280:
| 848.3
| 848.3
| [[13/8]], [[18/11]]
| [[13/8]], [[18/11]]
|
| {{UDnote|step=41}}
| {{UDnote|step=41}}
|-
|-
Line 240: Line 286:
| 869.0
| 869.0
| [[28/17]], [[33/20]]
| [[28/17]], [[33/20]]
|
| {{UDnote|step=42}}
| {{UDnote|step=42}}
|-
|-
Line 245: Line 292:
| 889.7
| 889.7
| [[5/3]]
| [[5/3]]
|''32/19''
| {{UDnote|step=43}}
| {{UDnote|step=43}}
|-
|-
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| 910.3
| 910.3
| [[17/10]], [[22/13]]
| [[17/10]], [[22/13]]
|
| {{UDnote|step=44}}
| {{UDnote|step=44}}
|-
|-
Line 255: Line 304:
| 931.0
| 931.0
| [[12/7]]
| [[12/7]]
|
| {{UDnote|step=45}}
| {{UDnote|step=45}}
|-
|-
Line 260: Line 310:
| 951.7
| 951.7
| [[26/15]]
| [[26/15]]
|19/11
| {{UDnote|step=46}}
| {{UDnote|step=46}}
|-
|-
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| 972.4
| 972.4
| [[7/4]]
| [[7/4]]
|
| {{UDnote|step=47}}
| {{UDnote|step=47}}
|-
|-
Line 270: Line 322:
| 993.1
| 993.1
| [[16/9]], [[30/17]]
| [[16/9]], [[30/17]]
|''34/19''
| {{UDnote|step=48}}
| {{UDnote|step=48}}
|-
|-
Line 275: Line 328:
| 1013.8
| 1013.8
| [[9/5]]
| [[9/5]]
|
| {{UDnote|step=49}}
| {{UDnote|step=49}}
|-
|-
Line 280: Line 334:
| 1034.5
| 1034.5
| [[20/11]]
| [[20/11]]
|
| {{UDnote|step=50}}
| {{UDnote|step=50}}
|-
|-
Line 285: Line 340:
| 1055.2
| 1055.2
| [[11/6]], [[24/13]]
| [[11/6]], [[24/13]]
|
| {{UDnote|step=51}}
| {{UDnote|step=51}}
|-
|-
Line 290: Line 346:
| 1075.9
| 1075.9
| [[13/7]], [[28/15]]
| [[13/7]], [[28/15]]
|
| {{UDnote|step=52}}
| {{UDnote|step=52}}
|-
|-
Line 295: Line 352:
| 1096.6
| 1096.6
| [[15/8]], [[17/9]], [[32/17]]
| [[15/8]], [[17/9]], [[32/17]]
|36/19
| {{UDnote|step=53}}
| {{UDnote|step=53}}
|-
|-
Line 300: Line 358:
| 1117.2
| 1117.2
| [[21/11]], [[40/21]], ''[[48/25]]''
| [[21/11]], [[40/21]], ''[[48/25]]''
|19/10
| {{UDnote|step=54}}
| {{UDnote|step=54}}
|-
|-
Line 305: Line 364:
| 1137.9
| 1137.9
| [[25/13]], [[27/14]], [[52/27]], [[64/33]]
| [[25/13]], [[27/14]], [[52/27]], [[64/33]]
|
| {{UDnote|step=55}}
| {{UDnote|step=55}}
|-
|-
Line 310: Line 370:
| 1158.6
| 1158.6
| [[35/18]], [[39/20]], [[49/25]], [[88/45]], [[96/49]], [[108/55]]
| [[35/18]], [[39/20]], [[49/25]], [[88/45]], [[96/49]], [[108/55]]
|
| {{UDnote|step=56}}
| {{UDnote|step=56}}
|-
|-
Line 315: Line 376:
| 1179.3
| 1179.3
| [[55/28]], [[63/32]], [[160/81]], [[180/91]], [[208/105]]
| [[55/28]], [[63/32]], [[160/81]], [[180/91]], [[208/105]]
|
| {{UDnote|step=57}}
| {{UDnote|step=57}}
|-
|-
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| 1200.0
| 1200.0
| [[2/1]]
| [[2/1]]
|
| {{UDnote|step=58}}
| {{UDnote|step=58}}
|}
|}
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