2460edo: Difference between revisions
Cleanup; style; clarify the title row of the rank-2 temp table |
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2460edo is uniquely [[consistent]] through to the [[27-odd-limit]], which is not very remarkable in itself ([[388edo]] is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-odd-limit intervals (see below). It is also a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta peak and zeta peak integer edo]], and it has been used in [[Sagittal notation]] to define the ''olympian level'' of JI notation. | 2460edo is uniquely [[consistent]] through to the [[27-odd-limit]], which is not very remarkable in itself ([[388edo]] is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-odd-limit intervals (see below). It is also a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta peak and zeta peak integer edo]], and it has been used in [[Sagittal notation]] to define the ''olympian level'' of JI notation. | ||
As a micro (or nano) temperament, it is a landscape system in the 7-limit, tempering out [[250047/250000]], and in the 11-limit it tempers out [[9801/9800]]. Beyond that, [[10648/10647]] in the 13-limit, 12376/12375 in the 17-limit, 5929/5928 and 6860/6859 in the 19-limit and 8281/8280 in the 23-limit. | As a micro- (or nano-) temperament, it is a landscape system in the 7-limit, tempering out [[250047/250000]], and in the 11-limit it tempers out [[9801/9800]]. Beyond that, [[10648/10647]] in the 13-limit, 12376/12375 in the 17-limit, 5929/5928 and 6860/6859 in the 19-limit and 8281/8280 in the 23-limit. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 20: | Line 20: | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" |[[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
8ve | ! colspan="2" | Tuning Error | ||
! colspan="2" |Tuning | |||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3 | ||
|{{monzo|-3899 4320}} | | {{monzo| -3899 4320 }} | ||
| | | {{mapping| 2460 3899 }} | ||
|0.001 | | 0.001 | ||
|0.001 | | 0.001 | ||
|0.24 | | 0.24 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{monzo|91 -12 -31}}, {{monzo|-70 72 -19}} | | {{monzo| 91 -12 -31 }}, {{monzo| -70 72 -19 }} | ||
| | | {{mapping| 2460 3899 5712 }} | ||
| | | -0.003 | ||
|0.006 | | 0.006 | ||
|1.29 | | 1.29 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|250047/250000, | | 250047/250000, {{monzo| 3 -24 3 10 }}, {{monzo| -48 0 11 8 }} | ||
| | | {{mapping| 2460 3899 5712 6096 }} | ||
|0.002 | | 0.002 | ||
|0.010 | | 0.010 | ||
|2.05 | | 2.05 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|9801/9800, 250047/250000, {{monzo|24 -10 -5 0 1}}, {{monzo|-3 -16 -1 6 4}} | | 9801/9800, 250047/250000, {{monzo| 24 -10 -5 0 1 }}, {{monzo| -3 -16 -1 6 4 }} | ||
| | | {{mapping| 2460 3899 5712 6096 8510 }} | ||
|0.007 | | 0.007 | ||
|0.014 | | 0.014 | ||
|2.86 | | 2.86 | ||
|- | |- | ||
|2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
|9801/9800, 10648/10647, 105644/105625, 196625/196608, 1063348/1063125 | | 9801/9800, 10648/10647, 105644/105625, 196625/196608, 1063348/1063125 | ||
| | | {{mapping| 2460 3899 5712 6096 8510 9103 }} | ||
|0.008 | | 0.008 | ||
|0.013 | | 0.013 | ||
|2.63 | | 2.63 | ||
|- | |- | ||
|2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
|9801/9800, 10648/10647, 12376/12375, 31213/31212, 37180/37179, 221221/221184 | | 9801/9800, 10648/10647, 12376/12375, 31213/31212, 37180/37179, 221221/221184 | ||
| | | {{mapping| 2460 3899 5712 6096 8510 9103 10055 }} | ||
|0.009 | | 0.009 | ||
|0.013 | | 0.013 | ||
|2.56 | | 2.56 | ||
|} | |} | ||
2460edo has lower 23-limit relative error than any edo until [[8269edo|8269]]. Also it has a lower 23-limit [[TE logflat badness]] than any smaller edo and less than any until [[16808edo|16808]]. | * 2460edo has lower 23-limit relative error than any edo until [[8269edo|8269]]. Also it has a lower 23-limit [[TE logflat badness]] than any smaller edo and less than any until [[16808edo|16808]]. | ||
* In addition, it has the lowest relative error in the 19-limit, being only bettered by [[3395edo]]. | |||
In addition, it has the lowest relative error in the 19-limit, being only bettered by [[3395edo]]. | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio | ||
! Temperaments | ! Temperaments | ||
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| 1021\2460<br>(1\2460) | | 1021\2460<br>(1\2460) | ||
| 498.049<br>(0.488) | | 498.049<br>(0.488) | ||
| 4/3<br>({{monzo|215 -121 -10}}) | | 4/3<br>({{monzo| 215 -121 -10 }}) | ||
| [[Niobium]] | | [[Niobium]] | ||
|- | |- | ||
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| [[Minutes]] | | [[Minutes]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
[[Category:Mina]] | [[Category:Mina]] | ||