2460edo: Difference between revisions
ArrowHead294 (talk | contribs) |
ArrowHead294 (talk | contribs) mNo edit summary |
||
| Line 3: | Line 3: | ||
== Theory == | == Theory == | ||
2460edo is [[consistency|distinctly 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 [[zeta edo|zeta peak | 2460edo is [[consistency|distinctly 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 [[zeta edo|zeta peak 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. | ||
| Line 18: | Line 18: | ||
In light of having a large amount of divisors and precise approximation of just intonation, 2460edo has been proposed as the basis for a unit, the [[mina]], which could be used in place of the cent. Moreover, a cent is exactly 2.05 [[mina]]s, and a mem, 1\205, is exactly 12 minas. | In light of having a large amount of divisors and precise approximation of just intonation, 2460edo has been proposed as the basis for a unit, the [[mina]], which could be used in place of the cent. Moreover, a cent is exactly 2.05 [[mina]]s, and a mem, 1\205, is exactly 12 minas. | ||
2460edo is also notable for being the smallest EDO that is [[Minimal consistent EDOs|purely consistent]] in the 15-odd-limit | 2460edo is also notable for being the smallest EDO that is a multiple of 12 to be [[Minimal consistent EDOs|purely consistent]] in the 15-odd-limit. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 35: | Line 35: | ||
| {{monzo| -3899 4320 }} | | {{monzo| -3899 4320 }} | ||
| {{mapping| 2460 3899 }} | | {{mapping| 2460 3899 }} | ||
| 0.001 | | +0.001 | ||
| 0.001 | | 0.001 | ||
| 0.24 | | 0.24 | ||
| Line 42: | Line 42: | ||
| {{monzo| 91 -12 -31 }}, {{monzo| -70 72 -19 }} | | {{monzo| 91 -12 -31 }}, {{monzo| -70 72 -19 }} | ||
| {{mapping| 2460 3899 5712 }} | | {{mapping| 2460 3899 5712 }} | ||
| | | −0.003 | ||
| 0.006 | | 0.006 | ||
| 1.29 | | 1.29 | ||
| Line 49: | Line 49: | ||
| 250047/250000, {{monzo| 3 -24 3 10 }}, {{monzo| -48 0 11 8 }} | | 250047/250000, {{monzo| 3 -24 3 10 }}, {{monzo| -48 0 11 8 }} | ||
| {{mapping| 2460 3899 5712 6096 }} | | {{mapping| 2460 3899 5712 6096 }} | ||
| 0.002 | | +0.002 | ||
| 0.010 | | 0.010 | ||
| 2.05 | | 2.05 | ||
| Line 56: | Line 56: | ||
| 9801/9800, 151263/151250, {{monzo| 24 -10 -5 0 1 }}, {{monzo| -3 -16 -1 6 4 }} | | 9801/9800, 151263/151250, {{monzo| 24 -10 -5 0 1 }}, {{monzo| -3 -16 -1 6 4 }} | ||
| {{mapping| 2460 3899 5712 6096 8510 }} | | {{mapping| 2460 3899 5712 6096 8510 }} | ||
| 0.007 | | +0.007 | ||
| 0.014 | | 0.014 | ||
| 2.86 | | 2.86 | ||
| Line 63: | Line 63: | ||
| 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 }} | | {{mapping| 2460 3899 5712 6096 8510 9103 }} | ||
| 0.008 | | +0.008 | ||
| 0.013 | | 0.013 | ||
| 2.63 | | 2.63 | ||
| Line 70: | Line 70: | ||
| 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 }} | | {{mapping| 2460 3899 5712 6096 8510 9103 10055 }} | ||
| 0.009 | | +0.009 | ||
| 0.013 | | 0.013 | ||
| 2.56 | | 2.56 | ||
| Line 141: | Line 141: | ||
| [[Minutes]] | | [[Minutes]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
[[Category:Mina]] | [[Category:Mina]] | ||