1848edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
1848edo is | 1848edo is an extremely strong 11-limit division, having the lowest 11-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any division until [[6079edo|6079]]. | ||
In the 5-limit it tempers out the minortone comma, {{monzo| -16 35 -17 }} and | In the 5-limit it tempers out the minortone comma, {{monzo| -16 35 -17 }} and [[Kirnberger's atom]], {{monzo| 161 -84 -12 }} and thus tunes the [[atomic]] temperament, for which it also provides the [[optimal patent val]] in the 11-limit. In the 7-limit it tempers out the [[landscape comma]], 250047/250000, so it supports [[domain]] and [[akjayland]]. In the 11-limit it tempers out [[9801/9800]], 151263/151250, [[1771561/1771470]], 3294225/3294172, and the [[spoob]]. | ||
It is distinctly [[consistent]] through the [[15-odd-limit]] (though just barely), and tempers out the 13-limit commas [[4225/4224]] and [[6656/6655]]. Higher-limit prime harmonics represented by 1848edo with less than 10% error are 37, 61, and 83, of which 61 is accurate to 0.002 edosteps (and is inherited from [[231edo]]). The harmonics represented by less than 20% error are 19, 47, 59, 67, 89. | It is distinctly [[consistent]] through the [[15-odd-limit]] (though just barely), and tempers out the 13-limit commas [[4225/4224]] and [[6656/6655]]. Higher-limit prime harmonics represented by 1848edo with less than 10% error are 37, 61, and 83, of which 61 is accurate to 0.002 edosteps (and is inherited from [[231edo]]). The harmonics represented by less than 20% error are 19, 47, 59, 67, 89, and the 2.3.5.7.11.19 subgroup is the simplest and most natural choice for using 1848edo with higher limits. In the 2.3.5.7.11.19, it tempers out [[5776/5775]]. | ||
1848edo is unique in that it consistently tunes both [[81/80]] and [[64/63]] to an integer fraction of the octave, 1 | 1848edo is unique in that it consistently tunes both [[81/80]] and [[64/63]] to an integer fraction of the octave, [[56edo|1\56]] and [[44edo|1\44]] respectively. As a corollary, it supports [[barium]] and [[ruthenium]] temperaments, which have periods 56 and 44 respectively. While every edo that is a multiple of 616 shares the property of directly mapping 81/80 and 64/63 to fractions of the octave, 1848edo is unique due to its strength in simple harmonics and it actually shows how 81/80 and 64/63 are produced. In 2.3.5.7.11.19, it also tempers [[96/95]] to [[66edo|1\66]], thus making it a valuable system where important raising or lowering commas are represented by intervals that fit evenly within the octave. | ||
=== Prime harmonics === | === Prime harmonics === | ||
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Since 1848 factors into {{factorization|1848}}, 1848edo has subset edos {{EDOs| 2, 3, 4, 6, 7, 8, 11, 12, 14, 21, 22, 24, 28, 33, 42, 44, 56, 66, 77, 84, 88, 132, 154, 168, 231, 264, 308, 462, 616, 924 }}. | Since 1848 factors into {{factorization|1848}}, 1848edo has subset edos {{EDOs| 2, 3, 4, 6, 7, 8, 11, 12, 14, 21, 22, 24, 28, 33, 42, 44, 56, 66, 77, 84, 88, 132, 154, 168, 231, 264, 308, 462, 616, 924 }}. | ||
[[5544edo]], which divides the edostep into three, | [[3696edo]], which divides the edostep into two, and [[5544edo]], which divides the edostep into three, provide decent corrections for the 13- and the 17-limit. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
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|- | |- | ||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br />8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 41: | Line 41: | ||
| {{monzo| -16 35 -17 }}, {{monzo| 129 -14 -46 }} | | {{monzo| -16 35 -17 }}, {{monzo| 129 -14 -46 }} | ||
| {{mapping| 1848 2929 4291 }} | | {{mapping| 1848 2929 4291 }} | ||
| | | −0.005705 | ||
| 0.011311 | | 0.011311 | ||
| 1.74 | | 1.74 | ||
| Line 48: | Line 48: | ||
| 250047/250000, {{monzo| -4 17 1 -9 }}, {{monzo| 43 -1 -13 -4 }} | | 250047/250000, {{monzo| -4 17 1 -9 }}, {{monzo| 43 -1 -13 -4 }} | ||
| {{mapping| 1848 2929 4291 5188 }} | | {{mapping| 1848 2929 4291 5188 }} | ||
| | | −0.004748 | ||
| 0.009935 | | 0.009935 | ||
| 1.53 | | 1.53 | ||
| Line 55: | Line 55: | ||
| 9801/9800, 151263/151250, 1771561/1771470, 67110351/67108864 | | 9801/9800, 151263/151250, 1771561/1771470, 67110351/67108864 | ||
| {{mapping| 1848 2929 4291 5188 6393 }} | | {{mapping| 1848 2929 4291 5188 6393 }} | ||
| | | −0.002686 | ||
| 0.009797 | | 0.009797 | ||
| 1.51 | | 1.51 | ||
| Line 65: | Line 65: | ||
| 0.029378 | | 0.029378 | ||
| 4.52 | | 4.52 | ||
|- | |- style="border-top: double;" | ||
| 2.3.5.7.11.19 | |||
| 5776/5775, 9801/9800, 10241/10240, 250047/250000, 233744896/233735625 | |||
| {{mapping| 1848 2929 4291 5188 6393 7850 }} | |||
| | | +0.002094 | ||
| 0.013936 | |||
| 2.15 | |||
|} | |} | ||
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! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br /> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 138: | Line 138: | ||
| [[Iridium]] | | [[Iridium]] | ||
|} | |} | ||
<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 | ||
== Music == | == Music == | ||