1848edo: Difference between revisions
Cleanup; clarify the title row of the rank-2 temp table; -redundant categories |
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1848edo is a super strong 11-limit division, having the lowest 11-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any division until [[6079edo|6079]]. It tempers out the 11-limit commas [[9801/9800]], 151263/151250, [[1771561/1771470]] and 3294225/3294172. In the 5-limit it tempers out the the [[atom]], {{monzo| 161 -84 -12 }} and thus tunes the [[atomic]] temperament, for which it also provides the [[optimal patent val]] in the 11-limit. and also the minortone comma, {{monzo| -16 35 -17 }}. It also tempers out the 7-limit [[landscape comma]], 250047/250000, so it supports [[domain]] and [[akjayland]]. | 1848edo is a super strong 11-limit division, having the lowest 11-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any division until [[6079edo|6079]]. It tempers out the 11-limit commas [[9801/9800]], 151263/151250, [[1771561/1771470]] and 3294225/3294172. In the 5-limit it tempers out the the [[atom]], {{monzo| 161 -84 -12 }} and thus tunes the [[atomic]] temperament, for which it also provides the [[optimal patent val]] in the 11-limit. and also the minortone comma, {{monzo| -16 35 -17 }}. It also tempers out the 7-limit [[landscape comma]], 250047/250000, so it supports [[domain]] and [[akjayland]]. | ||
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. | ||
1848edo is unique in that it consistently tunes both [[81/80]] and [[64/63]] to an integer fraction of the octave, 1/56th and 1/44th 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. Remarkably, on the patent val 1848edo tempers [[96/95]] also to [[66edo|1\66]], though it is not consistent in the 19-limit. | 1848edo is unique in that it consistently tunes both [[81/80]] and [[64/63]] to an integer fraction of the octave, 1/56th and 1/44th 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. Remarkably, on the patent val 1848edo tempers [[96/95]] also to [[66edo|1\66]], though it is not consistent in the 19-limit. | ||
=== Prime harmonics === | |||
{{Harmonics in equal|1848|columns=11}} | |||
=== Subsets and supersets === | === Subsets and supersets === | ||
| Line 13: | Line 16: | ||
[[5544edo]], which divides the edostep into three, provides a good correction for 13- and the 17-limit. | [[5544edo]], which divides the edostep into three, provides a good correction for 13- and the 17-limit. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 30: | Line 30: | ||
| 2.3 | | 2.3 | ||
| {{monzo| -2929 1848 }} | | {{monzo| -2929 1848 }} | ||
| | | {{mapping| 1848 2929 }} | ||
| 0.002192 | | 0.002192 | ||
| 0.002192 | | 0.002192 | ||
| Line 37: | Line 37: | ||
| 2.3.5 | | 2.3.5 | ||
| {{monzo| -16 35 -17 }}, {{monzo| 129 -14 -46 }} | | {{monzo| -16 35 -17 }}, {{monzo| 129 -14 -46 }} | ||
| | | {{mapping| 1848 2929 4291 }} | ||
| -0.005705 | | -0.005705 | ||
| 0.011311 | | 0.011311 | ||
| Line 44: | Line 44: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 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 }} | ||
| -0.004748 | | -0.004748 | ||
| 0.009935 | | 0.009935 | ||
| Line 51: | Line 51: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 9801/9800, 151263/151250, 1771561/1771470, 67110351/67108864 | | 9801/9800, 151263/151250, 1771561/1771470, 67110351/67108864 | ||
| | | {{mapping| 1848 2929 4291 5188 6393 }} | ||
| -0.002686 | | -0.002686 | ||
| 0.009797 | | 0.009797 | ||
| Line 58: | Line 58: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 4225/4224, 6656/6655, 9801/9800, 151263/151250, 1771561/1771470 | | 4225/4224, 6656/6655, 9801/9800, 151263/151250, 1771561/1771470 | ||
| | | {{mapping| 1848 2929 4291 5188 6393 6838 }} | ||
| +0.009828 | | +0.009828 | ||
| 0.029378 | | 0.029378 | ||
| Line 67: | Line 67: | ||
{| 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 | ||
| Line 126: | Line 126: | ||
| [[Iridium]] | | [[Iridium]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
== Music == | |||
; [[Eliora]] | |||
* [https://www.youtube.com/watch?v=pDCBMziEPko ''Nocturne for Strings in Major Arcana and Minortone''] | |||
[[Category:Akjayland]] | [[Category:Akjayland]] | ||
[[Category:Atomic]] | [[Category:Atomic]] | ||
[[Category:Listen]] | [[Category:Listen]] | ||