1848edo: Difference between revisions

Line 5:

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 is an atomic system, tempering out the [[atom]], {{monzo| 161 -84 -12 }} and also the minortone comma, {{monzo| -16 35 -17 }}. It also tempers out the 7-limit [[landscape comma]], 250047/250000. It is distinctly consistent through the 15-odd-limit, and tempers out the 13-limit commas [[4225/4224]] and [[6656/6655]].

In the 7-limit, it supports [[domain]] and [[akjayland]].

1848 factors as 2<sup>3</sup> × 3 × 7 × 11. It is a superabundant number in the no-fives subgroup, that is, if only numbers not divisible by 5 are counted. Its divisors are {{EDOs| 1, 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 }}.

=== Fractional-octave temperaments ===

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 this property, 1848edo is unique due to its strength in simple harmonics and it actually shows how 81/80 and 64/63 are produced.

=== Prime harmonics ===

@@ Line 5: / Line 5: @@
 edo 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 is an atomic system, tempering out the [[atom]], {{monzo| 161 -84 -12 }} and also the minortone comma, {{monzo| -16 35 -17 }}. It also tempers out the 7-limit [[landscape comma]], 250047/250000. It is distinctly consistent through the 15-odd-limit, and tempers out the 13-limit commas [[4225/4224]] and [[6656/6655]].
 In the 7-limit, it supports [[domain]] and [[akjayland]].
 factors as 2<sup>3</sup> × 3 × 7 × 11. It is a superabundant number in the no-fives subgroup, that is, if only numbers not divisible by 5 are counted. Its divisors are {{EDOs| 1, 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 }}.
+=== Fractional-octave temperaments ===
+edo 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 this property, 1848edo is unique due to its strength in simple harmonics and it actually shows how 81/80 and 64/63 are produced.
 === Prime harmonics ===