1848edo: Difference between revisions

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In the 5-limit it tempers out the minortone comma, {{monzo| -16 35 -17 }} and 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. 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~~/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]], ~~being~~ a ~~great 2.3.5.19 subgroup tuning~~.

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 ===

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| style="border-top: double;" | 0.002094

| style="border-top: double;" | 0.013936

| style="border-top: double;" | 2.15

| style="border-top: double;" | +2.15

|}

@@ Line 7: / Line 7: @@
 In the 5-limit it tempers out the minortone comma, {{monzo| -16 35 -17 }} and 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. 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]].
-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 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]], being a great 2.3.5.19 subgroup tuning.
+edo 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 ===
@@ Line 71: / Line 71: @@
 | style="border-top: double;" | 0.002094
 | style="border-top: double;" | 0.013936
-| style="border-top: double;" | 2.15
+| style="border-top: double;" | +2.15
 |}