1789edo: Difference between revisions
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=== Other === | === Other === | ||
1789edo can be used for the finite "French decimal" temperament | 1789edo can be used for the finite "French decimal" temperament—that is, where all the interval targets in just intonation are expressed as terminating decimals. For example, [[5/4]], [[25/16]], [[128/125]], [[32/25]], 625/512, etc. | ||
Since the 5/4 of 1789edo is on the 576th step, a highly divisible number, 1789edo can replicate a lot of [[ed5/4]] temperaments | Since the 5/4 of 1789edo is on the 576th step, a highly divisible number, 1789edo can replicate a lot of [[ed5/4]] temperaments—more exactly those which are divisors of 576, and that includes all from [[2ed5/4]] to [[9ed5/4]], skipping [[7ed5/4]]. One such scale which stands for [[4ed5/4]], is a tuning for the [[hemiluna]] temperament in the 1789bd val in the 13-limit. It is also worth noting that 1789bd val is better tuned than the patent val. | ||
1789edo has an essentially perfect [[9/8]], a very common interval. 1789edo supports the 2.9.5.11.13 subgroup temperament called ''commatose'' which uses the Pythagorean comma as a generator, which is excess of six 9/8s over the octave in this case. It is defined as a 460 & 1789 temperament. | 1789edo has an essentially perfect [[9/8]], a very common interval. 1789edo supports the 2.9.5.11.13 subgroup temperament called ''commatose'' which uses the Pythagorean comma as a generator, which is excess of six 9/8s over the octave in this case. It is defined as a 460 & 1789 temperament. | ||
Since 1789edo has a very precise 31/29, it supports tricesimoprimal miracloid | Since 1789edo has a very precise 31/29, it supports tricesimoprimal miracloid—a version of secor with 31/29 as the generator and a flat, meantone-esque fifth of about 692.23 cents. Using the maximal evenness method, we find a 52 & 1789 temperament. Best subgroup for it is 2.5.7.11.19.29.31, since both 52edo and 1789edo support it well, and the comma basis is 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688. | ||
On the patent val in the 7-limit, 1789edo supports 99 & 373 temperament called maviloid. In addition, it also tempers out [[2401/2400]]. | On the patent val in the 7-limit, 1789edo supports 99 & 373 temperament called maviloid. In addition, it also tempers out [[2401/2400]]. | ||
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== Table of selected intervals == | == Table of selected intervals == | ||
{| class="wikitable mw-collapsible mw-collapsed" | {| class="wikitable mw-collapsible mw-collapsed" | ||
|+ style=white-space:nowrap | Selected intervals in 1789edo | |+ style="font-size: 105%; white-space: nowrap;" | Selected intervals in 1789edo | ||
|- | |||
! Step | ! Step | ||
! Eliora's Naming System | ! Eliora's Naming System | ||
| Line 158: | Line 159: | ||
| 2/1 | | 2/1 | ||
|} | |} | ||
<nowiki>* | <nowiki />* Based on the 2.5.11.13.29.31 subgroup where applicable | ||
<sup>†</sup> 1046\1789 as 3/2 is the patent val, 1047\1789 as 3/2 is the 1789b val | |||
== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
|- | |- | ||
| 2.9 | | 2.9 | ||
| {{monzo| -5671 1789 }} | | {{monzo| -5671 1789 }} | ||
| {{mapping| 1789 5671 }} | | {{mapping| 1789 5671 }} | ||
| | | −0.00044 | ||
| 0.00044 | | 0.00044 | ||
| 0.06 | | 0.06 | ||
| Line 183: | Line 176: | ||
| {{monzo| -70 36 -19 }}, {{monzo| 129 -7 -46 }} | | {{monzo| -70 36 -19 }}, {{monzo| 129 -7 -46 }} | ||
| {{mapping| 1789 5671 4154 }} | | {{mapping| 1789 5671 4154 }} | ||
| | | −0.00710 | ||
| 0.00942 | | 0.00942 | ||
| 1.40 | | 1.40 | ||
| Line 193: | Line 186: | ||
| 0.04093 | | 0.04093 | ||
| 6.10 | | 6.10 | ||
|- | |- style="border-top: double;" | ||
| 2.5.11.13 | |||
| 6656/6655, {{monzo| 43 -18 5 -5 }}, {{monzo| -38 -32 10 21 }} | |||
| {{mapping| 1789 4154 6189 6620}} | |||
| | | −0.00490 | ||
| 0.01405 | |||
| 2.09 | |||
|- | |- | ||
| 2.5.11.13.29 | | 2.5.11.13.29 | ||
| 6656/6655, 371293/371200, {{monzo| -18 -6 -1 3 5 }}, {{monzo| 34 -20 5 0 -1 }} | | 6656/6655, 371293/371200, {{monzo| -18 -6 -1 3 5 }}, {{monzo| 34 -20 5 0 -1 }} | ||
| {{mapping| 1789 4154 6189 6620 8691 }} | | {{mapping| 1789 4154 6189 6620 8691 }} | ||
| | | −0.00591 | ||
| 0.01272 | | 0.01272 | ||
| 1.90 | | 1.90 | ||
| Line 211: | Line 204: | ||
| 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321 | | 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321 | ||
| {{mapping| 1789 4154 6189 6620 8691 8863 }} | | {{mapping| 1789 4154 6189 6620 8691 8863 }} | ||
| | | −0.00363 | ||
| 0.01268 | | 0.01268 | ||
| 1.89 | | 1.89 | ||
{{comma basis end}} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {{rank-2 begin}} | ||
|- | |- | ||
| 35\1789 | | 35\1789 | ||
| Line 279: | Line 266: | ||
| 6875/4914 | | 6875/4914 | ||
| [[Eternal revolutionary]] (1789bd) | | [[Eternal revolutionary]] (1789bd) | ||
{{rank-2 end}} | |||
{{orf}} | |||
== Music == | == Music == | ||