1789edo: Difference between revisions
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The '''1789 equal divisions of the octave''' ('''1789edo'''), or the '''1789-tone equal temperament''' ('''1789tet'''), '''1789 equal temperament''' ('''1789et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1789 [[equal]] parts of about 671 [[cent | The '''1789 equal divisions of the octave''' ('''1789edo'''), or the '''1789-tone equal temperament''' ('''1789tet'''), '''1789 equal temperament''' ('''1789et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1789 [[equal]] parts of about 0.671 [[cent]]s each. It is the 278th [[prime edo]]. | ||
== Theory == | == Theory == | ||
{{Harmonics in equal|1789|columns = 10}} | {{Harmonics in equal|1789|columns = 10}} | ||
1789edo can be adapted for use with the 2.5.11.13.29.31.47.59.61 subgroup. | 1789edo can be adapted for use with the 2.5.11.13.29.31.47.59.61 subgroup. Perhaps the most notable fact about 1789edo, is the fact that it tempers out the jacobin comma ([[6656/6655]]), which is quite appropriate for edo's number. Although there are temperaments which are better suited for tempering this comma, 1789edo is unique in that it's number is the hallmark year of the French Revolution, thus making the temperance of the Jacobin comma on topic. | ||
=== Jacobin temperament === | === Jacobin temperament === | ||
| Line 16: | Line 14: | ||
Since 1789edo contains the 2.5 subgroup, it can be used for the finite "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. This rings particularly true for the French attempts to decimalize a lot more things than we are used to today. This property of 1789edo is amplified by poor approximation of 3 and 7, allowing for cognitive separation of the intervals (or whatever is left of it at such small step size). | Since 1789edo contains the 2.5 subgroup, it can be used for the finite "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. This rings particularly true for the French attempts to decimalize a lot more things than we are used to today. This property of 1789edo is amplified by poor approximation of 3 and 7, allowing for cognitive separation of the intervals (or whatever is left of it at such small step size). | ||
Using the maximal evenness method of finding rank two temperaments, we get a 1524 & 1789 temperament. | Using the maximal evenness method of finding rank two temperaments, we get a 1524 & 1789 temperament.<sup>[which mapping?]</sup> | ||
=== Other === | === Other === | ||
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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]]. | ||
== Table of selected intervals == | == Table of selected intervals == | ||
{| class="wikitable mw-collapsible mw-collapsed" | {| class="wikitable mw-collapsible mw-collapsed" | ||
| Line 79: | Line 78: | ||
| 754/649 | | 754/649 | ||
|- | |- | ||
|523 | | 523 | ||
|Breedsmic neutral third | | Breedsmic neutral third | ||
|49/40, 60/49 | | 49/40, 60/49 | ||
|- | |- | ||
| 576 | | 576 | ||
| Line 91: | Line 90: | ||
| [[13/10]] | | [[13/10]] | ||
|- | |- | ||
|777 | | 777 | ||
|Maviloid generator | | Maviloid generator | ||
| | | | ||
|- | |- | ||
| Line 111: | Line 110: | ||
| [[3/2]]† | | [[3/2]]† | ||
|- | |- | ||
|1444 | | 1444 | ||
|Harmonic seventh | | Harmonic seventh | ||
|[[7/4]] | | [[7/4]] | ||
|- | |- | ||
| 1535 | | 1535 | ||
| Line 134: | Line 133: | ||
† 1046\1789 as 3/2 is the patent val, 1047\1789 as 3/2 is the 1789b val | † 1046\1789 as 3/2 is the patent val, 1047\1789 as 3/2 is the 1789b val | ||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 150: | Line 148: | ||
| {{monzo| -5671 1789 }} | | {{monzo| -5671 1789 }} | ||
| [{{val| 1789 5671 }}] | | [{{val| 1789 5671 }}] | ||
| -0. | | -0.00044 | ||
| 0. | | 0.00044 | ||
| 0.066 | | 0.066 | ||
|- | |- | ||
|2. | | 2.9.5 | ||
|{{monzo| | | {{monzo| -70 36 -19 }}, {{monzo| 129 -7 -46 }} | ||
|[{{val|1789 | | [{{val| 1789 5671 4154 }}] | ||
| | | -0.00710 | ||
| | | 0.00942 | ||
| | | 1.40 | ||
|- | |- | ||
|2. | | 2.9.5.7 | ||
|{{monzo| | | 420175/419904, {{monzo| 34 2 -21 3 }}, {{monzo| -55 15 2 1 }} | ||
|[{{val|1789 | | [{{val| 1789 5671 4154 5022 }}] | ||
| +0.01606 | |||
| 0.04093 | |||
| 6.10 | |||
|0. | |||
|0. | |||
| | |||
|- | |- | ||
| 2.5.11.13.29.31 | | 2.5.11.13.29.31 | ||
| 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321 | | 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321 | ||
| [{{val|1789 4154 6189 6620 8691 8863}}] | | [{{val| 1789 4154 6189 6620 8691 8863 }}] | ||
| | | -0.00363 | ||
| 0. | | 0.01268 | ||
| | | 1.89 | ||
|} | |} | ||
== Rank | == Rank-2 temperaments by generator == | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
!Generator | ! Generator<br>(reduced) | ||
(reduced) | ! Cents<br>(reduced) | ||
!Cents | ! Associated<br>ratio | ||
(reduced) | ! Temperaments | ||
! Associated | |||
ratio | |||
!Temperaments | |||
|- | |- | ||
| | | 576\1789 | ||
| | | 386.36 | ||
| | | 5/4 | ||
| | | French decimal | ||
|- | |- | ||
| | | 777\1789 | ||
| | | 521.18 | ||
| | | 875/648 | ||
| | | Maviloid | ||
|} | |} | ||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||