1789edo: Difference between revisions
→Jacobin temperament: Updated comma basis in accordance with Sintel's temperament finder. |
-jacobin (it a rank-5 temperament); cleanup and request for clarification |
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== Theory == | == Theory == | ||
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]]), and it is also consistent on the subgroup 2.5.11.13 of the comma, 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 | 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]]), and it is also consistent on the subgroup 2.5.11.13 of the comma, 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 its number is the hallmark year of the French Revolution, thus making the tempering of the jacobin comma on topic. | ||
1789edo is consistent in the no-threes 13-odd-limit. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|1789|columns = 10}} | |||
=== Jacobin temperament === | === Jacobin temperament === | ||
A "proper" jacobin scale in 1789edo, the maximum evenness scale that uses 822 as a generator, contains only 37 notes. The step sizes are 48 and 49, making them indistinguishable to human ear at this scale. | A "proper" jacobin scale in 1789edo, the maximum evenness scale that uses 822 as a generator, contains only 37 notes. The step sizes are 48 and 49, making them indistinguishable to human ear at this scale. | ||
Using the maximal evenness method of finding rank two temperaments, we get a 37 & 1789 temperament with the comma set 6656/6655 and {{ | Using the maximal evenness method of finding rank two temperaments, we get a 37 & 1789 temperament with the comma set 6656/6655 and {{monzo| -119 -46 15 47 }}. The first is the well known jacobin comma, and it means that a stack of three 11/8 is equated with 13/10. | ||
The monotonicity can be fixed by using divisors of 822 as a generaator, for example 137\1789 "6th root of 11/8" temperament having 222 notes. In addition, this can be re-interpreted by using 13/10 as a generator instead which produces a more vibrant 1205 out of 1789, and partitioning the resulting 13/5s in three around the octave. Addition of 29 and 31 harmonic intervals may also be suitable to spice up an otherwise monotonous scale. | The monotonicity can be fixed by using divisors of 822 as a generaator, for example 137\1789 "6th root of 11/8" temperament having 222 notes. In addition, this can be re-interpreted by using 13/10 as a generator instead which produces a more vibrant 1205 out of 1789, and partitioning the resulting 13/5s in three around the octave. Addition of 29 and 31 harmonic intervals may also be suitable to spice up an otherwise monotonous scale. | ||
=== French decimal temperament === | === French decimal temperament === | ||
1789edo 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 | Using the maximal evenness method of finding rank-2 temperaments, we get a 1525 & 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]]. | ||
Since it 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 | Since it 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.<sup>[which mapping?]</sup> | ||
1789edo supports the 2.9.5.11.13 subgroup temperament which Eliora proposes be called ''commatose'', and which uses the Pythagorean comma as a generator. It is defined as a 460 & 1789 temperament, and its comma basis is 62748517/62726400, 479773125/479756288, and 30530193408/30517578125. | 1789edo supports the 2.9.5.11.13 subgroup temperament which Eliora proposes be called ''commatose'', and which uses the Pythagorean comma as a generator. It is defined as a 460 & 1789 temperament, and its comma basis is 62748517/62726400, 479773125/479756288, and 30530193408/30517578125. | ||
1789edo supports the 2.5.11.13.19 subgroup temperament called '' | 1789edo supports the 2.5.11.13.19 subgroup temperament called ''estates general'' defined as 1789 & 3125. This is referencing the fact that Estates General were called by Louis XVI on 5th May 1789, written as 05/05, and 3125 is 5 to the 5th power and also provides an optimal patent val for tempering out the jacobin comma, contuing the lore. It is unambiguous in the 2.5.11.13.19.23.29.31 subgroup. | ||
== Table of selected intervals == | == Table of selected intervals == | ||
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| {{monzo| 65 -28 }} | | {{monzo| 65 -28 }} | ||
|- | |- | ||
|35 | | 35 | ||
|Pythagorean comma | | Pythagorean comma | ||
|[[531441/524288]] | | [[531441/524288]] | ||
|- | |- | ||
| 36 | | 36 | ||
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| [[3/2]]† | | [[3/2]]† | ||
|- | |- | ||
|1213 | | 1213 | ||
|Classical minor sixth | | Classical minor sixth | ||
|[[8/5]] | | [[8/5]] | ||
|- | |- | ||
| 1444 | | 1444 | ||
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== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | Subgroup | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning Error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
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|} | |} | ||
== Rank-2 temperaments | == Rank-2 temperaments == | ||
{| class="wikitable center-all left- | {| class="wikitable center-all left-4" | ||
! Generator<br>( | ! Generator<br>(Reduced) | ||
! Cents<br>( | ! Cents<br>(Reduced) | ||
! Associated<br> | ! Associated<br>Ratio | ||
! Temperament | ! Temperament | ||
|- | |- | ||
|35\1789 | | 35\1789 | ||
|23.48 | | 23.48 | ||
|531441/524288 | | 531441/524288 | ||
|Commatose | | [[Commatose]] | ||
|- | |- | ||
|172\1789 | | 172\1789 | ||
|115.37 | | 115.37 | ||
|31/29 | | 31/29 | ||
|Tricesimoprimal | | Tricesimoprimal miracloid<sup>[definition?]</sup> | ||
|- | |- | ||
| 576\1789 | | 576\1789 | ||
| 386.36 | | 386.36 | ||
| 5/4 | | 5/4 | ||
| French decimal | | French decimal<sup>[definition?]</sup> | ||
|- | |- | ||
| 777\1789 | | 777\1789 | ||
| 521.18 | | 521.18 | ||
| 875/648 | | 875/648 | ||
| Maviloid | | [[Maviloid]] | ||
|- | |- | ||
| | | 778\1789 | ||
| | | 521.86 | ||
| | | 80275/59392 | ||
| | | [[Estates general]] | ||
|} | |} | ||
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[[Category:Jacobin]] | [[Category:Jacobin]] | ||
{{Todo| review }} | {{Todo| review | clarify }} | ||