1789edo: Difference between revisions
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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. | ||
=== Jacobin temperament === | |||
The "proper" Jacobin temperament 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. This 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. | |||
Using the maximal evenness method of finding rank two temperaments, we get a 37 & 1789 temperament. | |||
=== Decimal temperament === | |||
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. | |||
=== Other === | |||
For its elaborate xenharmonicity, 1789edo has an essentially perfect [[9/8]], a very common interval. The associated comma is [5671 -3578⟩. This is a direct consequence of 1789edo being a [[dual-fifth system]]. 1046th and 1047th steps aren't associated with JI intervals by themselves, but satisfy 3 × 3II = 9, which represents 9/1, and therefore when octave reduced this leads to 9/8. | |||
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]]. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
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|15.7 | |15.7 | ||
|- | |- | ||
|2. | |2.3II | ||
|{{monzo| -2836 1789 }} | |{{monzo| -2836 1789 }} | ||
|[{{val|1789 2836}}] | |[{{val|1789 2836}}] | ||
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|1.9 | |1.9 | ||
|} | |} | ||
== Table of selected intervals == | == Table of selected intervals == | ||
{| class="wikitable collapsible mw-collapsed" | {| class="wikitable collapsible mw-collapsed" | ||