1789edo: Difference between revisions

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Decimal as a temperament name has been taken, so I add quotation marks to it; fix interval table (+notes, -approximations that don't make sense); rework RTT table (as 2.9); and misc
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'''1789 EDO''' divides the octave into equal steps of 0.67 cents each. It is the 278th [[prime edo]]. 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.
'''1789edo''' divides the octave into equal steps of 0.67 cents each. It is the 278th [[prime edo]]. 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.
{{Wikipedia|1789}}
 
==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.  


=== Jacobin temperament ===
=== 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.
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. 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.
Using the maximal evenness method of finding rank two temperaments, we get a 37 & 1789 temperament.


=== Decimal 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).
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.


=== Other ===
=== 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.  
For its elaborate xenharmonicity, 1789edo has an essentially perfect [[9/8]], a very common interval. The associated comma is {{monzo| 5671 -1789 }} in 2.9. This is a direct consequence of 1789edo being a [[dual-fifth system]]. 1046th and 1047th steps are not associated with JI intervals by themselves, but satisfy 3<sup>+</sup> × 3<sup>-</sup> = 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]].
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]].
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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]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal
! rowspan="2" | Optimal<br>8ve stretch (¢)
8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" |Tuning error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.9
|{{monzo| -2835 1789 }}
| {{monzo| -5671 1789 }}
|[{{val|1789 2835}}]
| [{{val| 1789 5671 }}]
|0.105
| -0.000441
|0.105
| 0.000441
|15.7
| 0.066
|-
|-
|2.3II
| 2.5.11.13.29.31
|{{monzo| -2836 1789 }}
| 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321
|[{{val|1789 2836}}]
|0.106
|0.106
|15.8
|-
|2.5.11.13.29.31
|6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321
| [{{val|1789 4154 6189 6620 8691 8863}}]
| [{{val|1789 4154 6189 6620 8691 8863}}]
|  -0.003
|  -0.003
|0.013
| 0.013
|1.9
| 1.9
|}
|}
== Table of selected intervals ==
== Table of selected intervals ==
{| class="wikitable collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+
|+ style=white-space:nowrap | Selected intervals in 1789edo
Selected intervals in 1789 EDO
! Step
!Step
! Eliora's Naming System
!Name
! JI Approximation or Other Interpretations*
!JI Approximation, Monzo, or another interpretation
<small>(based on the 2.5.11.13.29.31 subgroup where applicable)</small>
|-
|-
|0
| 0
|Unison
| Unison
|1/1 exact
| 1/1
|-
|-
|25
| 25
|28-thirds comma
| 28-thirds comma
|[65 -28]
| {{monzo| 65 -28 }}
|-
|-
|36
| 36
|
|
|145/143
| 145/143
|-
|-
|61
| 61
|Lesser diesis
| Lesser diesis
|[[128/125]]
| [[128/125]]
|-
|-
|74
| 74
|
|
|319/310
| 319/310
|-
|-
|122
| 122
|
|
|65/62
| 65/62
|-
|-
|172
| 172
|Tricesimoprimal Miracle semitone
| Tricesimoprimal Miracle semitone
|[[31/29]]
| [[31/29]]
|-
|-
|226
| 226
|
|
|440/403
| 440/403
|-
|-
|290
| 290
|Jacobin minor interval
| Jacobin minor interval
|160/143, 649/580
| 160/143, 649/580
|-
|-
|338
| 338
|Minor sqrt(13/10)
| Minor sqrt(13/10)
|[[Square root of 13 over 10]] I,
|  
|-
|-
|339
| 339
|Major sqrt(13/10)
| Major sqrt(13/10)
|[[Square root of 13 over 10]] II, (11/8)^20
| {{monzo| -69 0 0 0 20 }}
|-
|-
|387
| 387
|Jacobin major interval
| Jacobin major interval
|754/649
| 754/649
|-
|-
|576
| 576
|Major third
| Major third
|[[5/4]]
| [[5/4]]
|-
|-
|677
| 677
|Jacobin naiadic
| Jacobin naiadic
|[[13/10]]
| [[13/10]]
|-
|-
|822
| 822
|Jacobin superfourth, Mongolian fourth
| Jacobin superfourth, Mongolian fourth
|[[11/8]]
| [[11/8]]
|-
|-
|1032
| 1032
|Secor fifth, Tricesimoprimal Miracle fifth
| Secor fifth, Tricesimoprimal Miracle fifth
|(31/29)^6
| (31/29)<sup>6</sup>
|-
|-
|1046
| 1046
|Minor fifth
| Minor fifth
|<u>[[3/2]] I</u>
| [[3/2]]†
|-
|-
|1047
| 1047
|Major fifth
| Major fifth
|[[3/2]] II
| [[3/2]]†
|-
|-
|1535
| 1535
|29th harmonic
| 29th harmonic
|[[29/16]]
| [[29/16]]
|-
|-
|1579
| 1579
|59th harmonic
| 59th harmonic
|[[59/32]]
| [[59/32]]
|-
|-
|1707
| 1707
|31st harmonic
| 31st harmonic
|[[31/16]]
| [[31/16]]
|-
|-
|1789
| 1789
|Octave
| Octave
|2/1 exact
| 2/1
|}
|}
<nowiki>*</nowiki> based on the 2.5.11.13.29.31 subgroup where applicable


==Scales==
† 1046\1789 as 3/2 is the patent val, 1047\1789 as 3/2 is the 1789b val
 
== Scales ==
* Jacobin[37]
* Jacobin[37]
* Jacobin[74]
* Jacobin[74]
* Jacobin[111]
* Jacobin[111]
* Jacobin[222]
* Jacobin[222]
* Decimal[265]
* "Decimal"[265]
* Decimal[1524]
* "Decimal"[1524]


[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
[[Category:Prime EDO]]
[[Category:Prime EDO]]
{{Todo| review }}

Revision as of 18:56, 20 March 2022

1789edo divides the octave into equal steps of 0.67 cents each. It is the 278th prime edo. 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.

Theory

Approximation of odd harmonics in 1789edo
Harmonic 3 5 7 9 11 13 15 17 19 21
Error Absolute (¢) -0.334 +0.047 -0.240 +0.003 +0.052 -0.058 -0.287 -0.316 +0.307 +0.097
Relative (%) -49.8 +7.1 -35.8 +0.4 +7.7 -8.7 -42.7 -47.1 +45.8 +14.4
Steps
(reduced) 		2835
(1046) 		4154
(576) 		5022
(1444) 		5671
(304) 		6189
(822) 		6620
(1253) 		6989
(1622) 		7312
(156) 		7600
(444) 		7858
(702)

1789edo can be adapted for use with the 2.5.11.13.29.31.47.59.61 subgroup.

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. 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 -1789⟩ in 2.9. This is a direct consequence of 1789edo being a dual-fifth system. 1046th and 1047th steps are not associated with JI intervals by themselves, but satisfy 3+ × 3- = 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

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.9 [-5671 1789⟩ [⟨1789 5671]] -0.000441 0.000441 0.066
2.5.11.13.29.31 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321 [⟨1789 4154 6189 6620 8691 8863]] -0.003 0.013 1.9

Table of selected intervals

Selected intervals in 1789edo
Step Eliora's Naming System JI Approximation or Other Interpretations*
0 Unison 1/1
25 28-thirds comma [65 -28⟩
36 145/143
61 Lesser diesis 128/125
74 319/310
122 65/62
172 Tricesimoprimal Miracle semitone 31/29
226 440/403
290 Jacobin minor interval 160/143, 649/580
338 Minor sqrt(13/10)
339 Major sqrt(13/10) [-69 0 0 0 20⟩
387 Jacobin major interval 754/649
576 Major third 5/4
677 Jacobin naiadic 13/10
822 Jacobin superfourth, Mongolian fourth 11/8
1032 Secor fifth, Tricesimoprimal Miracle fifth (31/29)6
1046 Minor fifth 3/2†
1047 Major fifth 3/2†
1535 29th harmonic 29/16
1579 59th harmonic 59/32
1707 31st harmonic 31/16
1789 Octave 2/1

* based on the 2.5.11.13.29.31 subgroup where applicable

† 1046\1789 as 3/2 is the patent val, 1047\1789 as 3/2 is the 1789b val

Scales

  • Jacobin[37]
  • Jacobin[74]
  • Jacobin[111]
  • Jacobin[222]
  • "Decimal"[265]
  • "Decimal"[1524]
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