1619edo: Difference between revisions

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→Theory: rearranged the page contents so that they belong to a specific seciton, proposed the idea of the Vidarines
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1619edo is excellent in the 13-limit, where it tempers out [[4225/4224]], [[4375/4374]], [[6656/6655]], 78125/78078, and 117649/117612. It also notably tempers out [[quartisma]] (117440512/117406179) and [[123201/123200]].   
1619edo is excellent in the 13-limit, where it tempers out [[4225/4224]], [[4375/4374]], [[6656/6655]], 78125/78078, and 117649/117612. It also notably tempers out [[quartisma]] (117440512/117406179) and [[123201/123200]].   


It supports [[vidar]], which has the comma basis 4225/4224, 4375/4374, and 6656/6655, and other unnamed expansions of the [[ragismic]] temperament such as the 270 & 441 & 1619, tempering out 4225/4224, 4375/4374, 655473/655360, or the 72 & 270 & 494 & 1619 temperament tempering out 6656/6655 and 2912000/2910897.  
1619edo tunes [[keenanisma]] very finely, to 6 steps, and can use it as a microchroma. 1619edo has 7/6 on 360th step, a highly divisible number, 27/25 on 180th, and 33/32 on 72nd as a consequence of tempering out the commas. This means that 72ed33/32 is virtually equivalent to 1619edo. When it comes to using 33/32 as the generator, 1619edo supports the [[ravine]] temperament, which tempers out 196625/196608, 200000/199927, 2912000/2910897, and 3764768/3764475.


1619edo supports the rank-5 temperament tempering out the jacobin comma, 6656/6655 and its fifth-order maximal evenness scale is represented by every 3rd step of the 72 & 270 & 494 & 1619 temperament{{clarify}}.
Since 33/32 is close to 1\45, 7\6 is close to 1\9, and 385/384 is close to 1\270, 1619edo can be thought of as [[1620edo]] where one step was extracted and all others were moved into a more harmonically just position. It achieves this because 1620edo is contorted 270edo in the 11-limit, and its 13/8 is on the flat side coming from 324edo, and thus when it is octave stretched, steps sharpen enough to arrive at 1619edo's 13-limit excellence.


1619edo tunes [[keenanisma]] very finely, to 6 steps, and can use it as a microchroma. In addition, it can use the keenanisma as a generator in the keenanose temperament, 270 & 1619, in which it highlights the relationship between 270 keenanismas and the octave. It also achieves this since 270 × 6 = 1620, and 1619 is 1 short of that and also excellent in the 13-limit. 1619edo has 7/6 on 360th step, a highly divisible number, 27/25 on 180th, and 33/32 on 72nd as a consequence of tempering out the commas. This means that 72ed33/32 is virtually equivalent to 1619edo. When it comes to using 33/32 as the generator, 1619edo supports the [[ravine]] temperament, which tempers out 196625/196608, 200000/199927, 2912000/2910897, and 3764768/3764475.
=== The Vidarines ===
1619edo supports [[vidar]], which has the comma basis 4225/4224, 4375/4374, and 6656/6655. In addition, it contains a wealth of rank-two 13-limit temperaments that are produced by adding one comma on top of the vidar comma basis, such as decigrave, keenanose, moulin, etc. Eliora proposes the name ''The Vidarines'' for this collection of temperaments.  


Another temperament which highlights the interval relationships in 1619edo is 45 & 1619, and if it had a name, it would be called ''decigrave'', since 10 steps make a 7/6, which is referred to as the grave minor third sometimes. It has a comma basis 4225/4224, 4375/4374, 6656/6655, {{monzo|23  5 13 -23  1 0}} in the 13-limit. Its generator is 36 steps, which represents 65/64 and 66/65 tempered together, and 2 of them make 33/32. 5 of them make [[27/25]], and 10 of them make 7/6.
One such temperament is the keenanose temperament, which has comma basis 4225/4224, 4375/4374, 6656/6655, and 151263/151250. Keenanisma is the generator in the keenanose temperament, 270 & 1619, in which it highlights the relationship between 270 keenanismas and the octave. It also achieves this since 270 × 6 = 1620, and 1619 is 1 short of that and also excellent in the 13-limit.  


Since 33/32 is close to 1\45, 7\6 is close to 1\9, and 385/384 is close to 1\270, 1619edo can be thought of as [[1620edo]] where one step was extracted and all others were moved into a more harmonically just position. It achieves this because 1620edo is contorted 270edo in the 11-limit, and its 13/8 is on the flat side coming from 324edo, and thus when it is octave stretched, steps sharpen enough to arrive at 1619edo's 13-limit excellence.
Another temperament which highlights the interval relationships in 1619edo (and is also a member of The Vidarines collection) is 45 & 1619, and if it had a name, it would be called ''decigrave'', since 10 steps make a 7/6, which is referred to as the grave minor third sometimes. It has a comma basis 4225/4224, 4375/4374, 6656/6655, {{monzo|23  5 13 -23  1 0}} in the 13-limit. Its generator is 36 steps, which represents 65/64 and 66/65 tempered together, and 2 of them make 33/32. 5 of them make [[27/25]], and 10 of them make 7/6.
 
1619edo supports the 494 & 1619 temperament called moulin, which is also a member of The Vidarines collection. In this case, 91125/91091 has to be added to the three commas to produce a rank-2 temperament.
 
A quick summary is shown below.
{| class="wikitable"
|+The Vidarines in 1619edo (named and unnamed)
!Temperament
!Generator
associated ratio
!Completing comma
|-
|Keenanose
|385/384
|151263/151250
|-
|Decigrave
|66/65 ~ 65/64
|{{monzo|23  5 13 -23  1 0}}
|-
|Moulin
|13/11
|91125/91091
|-
|224 & 1619
|256/175
|18753525/18743296
|}
While [[abigail]] is a member of the vidarines, 1619edo does not support it because abigail is a period-2 temperament, and 1619 is an odd number.


=== Prime harmonics ===
=== Prime harmonics ===

Revision as of 16:31, 7 October 2022

← 1618edo 1619edo 1620edo →
Prime factorization 1619 (prime)
Step size 0.741198 ¢ 
Fifth 947\1619 (701.915 ¢)
Semitones (A1:m2) 153:122 (113.4 ¢ : 90.43 ¢)
Consistency limit 15
Distinct consistency limit 15

Template:EDO intro

Theory

1619edo is excellent in the 13-limit, where it tempers out 4225/4224, 4375/4374, 6656/6655, 78125/78078, and 117649/117612. It also notably tempers out quartisma (117440512/117406179) and 123201/123200.

1619edo tunes keenanisma very finely, to 6 steps, and can use it as a microchroma. 1619edo has 7/6 on 360th step, a highly divisible number, 27/25 on 180th, and 33/32 on 72nd as a consequence of tempering out the commas. This means that 72ed33/32 is virtually equivalent to 1619edo. When it comes to using 33/32 as the generator, 1619edo supports the ravine temperament, which tempers out 196625/196608, 200000/199927, 2912000/2910897, and 3764768/3764475.

Since 33/32 is close to 1\45, 7\6 is close to 1\9, and 385/384 is close to 1\270, 1619edo can be thought of as 1620edo where one step was extracted and all others were moved into a more harmonically just position. It achieves this because 1620edo is contorted 270edo in the 11-limit, and its 13/8 is on the flat side coming from 324edo, and thus when it is octave stretched, steps sharpen enough to arrive at 1619edo's 13-limit excellence.

The Vidarines

1619edo supports vidar, which has the comma basis 4225/4224, 4375/4374, and 6656/6655. In addition, it contains a wealth of rank-two 13-limit temperaments that are produced by adding one comma on top of the vidar comma basis, such as decigrave, keenanose, moulin, etc. Eliora proposes the name The Vidarines for this collection of temperaments.

One such temperament is the keenanose temperament, which has comma basis 4225/4224, 4375/4374, 6656/6655, and 151263/151250. Keenanisma is the generator in the keenanose temperament, 270 & 1619, in which it highlights the relationship between 270 keenanismas and the octave. It also achieves this since 270 × 6 = 1620, and 1619 is 1 short of that and also excellent in the 13-limit.

Another temperament which highlights the interval relationships in 1619edo (and is also a member of The Vidarines collection) is 45 & 1619, and if it had a name, it would be called decigrave, since 10 steps make a 7/6, which is referred to as the grave minor third sometimes. It has a comma basis 4225/4224, 4375/4374, 6656/6655, [23  5 13 -23  1 0⟩ in the 13-limit. Its generator is 36 steps, which represents 65/64 and 66/65 tempered together, and 2 of them make 33/32. 5 of them make 27/25, and 10 of them make 7/6.

1619edo supports the 494 & 1619 temperament called moulin, which is also a member of The Vidarines collection. In this case, 91125/91091 has to be added to the three commas to produce a rank-2 temperament.

A quick summary is shown below.

The Vidarines in 1619edo (named and unnamed)
Temperament Generator

associated ratio

Completing comma
Keenanose 385/384 151263/151250
Decigrave 66/65 ~ 65/64 [23  5 13 -23  1 0⟩
Moulin 13/11 91125/91091
224 & 1619 256/175 18753525/18743296

While abigail is a member of the vidarines, 1619edo does not support it because abigail is a period-2 temperament, and 1619 is an odd number.

Prime harmonics

Approximation of prime harmonics in 1619edo
Harmonic 2 3 5 7 11 13 17 19 23 29
Error Absolute (¢) +0.000 -0.040 -0.149 -0.080 +0.134 -0.009 +0.295 -0.293 +0.262 -0.053
Relative (%) +0.0 -5.4 -20.2 -10.8 +18.0 -1.2 +39.8 -39.5 +35.3 -7.1
Steps
(reduced) 		1619
(0) 		2566
(947) 		3759
(521) 		4545
(1307) 		5601
(744) 		5991
(1134) 		6618
(142) 		6877
(401) 		7324
(848) 		7865
(1389)

Miscellaneous properties

1619edo is the 256th prime edo.

Selected intervals

Table of intervals in 1619edo
Step Cents Ratio Name*
0 0.000 1/1 prime, unison
6 4.447 385/384 keenanisma
72 53.366 33/32 al-Farabi quarter-tone
360 266.831 7/6 septimal subminor third
1619 1200.000 2/1 perfect octave

* named in accordance to their most just 13-limit counterpart using the names accepted on the wiki.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-2566 1619⟩ [⟨1619 2566]] +0.0127 0.0127 1.71
2.3.5 [-69 45 -1⟩, [-82 -1 36⟩ [⟨1619 2566 3759]] +0.0299 0.0265 3.57
2.3.5.7 4375/4374, 52734375/52706752, [-67 14 6 11⟩ [⟨1619 2566 3759 4545]] +0.0295 0.0229 3.09
2.3.5.7.11 4375/4374, 117649/117612, 759375/758912, [24 -6 0 1 -5⟩ [⟨1619 2566 3759 4545 5601]] +0.0159 0.0341 4.60
2.3.5.7.11.13 4225/4224, 4375/4374, 6656/6655, 78125/78078, 117649/117612 [⟨1619 2566 3759 4545 5601 5991]] +0.0136 0.0315 4.26

Rank-2 temperaments

Periods
per Octave 		Generator
(Reduced) 		Cents
(Reduced) 		Associated
Ratio 		Temperaments
1 6\1619 4.447 385/384 Keenanose
1 36\1619 26.683 65/64 ~ 66/65 Decigrave
1 72\1619 53.366 33/32 Ravine
1 390\1619 289.067 13/11 Moulin
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