# 1619edo

 ← 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

1619 equal divisions of the octave (abbreviated 1619edo or 1619ed2), also called 1619-tone equal temperament (1619tet) or 1619 equal temperament (1619et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1619 equal parts of about 0.741 ¢ each. Each step represents a frequency ratio of 21/1619, or the 1619th root of 2.

## 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.

1619edo supports a very precise rank two temperament, 19 & 1619, which uses 6/5 as a generator and has a comma basis 4375/4374, 91125/91091, 196625/196608, and 54925000/54908469.

1619edo supports 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 is 45 & 1619, 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, with the comma basis of 4225/4224, 4375/4374, 6656/6655, 91125/91091. The 25-tone scale of moulin is capable of supporting the 8:11:13 triad, as it takes less than 25 notes to map the 11th and 13th harmonics.

### 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;. Temperaments described above such as decigrave, keenanose, moulin, are members of this collection. Eliora proposes the name The Vidarines for this collection of temperaments.

A quick summary is shown below.

The Vidarines in 1619edo (named and unnamed)
Temperament Generator
associated ratio
Completing comma
Keenanose (270 & 1619) 385/384 151263/151250
Decigrave (45 & 1619) 66/65 ~ 65/64 [23  5 13 -23  1 0
Moulin (494 & 1619) 13/11 91125/91091
46 & 1619 3328/3087 [-18 9 -2 8 -3 -1
178 & 1619 4429568/4084101 [-29 10 2 12 -3 -4
224 & 1619 256/175 18753525/18743296
764 & 1619 12375/8918 52734375/52706752
901 & 1619 104/99 34875815625/34843787264

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 31
Error Absolute (¢) +0.000 -0.040 -0.149 -0.080 +0.134 -0.009 +0.295 -0.293 +0.262 -0.053 +0.116
Relative (%) +0.0 -5.4 -20.2 -10.8 +18.0 -1.2 +39.8 -39.5 +35.3 -7.1 +15.6
Steps
(reduced)
1619
(0)
2566
(947)
3759
(521)
4545
(1307)
5601
(744)
5991
(1134)
6618
(142)
6877
(401)
7324
(848)
7865
(1389)
8021
(1545)

### Subsets and supersets

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, grave minor third
744 551.451 11/8 11th harmonic, undecimal superfourth
1134 840.519 13/8 13th harmonic, tridecimal neutral sixth
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* Cents* Associated
Ratio
Temperaments
1 6\1619 4.447 385/384 Keenanose
1 36\1619 26.683 65/64 Decigrave
1 72\1619 53.366 33/32 Ravine
1 112\1619 83.014 1573/1500 Acrosextilififths
1 390\1619 289.067 13/11 Moulin
1 426\1619 315.750 6/5 Oviminor
1 587\1619 435.083 9/7 Supermajor
1 672\1619 498.085 4/3 Counterschismic

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct