1619edo

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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 1619-tone equal temperament (1619tet), 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 of 1619edo 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 -5 -20 -11 +18 -1 +40 -39 +35 -7 +16
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