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 (1619edo), or 1619-tone equal temperament (1619tet), 1619 equal temperament (1619et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 1619 equal parts of about 0.741 ¢ each.

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.

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 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[clarification needed].

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.

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, [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.

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.

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 -5 -20 -11 +18 -1 +40 -39 +35 -7
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