1619edo

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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

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

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf