19edt

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19edt

20edt →

Prime factorization	19 (prime)
Step size	100.103 ¢
Octave	12\19edt (1201.23 ¢) (convergent)
Consistency limit	10
Distinct consistency limit	6

19 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 19edt or 19ed3), is a nonoctave tuning system that divides the interval of 3/1 into 19 equal parts of about 100 ¢ each. Each step represents a frequency ratio of 3^1/19, or the 19th root of 3. It is also known as Stopper tuning.

Theory

19edt is not a truly xenharmonic tuning; it is a slightly stretched version (with an octave of 1201.2 cents) of the normal 12-tone scale. Although it is really just the normal 12edo framed in a pure-3 tuning, it can still be used as a temperament with no twos like other tritave-equivalent tunings, although limited in accuracy, with 5/3 approximated as 9 steps and 7/3 approximated by 15 steps. It completely misses the next tritave-reduced prime harmonic, 11/9.

This approach can create very non-standard chords and scales such as the approximation of the 5:7:9 chord as 0–600–1000 cents. These could be considered xenharmonic in a sense, since they have little connection to standard 12-tone practice in spite of using the 12-tone interval set. The "default" approach to it is as a "macro-godzilla" temperament (with a generator of 400.4 cents and a 3:1 ratio 5L 4s⟨3/1⟩ scale, and it is an interesting coincidence how 17edt and 19edt tonality have the same "default" scheme with two tones more or less). Beyond this, it also contains the tritave twin of meantone temperament (with a generator of 700.7 or 1201.2 cents), producing a basic 8L 3s⟨3/1⟩ scale.

Harmonics

Approximation of harmonics in 19edt

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+1.2	+0.0	+2.5	+16.6	+1.2	+34.7	+3.7	+0.0	+17.8	-47.1	+2.5
	Relative (%)	+1.2	+0.0	+2.5	+16.6	+1.2	+34.6	+3.7	+0.0	+17.8	-47.1	+2.5
Steps (reduced)		12 (12)	19 (0)	24 (5)	28 (9)	31 (12)	34 (15)	36 (17)	38 (0)	40 (2)	41 (3)	43 (5)

Approximation of harmonics in 19edt (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	-36.0	+35.9	+16.6	+4.9	+0.1	+1.2	+7.7	+19.0	+34.7	-45.9	-22.7	+3.7
	Relative (%)	-36.0	+35.9	+16.6	+4.9	+0.1	+1.2	+7.7	+19.0	+34.6	-45.8	-22.7	+3.7
Steps (reduced)		44 (6)	46 (8)	47 (9)	48 (10)	49 (11)	50 (12)	51 (13)	52 (14)	53 (15)	53 (15)	54 (16)	55 (17)

Subsets and supersets

19edt is the 8th prime edt, following 17edt and before 23edt.

Intervals

Steps	Cents	Hekts	Approximate ratios
0	0	0	1/1
1	100.1	68.4	16/15, 17/16, 18/17, 19/18, 20/19, 21/20
2	200.2	136.8	9/8, 17/15, 19/17
3	300.3	205.3	6/5, 13/11, 19/16
4	400.4	273.7	5/4, 19/15, 24/19
5	500.5	342.1	4/3
6	600.6	410.5	10/7, 17/12, 24/17
7	700.7	478.9	3/2
8	800.8	547.4	8/5, 19/12
9	900.9	615.8	5/3, 22/13
10	1001	684.2	9/5, 16/9, 23/13
11	1101.1	752.6	15/8, 17/9, 19/10
12	1201.2	821.1	2/1
13	1301.3	889.5	17/8, 19/9, 21/10
14	1401.4	957.9	9/4
15	1501.5	1026.3	12/5, 19/8
16	1601.6	1094.7	5/2
17	1701.7	1163.2	8/3
18	1801.9	1231.6	17/6, 20/7
19	1902	1300	3/1

See also

• 7edf – relative edf
• 12edo – relative edo
• 28ed5 – relative ed5
• 31ed6 – relative ed6
• 34ed7 – relative ed7
• 40ed10 – relative ed10
• 43ed12 – relative ed12
• 76ed80 – close to the zeta-optimized tuning for 12edo
• AS18/17 – relative ambitonal sequence

External links

• Bernhard Stopper's OnlyPure tuning^{[dead link]}