5-Limit, 19-Tone Just Intonation

What is 5-Limit, 19-Tone Just Intonation?

It is an alternate tuning that is constructed from ratios using only the prime numbers 1, 2, 3, and 5 as factors. It uses 19 tones that are harmonically related to each other rather than the 12 equal-tempered tones (12-TET) used in most western music today. Since the tones are harmonically related, chords using them sound richer than the same chords using 12-TET. The designation “5-limit” signifies that 5 is the largest prime number used as a factor.

How to create MIDI sequences using 5-Limit, 19-Tone Just Intonation

To create your own sequences, follow these steps:

1. Create your sequence limiting each channel to a single note or pitch class at any time.
2. Add the pitch bends from the following table to each channel for each note or pitch class. Transpose the table to the appropriate key. These may be on a separate track to allow muting for 12-tone equal tempered playback.
3. After the end of the music place the pitch bend 8192 on each channel so the next sequence played will not sound out of tune. MIDI reset does not do this.
4. When there is more that one bend listed for a pitch refer to the row for the root of the chord for the proper chord members. For example F-A-C = 9-15-1, A-C-E = 15-1-8, but D-F-A = 5-10-16

No.	Ratio	Pitch	Bend	m3	M3	P4	P5	m6	M6	m7	M7
1	1/1	C	8192	7	8	9	12	14	15	18	19
2	25/24	C#	6991	8			13	15		19
3	135/128	C#	7872			11		16	17
4	16/15	Db	8673		9		14				1
5	9/8	D	8352	10	11	12	16	18	19		3
6	75/64	D#	7151	11		13	17	19		3
7	6/5	Eb	8833		12	14	18		1		5
8	5/4	E	7631	12	13	15	19	1	2	5	6
9	4/3	F	8112	14	15		1	4		7	8
10	27/20	F	8993		16	18			5
11	45/32	F#	7792	16	17	19	3	5	6
12	3/2	G	8272	18	19	1	5	7	8	10	11
13	25/16	G#	7071	19		2	6	8		11
14	8/5	Ab	8753		1	4	7		9		12
15	5/3	A	7551	1	2		8	9		12	13
16	27/16	A	8432		3	5		10	11
17	225/128	A#	7231	3		6		11
18	9/5	Bb	8913		5	7	10		12		16
19	15/8	B	7711	5	6	8	11	12	13	16	17
1	2/1	C	8192	7	8	9	12	14	15	18	19

How Is 5-Limit, 19-Tone Just Intonation Constructed?

First, ratios of 3/2 and 5/4 are selected for the perfect 5th and major 3rd, respectively. Fifths are taken then from 1 below to 3 above giving tones 9, 1, 12, 5, and 16. From these, thirds are taken from one below to 2 above, except that 2 above tone 16 is omitted. This results in the remaining 14 tones. The following table illustrates this process.

		Thirds
		-1	0	+1	+2
F i f t h s	+3	10	16	3
	+2	18	5	11	17
	+1	7	12	19	6
	0	14	1	8	13
	-1	4	9	15	2

Why do we stop at 19 tones? Well, the 20th tone would be an F lowered one tenth of a step. This difference is generally imperceptible and so we stop at 19.

How different does it sound?

To hear a comparison 5-19 JI to 12-TET, click on this short demonstration. First, a flute plays a series of chords switching from 12-TET to 5-19 JI at midpoint. This is then repeated by guitar. Then the flute plays a series of ascending fifths in 5-19 JI switching to 12-TET in the middle of the final chord.