Just Intonation

Ratio	Cents	Note
1/1	0.00	C +0
14/13	128.30	C# +28
13/12	138.57	C# +39
12/11	150.64	D -49
11/10	165.00	D -35
10/9	182.40	D -18
9/8	203.91	D +4
8/7	231.17	D +31
7/6	266.87	D# -33
13/11	289.21	D# -11
6/5	315.64	D# +16
11/9	347.41	D# +47
16/13	359.47	E -41
5/4	386.31	E -14
14/11	417.51	E +18
9/7	435.08	E +35
13/10	454.21	F -46
4/3	498.04	F -2
11/8	551.32	F# -49
18/13	563.38	F# -37
7/5	582.51	F# -17
10/7	617.49	F# +17
13/9	636.62	F# +37
16/11	648.68	F# +49
3/2	701.96	G +2
20/13	745.79	G +46
14/9	764.92	G# -35
11/7	782.49	G# -18
8/5	813.69	G# +14
13/8	840.53	G# +41
18/11	852.59	A -47
5/3	884.36	A -16
22/13	910.79	A +11
12/7	933.13	A +33
7/4	968.83	A# -31
16/9	996.09	A# -4
9/5	1017.60	A# +18
20/11	1035.00	A# +35
11/6	1049.36	A# +49
24/13	1061.43	B -39
13/7	1071.70	B -28
2/1	0.00	C +0

Just Intonation (or JI) uses the actual uncompromised intervals which our ears perceive to be consonant.  In short, small whole number ratios.  The table on the right lists of most of the intervals that our ears hear as consonant.  Specifically, the simplest 13 odd-limit ratios within an octave and their inversions.

The ear will "correct" for small deviations from consonant intervals, thus 301/200 will be heard as a slightly out of tune 3/2 and still sound consonant.  Consonant ratios generally involve numerators and denominators which are multiples of primes no greater than 13 but the definition seems to vary a bit from person to person.

Just Intonation is used in the music of Harry Partch, La Monte Young, Ben Johnston, and Jon Catler.  It is unconsciously used by good violinists and vocalists because it just sounds "right".  It is the rule in Indian Classical music and other Eastern Music.  It can pose some challenges when fretting stringed instruments because all the intervals are of different sizes, so the frets do not lie in the same place under differently tuned strings.  Often specialized frets are required.  The challenge is justified by the beauty and playability of such a neck.

Just Intonation Scale Design

Designing just intonation scales is particularly fun for people who enjoy playing with numbers, particularly integers, and their relationships to each other in terms of ratios, powers, and common divisors. When designing scales a useful analysis tool is the tonality diamond. That's diamond in the sense of a baseball diamond, not a 14 karat.

Arrange your just intonation scale so that the rows are numerators and columns are denominators as follows:

It isn't a diamond, but if you rotate it 45 degrees it would be. Consider the layout of the following instrument:

Harry Partch's Diamond Marimba with pitches labeled
Photo by Glen Peterson

Harry Partch often designed his scales around a grid like this one. He called it the "Incipient Tonality Diamond." Many of his instruments were designed around this shape as well. The Diamond Marimba was one (shown above), and the Quadrangularis Reversium was another - a mirror image of the Diamond Marimba (with some extra notes added for good measure!). This Tonality Diamond is more than just a mathematical model. Each row or column in this chart represents a consonant chord. Rows are Otonal or major chords (Dominant 7, 9, 11) and columns are Utonal or minor (sort-of). Don't be surprised if this concept seems strange. The Just Intonation Tonality Diamond has no parallel in Western musical notation or instrument design.

Another amazing feature of this diamond is that if you tune a guitar to any row and fret it to play the notes of the diamond, all the other rows can will be represented by frets going straight across the neck. If you tune to a column, all the other columns will be represented by straight frets. Fretting to a tuning of either a row or a column seems to yield more straight frets than any other tuning of a Just Intonation guitar.

Notice that the only even row or column heading is the number 2. A multiple of 2 is an octave and I consider A=440 and A=880 to sound so similar that I treat them as the same note. This allows me to multiply or divide any note by any power of 2 to bring it into the octave we are considering. Any even numbered row or column (except 2) would contain only fractions that would reduce to an existing row in the chart.

Suggested Listening

Harry Partch, Music Of Harry Partch - This music makes no apology. Harry Partch created his own tuning system, harmonic and melodic systems, and musical instruments to actualize these theories. He insisted on corporeal performance - involving the performer's body as well as mind. I like almost everything from Harry Partch. My personal favorites are Revelation in the Courthouse Park (recent recording), and Water Water, but this album is the most accessible.

Terry Riley, The Harp of New Albion - Two hours of delightfully microtonal piano pieces. The piano is tuned to a 12 note JI scale based on C#. Each piece is a planned improvisation. Very listenable, yet obviously microtonal.

The Catler Brothers, Crash Landing - Jon Catler's music has a distinctly microtonal sound without descending into squirming microtonal dissonance as so many composers do. A great progressive rock/fusion album in a JI tuning system.

Ben Johnston, Microtonal Piano - Ben Johnston retunes the piano to a Just Intonation system, but each octave is tuned to a slightly different set of notes. Really interesting stuff.