The Tuning of Lou Harrison's Adagio, Arioso

from the Mode CD Por Gitaro: Suites for Tuned Guitars performed by John Schneider

Notes by Bill Alves


The fingerboard used in John Schneider's recording of the Ditone Suite.		 The oldest tuning for which we have a record is ditone tuning. It is also known as Pythagorean tuning, but, as Harrison was quick to point out, Pythagoras himself lived many centuries after this tuning was first recorded on a Sumerian tablet. The tuning is based on the simplest ratio less than an octave (apart from the unity, 1/1), that is, 3/2, the ratio usually represented as a perfect fifth. A musician finds pitches in this tuning by repeatedly moving up by this 3/2 perfect fifth (moving the pitch into the correct octave when necessary). Music theory students know this pattern as the "circle of fifths," but it only forms a circle when the fifths are tempered (or, as Harrison would say, "tampered with"). Otherwise it just keeps spiraling upward until the tuner decides to stop (because no matter how many 3s you multiply over 2s, it will never become an octave).

From his earliest studies with Henry Cowell, Harrison sometimes composed using a technique he called "quintal counterpoint," where instead of thirds, perfect fifths, sixths, and octaves being considered the consonances as in traditional counterpoint, only perfect fourths, perfect fifths, and octaves are. Therefore thirds are treated as dissonances and have to be "resolved" to fourths or fifths. This perspective makes sense in a tuning system based on these 3/2 perfect fifths, because the major third, for example, is represented by the relatively complex ratio of 81/64 (instead of the sweet 5/4 found in other just intonation tunings on this disc). This ratio is created by the combination of two "tones" (9/8 whole steps), which is the origin of Harrison's prefered name for this tuning: "ditone." Harrison was well aware that composers of the 12th to 14th centuries in Europe, when this tuning was the standard, also treated the ditone third as dissonant. Harrison's Ditone Suite evokes the austere harmonies of this period. For this recording, John Schneider also used the ditone tuning for Adagio, Arioso, originally from Harrison's Harpsichord Sonata.

Here are the details for this form of tuning:

PitchC C# D Eb E F F# G G# A Bb B
Ratio from 1/11/1 2187/2048 9/8 32/27 81/64 4/3 729/512 3/2 6561/4096 27/16 16/9 243/128
Cents from 1/10 113.7 203.9 294.1 407.8 498 611.7 702 815.6 905.9 996.1 1109.8
Frequency261.6 279.4 294.3 310.1 331.1 348.8 372.5 392.4 419.1 441.5 465.1 496.7
Ratio between pitches2187/2048 256/243 256/243 2187/2048 256/243 2187/2048 256/243 2187/2048 256/243 256/243 2187/2048
Cents between pitches113.7 90.2 90.2 113.7 90.2 113.7 90.2 113.7 90.2 90.2 113.7

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