2014 Update from William Copper
I've been trying to put these thoughts in better order, in blog form at the intonalist.
Please visit The Intonalist by William Copper
These remarks apply to markings in some of the music published by Hartenshield Music. They all indicate adjustments from the tuning of a piano (equal temperament) to the correct tuning of an orchestra or chorus (pure or just tuning).
These notes are incomplete (June 2013) ... I'm working on them. Hints of my direction may be found in the O Danny Boy Page. William
There are footnotes for specific pieces at the bottom of this page. All the markings are in one sense approximations, like dynamics markings -- but in another sense, very specific: there is only one exactly pure major third, and it should always be tuned perfectly in an ideal chorus. Expressive variations in pitch are natural, expected, and desired -- but the basic intonation for the works using these markings is just. Each major triad has a pure fifth and a pure major third. Each minor triad has a pure fifth and a minor third tuned as between the third and fifth of the harmonic series. Where modifications are required -- and they frequently are, in any music -- they are indicated by means of the markings.
From a given tonic, the fifth higher (dominant) and fifth lower (subdominant) are each tuned slightly differently in just tuning: the dominant fifth is about two cents higher, and the subdominant about two cents lower. In the range around middle C, the difference makes an audible beat of around 1 per second. These are marked with a small arrow above the note, pointing up for the dominant and pointing down for the subdominant.
By extension, when the second scale degree is used as a dominant-of-the-dominant or as the fifth of a dominant chord, it too is tuned higher (about four cents); the subdominant of the subdominant, or flatted seventh, is tuned lower (about four cents). These are marked with the same arrow markings. Where there is a true dominant of dominant (V of V), with a raised third, then the fifth of this chord is raised yet again, by two more cents (about six cents).
The thirds of major triads generally are tuned significantly lower than the corresponding note in equal temperament. These are marked with a filled triangle, with the apex pointing down. The thirds of different triads are tuned by increasing amounts (relative always to equal temperament) as chords move in the subdominant direction around the circle of fifths, and by decreasing amounts as chords move in the dominant direction around the circle of fifths.
The third of a tonic triad should be tuned about 14 cents low; the third of a dominant triad about 12 cents low; the third of a subdominant triad about 16 cents low.
The thirds of minor triads, in a major key, generally are tuned as fifths from the tonic of the key, while the root and fifth are tuned in parallel an amount corresponding to the difference between a minor third in equal temperament and in the third between a pure major third and a pure fifth. For the chord build on the second scale step of a minor key, the third is tuned just as a subdominant, while the root is tuned down about 18 cents and the fifth is tuned down about 16 cents. The markings for the root and fifth of such a minor triad are the same as the major third marking: a filled triangle with apex down.
The thirds of minor triads in a minor key are tuned up by about 18 cents; they are marked with an open triangle with apex up.
Some markings show non-tonal notes with a down-pointing arrow and a number of cents for tuning.
(*1) Page 6. Non-tonal notes, when part of a clear chord from tonal theory, are often marked with standard markings. Here, in measure 4, the soprano F# is the third of an implied secondary dominant to the sixth scale step, so the third is tuned down (about 6 cents).
(*2) Page 6. Sometimes a part must change intonation on a held note; the baritone part, measure 7, should begin on a high second scale step (root of a secondary dominanant) and then mutate to a lowered second scale step (root of the minor triad on the second degree of the scale).
(*3) Page 7. Measure 16, non-tonal notes tuned as indicated (soprano, alto).
(*4) Page 7. Measure 27, tune to the third of the triad, the lowered seventh. It is tuned as a fourth from the subdominant (4 cents lower); the whole measure is tuned around the final chord, the minor chord on the second scale step. The C is tuned down about 18 cents, while the F is tuned down about 20 cents. (This location could be interpreted differently, but the given interpretation is based on the similar music in the recapitulation, where the first beat of the measure is on the second scale step, and tuned around the subdominant note.)
(*5) Page 9. Local tonic, Db, marked as a lowered tonic (lower by about 6 cents).
(*6) Page 9. Tune the Db as a minor third up from Bb tonic.
(*7) Page 10. Local tonic, D, marked as a raised tonic (higher by about 8 cents).