An excellent example of the way in which intervals of the harmonic series occur in music comes from the Prologue to Benjamin Britten’s Serenade for Tenor solo, Horn and Strings. Britten asked that the player use only natural harmonics, a simple enough task for the horn player who produces the overtones above a given fundamental by altering the vibrations of the lips. Britten exploits certain harmonics for their tonal effect in a most masterful way.
Example 1.1 shows the Prologue in its entirety. Taking a close look at the score we see the familiar interval of a perfect 5th (bar 1) establishing the tonality of F concert and in conjunction with the choice of rhythm creating a mood of openness and evocation. The horn player does this by producing the 8th harmonic then the 12th harmonic. Diverting our attention briefly to Ex. 1.2 we see harmonics 8 and 12 in the series and between them the same interval as between harmonics 4 and 6, and also 2 and 3. This demonstrates an important aspect of JI language, the language of ratios. The interval of the perfect fifth when written as a ratio reads 3:2 or 6:4 or 12:8. Furthermore the convention when using ratios for JI represents an interval in the smallest possible terms, therefore 12:8 (or 6:4) becomes 3:2 showing that these three fifths have exactly the same ratio.
To further demonstrate the harmonics used by Britten, the number of each harmonic in the harmonic series has been included underneath each pitch in Example 1.1. Some of them deserve special mention. In bar four we hear the evocative and haunting sound of the 11th harmonic. Part of this inherent sound quality arises because it bears almost no relation to our accepted diatonic scale. Once we gain a
familiarity with this pitch we can enjoy it for its own sake and not hear it as out of tune, for harmonics occur naturally. Here I echo Lou Harrison who chooses pitches
because he ‘knows’ their evocative nature although he cannot describe what they evoke other than “certain emotional feelings” (Harrison 1987: 7). We could say that harmonics represent natural intonation and that human beings have constructed scales on the basis of the natural intervals of the harmonic series. From this point of view some pitches in our diatonic scale should sound odd such as the fourth and sixth degrees in the key of C because they do not naturally occur in the harmonic series. For instance one type of pentatonic scale appears naturally as harmonics 8, 9, 10, 12, and 15 giving notes C, D, E, G, B. However we have become so used to the sound of a major scale (C, D, E, F, G, A, B, C) that pitches like the 11th and 13th and even the 7th harmonics can sound ‘wrong’ as we tend to associate them with scale degrees 4, 6 and 7. Of course we can get used to the inherent qualities of these harmonics.
Example 1.1. Britten, B. Serenade Op.31. The Prologue shows a use of harmonics.
CD Tracks 1 and 2.
Example 1.2. The harmonic series in staff form with arrows to show differences
higher or lower than Pythagorean intonation as explained in the Introduction and
Chapter 3. Having set the fundamental pitch we can determine all other pitches
accurately by number (ratio) and by production on a suitable instrument.
CD Track 3.
We still do not have a vocabulary to describe the emotional effect of the 11th harmonic and we rarely describe more common harmonic intervals, such as the 3:2 ratio (perfect 5th ); 5:4 (major 3rd); 6:5 (minor 3rd), in other than the simplistic terms of stable, happy and sad respectively. In bar four Britten moves from harmonic 8 to 11, back down in pitch through the 10th and 9th harmonics, with these leading to the last two notes of bar two followed by a repetition of bar three in the following bar. These notes in bar five seem different now for having made our strange journey to the non-diatonic interval of the 11th harmonic. Example 1.3 shows bars four and five of the Prologue for convenience.
Example 1.3. Bars four and five from Britten’s Prologue .
The more simplistic language of ratios unambiguously shows this group of intervals in bars four and five as:
8:11 11:10 10:9 9:6 (3:2) 6:10 (3:5) 10:8 (5:4)
However, staff notation becomes confused. The 11th harmonic could be written as an F or an F#. Obviously Britten chose F. The 11th harmonic sounds radically different to an F from Pythagoras’ scale. It sounds 53 cents higher (a 33:32 higher than 4:3) therefore 551 cents higher than the 8th harmonic (ratio 11:8). As Pythagoras’ whole tone has a distance of 204 cents and his semitone 90 cents this difference of 53 cents (considerably more than half his semitone) is quite radical. The Pythagorean F# sits 612 cents above C with ratio 729:512 therefore the 11th harmonic sits 59 cents lower than F#. The 11th harmonic is thereby six cents closer to F than to F# giving one reason to prefer F to F# for notation purposes.