Microtonality is one of those seemingly deeply esoteric terms that both mystifies and terrifies musicians. The purpose of this series is to open the door of understanding to the world of microtonality and, ultimately, intonation. The question of how to achieve intonation in choral music has plagued conductors for centuries. We struggle with teaching our singers to hear the alignment of harmonies, and training their technique to maintain the pitches they hear.
To make sure we’re all on the same page, here’s a list of terms I’ll be using in this series:
|Overtones/Harmonics||The upper harmonics present in all pitches|
|Hertz (hz)||The absolute frequency of any given pitch|
|Equal Temperament (ET)||The division of the octave into 12 equal parts|
|Cents||A Division of half-steps in Equal Temperament into 100 parts|
|Just Intonation||A tuning based on low, whole number ratios derived from the overtone series|
Part 1: What is Consonance?
To fully understand intonation, we must first understand consonance. In traditional music theory, we are taught certain intervals are dissonant (2nds, 7ths, and tritones), and others consonant (3rds, 6ths, perfect intervals). But within these we recognize degrees of consonance, with some intervals being more stable than others. In truth, there is no specific line of demarcation when consonance ends and dissonance begins; all intervals are on a spectrum of consonance.
Ultimately, what defines the level of an intervals consonance is a question of how the overtones of the two pitches overlap. The more overlap, the more consonant and stable we perceive the pitches to be. For an excellent introduction to harmonics (overtones) check out this site. Here’s a quick primer on the overtone series:
Sound is caused by vibration in the air and within each vibration are a series of smaller vibrations. These smaller, and higher, vibrations are the overtone series. This rising scale of subtle pitches emanating from the original pitch always follows the same pattern. So, if our first pitch is at 100 Hz (we’ll call it a C, though it’s not really) the series will look like this:
|Overtones of 100 Hz|
*These pitches differ substantially from the ones we would hear on the piano. The F# in particular is about a 1/4 step lower than on the piano.
This pattern of overtones is always the same, no matter what pitch you start on. The next harmonic is always increased by the amount of the original number of cycles per second, which we hear as an octave, then 5th, than octave again, then 3rd, and so on.
Here we can see the shared harmonics in a major 3rd. Notice there are only two shared harmonics in the first 10 overtones (though the other ones are close). This creates the kind of colorful consonance we associate with thirds.
|Harmonics of a Fundamental with a Major Third|
*Some note names are the same as the harmonics from C, but there is a different Hz value. We’ll go over this during the next article on Just Intonation.
By contrast, the perfect fifth is one of the most consonant intervals, below we can see why: within the first 8 harmonics of the fifth, half of them align with the lower pitch.
|Overlapping Harmonies of a Fundamental with a Perfect Fifth|
Interestingly, all of the first four harmonics in the intervals of a perfect 5th + an octave are shared with the lower pitch (as shown below), with none that are not shared. This means that an octave+5th is actually more consonant than a regular perfect 5th. This shows not all intervals have an equal amount of consonance when transposed by an octave, but can actually be more or less consonant depending on the size of the interval.
|Overlapping Harmonies of a Fundamental with a Perfect Octave and a Fifth|
So, what does that mean for us as singers? Well, check out the next article on Just Intonation.
Fahad is the director of See-A-Dot Music Publishing, Inc., a company devoted to the advocacy of new choral works and emerging composers. He is an active performer, composer, and conductor and specializes in contemporary and experimental music, particularly improvisation and the use of extended vocal techniques. He is co-artistic director for the interdisciplinary music/dance company The Resonance Collective, is a founding member of the C3LA: the Contemporary Choral Collective of Los Angeles, and a board member of C4: The Choral Composer/Conductor Collective. He is currently a faculty member, choral director, and doctoral candidate in the performer/composer program at the California Institute of the Arts.