Composer PETER MORAN explains the Ancient Greek microtonal tuning behind his recent work which was released last year on Benjamin Dwyer's Knowing/Unknowing on Farpoint Recordings.
Enharmonic Harmonics is a 2016 work for guitar and tape which came out of my investigation into the tuning systems of Ancient Greece. In this article, I will explain the nature of these tuning systems, and look at the challenges of adapting them to modern instruments.
Enharmonic Harmonics was written for fellow Irish composer and guitarist Benjamin Dwyer as a companion piece to my earlier electro-acoustic microtonal work Anois 's Arís (written in collaboration with Judith Ring), which also appears on this CD. But whereas Anois 's Arís might be described as “freely microtonal”, Enharmonic Harmonics is a deliberate effort to re-create an ancient tuning on a modern instrument.
The Greek Tetrachords
Greek music was organised into three types (or “genera”) of tetrachords: the diatonic genus, the chromatic genus, and the enharmonic genus. Each type of tetrachord contained four notes spanning a perfect fourth. The diatonic genus, for example, could be played on the notes A#, B, C# and D#, while the chromatic genus could be played A#, B, C, and D#. It was the intervals within the perfect fourth that defined each genus. (My examples here all begin on A#, as that is the starting pitch I used in my own composition, but of course we could begin our scale on any pitch).
Descriptions of how exactly these intervals were tuned vary from one source to another but, by and large, the diatonic and chromatic genera can reasonably be recreated using the tones and semitones of our modern system. It is the third genus, the enharmonic, with its distinctly microtonal intervals, which was the most appealing to me from a compositional perspective.
The enharmonic genus contained two intervals which were smaller than a semitone, and one major third. These smaller intervals are often referred to as quartertones, so the tetrachord might be written as A#, B-quarter flat, B, and D#, but the truth is a little more interesting than that. There were actually a variety of tunings proposed for the enharmonic genus (and indeed all genera), with different approaches being preferred at different times over the centuries.
One of the most famous music theorists of Ancient Greece, Archytas, used precise mathematical ratios to measure the intervals of each genus. Describing these intervals in cents, we can see that when we subtract a pure major third (386c) from a perfect fourth (498c), we are left with a semitone of 112c. Archytas then splits this semitone into two unequal halves, measuring approximately 66c and 46c each, or roughly a thirdtone and a quartertone. (See Figure 1)
[For those who are unfamiliar with this system, cents are just a handy way to measure intervals of different sizes. One regular semitone contains 100 cents, so you can see that the semitone in the above example is slightly wider than usual.]
Figure 1: The intervals of Archytas' enharmonic genus measured in cents:
Re-Tuning the Modern Guitar
Archytas' exact measurement of those intervals can give us a starting point for recreating the enharmonic genus on a modern instrument. Since other Greek writers were less precise in their measurements, we know that the exact tuning can be somewhat flexible. There is also the question of how well the instruments of the ancient world could hold their tuning at all, and whether these measurements were treated with such mathematical precision in performance practise.
And so, with this small degree of flexibility, it is possible to recreate the enharmonic genus on a modern guitar with only a simple re-tuning of the strings, and straying only slightly from Archytas' original measurements. The key lies in the seventh harmonic on String VI, the low E-string.
The seventh harmonic on any string will sound a minor seventh (and two octaves) above the open string, and it is always 31c flatter than the same pitch in equal temperament. (The seventh harmonic can be played by lightly touching the guitar string above the third fret). So, on String VI, the seventh harmonic will sound a D, slightly flatter than the D played on the 15th fret of String II (the B-string).
To re-tune the guitar to play the enharmonic genus, first flatten String II so the D on the 15th fret is in unison with the seventh harmonic on String VI. The B-string is now 31c flatter than an equal-tempered B. Or, to put it another way, it is 69c sharper than a A#. Borrowing from the language of tuning theory, I refer to this pitch as “B-septimal-flat”, since it is derived from the seventh harmonic. Playing these three pitches in order — A#, B-septimal-flat, B — we already have the first three steps in the enharmonic genus. And the intervals created — 69c and 31c — are extremely close to the 66c and 46c intervals proposed by Archytas. Adding a major third completes the tetrachord, A#, B-septimal-flat, B and D#.
To access this tuning across the full range of the guitar, we can also flatten String VI itself, until it is a perfect 12th below String II. This pitch then, would of course be E-septimal-flat.
The final adjustment made to the guitar tuning in my composition Enharmonic Harmonics is simply to sharpen Strings III (G) and IV (D) to G# and D# respectively (See Figure 2). This allows us to add an additional tetrachord beneath that already described. The pitches D#, E-septimal-flat, E and G#, can now be played on Strings IV and VI. This is the same enharmonic genus transposed down a fifth.
In fact, these two tetrachords can now be played back-to-back across a full octave: D#, E-septimal-flat, E, G#, A#, B-septimal-flat, B and D#. These are what the Greeks called “disjunct tetrachords”, with two tetrachords of the same genus placed side-by-side in this way. This, then, is the full set of pitches employed in Enharmonic Harmonics.
Figure 2: Guitar scordatura in Enharmonic Harmonics:
Being Practical in a Modern Context
For many composers and performers, alternative tuning systems seem at first glance to offer insurmountable practical challenges. Harry Partch built an entirely original collection of instruments just to tackle these problems, but I hope that I have demonstrated that we do not have to go so far. With a little careful planning and forethought, we can adapt our modern instruments to alternative systems without presenting too many difficulties for the performer.
Partch himself famously wrote his own short Study on Archytas' Enharmonic as part of his Eleven Intrusions. That was perhaps the first time that rare and beautiful tuning had been heard in many centuries. It is exciting to think that this piece could form part of such a feint historical thread, and hopefully, we will hear much more of it in the future.
Dr. Peter Moran is a composer and performer, and founder and co-editor of the AIC New Music Journal.
Enharmonic Harmonics is now available from Farpoint Recordings on Ben Dwyer's Knowing/Unknowing.