Harmony: the Interaction and Intonation of Tonal Sounds

Different frequencies of vibration coexisting – in a sounding body, a resonator or in the air itself – produce interferences, which are perceived as audible patterns: characteristic beatings and combination tones. Intonation is the art of learning to recognise, distinguish, produce and shape composite rhythms and inner melodies within any combination of sounds: namely, to make harmony. To explore harmony in an experimental way is to investigate how these interactions of tone are sensed, how minute changes and shadings of intonation affect the perceptions of beating, fusion, spatialisation, colouration, orchestration, consonance and dissonance within sound-aggregates.

Tonal sounds are any sounds heard as pitches, and are generally composed of a spectrum of harmonic partials, whole number multiples of a fundamental vibrating frequency. In the special case of an electronically generated sinewave, only the fundamental is sounding. Our brain perceives tonal sounds by grouping harmonically related partials with similar morphologies to reconstruct layerings of individual timbres. For example, it is possible to distinguish a clarinet playing while a human voice sings a particular vowel, even on the same pitch.

Some tonal sounds, like those produced by a piano, include partials that deviate from an ideal series of harmonics. However, even so-called inharmonic mixtures, like some multiphonics or gong-like sounds, in which several fundamentals are perceived, may be analysed by the hearing process as a superposition of (possibly detuned) intervals definable as frequency ratios.

Tonal sounds may be sensed as being proximate when some of their respective partials vibrate at the same, or nearly the same, frequencies. If this is true of their perceived fundamentals, the melodic distance between the sounds is small, that is, there is nearly a unison or alignment of all partials. When a unison takes place between some of the higher partials, even though the sounds' fundamentals differ, another kind of proximity may be perceived, called harmonic distance. The stronger the audible interaction between the proximate partials, the closer the relationship between fundamentals will seem. The more equal the proximate partials' volumes, the closer they are to their respective fundamentals, the more likely that a consonant intonation may be possible by minimising or eliminating beating.

Harmonic distance may be quantified in various ways. I think it is most usefully modelled by James Tenney's generalised harmonic space lattice, with dimensions generated by each prime number. Distances are measured by following the pathways along various axes connecting individual pitches in a network. Compactness and proximity of pitches represented in this harmonic space provide one way to imagine music composed in a microtonally extended Just Intonation, which is the practice of conceptualising intervals between tonal sounds as frequency ratios, tuning these intervals accordingly, and thereby searching for ways of hearing and composing the manifold tonal relationships between sounds.

Every timbre suggests harmonic constellations, but timbre and harmony ought not to be mistaken for each other; harmony is the more general musical principle. Timbre describes the composition, morphology and colour of a specific sound; harmony describes a world of potentially perceivable relationships within the interactions of any combination of tonal sounds. By changing or inflecting timbres, intensities, registers, etc. in composition and in the playing of music, different aspects of the harmonic relationship unfolding in time are highlighted or revealed.

1: Harmonic Space, the Pythagorean Diatonic and a Well-Tempered Just Intonation Tuning

Harmonic space may be extended infinitely and symmetrically from any given pitch. Its exploration is made possible by establishing a notational frame of reference, which enables various proximate relationships of different dimensions to be compared, composed and played.

Each dimension of harmonic space is generated by the ratio of a prime number to unity (1:p or p:1). Every other ratio may be written as a product of such elementary proportions, which may be interpreted as a collection of steps in harmonic space. The simplest harmonic step is the octave 1:2. An equal division of this ratio into many small intervals produces various atonal approximations of the infinite space of all tonal, rational pitch relationships. These irrational constructions are useful as measurement schemes, revealing enharmonic proximities and assisting musicians to come close to hearing and playing tonal pitch relationships that may actually be heard.

To approach any rational relationship within the margin of error of an acoustic instrument, the division of the octave into 1200 cents, with the frequency chosen for the tuning reference note A set to 0 cents, is the most practical and compatible with current music practice. On the other hand, to interpret arbitrary cents values harmonically as sequences of steps in harmonic space, it is necessary to make a reverse mapping. For any given pitch, what are the simplest rational relationships occurring within a small melodic distance? In other words: what simple combinations of steps in harmonic space return close to their starting point without backtracking?

The notation of harmonic space requires a representation of ratios from an arbitrary origin (1/1). For practical purposes, we may call this origin D, which has the advantage of being symmetrically positioned in the existing system of pitch notation, and on the standard keyboard layout. The simplest harmonic interval that generates different pitch-classes is the ratio 1:3 (octave plus fifth). Extended above and below D, and combined with octave transpositions, a Pythagorean series is obtained.

The first seven pitches of this series, symmetric around D, are F-8c C-6c G-4c D-2c A-0c E + 2c B + 4c. This sequence of pitches, ordered melodically, produces a Pythagorean diatonic division of the octave into five major whole-tones – C-D-E and F-G-A-B (all in the proportion 8:9 or 204c) – and two semitone limmas – E-F and B-C (both in the proportion 243:256 or 90c). Note that the two limmas, taken together, are approximately 180c or 24c (one Pythagorean comma) less than a whole-tone. Similarly, six Pythagorean whole-tones, each consisting of two fifths less one octave, when combined, exceed an octave by 24c.

This Pythagorean diatonic division is compatible with many currently used musical instruments: it matches the natural tuning of the open strings of violins, violas, cellos and basses, viols, guitars and lutes as well as traditional stringing and fretting systems of classical Turkish, Arabic and Persian instruments. So it is the most logical starting point for a common space of tonal sounds.

How may the Pythagorean diatonic be extended to map a more general harmonic space? Continuing the sequence of fifths until there are 13 pitches produces the small Pythagorean comma mentioned above. Therefore, a more complete tonal space with perfect 2:3 fifths will require, as a minimum condition, a microtonal division into steps approximating one comma.

Returning to a simpler approach, taking the diatonic pitches once again as starting point, it would also be possible to divide each of the whole-tones in two approximately equal parts, obtaining a sequence of 12 semitones and accepting some more or less ‘out-of-tune’ approximate fifths. This is the process that has produced the common 12-note keyboard layout used today. If the seven diatonic notes are established by the Pythagorean diatonic, producing a sequence of six perfect fifths, the remaining five pitches might usefully be chosen so that each additional ‘fifth’ produced is slightly too small by about 1/6 Pythagorean comma, resulting in a well-tempered ‘circle’ of fifths. Another way of looking at this would be to say that each successive semitone in the sequence of fifths – namely above E, A, D, G and below C, G, D, A – must slightly increase in size, ranging from a limma of 90c to a half-tone of 102c equally dividing the 204c G:A interval.

The simplest rational semitones of this size may be obtained from the harmonic series by considering the ratios 16:17 (105c), 17:18 (99c) and 18:19 (94c). To obtain a near-equal rational division of the whole-tone G:A (8:9 or 16:18), consider the simplest epimoric (n:n + 1) division 16:17:18 (G:A@:A) and also consider its inversion, produced by taking first 17:18 (G:G#) followed by 16:17 (G#:A). The first division is obtained by playing the natural harmonics 16, 17, and 18. The second division is easily produced on a monochord or fretted instrument by dividing the octave in 18 equidistant units. The proportion G:G#:A@:A, combining both divisions, may be written as (16*17):(16*18):(17*17):(17*18) or 272:288:289:306. By taking the arithmetic mean of 288 and 289, and doubling the numbers, the mean proportion 544:577:612 is obtained. 577 represents the mean semitone which may be interpreted as either G# or A@. G:G#=A@ with proportion 544:577 is 101.96c and G#=A@:A with proportion 577:612 is 101.95c, obtaining the two desired half-tones.

Thus, by tuning A:B@ and F#:G as 18:19, D:D# and D@:C as 17:18, and dividing G:A as above, it is possible to complete a well-tempered circle of ‘fifths’ in which all of the notes are harmonically related in a space defined by the mostly-smaller primes 2, 3, 17, 19 and 577 (!). The resulting tuning is excellent in all keys and offers precise and fascinating just harmonies. Of course, the most simple major triads: F, C, and G, are Pythagorean, since there are no 5-limit consonances, but this inversion of historical practice is in fact remarkable and refreshing to my ears. I highly recommend the adoption of this Just Intonation (JI) well temperament for solo and chamber music. The beating differences between the variously complex, rationally tuned thirds highlight, to my ears, the falsity at the heart of conventional equal temperament, which disguises and softens its dissonance by imposing a symmetrical uniformity.

The first of my three tables for Bob is a notation of this piano tuning, with a complementary extension into the ‘quartertone’ realm based on harmonics 5, 7, 11 and 13 for an optional second piano. The inspiration for this tuning was the traditional Vallotti well temperament with six Pythagorean fifths and six tempered fifths reduced by 1/6 Pythagorean comma. The more colourful but still well-tempered JI tuning of the second piano, with one fifth even tuned slightly larger than 2:3, reflects the fantasy of early historical systems like the French ‘tempérament ordinaire’.

Taken together, both pianos present a spectrum of just ‘near-thirds’ and ‘near-fifths’ producing intervals that echo the various comma-fractional alterations of historical temperaments, but also offering pitches that may be integrated without compromise in a larger mosaic of microtonally extended Just Intonation.

2: The Euler Lattice and Primary 23-Limit Enharmonic Proximities

How far is it useful to extend a representation of harmonic space beyond the Pythagorean division? The Extended Helmholtz-Ellis JI Pitch Notation, described elsewhere (plainsound.org), allows notation of the harmonic series from any fundamental as far as the 64th partial and beyond. In my experience, some intervals up to at least the 23rd partial may be directly tuned in an appropriately composed register, timbre and musical context. In my recent music, I generally have not exceeded the 23rd partial, and for the most part am exploring intervals and aggregates up to the 13th harmonic. Higher primes above 23 are perhaps more usefully notated by means of cents and a text indication rather than by using special accidentals, which I now prefer to reserve for the more easily perceived lower primes. Therefore, in defining a generally useful subset or region of harmonic space, I would like to provide for a precise notation of intervals that may be tuned directly by ear in a musical context.

Table 1:

Table 2a: 53-tone 5-limit Euler lattice (shaded) with 46 enharmonic border tones and 23-limit harmonics / subharmonics from D

Table 2b: Some 23-limit enharmonic proximities

At the same time, some pitches or ratios that may easily be heard and tuned in some circumstances end up requiring a visual notation that is excessively laden with signs. This happens because the musical point of reference is not always perceived as monophonic, in the sense of Harry Partch (a fabric derived from and always respecting one originating pitch), but rather as a shifting sequence of reference points, each related to one another. The simplest way to cope with this notational difficulty is to be able to move flexibly between a fixed absolute pitch notation and a floating relative pitch notation – after all, ancient Greek music had both systems concurrently. The smallest residual commas in a modulation, which then affect a range of subsequent pitches, might be more effectively subsumed into a slight shifting of the Kammerton, like a shifting of key. Another approach would be to substitute very near almost-equivalent pitches with a simpler notation, and to indicate the desired harmonic relationships as ratios in the score. In both situations, including exact desired cents values may provide the desired intonation without compromise.

The most consonant basis for harmonic space, and the simplest notational extension of the Pythagorean series, as mentioned above, is a comma-based division. The 5-limit Euler lattice based on primes 2, 3 and 5 produces the network of pitches that underlies both the Indian raga system of sruti regions and the European practice of consonant counterpoint. In my 2011 string quartet Euler Lattice Spirals Scenery the section ‘Harmonium for Ben Johnston’ maps a complete journey through a central region of 53 pitches combined with their enharmonic peripheral neighbours: 46 additional pitches completing just major and minor triads. I have composed a pathway through this space of 99 pitches by linking them in a chain of common-tone triadic modulations with one very small enharmonic transition of 4.2c (10460353203:10485760000).

In my second table for Bob, this Euler lattice subset is superimposed with 23-limit harmonic and subharmonic intervals calculated from D, to reveal some of the primary enharmonic proximities in the extended JI universe. The higher prime consonances may be considered as ways of more directly leaping to distant regions of the Euler lattice with slight enharmonic variations of intonation. These relationships are indicated in a notational shorthand for the relationship to a fundamental: D:7^o or D^7o or D⁷ may be used to indicate an otonal relationship, and _u7:D or _u7D or ₇D may be used to indicate a utonal one.

3: A map of 23-limit harmonic space

Investigating very small pitch variations produced by following different harmonic pathways inspired the third table, which is a rising map of 23-limit harmonic space reduced to the simplest combinations of The Extended Helmholtz-Ellis JI Pitch Notation (HE accidentals). The idea for this table evolved from my work on the micromelodeon, an algorithm created in 2008 for real-time harmonic microtuning based on projecting tuneable intervals two steps removed from the Pythagorean pentatonic.

In writing music for ensembles of musicians, I am seeking to open awareness of the subtle harmonic variations made possible by considering the intervals of a microtonally extended just intonation. To make the tuning and the harmonic motivation of such intervals clear, I would like to invite musicians to understand both a cents notation and a partial based accidental notation. Even though some pathways composed in my music lead to sequences of more than two accidentals, I find such combinations generally too difficult to read in real-time music making, especially for larger ensembles. So I decided to attempt a reduction of the infinite nesting of rational relationships by limiting the number of microtonal signs.

The comma-based division established by the Euler lattice divides the octave unequally in 53 comma-sized regions. Each diatonic whole-tone has approximately nine comma-sized regions, each apotome (sharp or flat) five comma-sized regions and each limma four comma-sized regions. Notes altered by three Syntonic commas are only needed as enharmonic completions of major or minor triads, so for the most part, two-comma alterations are sufficient, and the use of double-sharps and -flats is minimised.

Another way of constructing the Euler lattice, arguably a more compact one, would be to have rows of nine pitches, each spanning eight perfect fifths. For example, the central row could be restricted to the central nine fifths from B@ to F#. In the row above, the rightmost pitch would be A# lowered by a syntonic comma, lying one schisma (2c) higher than the leftmost B@ in the central row. Similarly, in the row below, the leftmost pitch would be G@ raised by a syntonic comma, one schisma lower than the rightmost F# of the central row. Expanding to a block of seven rows would obtain 63 pitches. Six pairs of pitches are a schisma apart, and four pairs of pitches are in the proportion 15552:15625 or 8.1c apart.

In addition to facilitating readability (fewer arrows, double-sharps and double-flats), there are several other reasons why I have chosen a different expansion of the 53-comma octave, favouring longer sequences of fifths in five rows set off from each other. My central row runs from E@ to C#, with enharmonic neighbours A@ and G# delineating the Pythagorean comma. The main 53 include, however, the schisma altered versions of these tones: G#-comma-down and A@-comma-up, producing the most natural transition from Pythagorean to Syntonic intervals when building the sruti regions around D. Limits of the single-comma rows are determined by completing the otonal triad on A-comma-up and the utonal trial under G-comma-down. The double-comma rows begin, therefore, from E#-double-down and C@-double-up, respectively. These are then extended to reach the elegant enharmonic seam dividing the limmas E–F and B–C in four commas and a small remainder of 4.2c (see the second interval noted in Table 2b and mentioned above).

To make the following map of harmonic space written in a simplified subset of HE accidentals I decided to limit the combinations to two signs, in a manner I expect any musician might readily learn to read intuitively. Many complex combinations may be closely approximated, within a few cents, by simpler ones. Small corrections could nevertheless produce any desired interval, which may be notated in the music as a ratio. Other pitches are perhaps most easily notated by either subsuming accidentals into a Kammerton shift, as noted above, or by means of ratio and cents indications. The large boxed numbers indicate 53 regions of approximately one comma, dividing the Pythagorean diatonic in five nine-comma whole-tones and two four-comma limmas. The Euler lattice is combined with an otonal and utonal expansion from D, including the most clearly perceptible tuneable intervals and epimoric ratios up to the Syntonic comma.

This third table is a just intonation counterpart to the approximation of pitches in cents: a reverse-lookup table mapping cents to frequency ratios. It suggests one model of how combinations of the simplest prime partials, taking only a few harmonic steps from the central Pythagorean region of harmonic space, establish manifold enharmonic proximities and create a flowing complex design of unfolding harmonic relationships.

Table 3: