Dynamic tonality

A vibrating string, a column or air, and the human voice all emit a specific pattern of partials called the Harmonic Series. ("Partials" are also called "harmonics" and/or "overtones.") Each musical instrument's unique sound is called its timbre, so we can call an instrument's timbre a "Harmonic timbre" if its partials are emitted as per the Harmonic Series.

Just Intonation is a system of tuning that adjusts a tuning's notes to maximize their alignment with a Harmonic timbre's partials. This alignment maximizes the consonance of music's tonal intervals, and is, arguably, the source of tonality.

Unfortunately, the Harmonic Series and Just Intonation share an infinitely-complex—i.e., rank-∞—pattern that is determined by the infinite series of prime numbers. A temperament is an attempt to reduce this complexity by mapping this rank-∞ pattern to a simpler—i.e., lower rank—pattern.

Throughout history, the pattern of notes in a tuning could be altered by humans but the pattern of partials sounded by an acoustic musical instrument was unalterably determined by the physics of the Harmonic Series. The resulting misalignment between pseudo-Just tempered tunings and fully-Harmonic untempered timbres made temperament "a battleground for the great minds of Western civilization."[4][5][6] This misalignment, in any tuning that is not fully Just (and hence infinitely complex), is the defining characteristic of the Static Timbre Paradigm.

Many of the pseudo-Just temperaments proposed during this "temperament battle" were rank-2 (two-dimensional)—such as quarter-comma meantone—that provided more than 12 notes per octave. However, the standard piano-like keyboard is only rank-1 (one-dimensional), affording at most 12 notes per octave. Piano-like keyboards affording more than 12 notes per octave were developed by Vicentino,[4]^:127 Colonna,[4]^:131 Mersenne,[4]^:181 Huygens,[4]^:185 and Newton,[4]^:196 but were deemed cumbersome and difficult to learn.[4]^:18

The endpoints of the valid 5-limit tuning range of the syntonic temperament, shown in Figure 1, are:

• P5=686 (7-TET): The minor second is as wide as the major second, so the diatonic scale is a seven-note whole tone scale. This is the traditional tuning of the traditional Thai ranat ek, in which the ranat's inharmonic timbre is maximally consonant.[8]^:303 Other non-Western musical cultures are also reported to tune their instruments in 7-TET, including the Mandinka balafon.[12]
• P5=720 (5-TET): The minor second has zero width, so the diatonic scale is a five-note whole tone scale. This is arguably the slendro scale of Java's gamelan orchestras, with which the gamelan's inharmonic timbres are maximally consonant.[8]^:73

The partials of a pseudo-Harmonic timbre are digitally mapped, as defined by a temperament, to specific notes of a pseudo-Just tuning. When the temperament's generator changes in width, the tuning of the temperament's notes changes, and the partials change along with those notes—yet their relative position remains invariant on the temperament-generated isomorphic keyboard. The frequencies of notes and partials change with the generator's width, but the relationships among the notes, partials, and note-controlling buttons remain the same: as defined by the temperament. The mapping of partials to the notes of the syntonic temperament is animated in Video 5.

On an isomorphic keyboard, any given musical structure—a scale, a chord, a chord progression, or an entire song—has exactly the same fingering in every tuning of a given temperament. This allows a performer to learn to play a song in one tuning of a given temperament and then to play it with exactly the same finger-movements, on exactly the same note-controlling buttons, in every other tuning of that temperament. See Video 3 (Same Shape).

For example, one could learn to play Rodgers and Hammerstein's Do-Re-Mi in its original 12-tone equal temperament (12-tet) and then play it with exactly the same finger-movements, on exactly the same note-controlling buttons, while smoothly changing the tuning in real time across the syntonic temperament's tuning continuum.

The process of digitally tempering a pseudo-Harmonic timbre's partials to align with a tempered pseudo-Just tuning's notes is shown in Video 6 (Dynamic tuning & timbre).[13]

Dynamic Tonality enables two new kinds of real-time musical effects: those that require a change in tuning, and those that affect the distribution of energy among a pseudo-Harmonic timbre's partials.

Dynamic Tonality's novel tuning-based effects[2] include:

• Polyphonic tuning bends, in which the pitch of the tonic remains fixed while the pitches of all other notes change to reflect changes in the tuning, with notes that are close to the tonic in tonal space changing pitch only slightly and those that are distant changing considerably;
• New chord progressions that start in a first tuning, change to second tuning (to progress across a comma which the second tuning tempers out but the first tuning does not), optionally change to subsequent tunings for similar reasons, and then conclude in the first tuning; and
• Temperament modulations, which start in a first tuning of a first temperament, change to a second tuning of the first temperament which is also a first tuning of a second temperament (a "pivot tuning"), change note-selection among enharmonics to reflect the second temperament, change to a second tuning of the second temperament, then optionally change to additional tunings and temperaments before returning through the pivot tuning to the first tuning of the first temperament.

Dynamic Tonality's novel timbre effects[3]^:39-40 include:

• Primeness: Partials 2, 4, 8, 16, …, 2ⁿ are factorised only by prime 2, and so these partials can be said to embody twoness. Partials 3, 9, 27, …, 3ⁿ are factorised only by prime 3, and so can be said to embody threeness. Partials 5, 25, 125, …, 5ⁿ are factorised only by prime 5, and so can be said to embody fiveness. Other partials are factorised by two, or more, different primes. Partials 12 is factorised by both 2 and 3, and so embodies both twoness and threeness; partial 15 is factorised by 3 and 5, and so embodies both threeness and fiveness. Primeness empowers the musician to manipulate any given timbre such that its twoness, threeness, fiveness, …, primeness can be enhanced or reduced. Adding a second comma to the syntonic temperament's comma sequence defines the 7th partial (see Video 5), thus similarly enabling sevenness.
• Conicality: Turning down twoness will lead to an odd-partial-only timbre – a “hollow or nasal” sound[14] reminiscent of cylindrical closed bore instruments (e.g. clarinet). As the twoness is turned up, the even partials are gradually introduced creating a sound more reminiscent of open cylindrical bore instruments (e.g. flute, shakuhachi), or conical bore instruments (e.g. bassoon, oboe, saxophone). This perceptual feature is called conicality.
• Richness: When richness is at minimum, only the fundamental sounds; as it is increased, the twoness gain is increased, then the threeness gain, then the fiveness gain, etc..

The 7th partial is cited by some[15][16] as being the essence of the "blue notes" played in the blues and related music.

Adding the starling comma to the syntonic temperament's comma sequence maps the 7th partial to the fundamental's augmented sixth (see Video 5). On the one hand, adding this narrows the valid tuning range of the syntonic temperament to the 7-limit range of a mere 5 cents (centered on 1/4-comma meantone, P5=696.58 cents; see Figure 1). On the other hand, it adds the 7th partial to the timbre, on a unique note, which gives musicians the option of emphasizing that partial when playing blues-inspired music. (See Primeness above. Real-time changes to the sevenness of a 7-limit timbre could prove to be musically useful.)

The augmented sixth is far to the right of the fundamental on the Wicki-Hayden keyboard (as shown in Video 5), so it is suitable for use in the I-IV-V blues progression in only C and keys flat-ward thereof.

One can use Dynamic Tonality to temper only the tuning of notes, without tempering timbres, thus embracing the Static Timbre Paradigm.

Similarly, using a synthesizer control such as the Tone Diamond,[17] a musician can opt to maximize regularity, harmonicity, or consonance—or trade off among them in real time (with some of the jammer's 10 degrees of freedom mapped to the Tone Diamond's variables), with consistent fingering. This enables musicians to choose tunings that are regular or irregular, equal or non-equal, major-biased or minor-biased—and enables the musician to slide smoothly among these tuning options in real time, exploring the emotional affect of each variation and the changes among them. Everything that the Static Timbre Paradigm offered, Dynamic Tonality can do—and more.

Imagine that the valid tuning range of a temperament (as defined in Dynamic Tonality) is a string, and that individual tunings are beads on that string. The microtonal community has typically focused primarily on the beads, whereas Dynamic Tonality is focused primarily on the string. Both communities care about both beads and strings; only their focus and emphasis differ.

An early example of Dynamic Tonality can be heard in "C to Shining C" C2ShiningC (composed and recorded by William Sethares in April 2008). This sound example contains only one chord, Dmaj (the piece is, oddly enough, recorded or transposed in the recording to D major, despite its name), played throughout, yet a sense of harmonic tension is imparted by a tuning progression and a timbre progression, as follows:

Cmaj 19-tet/harmonic -> Cmaj 5-tet/harmonic -> Cmaj 19-tet/consonant -> Cmaj 5-tet/consonant

• The timbre progresses from a harmonic timbre (with partials following the harmonic series) to a 'pseudo-harmonic' timbre (with partials adjusted to align with the notes of the current tuning) and back again.
• Twice as rapidly, the tuning progresses (via polyphonic tuning bends), within the syntonic temperament, from an initial tuning in which the tempered perfect fifth (P5) is 695 cents wide (19-tone equal temperament, 19-tet) to a second tuning in which the P5 is 720 cents wide (5-tet), and back again.

As the tuning changes, the pitches of all notes except the tonic change, and the widths of all intervals except the octave change; however, the relationships among the intervals (as defined by the syntonic temperament's period, generator, and comma sequence) remain invariant (i.e., consistent) throughout. This invariance among a temperament's interval relationships is what makes Dynamic Tonality possible.

In the syntonic temperament, the tempered major third (M3) is as wide as four tempered perfect fifths (P5's) minus two octaves—so the M3's width changes across the tuning progression

• from 380 cents in 19-tet (P5 = 695), where the Cmaj triad's M3 is very close in width to its just width of 386.3 cents,
• to 480 cents in 5-tet (P5 = 720), where the Cmaj triad's M3 is close in width to a slightly flat perfect fourth of 498 cents, making the Cmaj chord sound rather like a Csus4.

Thus, the tuning progression's widening of the Cmaj's M3 from a nearly-just major third in 19-tet to a slightly flat perfect fourth in 5-tet creates harmonic tension, which is relieved by the return to 19-tet.

This is an example of Dynamic Tonality's ability to expand the frontiers of tonality by offering new means of creating tension and release, even within a single chord.

Dynamic Tonality was developed primarily by a collaboration between Prof. William Sethares, Dr. Andrew Milne, and James "Jim" Plamondon (see papers cited below).

In late 2003, Plamondon was studying the economic forces that compelled the emergence of the QWERTY keyboard standard, which led him to study concertina note-layouts as a possible counter-example. That exposed him to the Wicki-Hayden note-layout. He reached out to dozens of academics in music theory asking "what deep property of music is exposed by this keyboard's invariant note-pattern?", but only Sethares and Milne dug into the problem, applying their knowledge of music and mathematics to publish a series of papers that solved the mystery.[3][7][1] Sethares' previous work, showing that consonance arose solely from the alignment of notes and partials, was a key input to Dynamic Tonality. Milne & Sethares' grad students did much of the work in developing electronic synthesizers and sequencers for Dynamic Tonality.[13]

A prototype of the Thummer.

Meanwhile, Plamondon formed Thumtronics Pty Ltd to develop an expressive, tiny, electronic Wicki-Hayden keyboard instrument: Thumtronics' "Thummer." However, he spent too much of the company's limited capital on researching motion-sensing (which is now available in a single-chip solution) and polyphonic aftertouch, so the company failed before it could bring the Thummer to market. The generic name for a Thummer-like instrument is "jammer." With two thumb-sticks and internal motion sensors, a jammer would afford 10 degrees of freedom, which would make it the most expressive polyphonic instrument available. Without the expressive potential of a jammer, musicians lack the expressive power needed to exploit Dynamic Tonality in real time, so Dynamic Tonality's new tonal frontiers remain largely unexplored.

Dynamic Tonality is the foundation of a broad research project named Musica Facta—meaning created music—that unites a loose association of collaborators in their explorations of Dynamic Tonality's invariances, isomorphisms, and the implications thereof.