Dynamic tonality


Dynamic Tonality is a new paradigm for music which generalizes the special relationship between Just Intonation and the Harmonic Series to apply to a much wider set of pseudo-Just tunings and pseudo-Harmonic timbres.
Dynamic Tonality enables many new musical effects that could expand the frontiers of tonality, including polyphonic tuning bends, tuning modulations, new chord progressions, temperament modulations and progressions, and novel timbre effects such as dynamic changes to primeness, conicality, and richness.

The Static Timbre paradigm

Harmonic timbres

A vibrating string, a column or air, and the human voice all emit a specific pattern of partials called the Harmonic Series. 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 tunings

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.

Temperament

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." This misalignment, in any tuning that is not fully Just, is the defining characteristic of the Static Timbre Paradigm.

Instruments

Many of the pseudo-Just temperaments proposed during this "temperament battle" were rank-2 —such as quarter-comma meantone—that provided more than 12 notes per octave. However, the standard piano-like keyboard is only rank-1, affording at most 12 notes per octave. Piano-like keyboards affording more than 12 notes per octave were developed by Vicentino, Colonna, Mersenne, Huygens, and Newton, but were deemed cumbersome and difficult to learn.

The Dynamic Tonality paradigm

The defining characteristic of Dynamic Tonality is that a given rank-2 temperament is used to generate, in real time during performance, the same set of intervals among:
  1. A pseudo-Just tuning's notes;
  2. A pseudo-Harmonic timbre's partials; and
  3. An isomorphic keyboard's note-controlling buttons.
Generating all three from the same temperament solves two problems and creates three opportunities.
  1. Dynamic Tonality solves the problem of maximizing the consonance of tempered tunings, and extends that solution across a much wider range of tunings than were previously considered to be consonant.
  2. Dynamic Tonality the "cumbersome" problem cited by Isacoff by generating a keyboard that is isomorphic with its temperament, and yet is tiny. The creators of Dynamic Tonality could find no evidence that any of Isacoff's Great Minds knew about isomorphic keyboards or recognized the connection between the rank of a temperament and the dimensions of a keyboard.
  3. Dynamic Tonality gives musicians the opportunity to explore new musical effects.
  4. Dynamic Tonality creates the opportunity for musicians to explore rank-2 temperaments other than the syntonic temperament easily and with maximum consonance.
  5. Dynamic Tonality creates the opportunity for a significant increase in the efficiency of music education.
A rank-2 temperament defines a rank-2 note-space, as shown in Video 1.
The syntonic temperament is a rank-2 temperament defined by its period, its generator and its comma sequence. The construction of the syntonic temperament's note-space is shown in Video 2.
The valid tuning range of the syntonic temperament is show in Figure 1.
A keyboard that is generated by a temperament is said to be isomorphic with that temperament. Isomorphic keyboards are also known as generalized keyboards. Isomorphic keyboards have the unique properties of transpositional invariance and tuning invariance when used with rank-2 temperaments of just intonation. That is, such keyboards expose a given musical interval with "the same shape" in every octave of every key of every tuning of such a temperament.
Of the various isomorphic keyboards now known, the Wicki-Hayden keyboard is optimal for dynamic tonality across the entire valid 5-limit tuning range of the syntonic temperament. The isomorphic keyboard shown in this article's videos is the Wicki-Hayden keyboard, for that reason. It also has symmetries related to Diatonic Set Theory, as shown in Video 3.
The Wicki-Hayden keyboard embodies a tonnetz, as shown in Video 4. The tonnetz is a lattice diagram representing tonal space first described by Leonhard Euler in 1739, which is a central feature of Neo-Riemannian music theory.

Non-Western tunings

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

Dynamic tuning

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.
For example, one could learn to play Rodgers and Hammerstein's Do-Re-Mi in its original 12-tone equal temperament 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.

New musical effects

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.

Tuning-based effects

Dynamic Tonality's novel tuning-based effects include:
Dynamic Tonality's novel timbre effects include:
The 7th partial is cited by some 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. 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. 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.
The augmented sixth is far to the right of the fundamental on the Wicki-Hayden keyboard, so it is suitable for use in the I-IV-V blues progression in only C and keys flat-ward thereof.

Superset of static timbre paradigm

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, a musician can opt to maximize regularity, harmonicity, or consonance—or trade off among them in real time, 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.

Compared to microtonality

Imagine that the valid tuning range of a temperament 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.

Example: C2ShiningC

An early example of Dynamic Tonality can be heard in "C to Shining C" .
This sound example contains only one chord, Dmaj, 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

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 remain invariant throughout. This invariance among a temperament's interval relationships is what makes Dynamic Tonality possible.
In the syntonic temperament, the tempered major third is as wide as four tempered perfect fifths minus two octaves—so the M3's width changes across the tuning progression
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.

History

Dynamic Tonality was developed primarily by a collaboration between Prof. William Sethares, , and .
In late 2003, Plamondon was studying the economic forces that compelled the emergence of the QWERTY keyboard standard, which led him to study concertina 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. 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.
Meanwhile, Plamondon formed 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 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.

''Musica Facta''

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