Dynamic tonality

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Dynamic tonality is tonal music which uses real-time changes in tuning to perform new musical effects such as polyphonic tuning bends, new chord progressions, and temperament modulations. The performance of dynamic tonality requires an isomorphic keyboard driving a music synthesizer which implements dynamic tuning and dynamic timbres.

Contents

[edit] Isomorphic keyboards and tuning invariance

Isomorphic keyboards have the unique properties of transpositional invariance[1] and tuning invariance[2] when used with rank-2 temperaments of Just Intonation. That is, they expose a given musical interval with "the same shape" in every octave of every key of every tuning of such a temperament. For example, a given tonal scale and its chords will have the same shape, and hence consistent fingering, all across the tuning range of the syntonic temperament. Of the various isomorphic keyboards (e.g., the Bosanquet, Janko, Fokker, and Wesley), the Wicki keyboard is optimal for dynamic tonality.[3]

[edit] Dynamic Tuning

On a tuning invariant 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 the exactly the same note-controlling buttons, in every other tuning of that temperament. 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 the exactly the same note-controlling buttons, while smoothly changing the tuning in real time across the syntonic temperament's tuning continuum.

[edit] Piano keyboard

The piano keyboard is one-dimensional, and therefore cannot present the intervals of a rank-2 (two-dimensional) temperament isomorphically. As it is not isomorphic, the piano keyboard is also not tuning invariant, and hence cannot control dynamic tuning with consistent fingering. Without consistent fingering, a pianist would have to change a given interval's fingering constantly as the tuning changed, which would be impractical. Whereas an isomorphic keyboard's tuning invariance makes dynamic tonality easy to control, the piano keyboard's lack of tuning invariance makes it impractical.

[edit] New musical effects

Dynamic tuning enables musical effects which require a change in tuning, such as

  1. 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;
  2. new chord prgressions 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
  3. 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.

Polyphonic tuning bends are demonstrated in this video.

[edit] Timbre & tonality

The consonance of musical intervals is maximized when the partials of a given timbre (which for harmonic timbres are also called harmonics or overtones) align with the notes of a given tuning.[4] Many cultures' tuning systems can be seen as attempts to maximize the consonance of music played with the timbres of their dominant instruments. For example, the non-tonal tuning systems of Indonesia, Thailand, and Mandinka Africa maximize the consonance of their dominant instruments' timbres (gongs, ranats, and balafons. respectively).

Likewise, the West's system of tonality can be seen as arising from an attempt to maximize consonance with its dominant instruments' timbres, which are almost universally harmonic. The system of tuning that most closely aligns fundamentals with harmonic partials is Just Intonation.

However, the benefits of consonance are not the only consideration in tonality. The utility of being able to modulate freely among keys is also highly valued, and attaining this benefit -- especially on fixed-pitch instruments such as the piano -- requires tempering a tuning away from Just Intonation. Such tempering moves notes out of alignment with their corresponding harmonics, thereby reducing consonance. This reduction in consonance has been vigorously opposed by many musicians and music theorists throughout history, and this opposition continues today.[5][6][7]

[edit] Dynamic timbre

One solution to the problem of temperament is to temper a timbre's partials to align with a tempered tuning's notes.[8] The laws of physics make it impossible for acoustic instruments to temper their partials in real time, but acoustically-produced timbres can be tempered electronically in real time, and electronic music synthesis can be used to generate tempered timbres in real time, too.

The tempering of timbres to match tunings can be seen as a generalization of the relationship between Just Intonation and the Harmonic Series -- the relationship from which all of tonality arguably springs -- to a much wider range of 'pseudo-tonal' tunings and their related 'pseudo-harmonic' timbres. As such, it can be seen as a general solution to the problem of temperament, at least for music that is produced or processed electronically.

[edit] vs. Microtonality

While microtonal music usually involves a single fixed tuning that is something other than 12-tone equal temperament (12-tet), dynamic tuning involves a real-time change among two or more tunings of which one may be 12-tet. Further, Wikipedia's entry for microtonal music states that "the following systems are not microtonal: a diatonic scale in any meantone tuning...", whereas the syntonic temperament's tonally-valid tuning range includes all of the meantone tunings, facilitating the use of the diatonic scale (or any other) therein through its consistent fingering on an isomorphic keyboard.

Tuning invariance facilitates the exploration of static microtonal tunings, while dynamic tuning incorporates microtonality's static tunings into tuning continua along which tuning can be changed dynamically in real time.

[edit] Example: C2ShiningC

An example of Dynamic Tonality can be heard in "C to Shining C" C2ShinigC (composed and recorded by Prof. W.A. Sethares in April 2008). This sound example contains only one chord, Cmaj, played throughout, yet a sense of harmonic tension is imparted by a tuning progression and a timbre progression:

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 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 temperament's comma sequence, remain constant throughout. This consistency among a temperament's interval relationships is what makes tuning invariance possible (Milne, 2007).

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 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.

[edit] References

  1. ^ Keislar, D., History and Principles of Microtonal Keyboard Design, Report No. STAN-M-45, Center for Computer Research in Music and Acoustics, Stanford University, April 1988.
  2. ^ Milne, A., Sethares, W.A. and Plamondon, J., Invariant Fingerings Across a Tuning Continuum, Computer Music Journal, Winter 2007, Vol. 31, No. 4, Pages 15-32.
  3. ^ Milne, A., Sethares, W.A. and Plamondon, J., Tuning Continua and Keyboard Layouts, Journal of Mathematics and Music, Spring 2008 (forthcoming).
  4. ^ Sethares, W.A., 2004 Tuning, Timbre, Spectrum, Scale.
  5. ^ Barbour, J.M., 2004, Tuning and Temperament: A Historical Survey.
  6. ^ Duffin, R.W., 2006, How Equal Temperament Ruined Harmony (and Why You Should Care).
  7. ^ Isacoff, Stuart, 2003, Temperament: How Music Became a Battleground for the Great Minds of Western Civilization.
  8. ^ Sethares, W.A., Relating Tuning and Timbre, Experimental Musical Instruments, September 1992.