Isomorphic keyboard

An isomorphic keyboard is a musical input device consisting of a two-dimensional array of note-controlling elements (such as buttons or keys) on which any given sequence and/or combination of musical intervals has the “same shape” on the keyboard wherever it occurs – within a key, across keys, across octaves, and across tunings.

Examples

Isomorphic keyboards were developed by Bosanquet (1875), Janko (1882), [http://www.concertina.com/gaskins/wicki Wicki] (1896), Fokker (1951), and [http://www.thearraymbira.com/arraysystem.php Wesley] (2001). The keyboards of Bosanquet and Erv Wilson are also known as generalized keyboards.

Invariance

Isomorphic keyboards expose, through their geometry, two invariant properties of music theory:
# transpositional invariance [Keislar, D., [http://ccrma.stanford.edu/STANM/stanms/stanm45/stanm45.pdf "History and Principles of Microtonal Keyboard Design"] , Report No. STAN-M-45, Center for Computer Research in Music and Acoustics, Stanford University, April 1988.] , in which any given sequence and/or combination of musical intervals has the same shape when transposed to another key, and
# tuning invariance [Milne, A., Sethares, W.A. and Plamondon, J., [http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15 Invariant Fingerings Across a Tuning Continuum] , "Computer Music Journal", Winter 2007, Vol. 31, No. 4, Pages 15-32.] , in which any given sequence and/or combination of musical intervals has the same shape when played in another tuning of the same musical temperament.

Basis vectors

All isomorphic keyboards derive their invariance from their relationship to rank-2 regular temperaments of Just Intonation.

A is generated by two basis vectors, and so is a rank-2 regular temperament of Just Intonation. A keyboard lattice generated by two given musical intervals will be isomorphic in any rank-2 temperament that is also generated by those same two intervals. For example, an isomorphic keyboard generated by the octave and tempered perfect fifth will be isomorphic with both the syntonic and schismatic temperaments, which are both generated by those same two intervals.

Benefits

Two primary benefits are claimed by the inventors and enthusiasts of isomorphic keyboards:
# Ease of teaching, learning, and playing
#: isomorphic keyboards' invariance facilitates music education and performance. [ [http://www.thummer.com/blog/2007/09/thummusic-system.html ThumMusic System.] ] [ [http://chroma.jp/main.jsp Wholetone Revolution.] ] [ [http://www.c-thru-music.com/cgi/?page=layout C-Thru Music.] ] This claim has not been rigorously tested, so its validity has been neither proven or disproven.
# Microtonality
#: isomorphic keyboards' provision of more than the usual 12 note-controlling elements per octave facilitate the performance of music that requires more than 12 notes per octave.

A third potential benefit of isomorphic keyboards, dynamic tonality, has recently been demonstrated, but its utility is not yet been proven. Using a continuous controller, a performer can vary the tuning of all notes in real time, while retaining invariant fingering on an isomorphic keyboard. Dynamic Tonality has the potential to enable new real-time tonal effects such as polyphonic tuning bends, new chord progressions, and temperament modulations, but the musical utility of these new effects has not yet been demonstrated.

Comparisons

Isomorphic keyboards can be compared and contrasted using metrics such as the thickness of an octave's "swathe" of buttons on the keyboard and the number of "repetitions" of a given note on the keyboard. Different isomorphic keyboards are suited for different uses. Within the syntonic temperament's broad tuning continuum, for example, the Fokker keyboard is well-suited to tunings of the syntonic temperament in which the tempered perfect fifth stays in a narrow range around 700 cents, whereas the Wicki keyboard is useful over the syntonic temperament's entire tuning range. [Milne, A., Sethares, W.A. and Plamondon, J., [http://www.thummer.com/ThumTone/Tuning_Invariant_Layouts_Last_Draft.pdf Tuning Continua and Keyboard Layouts] , "Journal of Mathematics and Music", Spring 2008 (forthcoming).]

Commercially availabile

Many isomorphic keyboards are commercially available, including:
# [http://www.chroma.jp Chromatone CT-312] , uses the Janko note-layout
# [http://www.cortex-design.com/body-project-terpstra-1.htm Terpstra] , uses the Bosanquet note-layout
# [http://www.starrlabs.com/keyboards.php U990] , uses the Bosanquet note-layout
# [http://www.c-thru-music.com/cgi/?page=prod_axis-64 AXIS-64] , uses the [http://www.c-thru-music.com/cgi/?page=layout Harmonic Table] note-layout
# [http://www.hobgoblin.com/local/bigpic.php?L=US&ID=GR4901 Stagi Hayden Duet Concertina] , uses the [http://www.concertina.com/gaskins/wicki Wicki] note-layout, shown in Figure 1 above (known as the 'Hayden' layout by concertina players)

In development

# [http://www.thummer.com Thummer] , uses the [http://www.concertina.com/gaskins/wicki Wicki] note-layout, shown in Figure 1 above.
# [http://www.wicki.org.uk Wicki.org.uk] , free UK site containing Java, Flash, and PC applications to enable users to play their alpha-numeric keyboard to sound 12 equal tempered pitches using Wicki/Janko layout.

References