Isomorphic keyboard
Encyclopedia
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.
(1882), Wicki
(1896), Fokker
(1951), Erv Wilson
(1975-present) and Wesley (2001). The keyboards of Bosanquet and Erv Wilson
are also known as generalized keyboards
.
:
regular temperaments
of Just Intonation
.
A two-dimensional lattice 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.
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.
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.
Input device
In computing, an input device is any peripheral used to provide data and control signals to an information processing system such as a computer or other information appliance...
consisting of a two-dimensional array
Array
In computer science, an array data structure or simply array is a data structure consisting of a collection of elements , each identified by at least one index...
of note-controlling elements (such as buttons or keys) on which any given sequence and/or combination of musical intervals
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
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), JankoJanko keyboard
The Jankó keyboard is a musical keyboard layout for a piano designed by Paul von Jankó in 1882.Based on the premise that the hand can barely stretch more than a 9th on the piano, and that all scales are fingered differently, Jankó's new keyboard had two interlocking 'manuals' with three...
(1882), Wicki
Wicki-Hayden note layout
In music, the Wicki-Hayden note layout is a key layout for musical instruments that offers some advantages over the traditional keyboard layout.-History:...
(1896), Fokker
Adriaan Fokker
Adriaan Daniël Fokker , was a Dutch physicist and musician.Fokker was born in Buitenzorg, Dutch East Indies ; he was a cousin of the aeronautical engineer Anthony Fokker...
(1951), Erv Wilson
Erv Wilson
Ervin Wilson is a Mexican/American music theorist. Despite his avoidance of academia, Wilson has been influential on those interested in microtonal music and just intonation, especially in the areas of scale, keyboard, and notation design...
(1975-present) and Wesley (2001). The keyboards of Bosanquet and Erv Wilson
Erv Wilson
Ervin Wilson is a Mexican/American music theorist. Despite his avoidance of academia, Wilson has been influential on those interested in microtonal music and just intonation, especially in the areas of scale, keyboard, and notation design...
are also known as generalized keyboards
Generalized keyboard
Generalized keyboards are musical keyboards with regular, tile-like arrangements usually with rectangular or hexagonal keys, and were developed for performing music in different tunings...
.
Invariance
Isomorphic keyboards expose, through their geometry, two invariant properties of music theoryMusic theory
Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...
:
- transpositional invariance, in which any given sequence and/or combination of musical intervals has the same shape when transposed to another key, and
- tuning invariance, in which any given sequence and/or combination of musical intervals has the same shape when played in another tuning of the same musical temperamentMusical temperamentIn musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament system...
.
Basis vectors
All isomorphic keyboards derive their invariance from their relationship to rank-2Rank of an abelian group
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the...
regular temperaments
Regular temperament
Regular temperament is any tempered system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios...
of Just Intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
.
A two-dimensional lattice is generated by two basis vectors
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
, 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
Syntonic temperament
The syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect fifth, with the width of the tempered major third being equal to four tempered perfect fifths minus two octaves and the width of...
and schismatic
Schismatic temperament
The schismatic temperament is a musical tuning system that results from tempering the schisma of 32805:32768 to a unison. It is also called the schismic temperament or Helmholtz temperament.-Comparison with other tunings:...
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. This claim has not been rigorously tested, so its validity has been neither proven or disproven.
- MicrotonalityMicrotonal musicMicrotonal music is music using microtones—intervals of less than an equally spaced semitone. Microtonal music can also refer to music which uses intervals not found in the Western system of 12 equal intervals to the octave.-Terminology:...
- 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
Dynamic tonality
Dynamic tonality is tonal music which uses real-time changes in tuning and timbre to perform new musical effects such as polyphonic tuning bends, new chord progressions, and temperament modulations, with the option of consonance. The performance of dynamic tonality requires an isomorphic keyboard...
, 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'sSyntonic temperament
The syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect fifth, with the width of the tempered major third being equal to four tempered perfect fifths minus two octaves and the width of...
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.
Software
- 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.
- Relayer - a free application that enables musicians who play the AXiS-49, the QWERTY computer keyboard, or the Thummer, to play in a wide variety of isomorphic note layouts and tunings (even with a standard multitimbral synth).
- Musix - iPhone and iPad app. Musix is a fully customizable multiple-layout isomorphic musical keyboard.