Isomorphic keyboard

An isomorphic keyboard is a musical input device consisting of a two-dimensional grid 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
Helmholtz's 1863 book On the Sensations of Tone gave several possible layouts. Practical isomorphic keyboards were developed by Bosanquet (1875), Janko (1882), Wicki (1896), Fokker (1951), Erv Wilson (1975–present), Wesley (2001) and Antonio Fernández (2009). Accordions have been built since the 19th century using various isomorphic keyboards, typically with dimensions of semitones and tones. The keyboards of Bosanquet and Erv Wilson are also known as generalized keyboards. The keyboard of Antonio Fernández is also known as Transclado. The Ragzpole is a recently developed cylindrical MIDI controller having dimensions in fifths and major thirds. The Harpejji, while not strictly a keyboard, uses an isomorphic pattern of frets and tapped strings. It has been adopted by many keyboard players, most notably Stevie Wonder.

Invariance
Isomorphic keyboards expose, through their geometry, two invariant properties of music theory:
 * 1) transpositional invariance, in which any given sequence and/or combination of musical intervals has the same shape when transposed to another  key, and
 * 2) 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 temperament.

Theory
All isomorphic keyboards derive their invariance from their relationship to rank-2 regular temperaments of just intonation. A two-dimensional lattice is generated by two basis vectors. A keyboard lattice generated by two given musical intervals, which are mapped to those basis vectors, is isomorphic with 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:
 * 1) Ease of teaching, learning, and playing
 * According to some authors,   the invariance of isomorphic keyboards facilitates music education and performance. This claim has not been rigorously tested, so its validity has been neither proven nor disproven.
 * 1) Microtonality
 * Isomorphic keyboards' provision of more than the usual 12 note-controlling elements per octave may 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 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 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; 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 both this and a much broader range of tunings.

Software

 * Relayer – a free application for Windows and Mac 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 using either a Dynamic Tonality synth or a standard multitimbral synth.
 * Hex – a free software MIDI sequencer, which uses a generalized keyboard in place of the standard piano keyboard. Lanes are extended from the keys and MIDI notes can be drawn into each lane, and edited, with the mouse (as in a standard MIDI sequencer like Logic, Reaper, SONAR, etc.). The layout can be sheared to ensure that the vertical height of each key (and note lane) is proportional to its pitch height—regardless of the tuning used. A wide variety of isomorphic layouts are possible, including Bosanquet and Wicki.
 * Musix – iPhone and iPad app. Musix is a fully customizable multiple-layout isomorphic musical keyboard.
 * IsoKeys – Android app. IsoKeys is a free customizable isomorphic musical keyboard that supports several popular layouts.
 * Hexiano – Android app. Hexiano is a free and open-source customizable isomorphic musical keyboard that supports several popular layouts.
 * Navichord – iPad app. Uses isomorphic keyboard for chords and piano keyboard for melody creation.
 * Qwertonic – 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.
 * Xenharmonic Keyboard - Windows 10 app. Xenharmonic Keyboard is an isomorphic microtonal MIDI keyboard. It supports all possible regular two dimensional isomorphic keyboards with any shaped key. It also supports any regular musical temperament.
 * Dodeka Music - iPad app. Dodeka Music is a free mobile application that allows users to play several songs on Dodeka's linear keyboard layout.