Tripod Notation

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Tripod Notation

Tripod Notation is an adaptation of staff notation to work with 225:224 tempered out (7-limit Marvel Temperament and more specifically Magic Temperament. More generally, the umbrella of Tripod Notation covers the related ideas of Tricycle Notation and Tricolour keyboards that involve a division of the octave into three regions corresponding to three pitches of a triad.

The Tripod Notation

Tripod Notation solves the problem that Magic Temperament doesn't have a proper MOS with or close to 7±2 notes. The 9 note "tripod scale", taken from a rank 3 temperament, is used instead.

Tripod Notation was originally described in 2009 and involves a three line octave staff. Each line is separated by a major third approximating 5:4 and the line is the centre of a pitch region called a "foot". Other staff positions are an approximate 16:15 from the adjacent line. With 225:224 tempered out, 16:15 and 15:14 are identical and called a "toe". Accidentals move pitches by 25:24 or 36:35, intervals called a "semitoe". In magic temperament, a toe divides into two equal semitoes and different accidentals become equivalent. As a major third doesn't divide into equal toes, this is in the general case a notation for a rank 3 temperament or 5-limit just intonation.

Accidental symbols are taken from the Sagittal set. Typesetting and MIDI files are generated by LilyPond.

The three lines of the Tripod staff can be thought of as three strings. A stringed instrument tuned in thirds with semitoes down the fretboard will naturally work with Tripod Notation and look a lot like a Kite Guitar.

A generalization of the Halberstadt keyboard for Tripod Notation can also be imagined: divide the octave into three regions each with a pattern of black and white keys that looks like F to B in the Halberstadt mapping. The black keys then correspond to staff positions and the white keys to accidentals. You need to add one more key for the 22 note MOS. For use on a retuned Halberstadt keyboard, the Tripod staff positions must alternate between F♯, G♯, and B♭ keys for the ideal Tripod adaptation and C, E, and G keys for alternate feet. This adds up to 18 notes per octave.

Other resources: Extended Tripod Notations taking it to the 13-limit and Trippy Trojans showing how to get more conventional staff output from LilyPond with Tripod Notation as input.

Tricycle Notation

As a notation for Magic Temperament, Tricycle Notation allows many pitches to be written as if they were in conventional meantone-based notation. As a notation for 5-limit just intonation or more general rank 3 temperaments Tricycle Notation solves the problem of the choice of nominals in an unusual way. Instead of choosing the pitch of each nominal and either having thirds out of tune with just intonation or placing a wolf in the middle of the natural scale, Tricycle Notation assumes simple 5-limit harmony by default and requires all comma ambiguities to be resolved.

Tricycle Notation follows from Tripod Notation but uses a different metaphor: the octave is divided into three wheels rather than three feet. Tricycle Notation is a simple extension of staff notation or the related pitch names where the octave is divided into three wheels instead of feet. In the original LilyPond implementation, each wheel is notated by a fingering with a number of 1, 2, or 3. Other notations could include shaped noteheads or colours.

Tricycle Notation is also described in the Tripod Notation PDF linked above and a web page specific to Tricycle Notation.

Tricycle Notation is cyclic in the sense that the three pitch regions are constantly shifting and a pitch can modulate through syntonic commas and end up on the wheel it started although a tuning to just intonation or a rank 3 temperament will leave it shifted by 3 syntonic commas. The rule for assigning wheels is simple: each pitch of a major or minor triad sits on a different wheel. As an extension, each pitch of a 6:7:9 subminor triad and the related supermajor triad sits on a different wheel. Furthermore, each pitch of an extended Pythagorean sus4 or sus2 triad sits on a different wheel. Dominant seventh chords have different wheel assignments corresponding to different tunings.

Tricolour Keyboards

Tripod Notation can be shown on a generalized keyboard by giving each foot a different colour giving a Tricolour metaphor relating to certain national flags. A generalized layout consistent with Meantone Temperament and correct choice of colour specify a rank 3 tuning consistent with 5-limit just intonation or Marvel Temperament.

Tricolour keyboards are implemented on the Lumatone. The standard colours don't follow any national flag, but are chosen to be colour-blind safe and look good and distinct with LED keys. The standard colours, with RGB hex, put B and C on (bluish) green (009e73), E and F on pink or reddish purple (CC79A7) and G and A on orange (D55E00).

Tricolour colouring could also work on the staff as a form of tricycle notation. Here, each notehead can be given a Tricolour colour when needed to resolve ambiguity, or left black by default. The standard colours will need to be darkened to work against white paper.

Spelling and pronunciation guide: the word "Tricolour" specifically refers to the national flag of the Republic of Ireland. The standard pronunciation is a variation on the French "Tricolore". The spelling "Tricolour" is preferred to "Tricolore" because the standard colours reflect the Irish flag better than the French. An Americanized spelling of "Tricolor" is also possible.