The Helmholtz-Ellis 31-Limit Harmonic Space Calculator is a tool for composers and musicians who are interested in discovering and working with the properties of intervals tuned in just intonation. It makes use of the Extended Helmholtz-Ellis JI Pitch Notation developed by Marc Sabat and Wolfgang von Schweinitz. HEJI explicitly notates the raising and lowering of pitches by specified microtones and provides visually distinctive “logos” distinguishing “families” of natural intervals based on the harmonic series.

The calculator’s default workflow is to input pitches using the HEJI notation palette by selecting octave (default = 4), diatonic note name (default = A), and just intonation accidental (default = natural). The first row represents the accidentals historically used to notate a series of perfect fifths. Subsequent rows represent various prime dimensions of harmonic space up to 31 with increasing amounts of pitch deviations arranged from left to right. The direction of otonal (harmonic) or utonal (subharmonic) movement is indicated by an overline (for otonal) and an underline (for utonal) in the prime’s row label on each side of the palette.

Pitches may also be entered by inputting any ratio, representing an interval above or below the *reference*. In addition, an *offset* ratio (default = 1/1) may be specified. The *input* is multiplied by the *offset* before being expressed in terms of the *reference*. The *offset* may be entered directly or the current output of the calculator may be saved by clicking “save current”; “reset” restores the *offset* to 1/1. By successively clicking “save current”, it is possible to compound the *input* ratio. Ratio input allows for primes beyond 31.

Target cent values between 0 and 1200 may be entered into the cent input box. Alternately, they may be automatically transferred from either the current calculator output or the current melodic step value. If frequency input is selected, the chosen Hz value will be evaluated as a rational interval measured from the reference frequency (1/1). In this way, complex intervals may be tested to find close rational approximations. All intervals larger than 2/1 are automatically normalised.

The output list, which is generated by clicking "search", may be filtered by a number of parameters, such as:

- which primes to include (default = 13-limit)
- the search range around the target value (default = 8 cents)
- minimum and maximum allowable Harmonic Distance values from the reference note (default range = between 0 and 20)
- the size of the output list (default = 48)
- the maximum number of accidentals needed to express a note (default = 2)
- whether the list is ordered by increasing Harmonic Distance values or increasing absolute value cent deviations from the target note (default = sort by HD)

- octave reduced 23-limit monzo
- ratio
- interval cents (from reference 1/1)
- amount by which it differs from the target note in cents
- Harmonic Distance (from reference 1/1)
- HE notation

For a version of this calculator adapted by Kite Giedraitis for his Color Notation, visit his website.