About the calculator


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 origin or reference of the notation and harmonic space can be defined by octave (default = 4, in scientific pitch notation), by diatonic note name (default = A), and by chromatic alteration of the diatonic pitch (default = natural). The reference can be further specified by altering the frequency of the "tuning note" A4 (default = 440 Hz) or by altering the frequency of 1/1 itself (default = 440 Hz). The default relationship between these notes is defined in terms of 12 equal divisions of the octave, to facilitate working with scores based on different or changing equal tempered fundamentals.


Input of pitches can follow one of two workflows - by notation or by ratio - selected prior to making calculations.

input by notation

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.

input by ratio

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.


The output shows information about the input pitch in relation to the reference: its ratio, spelling in 31-limit Helmholtz-Ellis notation, cent deviation from the nearest MIDI pitch class (12-ED2), total cents value, frequency, and Tenney harmonic distance. Toggling the normalize function reduces the output ratio to a value between 1/1 and 2/1, changing harmonic distance to projection distance. If the output ratio includes primes beyond 31 or, generally, more than 3 steps in ether direction along a prime axis in 31-limit harmonic space, then its notation will be “undefined”; all other information is nevertheless calculated and displayed.


The melodic distance function allows a user to compare pitches by determining the interval between them. The current output of the calculator may be saved by clicking “save current” or restored to 1/1 by clicking “reset” the current value of the calculator may be checked against the saved interval by clicking “check current”.


The 23-Limit Auto-Normalised Enharmonic Pitch-Class Comparison feature allows a user to search a database generated by Marc Sabat of approximately 25000 pitch-classes for possible respellings of a given input. The database consists of intervals expressible in terms of a string of no more than three HE accidentals. The accidentals range from double-flats through double-sharps, modified by at most three syntonic commas, two septimal and undecimal commas, and/or single commas in all other prime dimensions of harmonic space. The search finds ratios whose cent values fall within a specified range around the input.
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: The resulting list is a sub-database of all the enharmonic pitches that meet the the user defined parameters. The following data is displayed for each note:

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