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.

By default, the calculator’s harmonic

The reference pitches are related by 12-tone equal temperament (12-ED2). If a just intonation reference of E3 (3/8 in relation to A4) were desired, then the frequency of 1/1 may be adjusted accordingly, to 165 Hz. The frequency of A4 automatically adjusts itself to 440.497152 Hz because its relationship to E3 remains tempered within the reference framework. The result, however, is that an input of A4 (8/3 above the reference) in the calculator gives the

The

The

This last element is the ratio’s particular prime factorisation, where each number is a

By toggling the

Selecting

Example 6 shows an inputted ratio of 3/5 from the default reference of A4. By means of the offset ratio, however, this input method may be used to find both the ratios and notations of more distant and complex harmonic relationships, which must be constructed. For instance, if the desire were to find the normalised 11th harmonic (11/8) of the previously calculated 3/5, one may click

Note that, because calculations occur automatically when a change takes place in any element of the calculator, an E-flat (9/25) first appears in the output before 11/8 has been entered into the input. This demonstrates how cycles of a single interval may be easily calculated by repeatedly clicking

The

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.