About the Johnston to HE converter


This converter was created as a tool for conversion of Ben Johnston’s just intonation notation into Helmholtz-Ellis accidentals. Johnston’s “basic” C major scale (in which all of the notes are “natural” or carry no accidentals) refers to the Ptolemaic sequence of sucessive melodic ratios C_8:9_D_9:10_E_15:16_F_8:9_G_9:10_A_8:9_B_15:16_C. More information about Johnston's notation may be found here.


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 as defined by Johnston (default = C), and by chromatic alteration of the diatonic pitch as defined by Johnston (default = natural). These diatonic and chromatic pitches bear little relation to the standard notes of equal temperament and merely serve to recenter the harmonic space to a desired 1/1. The reference can be further refined 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 relationship between these notes is defined by the just intonation interval of the chosen 1/1 and A4.
Toggling C (Johnston) = C (HE) offsets the HE output by one syntonic comma, allowing C in Johnston notation to be equivalent to C in HE (as opposed to C-arrow-up). This may prove useful to some composers in their own creative work, though note that Johnston generally tuned the open strings of stringed instruments in relation to A (not A+); the C string is thus tuned as C- (i.e. Pythagorean C natural in HE). For additional options (calculating Just Intonation intervals with respect to different reference pitches or by ratio input) please return to the HE-Calc page.


The output shows information about the input pitch in relation to the reference: its ratio in terms of the reference, conversion into 31-limit Helmholtz-Ellis notation, cent deviation of the HE pitch from the nearest MIDI pitch class (12-ED2), total cents size from the reference pitch, frequency, and Tenney distance.


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”.