jot

A term coined by Augustus de Morgan to designate the tiny interval which represents 100,000 * log₁₀(2), thus obviating the need to calculate logarithms.

A jot is calculated as the 30103rd root of 2, or 2^(1/30103), with a ratio of approximately 1:1.000023026. It is an irrational number.

This interval therefore divides the "octave", which is assumed to have the ratio 2:1, into 30103 equal parts. Thus a jot represents one degree in 30103-EDO tuning.

One potential defect of using jots is that the familiar 12-EDO semitone does not come out with an integer number of jots, since 30103 does not divide evenly by 12. Thus, the 12-EDO semitone is ~2508.583333 = exactly 2508⁷/₁₂ jots.

a jot is ~0.039863137 cent, or just under ¹/₂₅ of a cent. The exact value is ¹/_{(25.08583333)} = ¹/_{(25 & 103/1200)} = ¹²⁰⁰/₃₀₁₀₃ of a cent.


The formula for calculating the jot-value of any ratio is:

jots = log₁₀(ratio) * [30103 / log₁₀(2)]


Ellis states that John Curwen, a great reformer of a cappella singing who intended to acheive just-intonation in his performances, used jots for his measurements.

Note that Sauveur's "heptamerides" of 2^(1/301) are related to the "jots", being simply a slightly less accurate rounding.


REFERENCES

Ellis, Alexander. 1885.
Appendix XX, in his translation of
Helmholtz, On the Sensations of Tone, p 437.
Dover reprint 1954.

De Morgan, Augustus. 1864.
"On the beats of imperfect consonances",
Transactions of the Cambridge Philosophical Society vol. 10, pp. 129-145.

[from Joe Monzo, JustMusic: A New Harmony]

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