music theory online : interval calculator lesson 27C
Dr. Brian Blood

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Sweet is every sound, Sweeter thy voice, but every sound is sweet; Myriads of rivulets hurrying through the lawn, The moan of doves in immemorial elms, With murmuring of innumerable bees.
Alfred, Lord Tennyson (1809-1892) from "The Princess" Part vii. Line 203

return to lesson 27 :: open calculator in a separate window

Interval calculator
  1. The first utility expresses in cents the interval between two notes whose frequencies are known. If the frequency ratio is known (for example, 3:2) then the interval this ratio defines will be displayed, also in cents.
  2. The second utility raises or lowers a chosen frequency by an interval expressed in cents.
  3. The third utility calculates the frequency of a particular harmonic of a fundamental, while also estimating the note nearest to the harmonic drawn from an equal-tempered chromatic scale of which the fundamental is a member.
  4. The fourth utility calculates pure and tempered 5ths, the latter by fractions of the syntonic (or Ptolemaic) comma (a.k.a. the comma of Didymus) an interval defined by the ratio 81:80 or of the ditonic (or Pythagorean) comma an interval defined by the ratio 531441:524288
1. frequency ratio to cents
frequency of higher note (in Hz/cps) or larger term in a ratio
frequency of lower note (in Hz/cps) or smaller term in a ratio
2. add cents to or subtract cents from a frequency
frequency of note (in Hz/cps)
cents shift
3. calculate harmonic and find nearest (in ET) resultant note
fundamental frequency (in Hz/cps)
note name
harmonic (where 1 is the fundamental)
4. fractional comma tempering (+ widens : - narrows)
frequency of fixed base note (in Hz/cps)
note name
tempering of 5th above base note
tempering of 5th below base note
Frequency is the number of vibrations or cycles per second (cps) of a musical pitch, usually measured in hertz (Hz), where one hertz is one vibration per second. The human frequency range is divided into three rough areas or bands. High frequencies (between about 5 kHz and 20 kHz), mid frequencies (between about 200 Hz and 5 kHz) and low frequencies (between about 20 Hz and 200 Hz)
The cent, or cyclic cent, is an interval measurement invented by Alexander Ellis which appears in his appendix to his translation of Hermann von Helmholtz's On the Sensations of Tone [1875]. A cent is a logarithmic unit used when measuring the difference between two pitches. One cent corresponds to a frequency ratio of the 1200th root of 2, or conversely, a pure octave is comprised of 1200 cents
Harmonics are notes, called partials, that accompany the fundamental (prime tone, generator) when it is produced with a string, a pipe, the human voice, etc. The fundamental frequency is called 'first harmonic', the 'first partial' or the 'first mode'. If all the partials are harmonic then the numbering of the harmonics matches that of the partials
ET is an abbreviation for equal temperament. In equal temperament, the interval ratio between successive semitones is equal to the twelfth root of 2, or 100 cents
The ditonic comma (also called the Pythagorean comma or the comma ditonicum) is the difference between seven octaves and twelve perfect fifths and is expressed as the ratio (531441:524288)
The syntonic comma (also called the comma of Didymus or Ptolemaic comma) is the difference between four perfect fifths and two major thirds plus two octaves and is expressed as the ratio (81:80)
A note about precision: we have allowed input and display of frequencies to the fourth decimal place. However, we have restricted input and display of cents to only the third decimal place. Many commentators recommend working to the nearest cent as this is within the threshold of human pitch discrimination. However, although we may not be able to tell two notes a cent or so apart, it is useful to work to a higher accuracy when one wishes to avoid rounding errors in a chain of calculations, for example, when extracting a scale from a recipe of tempered fifths expressed in cents. It is also worth remembering that when the first calculator displays the ratio of two frequencies as a number, the user should be prepared to round the result to 2 or 3 decimal places in order to extract a more useful result. Thus, the ratio 1.2499943005 would become 1.250 which is equivalent to 5:4