Base 40

The base-40 system is a method to encoded pitches as integers in such a way that the diatonic interval with chromatic alteration is represented as the difference between any two numbers in the system. It is a method of encoding diatonic pitches with accidentals up to double sharps or flats. The base-40 chroma for pitches can be built up from the property that a minor second is a difference of 5 between pitch numbers in the base-40 system, and major seconds are a difference of 6. The C chroma value is set to the value 2 rather than 0 so that the octave values for C♭ and C♭♭ remain in the same octave as C♮ just above these pitches when using division by the base to extract the octave number. Here is a complete table of the base-40 chroma, using C=2 as the reference:

Base-40 chroma values (pitch classes)
0 = C♭♭ 1 = C♭ 2 = C♮ 3 = C♯ 4 = C♯♯	5 = [D♭♭♭] 6 = D♭♭ 7 = D♭ 8 = D♮ 9 = D♯ 10 = D♯♯	11 = [E♭♭♭] 12 = E♭♭ 13 = E♭ 14 = E♮ 15 = E♯ 16 = E♯♯	17 = F♭♭ 18 = F♭ 19 = F♮ 20 = F♯ 21 = F♯♯	22 = unused 23 = G♭♭ 24 = G♭ 25 = G♮ 26 = G♯ 27 = G♯♯	28 = [A♭♭♭] 29 = A♭♭ 30 = A♭ 31 = A♮ 32 = A♯ 33 = A♯♯	34 = [B♭♭♭] 35 = B♭♭ 36 = B♭ 37 = B♮ 38 = B♯ 39 = B♯♯

To calculate an absolute pitch such as G4 (The G above middle C), multiply the octave by 40 and add the chroma value for the G pitch-class:

	 G4 = 4 * 40 + 25 = 160 + 25 = 185

Alternatively, G4 can be calculated by adding 3 half-steps (major seconds) and one minor second to C4 (represented as the number 162 in base-40):

	G4 = C4 + 3 * M2 + m2 = 162 + 3 * 6 + 5 = 162 + 23 = 185

Note that base-40 pitch representations preserve chromatic alterations of diatonic pitch-classes (up to double sharps/flats). Therefore F♯4 (180) is not in the same as the base-40 pitch G♭4 (184).

Intervals

A useful property of the base-40 system is that the difference between pitch numbers in the base-40 system represent one-to-one mappings with diatonic intervals with chromatic alterations:

Base-40 interval classes (d=diminished, m=minor, M=major, P=perfect, A=augmented)
-2 = dd1 -1 = d1 0 = P1 1 = A1 2 = AA1	3 = [dd2] 4 = d2 5 = m2 6 = M2 7 = A2 8 = AA2	9 = [dd3] 10 = d3 11 = m3 12 = M3 13 = A3 14 = AA3	15 = dd4 16 = d4 17 = P4 18 = A4 19 = AA4	20 = unused 21 = dd5 22 = d5 23 = P5 24 = A5 25 = AA5	26 = [dd6] 27 = d6 38 = m6 29 = M6 30 = A6 31 = AA6	32 = [dd7] 33 = d7 34 = m7 35 = M7 36 = A7 37 = AA7 (40 = octave)

As an example, consider the interval between E4 (174) and C4 (162) which is 174-162 = 12, representing a major third according to the table shown above. All intervals can be constructed by observing that major seconds are 6 and minor seconds are 5, so note that a major third, consisting of two major seconds, is 6 + 6 = 12. Below is an example segment of music with the pitches labeled in the base-40 system. Try subtracting values and comparing the results to numbers in the base-40 interval-class table above.

Transposition

Transposition of notes within the base-40 system is simple: just add a constant interval to the base-40 pitch numbers. Below is an example which transpose the music up a major second. In the base-40 system, a major second is represented by the integer 6, so add 6 to all of the pitch numbers in the original music to transpose them up a major second.

References

• A Base-40 Number-line Representation of Musical Pitch Notation, Walter Hewlet, Musikometrika 4 (1992), 1-14.
• http://extra.humdrum.org/man/notearray/#base40
• http://extra.humdrum.org/man/transpose