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
