Base 40

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The base-40 system is a method for encoding 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 can be used to encoding diatonic pitches names with chromatic alterations up to double sharps or flats. The base-40 pitch space can be built up from the property that a minor second is a difference of 5 between two pitch numbers, 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♯♯

The "base-40" name for the system refers to the property that an octave is represented by the integer 40. C4 is 162 while C5 is 162+40=202. To extract the octave number from a pitch value, just divide by 40. To extract the chroma from the pitch value, subtract the octave times 40.

      octave = value / 40
      chroma = value - octave * 40

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:

        pitch = octave * 40 + chroma
	 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).

1 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
28=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.

Intervals.svg

2 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.

Transpose.svg


3 References

4 Exercises

Base40exercise.svg