**Construction of the Pythagorean Chromatic Scale**

Step 1: As with the Pythagorean diatonic scale, begin at some arbitrary point,
but this time ascend **and** descend in perfect 5ths.

Step 2: Reduce the ratios obtained in step 1 into the range of a single octave:

Step 3: Arrange the notes obtained in step 2 in ascending order:

**Discussion**

There are two half-step intervals: Firstly, 256/243 = 1.053 between degrees 1 &
2, 3 & 4, 5 & 6, 8 & 9, 10 & 11, and 12 & 1. Secondly 2187/2048
= 1.068 between degrees 2 & 3, 4 & 5, 9 & 10, 11 & 12.

There is also an ambiguity concerning the tritone in that there are two different ratios for that interval: 729/512 = 1.4238, and 1024/729 = 1.4047