This table demonstrates some of the logic behind the Lambda scale.  It shows the 17-limit just intonation (JI) intervals that are closest to the equally tempered Bohlen Pierce scale.  Notice that there are no close JI intervals for the F#, H#, and A# notes.  Also, melody fragments in the D E F G range are virtually equal tempered with respect to one another.

The original JI scale has accidentals that are beyond the 17 limit: C# is 25/21, F# is 75/49, H# is 49/25, and A# is 63/25. Using those ratios would bring the accidentals within 10 cents of the ET notes, but the ear gravitates towards ratios with lower denominators.  In any case, it’s good to avoid H# because of its proximity to the octave. 49/25 can’t compete with 2/1. Same with A# against a strong C – most listeners will hear it as a too-sharp 10th.

Bohlen Pierce ET to JI intervals
best match, 17 limit
  ET cents JI  JI cents diff  
0 0.0 1/1 0.0 +0.0 C
1 146.3 12/11 150.6 +4.3  
2 292.6 13/11 289.2 -3.4 D
3 438.9 9/7 435.1 -3.8 E
4 585.2 7/5 582.5 -2.7 F
5 731.5 17/11 753.6 +22.1  
6 887.8 5/3 884.4 -3.4 G
7 1024.1 9/5 1017.6 -6.5 H
8 1170.4 2/1 1200.0 +29.6  
9 1316.7 15/7 1319.4 +2.7 J
10 1463.0 7/3 1466.9 +3.9 A
11 1609.3 5/2 1586.3 -23.0  
12 1755.7 11/4 1751.3 -4.4 B
13 1902.0 3/1 1902.0 +0.0 C

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