The Bohlen-Pierce Site
Other Unusual Scales

Last updated: February 8, 2009

In 1972/73 Heinz Bohlen found that the same principles that had resulted in the discovery of the "13-Step Scale" (now Bohlen-Pierce) could be used to create other harmonic scales. The emphasis was on harmonic properties and tonality; equal temperament of these scales was only considered as a check on sufficient equidistance. Bohlen published them, in the form of a footnote (footnote 26, on page 84), in his account on the "13-Step Scale", in 1978 [1]. They are listed below.

12 Steps in The Twelfth

Based on a triade 4:7:10, with the twelfth (3/1) as a frame interval, the following scale unfolds:

 Step No. Ratio Cents Cents (Equal Temp.) Defect (Cents) 0 1/1 0 0 0 1 11/10 165.00 158.50 - 6.50 2 6/5 315.64 316.99 1.35 3 30/23 459.99 475.49 15.50 4 10/7 617.49 633.99 16.50 5 11/7 782.49 792.48 9.99 6 7/4 968.83 950.98 - 17.85 7 21/11 1119.46 1109.47 - 9.99 8 21/10 1284.47 1267.97 - 16.50 9 23/10 1441.96 1426.47 - 15.49 10 5/2 1586.31 1584.96 - 1.35 11 11/4 1751.32 1743.46 - 7.86 12 3/1 1901.96 1901.96 0

This scale has been used by Enrique Moreno [2] for some fine compositions. Moreno named it A12.

It is obvious that Bohlen, when developing this scale, didn't spend much time on it, mainly because he deemed it much less logically consistent than the "13-Step-Scale". If he would review the scale today it would probably become modified, but it might loose some of its impertinence (he let it go up to 23-limit!) and freshness in the process.

7 Steps in The Octave (11 Steps in The Twelfth)

Bohlen based this scale on a triade 2:3:5 and found that it fitted both the frames of an octave or a twelfth equally well. Thus it got 2 possible equal temperaments: the well known 7TET of the octave and the not fully identical 11 step equal temperament of the twelfth. The equal temperament in the following table refers to the latter.

 Step No. Ratio Cents Cents (Equal Temp.) Defect (Cents) 0 1/1 0 0 0 1 10/9 182.40 172.91 - 9.49 2 6/5 315.64 345.81 30.17 3 4/3 498.04 518.72 20.68 4 3/2 701.96 691.62 - 10.34 5 5/3 884.36 864.53 - 19.83 6 9/5 1017.60 1037.43 19.83 7 2/1 1200.00 1210.36 10.36 8 9/4 1403.91 1383.24 - 20.67 9 5/2 1586.31 1556.15 - 30.17 10 27/10 1719.55 1729.05 9.50 11 3/1 1901.96 1901.96 0

The emphasis here is clearly on just intonation; the just scale shares little relationship with the equal temperament version.

8 Steps in The Octave

This 7-limit scale is based on the triad 5:7:9; it is entirely depending on just intonation and cannot be represented by 8TET:

 Step No. Ratio Cents Cents (8TET) Defect (Cents) 0 1/1 0 0 0 1 10/9 182.40 150 - 32.4 2 6/5 315.64 300 - 15.64 3 9/7 435.08 450 14.92 4 7/5 582.51 600 17.09 5 14/9 764.92 750 - 14.92 6 5/3 884.36 900 15.64 7 9/5 1017.60 1050 32.40 8 2/1 1200.00 1200 0

The scale's intention is to bind intervals like 7/5, 9/5 and 9/7 harmonically into the octave.

Literature

 [1] Bohlen, Heinz: 13 Tonstufen in der Duodezime. Acustica, vol. 39 no. 2, S. Hirzel Verlag, Stuttgart, 1978, pp. 76 - 86. [2] Moreno, Enrique Ignacio: Embedding Equal Pitch Spaces and The Question of Expanded Chromas: An Experimental Approach. Dissertation, Stanford University, Dec. 1995, pp. 12 - 22.