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Tritave
BP intervals
Enharmonics
Pentachords

The framework interval of the Bohlen-Pierce scale is 3:1. The question how to name it arose early on. In the traditional Western scale, this ratio represents the perfect twelfth. But since in none of the BP scales, be it diatonic or chromatic, 3:1 would mark the twelfth step, nor the twelfth note, this name didn't seem to be applicable in connection with BP.

So in analogy to the traditional Western scale, where the framework interval 2:1 is called "octave", as representing the eighth tone in a diatonic scale, Heinz Bohlen decided in 1972 to name 3:1 "Dekade" or "decade", respectively, because BP diatonic scales possess 10 tones. He was not really happy with that expression, though, being aware that the correct analogy to "octave" should have been "decime". But "Dezime" was already occupied, because it stands for the tenth (5:2) in German music language.

Nevertheless, he also had a hard time getting used to "tritave", an artificial word that John Pierce invented when he came across BP. But despite Bohlen's aversion against verbal contraptions, he had to admit that it served well the claim that 3:1 has the same pivotal importance for BP, which the octave has for the traditional Western scale. Thus "tritave" has become the generally accepted expression for 3:1.

- BP intervals

Most of the intervals appearing in the Bohlen-Pierce scale are not at all novel ones. But only two of them are also used in just Western tuning:
· 5/3, the just major sixth,
· 3/1, the perfect twelfth.

Two others resemble old acquaintances out of the equal-tempered western scale (12tET):
· 25/21 (302 cents) is almost identical with the equal-tempered minor third (300 cents),
· 63/25 is practically identical with the equal-tempered major tenth (1600 cents both).

Others are known as consonant, but are incompatible with traditional western tuning:
· 9/7, the supermajor or septimal major third,
· 7/5, the septimal diminished or subminor fifth (Huygen's tritone),
· 9/5, the just or acute (or natural) minor seventh,
· 15/7, the septimal minor ninth,
· 7/3, the minimal or septimal minor tenth. And
· 27/25, though certainly not consonant, is known as great (or large) limma.

When used in context with the BP scale, these names are meaningless, however. Thus, as long as we discuss these intervals in connection with BP, we give them (and their enharmonics) simple names as follows:

Span of interval (ratio)	Name
27/25	BP first
25/21	BP second
9/7	BP third
7/5	BP fourth
75/49	BP fifth
5/3	BP sixth
9/5	BP seventh
49/25	BP eighth
15/7	BP ninth
7/3	BP tenth
63/25	BP eleventh
25/9	BP twelfth
3/1	BP thirteenth, or tritave

The BP scale is what theorists call "7-limit". Each single interval (and all its enharmonics) can be mathematically expressed as

the exponents a, b, c being integers of the form -5,...-2,-1,0,+1,+2,...+6. The enharmonics table in the following chapter shows the exponents for all BP enharmonic intervals.

(For comparison: the traditional Western scale is "5-limit": 2^a 3^b 5^c, while the "triple BP scale" is "13-limit": 3^a 5^b 7^c 11^d 13^e.)

Following suggestions of Manuel Op de Coul we can address the intervals contained in a diamond as an interval class. Adding interval names we arrive at the table below (N_i stands for the number of co-incidences, i.e. how many times each enharmonic appears in the circle of possible scales):

Pentachords

In the same way the traditional Western scale can be divided into two tetrachords, any diatonic BP scale can be considered as to consist of two pentachords, as for instance

In all basic modes, including van Prooijen's 7-step modes, both pentachords consist of six semitone steps, with the highest and lowest tone forming the ratio 5/3 (just major sixth, BP sixth), and they are separated by the interval 27/25 (great limma, BP small semitone).

All four Walker modes possess two different pentachords, one again consisting of six semitone steps with a total span of 5/3, the other one having only five semitone steps and a span of 75/49 (BP fifth). In these modes the pentachords are separated by 147/125, a BP enharmonic whole tone.

Benson's Pythagorean BP scale consists of two equal pentachords 81/49 (enharmonic BP sixth), separated by 2401/2187, an interval exactly half ways between a BP major semitone and a BP great semitone.

Regarding pentachords, Stearn's BP Meantone Rotations behave in accordance with what has been said about the basic BP modes and the Walker modes above, just replacing the pentachords 5/3 and 75/49 with 868 and 766 cents, respectively, and the separating tones 27/25 and 147/125 with 166 and 268 cents, respectively.

Similarly, in Keenan's Minimum Error modes the larger pentachord is represented by 879.6 cents, the smaller one by 725.2 cents, and the two separating intervals by 142.8 cents and 297.2 cents, respectively.

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