On this page:
Tritave
BP
intervals
Enharmonics
Pentachords
The compass 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 compass 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.
Funny enough, John Pierce's creation has meanwhile invaded his (and also Heinz Bohlen's) original field of expertise, microwave technology. A recent paper on a 6-18 GHz solid-state receiver uses the expression "tritave" for just this frequency relation.
- 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 conventional
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), and is therefore called "quasi-tempered
minor third";
· 63/25 is practically identical with the equal-tempered
major tenth (1600 cents both) and consequently named "quasi-tempered
major tenth".
Others are known as consonant and not entirely "unheard of",
but are incompatible with conventional western tuning:
· 9/7, the supermajor or septimal major third, may occur
in bagpipe tuning between low G (7/8) and B (9/8) [*],
· 7/5, the septimal diminished or subminor fifth (Huygen's
tritone), also sometimes appearing in bagpipe tuning between C
(5/4) and high G (7/4) [*],
· 9/5, the just or acute (or natural) minor seventh, is
sometimes used as high G in bagpipe tuning [*],
· 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, and can appear in bagpipe tuning between F (5/3)
and high G (9/5) as well as between C (5/4) and D (27/20) [*].
[*]
Ewan Macpherson:
The
Pitch and Scale of the Great Highland Bagpipe.
Reference
is low A (1/1).
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":
2a 3b 5c, while the "triple
BP scale" is "13-limit": 3a 5b
7c 11d 13e.)
- Enharmonics
If
we try other chromatic scale members than 1/1 as base tones for
the intervals listed above, we encounter various enharmonic tones,
i.e. tones that are close to those already known, however not
exactly identical. Not surprisingly they show the same relationship
among each other that we detected when investigating the different
semitones: they form a chain, consisting of diamonds similar to
the one we know already (and including this one). The following
table tries to illustrate that chain (interval spans in brackets
are theoretical only):
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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 (Ni stands for the number of co-incidences, i.e. how many times each enharmonic appears in the circle of possible scales):
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unison |
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great limma, BP small semitone |
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BP minor semitone | |
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BP major semitone | |
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BP great semitone | |
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BP second, quasi-tempered minor third |
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septimal major third, BP third |
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9/4-tone, septimal semi-diminished fourth | |
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septimal tritone, BP fourth |
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BP fifth | |
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major sixth, BP sixth | |
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just minor seventh, BP seventh |
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octave minus maximal diesis |
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BP eighth | |
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septimal minor ninth, BP ninth |
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Al-Hwarizmi's lute middle finger |
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minimal tenth, BP tenth |
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BP eleventh, quasi-equal major tenth |
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classic augmented eleventh, BP twelfth | |
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just twelfth, BP thirteenth, "tritave" |
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.