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See also Heinz Bohlen's work:
http://www.huygens-fokker.org/bpsite/otherscales.html
Division of the tritave (3/1) into n equal parts | Rank two temperaments | Individual pages for EDT's | Multiples of 13EDT which approximate EDO

Division of the tritave (3/1) into n equal parts


Western music generally revolves around the principle of octave equivalence: notes an octave apart are often perceived in western music as being the same chroma but differing in pitch height. As the octave corresponds to a 2/1 frequency ratio, it has been proposed that the next-simplest after the octave, the 3/1, can also be used to evoke a sense of chroma equivalence. This interval corresponds to a perfect twelfth in the diatonic scale, but when used to refer to an equivalence interval it is often called the "tritave".

It has been argued that pitches a tritave apart can never truly be heard as equivalent in all of the ways that octaves are, with some claiming that the tonotopic representation of the mammalian auditory system is inherently biased towards octave-equivalence. With proper context, experience, and training, however, at least some people find that they can experience some degree of tritave equivalence especially using timbres restricted to odd harmonics. It is not known whether odd harmonics actually facilitate the ability to hear in tritave-equivalence. Either way, it is certain that musically valuable organizations of pitch can arise through the equal division of non-octave intervals, regardless of whether the period is perceived as being truly chroma-equivalent, and as such the multitude of equal divisions of the tritave are rich and ripe for exploration.

The Bohlen-Pierce (BP) scale, most commonly consisting of 13 equal divisions of the tritave (although a justly-intoned version exists as well), seems to have been the first such arrangement to be seriously studied and made into music. The BP scale was independently discovered by Heinz Bohlen, John Pierce and Kees Van Prooijen. Bohlen found it while looking for triads with equal-difference tones, Prooijen uncovered it while searching for equally-tempered scales with accurate higher harmonics, and Pierce stumbled upon it trying to find consonant chords other than 4:5:6. Though they all started with different goals in mind, each of them amazingly ended up at the same destination.

Rank two temperaments

If factors of two are eliminated, the simplest possible triad is (1):(3):5:7:(9), with 1, 3 and 9 in parentheses as they're all tritave-equivalent to 1. Hence, 3:5:7 can be viewed as the fundamental consonant triad of no-twos music. The linear temperament that best approximates these chords in the moderate complexity range is the Bohlen-Pierce linear temperament eliminating 245/243, which has a no-twos mapping of [<1 1 2|, <0 2 -1|] and a pure-tritaves TE generator which is a sharp 9/7 of 440.488 cents. It possesses MOS of the forms 4L1s (pentatonic) and 4L5s (nonatonic), and larger MOS of size 13, 17, 30, 43, 56, 69 and 82. This temperament serves a function analogous to meantone in the 5-limit.

At higher complexities, the rank two 3.5.7 temperament tempering out 16875/16807 called Canopus begins to predominate. This has a mapping [<1 3 3|, <0 -5 -4|] and a pure-tritaves TE generator a slightly flat 7/5 at 581.512 cents. This has MOS of size 3, 4, 7, 10, 13, 23, 36, etc, with the 36 note MOS being particularly even.

The other no twos rank two temperament which 13edt "supports" is Arcturus, which takes an ~5:3 as a generator. I speak advisedly of 13edt supporting this temperament because the lowest-error proper MOS of it is a 2L 11s triskaidecatonic scale. However, if you do not mind having a smeary 5, you will need only a 2L 7s (nonatonic) scale to make an understandable rendition of it.

The named but not necessarily no twos rank two temperament which 13edt "supports" is Sirius, which takes a generator between ~7:6 and ~6:5. Like Arcturus, I speak advisedly of 13edt supporting it because the most proper small MOS of it is triskaidecatonic. Unlike Arcturus, there is a smaller MOS of it than this which is technically proper. However, this MOS is the Grumpy heptatonic scale the use of which is made problematic by the uniqueness of the step of the second size. It is problematic to have the step of the second size be unique in a subscale of an edx because it creates a strong sense of a second equal division of a y strictly less than x, and this sense of two different equal divisions trying to happen in the same scale causes ordinary concepts of equivalence to break down in spectacular ways. If this "problem" has not been named yet, "cross-equivalence artifacting" would be a perfect name for it.

The final interval which 13edt can reasonably use to generate a rank two temperament is its false 3\2 of 5 degrees. By a weird coincidence, it will generate the 5L 3s unfair father octatonic scale just as if it were an interval of an edo, except that the scale will not always contain a false 4\3 as it must in an edo. This means, most importantly, that 16\15 cannot be assumed to be a "comma" tempered out by this false Father temperament when it is taken as a temperament of full just intonation. By a second, and totally separate, weird coincidence, the well-known Bohlen-Pierce temperament is its index-2 subtemperament.

Due to the fact of its 9/7 generator, the temperament which is to BP what neutral temperaments are to syntonic temperaments does not become intelligibly a division of the tritave until extended to 17 tones whereas EDOs supporting various neutral temperaments have an "ordinary" heptatonic scale which is intelligibly a division of the octave. Additionally, 7 and 9 being consecutive odd numbers means that trying to force this temperament into a no-twos subgroup induces very poor "approximations" of less intelligible higher harmonics. To avoid this, this temperament should be assumed to be a temperment of the 3.5.7.8 subgroup tempering out 245/243 and 64/63, the familiar comma from EDOs supporting the Superpythagorean or Parapythagorean diatonic scale.

Among the EDTs tempering out 245/243, 13EDT stands out. An apt analogy can be drawn with EDOs supporting meantone: 4EDT and 9EDT are to BP what 5EDO and 7EDO are to meantone. However, in contrast to meantone, the simplest EDT supporting the BP nonatonic scale - 13EDT, the traditional tempered BP scale - is the most accurate and lowest in tuning error until 56EDT. However, there are many EDTs supporting BP temperament which also support extensions to higher-limit temperaments; in particular 2, 3, and 4 times 13 in the form of 26EDT, 39EDT and 52EDT as well as 56EDT. For tempering out 16875/16807, 13EDT again stands out, though much better accuracy can be found in more complex divisions such as 114EDT or 127EDT. All of this explains the focus on 13EDT to the exclusion of other EDTs among practitioners of the art of nonoctave composition, but it must be noted that the analysis is only valid if consideration is confined to the 7-limit, which is exactly analogous to confining it to the 5-limit with EDOs. There's a whole other world out there which has not been much explored.

For example, 15EDT very well approximates the 5th and 13th harmonics, and 12EDT, the 13th and 17th. 39EDT makes for a fine 3.5.7.11.13 system, tempering out 245/243, 275/273, 847/845 and 1331/1343, and so supporting among other things the [<13 19 23 0 2|, <0 0 0 1 1|] temperament supported by the whole suite of 13nEDTs: 13, 26, 39, 52, 65, 78 etc.

One should bear in mind that, assuming tritave equivalence, when determining which harmonics are represented, the ratios of 3 in the denominator are fungible instead of those of 2. For example making the fifth harmonic 5:3 a "major sixth" by conventional (and arbitrarily silly for the purposes of xenharmony, even with octaves) pitch class terminology.

There are other uses, or conceptualizations, of tritave-based tunings. Purely intuitive use of these myriad, assuredly xenharmonic structures comes to mind (see "EDO" versus "equal temperament"). Another intent might be to find or define temperaments (such as Magic, Hanson, etc.), or to provide exact formulae for stretching/compressing what would musically be used as an "ordinary" octave of ~2:1. (And given the stable nature of octave-based systems, some aesthetic overlap even in the most tritave-equivalent of music, would be forseeable.) For instance, the Bernhard-Stopper (19edt) temperament, might for instance be found useful in tuning pianoforti, being equivalent to 12edo, except for a 2c sharp octave which is relevant to inharmonicity.

Below is a large list of EDT's; additionally, some equal divisions of the tritave are known by alternate names or have special interest:

3edt (Liese generator)
4edt (Vulture generator)
5edt (Tritave counterpart of Magic)
6edt (Tritave counterpart of Hanson)
7edt (Tritave counterpart of Orwell)
8edt (Tritave counterpart of Vulture)
11edt "Euler Temperament"
"Bohlen-Pierce" or "BP"
"Bernhard Stopper"
39edt Triple Bohlen-Pierce (Erlich)

Individual pages for EDT's


1edt
2edt
3edt
4edt
5edt
6edt
7edt
8edt
9edt
10edt
11edt
12edt
13edt
14edt
15edt
16edt
17edt
18edt
19edt
20edt
21edt
22edt
23edt
24edt
25edt
26edt
27edt
28edt
29edt
30edt
31edt
32edt
33edt
34edt
35edt
36edt
37edt
38edt
39edt
40edt
41edt
42edt
43edt
44edt
45edt
46edt
47edt
48edt
Larger divisions of the tritave include 52edt, 56edt, 71edt, 75edt, 88edt, 131edt, 245edt and 316edt.

A list of tritave reduced harmonics for easy comparison of JI and temperaments in tritave based systems.

Also may be found convenient: http://www.nonoctave.com/tuning/twelfth.html

EDO-EDT correspondences
EDO
EDT
Comments
5edo
8edt
8edt is equivalent to 5edo with ~11 cent octave compression.
Equivalently, 5edo is 8edt with ~18 cent stretched tritaves.
Patent vals match through the 13 limit.

9edt

6edo



10edt

7edo
11edt
11edt is equivalent to 7edo with ~10 cent stretched octaves.
Patent vals differ in the 7 limit, but neither can really be said
to represent the 7th harmonic with a straight face.

12edt

8edo



13edt
The equal-tempered BP scale cannot be considered equivalent to 8edo.
9edo
14edt
There is a lot of mismatch between the pure-octave and pure-tritave tunings,
but the patent vals match through the 13 limit. Great for stretched-octave pelog!

15edt

10edo
16edt
Similar situation to 5edo~8edt. Patent vals match through the 17 limit.

17edt

11edo



18edt

12edo
19edt
19edt is 12edo with ~1.2 cent octave stretch. Patent vals match
only through the 7 limit, but neither can be said to include 11 at all.

20edt

13edo



21edt


22edt

14edo



23edt

15edo
24edt
This is only a rough correspondence, as the (5n)edo ~ (8n)edt sequence
begins to break down. The patent vals match only through the 5 limit.
16edo
25edt
Also only a rough correspondence; 25edt corresponds to 16edo
with ~17 cent octave stretch. Patent vals match through the 5 limit.

26edt
Double BP
17edo
27edt
27edt is 17edo with ~2.5 cent compressed octaves. With the exception of
5 (which neither represents well), patent vals match through the 13 limit.

28edt

18edo



29edt

19edo
30edt
30edt is 19edo with ~5 cent stretched octaves.
Patent vals match through the 7 limit.

31edt

20edo



32edt


33edt

21edo



34edt

22edo
35edt
35edt is 22edo with ~4 cent compressed octaves.
Patent vals match through the 11 limit.

36edt

23edo



37edt

24edo
38edt
Same ~1.2 cent octave stretch as 12edo~19edt.
Patent vals match through the 19 limit.

39edt
Triple BP
25edo



40edt

26edo
41edt
41edt is 26edo with ~6 cent stretched octaves.
Patent vals match through the 7 limit.

42edt

27edo
43edt
43edt is 27edo with ~6 cent compressed octaves.
Patent vals match through the 7 limit.

44edt

28edo



45edt

29edo
46edt
46edt is 29edo with ~0.9 cent compressed octaves.
Patent vals match through the 89 limit. (Really! I checked!)

Multiples of 13EDT which approximate EDO


Also, on the topic of multiples of 13edt, 26 (double) and 39 (triple) offer very good harmonic approximations, the former of the 8th, 13th and 17th partials, and the latter of the 11th and 13th. However, quadruple and quintuple, ie. 52 and 65edt, also exist offering good approximations of the octave. 52edt is very nearly 33edo, and 65edt is practically identical to 41edo.