## Bohlen-Pierce

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### Bohlen-Pierce

I apologize for posting this while @Dave Keenan is still preoccupied, since I'm sure he'll want to weigh in, but I'm pretty excited about sharing my thoughts on this topic, and there's certainly no rush for him to respond.

So... there are no hits for Bohlen Pierce in the forum anywhere. Or there weren't until very recently.

I Googled for BP and Sagittal and found only a couple hits, both involving a fellow named Georg Hajdu: In the former, a reference is made to a Max/MSP plugin Hajdu worked on which provides Sagittal notation for BP. Here's the page about it:
http://www.computermusicnotation.com/microtones/
I have a Max/MSP license so I went ahead and played around with it a bit. For the life of me I couldn't get it to show anything fancier than semi- and sesqui- conventional accidentals. I couldn't get it to show EHEJIPN as it advertises, nor did I find any evidence that it includes Sagittal (anymore?).

In the latter link, Hajdu laments:
It is possible to notate BP pitches by using eighth-tone or sagittal accidentals, but it is unsatisfying because musicians do not easily recognize the structure of the non-octave BP scale when mapped onto a diatonic octave-based notation system.
The link in this footnote goes to the Sagittal main site, not to anywhere specific.

I decided I wanted to assuage Hajdu's concerns and try my hand at notating BP in Sagittal. Here's what I've got so far.

Basics

BP is often thought of strictly as 13EDT, but I prefer to think of BP as a system of scales in the 3.5.7 subgroup which specifically eschew the prime 2. We will get to 13EDT soon, but first I wanted to get at the underlying just intoned motivation, much as Sagittal does for EDOs.

The first thing we need to find for BP is its equivalent of the chain of Pythagorean fifths (3/2) which is so core to other Sagittal notations. On this page http://www.huygens-fokker.org/bpsite/notation.html it is described how BP's diatonic scale is instead generated by a chain of septimal major thirds (9/7).

At ~435.084¢, this generator leads to 9-note MOS with L=273.465 and s=161.619 and Ls pattern of LssLsLsLs, or a 13-note chromatic scale with with L=161.619 and s=111.847, and Ls pattern of LsLLsLLLsLLsL. Just like with a diatonic Pythagoran scale of 2/1, we have a limma, an apotome, and a whole tone which is their sum. The 273.465 is our whole tone, the 161.619 is our limma, and the 111.847 is our apotome (quite nice how closely that comes to the 113.685 apotome for Pythagorean scale of 2/1).

The apotome is equal to moving by the septimal major third upward 9 times; the monzo for this third is |0 2 0 -1>, so we first move to |0 18 0 -9>, then tritave-reduce so that the monzo for the BP apotome is |0 16 0 -9>, or in ratio form, 43046721/40353607. The BP limma is equal to moving by this third downward 5 times (interesting that it's the same number of times as the Pythagorean scale of 2/1's limma), so first we move to |0 10 0 -5>, then tritave-reduce so that the monzo for the BP limma is |0 9 0 -5>, or 19683/16807.

Moving by the BP limma increments one letter along BP's cycle of nominals CDEFGHJAB, much in the same way we normally increment a letter along the nominals CDEFGAB. The BP apotome is conventionally represented by . Can we represent it as ? I want to immediately shout "yes," but let's be cautious. If I were to describe the secondary comma zone for the apotome, I would say it should be equivalent to the secondary comma zone for the natural symbol. In other words, since at the first standard JI precision level where the natural appears (naturally, the lowest: medium) the capture zone for the natural ranges from -2.740¢ to 2.740¢, that is its secondary comma zone. And since the BP apotome is off from the standard apotome by only 113.685 - 111.847 = 1.838¢, the answer is yes: it is within that capture zone.

So then 13EDT BP can be notated using just  .

Commas

My instinct is to not throw out all of the lovely symbols and commas Sagittal already has going on outside of BP, even though almost all of them have 2's in them (only three don't: the 245C, 25:77M, and 7:65M).

If one felt so compelled, one could make 3.5.7 subgroup variants by counteracting any 2 with one of the following intervals:

ragisma			|-1 -7 4 1>	4375/4374					0.39576¢
laleruyo		|-1 4 11 -11>	3955078125/3954653486				0.18588¢
171&1547&4973 comma	|1 -15 -18 23>	54737494680161832686/54736736297607421875	0.023986¢


Perhaps if you're lucky, like the 25C , to have eleven 2's in your monzo, you could adjust yourself by the difference between the apotome and the BP apotome:
|-11 -9 0 9>	40353607/40310784	1.838150434¢

This issue aside, we would probably want a set of tailored-for-BP secondary commas helpfully provided for users, i.e. specific to common 3.5.7 subgroup JI pitches deviating from a chain of just septimal major thirds. It turns out that only two unique commas are necessary to achieve this (and multiples and combinations of them):

|0 -8 -3 7>	= 823543/820125	≈ 7.200¢
|0 -5 1 2>	= 245/243	≈14.191¢


The latter of these two is the 245C, which is already defined in Sagittal in the standard extreme precision level JI notation with the symbol  ; this is also the comma which 13EDT BP tempers out, and it is apparently known as the sensamagic comma.

The former of these two is not defined in Sagittal, but it could be a secondary comma of  ,  ,  , or  .

The flag arithmetic gets interesting here. The symbol for the ~14¢ comma has a left boathook and a right scroll. The symbol for the ~7¢ comma might have either a left boathook or a right scroll. The ~14¢ comma is plainly about twice the size of the ~7¢ comma, so this looks promising. It gets even more interesting when you observe that it is not exactly twice the size. So that suggests that there is actually another ~7¢ comma here which sums with the one we've already found to make this 14¢ comma. That comma is:

|0 3 4 -5>	16875/16807	≈ 6.990¢


The difference between them is:

|0 -11 -7 12>	13841287201/13839609375	≈ 0.209871¢


which is about 1.5 tinas, or half a mina. I named it the "bapbo schismina" for now.

So while it feels a bit odd to have two symbols so close to each other, it seems like it could be an interesting possibility to assign to 16875/16807 and to 823543/820125 so that the flag arithmetic for works out exactly.

Four other commas turned up frequently for me. Here they are with the symbols I think they should map to:

|0 -13 -2 9>	40353607/39858075		21.391¢
|0 -10 2 4>	60025/59049			28.381¢
|0 -15 3 6>	14706125/14348907		42.572¢
|0 -28 1 15>	23737807549715/22876792454961	63.962¢


Again, these are all merely combinations of multiples of the two previously introduced commas; e.g. is twice  , is thrice  , and is plus  . So the symbol's comma is not actually twice that of the comma, so maybe that situation could be massaged a bit using the bapbo schismina. The fact that the 25S which primarily represents is one of the few commas in Sagittal which is exactly twice the size of another comma (the other being the 49M which is exactly the 7C, and for that one the flag arithmetic is not as pretty... I guess fitting two arcs on one side of a shaft didn't work visually, or any other myriad of reasons) gives me extra impetus to make this so.

All of these commas are approximately multiples of 7¢. This leads me to believe that for 273EDT may be of some use to us. That's 13EDT with each step divided into 21 equal parts of ~6.96686813504¢ each.

So maybe we have a symbol at 7, 14, 21, 28, 35, 42, 49, 56, and 63, each of which covers a range of about 7¢, ±3.5¢ in either direction, and that's like the medium precision level in BP.
Those symbols and zones would be:

or from 3.483 to 10.450
from 10.450 to 17.417
from 17.417 to 24.384
from 24.384 to 31.351
from 31.351 to 38.318
from 38.318 to 45.285
from 45.285 to 52.252
or from 52.252 to 59.218
from 59.218 to 66.185

Because BP, in this sense, is 7-limit, this level of precision may be all we need.

Mixed (Evo) and Pure (Revo)

A key issue to consider with respect to a slightly smaller apotome is that of apotome complements.

Let's first consider the impact on the Evo (Mixed) flavor, because it's simpler.

The simplest answer is: no impact. Get your apotome complement as you did before, by apotome plus negated orientation of self.

However, if one wants to respect the special apotome complement rule inside that zone centered in the middle of an apotome interval where the ~68¢ zones emanating from the neighboring apotome anchors overlap and a simpler-to-write (not simpler-to-understand) option exists as the mirrored symbol across that zone, then the situation could change.

But does/should it? I think: no. A key question to ask is: is still the very center of the apotome and therefore the symbol which is its own apotome complement? To answer that, we must find the new center point of the apotome. 111.847 / 2 = 55.9235¢. So, what is the closest symbol to that? The two neighboring symbols are the incumbent center at 56.482¢ and the contender center at 54.528¢.

|56.482-55.9235|=0.5585
|54.528-55.9235|=1.3955

So actually we're still closer to  , so I suggest we change nothing with respect to apotome complements in the Evo flavor.

But now how about the Revo (Pure) flavor?

Probably the best place to start when thinking about how the Revo flavor would be affected is the relationship between the apotome complements symbol and the symbol which have low limit JI compositions and are near the center of the apotome. In Revo, one starts the multi-shaft symbols right after the symbol, which essentially gets transformed into the extra shaft , then working up from the double-shafted version of the smallest symbol right above the natural,  , which becomes : )||: , up to the which becomes , the apotome, at the point where the two aforementioned apotome complements are combined (with the notable exception that because a few single-shafted symbols exist which are larger than   (e.g. ) these override symbols which might-have-been but-aren't-actually such as : )||: , hence the lack of smiley for it. Anyway, this is the anatomy of a Revo apotome.

Because of the special structural importance of and to Revo, I think we must take special care in assigning BP commas to these symbols. We want commas that are in the 3.5.7 subgroup (with no 2), which are exactly apotome complements (to the BP apotome |0 16 0 -9>), and which are very close to the primary values for and  . Another issue arises here which is that the primary values for and are 11-limit, which is beyond the 3.5.7 subgroup. So while we could certainly squish the 2's out of these intervals using methods previously discussed, we'd then need to squish the 11's out without reintroducing 2's. Anyway, there's probably a simpler way to approach the problem - write an to find a 3.5.7 subgroup comma which is close to either 53.273 or 60.412.

EDTs

We would want to come up with a stack of 13R edos: the Trojan notation of BP.

Rather than dive straight into parceling the 146.304¢ of a 13EDT step into 20 or so symbols, though, let's start with just one 13N-EDT of particular importance: 39, known as Triple Bohlen-Pierce. With two steps in-between each step already labeled with a nominal and apotome symbol combination, we'll need just one more symbol to hit the other two (its up version, and its down version). So this symbol will represent one step of 39EDT then, so we're looking for something around 48.768¢.

In JI, that'd be very close to the symbol, but this is tempered town. In 39EDT, the 3 is just, but the 7 is tempered; it would normally be 1466.871¢ (tritave-reduced) but here is mapped to 10°13 or 1463.042¢. The 5 is also tempered; it would normally be 884.359¢ (again, tritave-reduced) but here is mapped to 6°13 or 877.825¢. However, I do not think we should let the tempering of the 5 affect the comma for the same reason that when choosing commas for EDOs we only temper the 3.

So we'll need to adjust our BP commas so that every 7 gets damaged by the proper amount (1466.871 - 1463.042 = 3.829¢). For example, this is what stretches the 111.685¢ just BP apotome out to 146.304¢; you've got nine 7's in there, so moving up 3.829¢ for each one gets you = 34.461 which equals 146.308 - 111.847 (well, within the expected margin of error for this level of decimal precision).

So what symbol would work then? There is one version of the 49¢ interval in BP with the monzo |0 -12 7 1> = 546875/531441 ≈ 49.562¢. Since it has only the one 7, it doesn't get damaged too badly. So I think it is acceptable to use the symbol for one step of 39EDT. This does also resonate with existing EDO notations such as 50, 57, and 64, where is one-third of  .

Conclusion

This has been a lot of work already, but it has been a lot of fun to think about. Please feel free to shoot down any/all of these ideas. I feel a bit out of my depth.

Dave Keenan
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Location: Brisbane, Queensland, Australia
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### Re: Bohlen-Pierce

I just want you to know that I'm finally looking into this. So, if you've had any new ideas in the meantime, or want to delete any old ones, now would be the time.

There are several different ways of approaching the notation of BP. Thanks for finding those articles.

I assume the notation that uses "1/8th-tone" symbols, notates BP as every 5th note of 41-edo on a conventional staff.

You seem to be trying to notate a lot more than Bohlen-Pierce here, but without clearly establishing what. You say the 3.5.7 subgroup, but that seems way more general than what you dealing with in this post.

You adopt the 9 MOS nominals CDEFGHJAB and therefore a BP staff, not a conventional staff. Therefore you're done when you substitute Sagittal sharps and flats for conventional ones. Then I'm reading on and thinking, why does he want all these other symbols for small commas? There are no intervals that small in BP. Then I learn you're trying to make like a "BP Trojan" notation to notate anything relative to 13-EDT (13-ED3), in particular, multiples of 13-EDT.

But I've never thought of Trojan (12-EDO Relative) notation as any kind of ideal. More of a sop for those who can't wean themselves off their dependence on 12-EDO. And that can be very hard because it's so well entrenched. But 13-EDT is hardly entrenched.

You mention the approximate 7:9 generator, with a tritave period. The Bohlen-Pierce-Stearns (BPS) temperament optimises that to around 442.14 cents (my current favourite). There's some info about it here.
https://robertinventor.com/tuning-math/ ... html#udpm5
It's better approximated as a subset of 30-EDT or 43-EDT or 116-EDT, rather than 13-EDT. It's kind of the "meantone" of BP.

For notating BP on a conventional staff, instead of every fifth step of 41-EDO, another option is to notate it like 8-EDO where the octave is compressed to 8/13 of a tritave (about 1170.4¢). On the periodic table it says to notate 8-EDO as a subset of 24-EDO, but ignore that and use the same symbols as 15-EDO, Revo:  , Evo:  . 2 steps to D:E. With 5 steps to the "fifth", the "fourth" of 3 steps turns out to be the approximate 7:9 generator. And we don't use any limmas, so no E:F or B:C steps, since the limma is negative in 8-EDO. That needs a lot more fleshing out, and of course has the problem that the nominals change as you go to higher tritaves.

I note that I'll be mostly working on my car for the next 2 days.