Sagittal notation for beatles[27]

I've started to take an interest in the beatles 7-limit temperament

[<1, 1, 5, 4], <0, 2, -9, -4]>
TOP-MAX P = 1197.104145, G = 354.720338
TOP-RMS P = 1196.641985, G = 354.908182

Since this is practically 27-ET, it would be nice to use 27-ET Sagittal notation for it. Unfortunately, Sagittal notation uses /| for 1 step of 27-ET and /|) for 2 steps, but these both represent the same interval in beatles temperament since |) is tempered out.

So, one of these has to be replaced. /| turns out to be one step of the beatles[27] MOS; the other step could be ~| or ~|)

One of the 2-step intervals is )||( which, when combined with the /| step results in ||\ for a 3-step interval. This fits well into the notation, where for instance E!!/ is a major third below G.

The other 2-step interval is less obvious, but here's one that works.

~|) + )/|\ = ||\

There's another 3-step interval, although it's not as useful for actual notation. But for the sake of completeness here it is.

/| + )/|\ = ~~||

So far we've got:

(+3, -10) ~|) [-4, -1, 0, 2>
(-5, +17) /| [-4, 4, -1>
(+6, -20) )||( [-3, -1, 2>
(-2, +7) )/|\ [-3, 4, 1, -2>
(+1, -3) ||\ [-7, 3, 1>
(-7, +24) ~~|| [-7, 8, 0, -2>

I haven't looked much into larger intervals, although /||\ would be a 4-step interval twice the size of )/|\ (-4, +14). It would be convenient for the notation to use /|\ in place of )/|\ (which would imply an 11-limit temperament).

Herman Miller wrote:
> I've started to take an interest in the beatles 7-limit temperament
> > [<1, 1, 5, 4], <0, 2, -9, -4]>
> TOP-MAX P = 1197.104145, G = 354.720338
> TOP-RMS P = 1196.641985, G = 354.908182
> > Since this is practically 27-ET, it would be nice to use 27-ET Sagittal > notation for it. Unfortunately, Sagittal notation uses /| for 1 step of > 27-ET and /|) for 2 steps, but these both represent the same interval in > beatles temperament since |) is tempered out.

After choosing symbols for magic tripod notation ... and getting it wrong ... I've decided that I don't like the Athenian ET notations. It's better to work with the JI logic and find a symbol of about the right size. But there could be all kinds of subtleties I'm not aware of.

> So, one of these has to be replaced. /| turns out to be one step of the > beatles[27] MOS; the other step could be ~| or ~|)

/| is the syntonic comma, isn't it? That's too small to be a step from 27-equal or anything close to it.

~|) is large in the PDF, and so deemed important, but doesn't belong to any of the subsets. Again, my opinion, but I'd rather ignore these non-subsetted characters altogether. The full Sagittal set is too large. Athenian symbols should be given priority, or Trojan ones where relevant (I prefer the Trojan logic to Athenian ETs). A 7-bit font would be very convenient.

> One of the 2-step intervals is )||( which, when combined with the /| > step results in ||\ for a 3-step interval. This fits well into the > notation, where for instance E!!/ is a major third below G.
> > The other 2-step interval is less obvious, but here's one that works.
> > ~|) + )/|\ = ||\

||\ is a good Spartan symbol so use it if it fits.

> There's another 3-step interval, although it's not as useful for actual > notation. But for the sake of completeness here it is.
> > /| + )/|\ = ~~||
> > So far we've got:
> > (+3, -10) ~|) [-4, -1, 0, 2>
> (-5, +17) /| [-4, 4, -1>
> (+6, -20) )||( [-3, -1, 2>
> (-2, +7) )/|\ [-3, 4, 1, -2>
> (+1, -3) ||\ [-7, 3, 1>
> (-7, +24) ~~|| [-7, 8, 0, -2>
> > I haven't looked much into larger intervals, although /||\ would be a > 4-step interval twice the size of )/|\ (-4, +14). It would be convenient > for the notation to use /|\ in place of )/|\ (which would imply an > 11-limit temperament).

Do you care about the prime limit? I think sometimes you have to use a symbol from outside the limit for a tempered notation, because it comes in between two intervals you're tempering to be the same.

In this case, /|\ is one of the main symbols, the quarter tone. And then )/|\ is an ugly symbol that isn't in any subsets ... but will look like /|\ with a bit of dirt on the screen so it'll be easily understood.

Graham

Graham Breed wrote:
> Herman Miller wrote:
>> I've started to take an interest in the beatles 7-limit temperament
>>
>> [<1, 1, 5, 4], <0, 2, -9, -4]>
>> TOP-MAX P = 1197.104145, G = 354.720338
>> TOP-RMS P = 1196.641985, G = 354.908182
>>
>> Since this is practically 27-ET, it would be nice to use 27-ET Sagittal >> notation for it. Unfortunately, Sagittal notation uses /| for 1 step of >> 27-ET and /|) for 2 steps, but these both represent the same interval in >> beatles temperament since |) is tempered out.
> > After choosing symbols for magic tripod notation ... and > getting it wrong ... I've decided that I don't like the > Athenian ET notations. It's better to work with the JI > logic and find a symbol of about the right size. But there > could be all kinds of subtleties I'm not aware of.

Another thing to consider is the size of fifths in beatles temperament. The fifths at the far ends of the chain are far enough off that "B" sounds sharp and "F" sounds flat. So B\! and F/| are actually reasonable notations for those notes, while B|) and F!) are better representations of the actual pitches from the chain of tempered fifths than just a plain B or F would be.

>> So, one of these has to be replaced. /| turns out to be one step of the >> beatles[27] MOS; the other step could be ~| or ~|)
> > /| is the syntonic comma, isn't it? That's too small to be > a step from 27-equal or anything close to it.

Yes; in fact /|) is a much better match. If the conflict with 27-ET notation can be ignored, it's a potential choice for notating (-5, +17). There are so many criteria to consider though, like symbol arithmetic.

/| + ||\ = /||\
/|) + (|\ = /||\

Since there's not any good reason to use (|\ for (+1, -3) in place of ||\ , that's another reason in favor of /| .

I'm wondering though how the 27-ET notation in the Sagittal paper came about. [2, 2, -1, -1> is about 1.1 steps of 27-ET, not anywhere near 2. Better choices for (+2) of 27-ET include )||( [-3, -1, 2>, which happens to match the 2-step (+6, -20) of beatles[27] that I mentioned.

> ~|) is large in the PDF, and so deemed important, but > doesn't belong to any of the subsets. Again, my opinion, > but I'd rather ignore these non-subsetted characters > altogether. The full Sagittal set is too large. Athenian > symbols should be given priority, or Trojan ones where > relevant (I prefer the Trojan logic to Athenian ETs). A > 7-bit font would be very convenient.

A~|) is 49/32 relative to D, which is a pretty useful ratio to be able to notate in a 7-limit temperament. A simpler alternative is A.(|\ 14/9, but that uses an accent and the alternative A|\\ is less well known. The only other one that might be under consideration is A'(| 75/49.

> Do you care about the prime limit? I think sometimes you > have to use a symbol from outside the limit for a tempered > notation, because it comes in between two intervals you're > tempering to be the same.
> > In this case, /|\ is one of the main symbols, the quarter > tone. And then )/|\ is an ugly symbol that isn't in any > subsets ... but will look like /|\ with a bit of dirt on the > screen so it'll be easily understood.

Actually, the 11-limit generator mapping that makes the most sense in this case is <0, 2, -9, -4, 5] which does have /|\ as a notation for (-2, +7), along with (|) )||~ and others. (|( shows up as an alternative for ~|) but that's another one of those unfamiliar symbols.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> I've started to take an interest in the beatles 7-limit temperament
>
> [<1, 1, 5, 4], <0, 2, -9, -4]>
> TOP-MAX P = 1197.104145, G = 354.720338
> TOP-RMS P = 1196.641985, G = 354.908182
>
> Since this is practically 27-ET, it would be nice to use 27-ET
Sagittal
> notation for it. Unfortunately, Sagittal notation uses /| for 1
step of
> 27-ET and /|) for 2 steps, but these both represent the same
interval in
> beatles temperament since |) is tempered out.

Hi Herman,

As I recall, Dave & I chose symbols for the 27-equal notation that
corresponded to the lowest primes that 1) are most accurately
represented, and 2) don't vanish, which would be 5 and 13, giving the
following sequence:

27: /| /|) ||\ /||\

Since |), which vanishes, doesn't appear in the symbol set to upset
the symbol arithmetic, we decided that it would be okay to use /|)
for the medium-size 13-diesis (1024:1053), although it's technically
a 35-diesis (35:36). Now it appears that our decision has come back
to bite us. :-(

As you point out, it would nice to use the same accidentals for both
27-edo & beatles[27]. How about if we we to notate 27-edo this way?

27: /| )/|\ ||\ /||\

Since )/|\ is defined as 5:49M (392:405), this would use 7-limit
symbols. I've come around to the idea that it's good to use )/|\ for
half of /||\ in places where /|\ isn't valid for that purpose.

--George

George D. Secor wrote:
> As you point out, it would nice to use the same accidentals for both > 27-edo & beatles[27]. How about if we we to notate 27-edo this way?
> > 27: /| )/|\ ||\ /||\
> > Since )/|\ is defined as 5:49M (392:405), this would use 7-limit > symbols. I've come around to the idea that it's good to use )/|\ for > half of /||\ in places where /|\ isn't valid for that purpose.

Yes, all of those are consistent with beatles. The full list of symbols my program came up with for the (-2, +7) interval is:

)| '|) )/|\ .||)

of which )| is clearly unsuitable due to its size, and the other two are accented (which is inconvenient if unaccented symbols are available).

The other 2-step interval in beatles[27], (+6, -20), has only the )||( symbol as an option.

It might be worth checking other temperaments with 27-note MOS for consistency: e.g. augene, superpyth, sensisept, myna, ennealimmal.

I've already got a list for myna.

(-8, +31) /|
(-9, +35) )/|\
(-10, +39) ||\
(-18, +70) /||\

None of these are really relevant in a 27-note subset of myna, so there's no real conflict. But if for some reason we want to notate 27-ET as myna, with a 7/27 generator, the math works out.

(31 * 7) - (8 * 27) = 1
(35 * 7) - (9 * 27) = 2
(39 * 7) - (10 * 27) = 3
(70 * 7) - (18 * 27) = 4