Sagittal accidentals for Decimal/MIRACLE notation

I wrote: "You can check whether they are valid accidentals for decimal notation by running their prime exponents thru the MIRACLE mapping to find out how many generators each corresponds to."

When I did so, I found that I'd got it wrong for the multi-shaft sagittals (by wrongly assuming 72-ET).

I'm now using the following 23-limit mapping to secors, [0, 6, -7, -2, 15, -34, -30, 54, 26]
based on Gene's 19-limit mapping from
/tuning-math/message/6450
<chew, chew, gulp>
That was me eating my words.

I was silly enough back then, to say that this mapping was uninteresting. Sorry.

So here's another attempt at sagittal versions of Graham's accidentals at
http://x31eq.com/decimal_notation.htm

Graham's ASCII-Sagittal Comma interp. secors
short long (prime exp, no 2)
-----------------------------------------------------
^^^ or m^ $# ~||| [5,0,0,0,1,0,0,-1] -30
^^ or m _# )||( [-1,2] apotome-25S -20
/^ ^ /|\ [1,0,0,1] 11-M-diesis 21
^ f |) [-2,0,-1] 7-comma -10
/ / /| [4,-1] 5-comma 31
\ \ \! [-4,1] -31
v t !) [2,0,1] 10
\v v \!/ [-1,0,0,-1] -21
vv or w =b )!!( [1,-2] 20
vvv or wv sb ~!!! [-5,0,0,0,-1,0,0,1] 30

These single (multishaft) symbols for ^^ and ^^^ and their inverses are fairly unsatisfying (particularly the apotome+13:23_kleisma for ^^^), so I tend to agree that it may be better to just use multiple copies of the sagittal 7-comma symbol.

Graham's /^ could also be represented by the single sagittal /|), but I chose /|\ for compatibility with 72-ET.

I attempted to push on and find possible single accidentals for degrees of 175-ET in decimal/MIRACLE:
~| /| )|( //| |) ? /|\ ? ||) )||( //|| ||\ ||~) /||\ ~||| /|||
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 deg
-72 31 -41 62 -10 -82 21 -51 52 -20 83 11 -61 42 -30 73 sec

17 degrees of 175-ET is one secor.

You'll see I failed to find single symbols for 6 and 8 degrees (-82 and -51 secors). But I've never thought about using sagittal like this before and it's early days.

If you or Graham can find simpler kommas corresponding to these numbers of secors, I can tell you the sagittal symbols for them. However I rejected several because they either,
(a) had a very large exponent for some prime (usually for 3 or 5), or
(b) required an accented sagittal symbol, or
(c) introduced inconsistencies in flag-degree arithmetic in 175-ET.

Regards,

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com/

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
> BTW (Dave, are you listening?), back when these messages were
> exchanged, I actually tried figuring out what sagittal accidentals
> might be used for 10 nominals for Miracle. What I ended up with was
> so unconventional, both with respect to our present way of thinking
> harmonically (by prime factors in ratios) and to the harmonic
> meanings of the sagittal symbols themselves, that I could only
> conclude that it was not worth the trouble even for a composer to use
> decimal notation.

I don't understand this at all. Where's the problem? Here's the
simplest way to do it. The basic chromatic comma is simply the 7-comma
63:64 (In ASCII shorthand "t" down, "f" up), and the enharmonic
equivalent of 3 stacked 7-commas uses the 5-comma 80:81 in the
opposite direction ("\" down, "/" up).

Here are the degrees of 72-ET between the nominals 0 and 1.

0 0/ 0f 0f/ 1t\ 1t 1\ 1
1tt 0ff

Carl:
> > But also in that thread, I assert that by forcing 7 nominals, you
> > ruin the mapping from commas to accidentals. Care to rebut that?

George:
> I answer with the following: In a Pythagorean sequence of tones
> (with 7 nominals), the position in the sequence determines the power
> of 3, and the prime factors above 3 may all be expressed in the
> accidentals.

Yes, but almost all accidental commas also contain powers of 2 and 3.

> In a sequence of tones separated by some other
> generator (such as a secor), the prime factors contained or implied
> in the relationships of those tones vary from one tone to another, so
> any comma-to-accidental relationship is much less useful (and much
> more complicated.)

Ah. I used a different method. My approach was to find out how many
generators the desired accidentals would need to correspond to, (-10
and 31). Then I calculated the MIRACLE mappings of the available
commas in order of decreasing popularity and chose the first with
unaccented symbols that corresponded to these numbers of generators.
|) and /|.

But I see now that this just notates the MIRACLE temperament itself
(maybe from 3/31 oct to 4/41 oct generators), i.e. it lets one extend
the chain beyond the 10 nominals, but doesn't say how to notate things
that are _off_ the chain. For example strict rational pitches, or
22-ET using twin chains of rather extreme-valued secors (1/11 octave).

This is analogous to looking at meantone with 7 nominals and seeing
that the basic chromatic comma corresponds to 7 generators (the
apotome) and that an enharmonic equivalent to the double apotome must
use a comma corresponding to -12 generators. This only says how to
notate meantone.

To notate rational pitches (e.g. JI), we must choose a rational value
for our notational generator, and it should be as simple a ratio as
possible (2:3). What value of secor would one choose for notating
rational pitches in decimal? It would have to be either 14:15 or
15:16. I suppose we should use 15:16 since it has the lower prime limit.

Then to notate any ET in decimal we should determine what the best
approximation is to 15:16 in that ET and if this is not between 1/10
and 1/11 octave, we should look at multiples of the ET. We should also
consider multiples if the tuning is not consistent in the sense that
two of the best 15:16 approximations are not the same as the best
225:256 approximation. Notice that this is violated in 72-ET. Maybe
14:15 would work better, but the problem is that the secor is intended
to approximate _both_ of these ratios.

This looks very messy to me, but maybe Carl and Graham can work on it
and come up with a list of commas they would like to use for various
purposes, and we can supply the sagittal symbols for them.

So I now see that, even if diatonic scales had never been invented or
had never been so popular, but assuming only that we agreed we wanted
to reuse the same nominals for pitches an octave apart, we would still
find that the fifth was the best overall choice of notational
generator simply because it has the simplest rational value possible
within the octave. Then, given human short-term memory, and the good
Dr Miller to tell us about "The magical number 7 plus or minus 2", we
would still have chosen 7 nominals, although we might have been
tempted to use 12.

Carl:
> >> But also in that thread, I assert that by forcing 7 nominals, you
> >> ruin the mapping from commas to accidentals. Care to rebut that?

George:
> >I answer with the following: In a Pythagorean sequence of tones
> >(with 7 nominals), the position in the sequence determines the power
> >of 3, and the prime factors above 3 may all be expressed in the
> >accidentals. In a sequence of tones separated by some other
> >generator (such as a secor), the prime factors contained or implied
> >in the relationships of those tones vary from one tone to another, so
> >any comma-to-accidental relationship is much less useful (and much
> >more complicated.)

Carl:
> I disagree. In the meantone diatonic scale, the position in the
> sequence of nominals tells you the powers of 3 and 5. There's
> one accidental, representing 25:24.

But these are not uniquely defined since syntonic commas may be
included or removed at will, and in any case, sagittal is not based on
meantone.

>To notate rational pitches (e.g. JI), we must choose a rational value
>for our notational generator,

Why?

>and it should be as simple a ratio as
>possible (2:3).

Why?

>What value of secor would one choose for notating
>rational pitches in decimal?

Huh?

Steps to make a notation:

1. Pick your nominals and put them on a lattice. This is equivalent
to selecting a basic scale as far as I'm concerned.

2. Pick accidentals that allow your basic scale to tile the lattice.

If your nominals correspond to a linear temperament, the value of
the generator shouldn't matter within the bounds set by the mapping
to primes that your tiling is based upon.

>This looks very messy to me, but maybe Carl and Graham can work on it
>and come up with a list of commas they would like to use for various
>purposes, and we can supply the sagittal symbols for them.

Just work your way down any of the comma lists that Gene and Paul
have published.

>So I now see that, even if diatonic scales had never been invented or
>had never been so popular, but assuming only that we agreed we wanted
>to reuse the same nominals for pitches an octave apart, we would still
>find that the fifth was the best overall choice of notational
>generator simply because it has the simplest rational value possible
>within the octave.

Why? As with temperaments, the simplicity of the overall map is the
important thing. Using the generators to hit primes seems a good
strategy for reducing complexity, but we still see a host of excellent
temperaments with half-octave generators; in MIRACLE itself the secor
isn't a rational approximation.

>George:
>> >I answer with the following: In a Pythagorean sequence of tones
>> >(with 7 nominals), the position in the sequence determines the power
>> >of 3, and the prime factors above 3 may all be expressed in the
>> >accidentals. In a sequence of tones separated by some other
>> >generator (such as a secor), the prime factors contained or implied
>> >in the relationships of those tones vary from one tone to another, so
>> >any comma-to-accidental relationship is much less useful (and much
>> >more complicated.)
>
>Carl:
>> I disagree. In the meantone diatonic scale, the position in the
>> sequence of nominals tells you the powers of 3 and 5. There's
>> one accidental, representing 25:24.
>
>But these are not uniquely defined since syntonic commas may be
>included or removed at will, and in any case, sagittal is not based
>on meantone.

I didn't say it was. I was providing a counterexample to George's
statement that the simplicity of the accidental system depends on
the generator. It depends on the entire map.

-Carl