Octave-reduced basis for notating temperaments

I asked on XA if people might find it easier to read a notation for
regular temperaments where the JI basis is arranged in terms of
octave-equivalent primes rather than just primes. For example, an
11-limit mapping matrix would be notated as having a basis of [2/1,
3/2, 5/4, 7/4, 11/8] rather than [2/1, 3/1, 5/1, 7/1, 11/1]. People
seemed to like the notion. I suggested it for the following reasons:

1) I don't even really look at the period mapping, I just check to see
what the period coefficient is for 2/1 and then just look at the
generator mapping. Igs said he does the same thing.
2) It's a pain in the ass, when looking at larger EDOs, to modulo
things out in my head to figure out where 11/8 or 13/8 is
3) I don't even care about 7/1 or 11/1; I don't ever intentionally use
those intervals. If I ever do, it's only indirectly because I'm
doubling something in the bass, and I'm probably thinking more about
the combination of "an 11/8" and "octave doubling the lowest note"
than something like 11/2 directly.
4) In fact, continuing #3, I sometimes even will confuse which octave
I'm in (which I note is common), meaning I tend to screw up a lot.

For mathematical reasons, it's probably still simpler to notate things
out in terms of 2.3.5, but for pedagogical reasons or to notate this
to actual musicians, this might be worth considering. Here are some
examples of things notated both ways:

Porcupine - 2.3.5:
[<1 2 3]
<0 -3 -5]>

Porcupine - 2/1.3/2.5/4:
[<1 1 1]
<0 -3 -5]>

--

Orwell - 2.3.5.7.11:
[<1 0 3 1 3]
<0 7 -3 8 2]>

Orwell - 2/1.3/2.5/4.7/4.11/8:
[<1 -1 1 -1 0]
<0 7 -3 8 2]>

--

Meantone - 2.3.5:
[<1 1 0]
<0 1 4]>

Meantone - 2/1.3/2.5/4:
[<1 0 -2]
<0 1 4]>

--

Slendric - 2.3.7:
[<1 1 3]
<0 3 -1]>

Slendric - 2.3/2.7/4:
[<1 0 1]
<0 3 -1]>

--

Negri - 2.3.5.7.11.13:
[<1 2 2 3 4 4]
<0 -4 3 -2 -5 -3]>

Negri - 2/1.3/1.5/4.7/4.11/4.13/8:
[<1 1 0 1 1 1]
<0 -4 3 -2 -5 -3]>

--

Sensi - 2.3.5.7:
[<1 -1 -1 -2]
<0 7 9 13]>

Sensi - 2/1.3/2.5/4.7/4:
[<1 -2 -3 -4]
<0 7 9 13]>

I find the octave-reduced versions considerably easier to parse,
especially for orwell and porcupine.

-Mike

Also, some random 13-limit patent vals - labels are arbitrary in each case:

Diatonic sized:
<7 4 2 6 3 5] vs <7 11 16 20 24 26]
<9 5 3 7 4 6] vs <9 14 21 25 31 33]
<10 6 3 8 5 7] vs <10 16 23 28 35 37]

Chromatic sized:
<12 7 4 10 5 8] vs <12 19 28 34 42 44]
<15 9 5 12 7 11] vs <15 24 35 42 52 56]
<16 9 5 13 7 11] vs <16 25 37 45 55 59]

Enharmonic sized:
<17 10 5 14 8 12] vs <17 27 39 48 59 63]
<19 11 6 15 9 13] vs <19 30 44 53 66 70]
<22 13 7 18 10 15] vs <22 35 51 62 76 81]
<26 15 8 21 12 18] vs <26 41 60 73 90 96]

Beyond enharmonic sized:
<31 18 10 25 14 22] vs <31 49 72 87 107 115]
<41 24 13 33 19 29] vs <41 65 95 115 142 152]
<46 27 15 37 21 32] vs <46 73 107 129 159 170]
<53 31 17 43 24 37] vs <53 84 123 149 183 196]

Really really big:
<68 40 22 55 31 48] vs <68 108 158 191 235 252]
<72 42 23 58 33 50] vs <72 114 167 202 249 266]
<84 49 27 68 39 59] vs <84 133 195 236 291 311]
<87 51 28 70 40 61] vs <87 138 202 244 301 322]

The following are dedicated to Gene:
<171 100 55 138 79 120] vs <171 271 397 480 592 633]
<224 131 72 181 103 157] vs <224 355 520 629 775 829]
<441 258 142 356 203 309] vs <441 699 1024 1238 1526 1632]
<494 289 159 399 227 346] vs <494 783 1147 1387 1709 1828]
<940 550 303 759 432 658] vs <940 1490 2183 2639 3252 3478]

Cents:
<1200 702 386 969 551 841] vs <1200 1902 2786 3369 4151 4441]

100*EDO cent alternatives:
<1000 585 322 807 459 700] vs <1000 1585 2322 2807 3459 3700]
<1500 877 483 1211 689 1051] vs <1500 2377 3483 4211 5189 5551]
<1700 994 547 1373 781 1191] vs <1700 2694 3947 4773 5881 6291]
<1900 3011 4412 5334 6573 7031] vs <1900 1111 612 1534 873 1331]
<2200 1287 708 1776 1011 1541] vs <2200 3487 5108 6176 7611 8141]
<2600 1521 837 2099 1195 1821] vs <2600 4121 6037 7299 8995 9621]
<3100 1813 998 2503 1424 2171] vs <3100 4913 7198 8703 10724 11471]
<9900 5791 3187 7993 4548 6934] vs <9900 15691 22987 27793 34248 36634]

Other cent alternatives:
<612 358 197 494 281 429] vs <612 970 1421 1718 2117 2265]
<2460 1439 792 1986 1130 1723] vs <2460 3899 5712 6906 8510 9103]
<8539 4995 2749 6894 3923 5981] vs <8539 13534 19827 23972 29540 31598]
<30103 17609 9691 24304 13830 21085] vs <30103 47712 69897 84510 104139 111394]
<46032 26927 14819 37164 21149 32243] vs <46032 72959 106883 129228
159245 170339]

I immediately like the octave-reduced mapping much more than the
prime-based one.

I assume it'll still be easier to use the prime-based one for various
mathematical reasons, so perhaps we could just have two different
conventions for each of these, and just call them different "forms" or
"normal forms" or something. Since we on this list are obviously still
using this all for mathematical purposes, I assume we'll stick to the
non-reduced prime notation. And then we can keep in mind that there
might be presentational or interpretational advantages for the
octave-reduced form if that ever coes up.

-Mike

On Sun, Nov 13, 2011 at 6:00 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
> I asked on XA if people might find it easier to read a notation for
> regular temperaments where the JI basis is arranged in terms of
> octave-equivalent primes rather than just primes. For example, an
> 11-limit mapping matrix would be notated as having a basis of [2/1,
> 3/2, 5/4, 7/4, 11/8] rather than [2/1, 3/1, 5/1, 7/1, 11/1]. People
> seemed to like the notion. I suggested it for the following reasons:

Mike Battaglia <battaglia01@gmail.com> wrote:

> I assume it'll still be easier to use the prime-based one
> for various mathematical reasons, so perhaps we could
> just have two different conventions for each of these,
> and just call them different "forms" or "normal forms" or
> something. Since we on this list are obviously still
> using this all for mathematical purposes, I assume we'll
> stick to the non-reduced prime notation. And then we can
> keep in mind that there might be presentational or
> interpretational advantages for the octave-reduced form
> if that ever coes up.

The Karp paper from 1984, which is one of the independent
applications of matrices to temperament theory, uses this
reduced basis. It would make the equal temperaments easier
to read. I don't find the vals at all simple, and I mostly
use them to either check the mappings or as input to a
function that can calculate something more digestible.

Instead of defining one alternative basis, we could show
the mappings of various different intervals. So far they
tend to be the unison vectors. To properly understand a
mapping, it helps to write the scale out with its
approximations, maybe a keyboard mapping, maybe a notation
system.

The prime intervals, corresponding to prime numbers, are
privileged in that they're of minimal complexity. They're
also the easiest basis to factorize a ratio into. Like
with 16:15, if you remember that 16 = 2**4, you know it's
[4, -1, -1>. It's not a great problem to work it out in
terms of the reduced basis. You know 3:2 gets from C to G,
then 5:4 from G to B. The semitone, B-C, is the difference
between this and the octave, so [1, -1, -1>. But still,
it's harder than the simple factorization, and as it
follows from octave-equivalent thinking you may as well
stay octave equivalent all the way.

With reduced mappings of higher rank temperaments, I don't
think the choice of basis is that important. The period
mapping is only there as a check. Most of us will think of
the octave-equivalent generator and only look at the
generator mapping.

Graham

On 11/13/2011 6:00 AM, Mike Battaglia wrote:
> I asked on XA if people might find it easier to read a notation for
> regular temperaments where the JI basis is arranged in terms of
> octave-equivalent primes rather than just primes. For example, an
> 11-limit mapping matrix would be notated as having a basis of [2/1,
> 3/2, 5/4, 7/4, 11/8] rather than [2/1, 3/1, 5/1, 7/1, 11/1]. People
> seemed to like the notion. I suggested it for the following reasons:
>
> 1) I don't even really look at the period mapping, I just check to see
> what the period coefficient is for 2/1 and then just look at the
> generator mapping. Igs said he does the same thing.
> 2) It's a pain in the ass, when looking at larger EDOs, to modulo
> things out in my head to figure out where 11/8 or 13/8 is
> 3) I don't even care about 7/1 or 11/1; I don't ever intentionally use
> those intervals. If I ever do, it's only indirectly because I'm
> doubling something in the bass, and I'm probably thinking more about
> the combination of "an 11/8" and "octave doubling the lowest note"
> than something like 11/2 directly.
> 4) In fact, continuing #3, I sometimes even will confuse which octave
> I'm in (which I note is common), meaning I tend to screw up a lot.

It's true that the period mapping isn't very meaningful in itself. You could almost get by without it in cases where the period is the octave. (You could view standard musical notation as a way of notating meantone that ignores the period mapping.) But even if you don't care about 7/1 or 11/1, you probably care about intervals like 7/5, 7/6, or 11/9. It's easier to get the mapping for those if you have the period mapping based on primes.

Still, it's an appealing idea to use musically useful intervals as the basis for the mapping, and the octave-reduced primes are probably the ones that make the most sense. Another possibility would be to use superparticular intervals with a prime number in the numerator (2/1, 3/2, 5/4, 7/6, 11/10).

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> I find the octave-reduced versions considerably easier to parse,
> especially for orwell and porcupine.

I found them confusing, perhaps because I'm not used to them.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> With reduced mappings of higher rank temperaments, I don't
> think the choice of basis is that important. The period
> mapping is only there as a check. Most of us will think of
> the octave-equivalent generator and only look at the
> generator mapping.

The period mapping is recoverable from the generator mapping in any case. I've explained how to do that with wedgies, but that's not the only way. With octave periods you can find the commas for the generator mapping, reduce to 1 < comma < sqrt(2), and then reconstitute the mapping from the commas, for instance. It might be worth thinking about the most expeditious way to accomplish this task.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> The period mapping is recoverable from the generator mapping in any case. I've explained how to do that with wedgies, but that's not the only way. With octave periods you can find the commas for the generator mapping, reduce to 1 < comma < sqrt(2), and then reconstitute the mapping from the commas, for instance. It might be worth thinking about the most expeditious way to accomplish this task.

The method involving wedgies seems simpler, but it only works when the temperament is reasonably sane. This can be used in general:

(1) Find the commas of the generator val
(2) Remove 2 from the list of commas
(3) Raise everything to the power of the number of periods in an octave
(4) Octave reduce to the range 1 < c < sqrt(2)
(5) Saturate the result
(6) Compute the mapping from the resultant commas

For an example, consider <0 2 3 2| with period 1/9 octave:

Commas: 2, 3/7, 25/63
Ninth powers reduced to 1<c<sqrt(2): |-11 -9 0 9>, |-12 18 -18 9>
Saturated: |-11 -9 0 9>, |6 8 -2 -5>
Mapping: [<9 1 1 12|, <0 2 3 2|]