Notation of non-3 temperaments

Petr Pařízek's recent example on the tuning list of music using a temperament without prime factors of 3 or 7 brings up an interesting problem when it comes to notation. How *do* you notate a set of pitches tuned in such a temperament when our whole system of notation is based on factors of 3?

Hmmm......

I picked a fairly arbitrary temperament without factors of 3 or 5 to test. This should really bring out the weak points of any notation system based on chains of fifths and traditional 5-limit harmony.

[<1, x, x, 1, 3], <0, x, x, 4, 1]>
P = 1201.318349, G = 542.802158

This has MOS of 9 and 11 notes, so a reasonable notation might use 9 or 11 nominals with accidentals. Of course, this means we need an accidental for something like 539/512 or 343/176. Since none of the usual meanings of the Sagittal accidentals will be meaningful in this temperament, we might as well just pick one that's close to the right size.

How about using the standard chain of fifths nominals along with whatever accidentals we might need for each one? Obviously some notes do work out: if you start with D as 1/1, the next notes are G/|\ 11/8, D.~!!)' 121/64, G)!!~ 14/11, C!) 7/4, and F.|)' 77/64. But it doesn't take long to run into problems, and this is one of the simpler temperaments.

One alternative would be to find a nearby ET and just notate that. This tuning of the temperament falls close to 31-ET, 135-ET, and 571-ET. Other tunings might suggest different ET's.

How about this: fill in the x's in the mapping with numbers for the missing primes that don't change the TOP error?

[<1, -7, 10, 1, 3], <0, 19, -17, 4, 1]>

This might end up being a reasonable solution, although those missing numbers might end up being too far out for more complex temperaments.

At 06:40 PM 3/24/2008, you wrote:
>Petr Pařízek's recent example on the tuning list of music using a
>temperament without prime factors of 3 or 7 brings up an interesting
>problem when it comes to notation. How *do* you notate a set of pitches
>tuned in such a temperament when our whole system of notation is based
>on factors of 3?

Whoops.

-Carl

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
> I picked a fairly arbitrary temperament without factors of 3 or 5 to
> test. This should really bring out the weak points of any notation
> system based on chains of fifths and traditional 5-limit harmony.
>
> [<1, x, x, 1, 3], <0, x, x, 4, 1]>
> P = 1201.318349, G = 542.802158
>
> This has MOS of 9 and 11 notes, so a reasonable notation might use 9 or
> 11 nominals with accidentals. Of course, this means we need an
> accidental for something like 539/512 or 343/176. Since none of the
> usual meanings of the Sagittal accidentals will be meaningful in this
> temperament, we might as well just pick one that's close to the
right size.
>

A reasonable solution.

> How about using the standard chain of fifths nominals along with
> whatever accidentals we might need for each one? Obviously some
notes do
> work out: if you start with D as 1/1, the next notes are G/|\ 11/8,
> D.~!!)' 121/64, G)!!~ 14/11, C!) 7/4, and F.|)' 77/64. But it doesn't
> take long to run into problems, and this is one of the simpler
temperaments.
>

The first reasonable approximation of prime 3 occurs after 19
iterations of this generator, period-reduced. So you would need 9
accidentals to do it this way. 4 accidentals for the 9-MOS and 5 for
the 11-MOS.

> One alternative would be to find a nearby ET and just notate that. This
> tuning of the temperament falls close to 31-ET, 135-ET, and 571-ET.
> Other tunings might suggest different ET's.

If there were accidentals in common between the standard notations for
these systems then they should probably be used for the linear
temperament. The generator is 14deg31 and it takes 19 of these to make
a 31-ET fifth, so that relates to the above.

>
> How about this: fill in the x's in the mapping with numbers for the
> missing primes that don't change the TOP error?
>
> [<1, -7, 10, 1, 3], <0, 19, -17, 4, 1]>

Aha! That is of course exactly what I did above, re prime 3. But I
like the idea of doing it for other missing primes too, to make life
easier in finding accidentals.

> This might end up being a reasonable solution, although those missing
> numbers might end up being too far out for more complex temperaments.
>

I think that is the right way to do it, if you choose not to go down
the MOS nominals path.

Dave Keenan wrote:

> The first reasonable approximation of prime 3 occurs after 19
> iterations of this generator, period-reduced. So you would need 9
> accidentals to do it this way. 4 accidentals for the 9-MOS and 5 for
> the 11-MOS.

An interesting thought, but wouldn't that be very sensitive to the size of the generator? You could have a non-3 temperament with two different approximations to prime 3 depending on the exact tuning, and that could be a problem if the switchover point comes near an optimal tuning.

>> One alternative would be to find a nearby ET and just notate that. This >> tuning of the temperament falls close to 31-ET, 135-ET, and 571-ET. >> Other tunings might suggest different ET's.
> > If there were accidentals in common between the standard notations for
> these systems then they should probably be used for the linear
> temperament. The generator is 14deg31 and it takes 19 of these to make
> a 31-ET fifth, so that relates to the above.
> >> How about this: fill in the x's in the mapping with numbers for the >> missing primes that don't change the TOP error?
>>
>> [<1, -7, 10, 1, 3], <0, 19, -17, 4, 1]>
> > Aha! That is of course exactly what I did above, re prime 3. But I
> like the idea of doing it for other missing primes too, to make life
> easier in finding accidentals.

Well, that gets around the problem of finding the appropriate 3/1 approximation. But I'm concerned that with more complex temperaments it could end up being so far out there aren't enough accidentals to fill the gap.

>> This might end up being a reasonable solution, although those missing >> numbers might end up being too far out for more complex temperaments.
>>
> > I think that is the right way to do it, if you choose not to go down
> the MOS nominals path.

MOS nominals are appealing if you use a MOS-based staff notation in general, but I find those can be hard to read. I've been considering a standard set of 5-based and 7-based nominals. Either of them could use compound nominals based on 31-ET notation, but that would be a fallback if I can't find anything better. Anyone crazy enough to try a non-3, non-5, non-7 temperament probably knows enough to come up with their own notation! :-)