Generators as names

We've been talking a little about this. I thought I'd point out that
not only can you get a generator from two numbers specifying an equal
division range, you can invert the process, deriving from a fraction
two integers; simply take the next and the previous numbers on the row
of the Farey sequence for that fractional generator, and the
denominators are your two numbers. So from 18/31 one gets, for
instance, [19, 12]; where the ordering means that 19 is on the flat
side for this generator, and 12 on the sharp side. We can also
interpret things in terms of periods other than an octave; 3/34 gives
us [23, 11], which we can interpret in terms of 34-equal, but which
when multiplied by two gives us [46, 22]; and 2/23 < 3/34 < 1/11,
using half-octave periods, become shrutar generators. Hence if we have
generator and period in the
form [1/2, 3/31], we can obtain from them [46, 22], and if we have
[46, 22], we can obtain from that [1/2, 3/31]. If now we interpret
these as defining vals in some p-limit, we can obtain a temperament.

Of course, this might not work out the way you want; from [19, 12]
which we can derive from 18/31, we get a fine 5 and 7 limit version of
the meantone mapping of 31, but we don't get either accurate version
in the 11-limit. Instead we map 11 to six fifths, which means
(3/2)^6/11 = 729/704 is a comma. This boils down to the system where
45/44 is a comma, which might be worth putting on Joe's meantone page,
I suppose; 88 does better for this as a tuning than 31 in case anyone
feels inspired to try it. This suggests to me that to name an 11-limit
version of meantone, we don't use 18/31; 29/50 will give us meanpop,
and 25/43 suffices for huygens. Hence "11-limit 29/50" could be a
possible name or way of denoting meanpop, and likewise "11-limit
25/43" could denote huygens. It's really just another form of the
19&31, 12&31 method.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> Of course, this might not work out the way you want;
> from [19, 12] which we can derive from 18/31, we get a
> fine 5 and 7 limit version of the meantone mapping of 31,
> but we don't get either accurate version in the 11-limit.
> Instead we map 11 to six fifths, which means (3/2)^6/11 =
> 729/704 is a comma. This boils down to the system where
> 45/44 is a comma, which might be worth putting on Joe's
> meantone page, I suppose;

hmm ... i remember that 45/44 popping up all over the
place when i was trying to analyze Schoenberg's mappings
to 12edo, but basing my research on the typo in Carter's
translation of _Harmonielehre_.

in that mistake, the 11th harmonic of "F" is given as "B"
(+6 generators) instead of the correct "Bb" (-1 generator)
that's in the German version (which is notated "B" ...
i know, confusing...).

> 88 does better for this as a tuning than 31 in case anyone
> feels inspired to try it.

88edo is LucyTuning ... close enough, anyway.

-monz

Gene Ward Smith wrote:

> This suggests to me that to name an 11-limit
> version of meantone, we don't use 18/31; 29/50 will give us meanpop,
> and 25/43 suffices for huygens. Hence "11-limit 29/50" could be a
> possible name or way of denoting meanpop, and likewise "11-limit
> 25/43" could denote huygens. It's really just another form of the
> 19&31, 12&31 method.

More or less. One advantage of the generator/period ratio method is that you don't need two consistent ET's, only one.

Herman Miller wrote:

> More or less. One advantage of the generator/period ratio method is that > you don't need two consistent ET's, only one.

Ah, but it gets hairy. You then have to decide which of the two different generator mappings to use for each prime. The point of insisting on consistent ETs is that it removes ambiguity.

Graham

Graham Breed wrote:

> Herman Miller wrote:
> > >>More or less. One advantage of the generator/period ratio method is that >>you don't need two consistent ET's, only one.
> > > Ah, but it gets hairy. You then have to decide which of the two > different generator mappings to use for each prime. The point of > insisting on consistent ETs is that it removes ambiguity.

You mean like mapping 7/1 as +2 or -10 in 12-ET? Just take the one with the smallest absolute value. The advantage of specifying 5/12 over 5&7 is that 12-ET is 7-limit consistent, but 7-ET isn't. I guess you could run into a case where some prime is mapped to +6 or -6, though, which would result in two different temperaments.

Herman Miller wrote:

> You mean like mapping 7/1 as +2 or -10 in 12-ET? Just take the one with > the smallest absolute value. The advantage of specifying 5/12 over 5&7 > is that 12-ET is 7-limit consistent, but 7-ET isn't. I guess you could > run into a case where some prime is mapped to +6 or -6, though, which > would result in two different temperaments.

You could do that, but there are other ways as well. Like the mapping with the minimax interval complexity, or the RMS, or the weighted RMS.

The nearest-prime 7-mapping for 8/19 is +9, consistent with 19&26. But the interval 7:5 maps as 13 generators. You can also map the 7 as -10 generators to give the usual septimal meantone consistent with 12&19. Here, the most complex interval is the 8:7 at 10 generators. So the overall mapping is simpler than the one you would have chosen.

Similiarly, 13/31 can map 11 to either -18 (31&43) or +13 (50&31 with 11:7 mapping to 23 generators).

And how would you resolve 2/58?

If you define by two consistent ETs, you get the same result whatever badness you're minimizing.

Graham

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

> If you define by two consistent ETs, you get the same result whatever
> badness you're minimizing.

I've been using standard vals to dismbiguate things; other choices,
such as using TOP to define the vals, would be possible. This gives
dominant for 5/12 and 8/19, and 2/58 would become 2&56, which gives the
temperament tempering out 2048/2025 and 50421/50000, which has a high
badness but which would be usable if someone really wanted to use it.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> Graham Breed wrote:
>
> > Herman Miller wrote:
> >
> >
> > > More or less. One advantage of the generator/period
> > > ratio method is that you don't need two consistent ET's,
> > > only one.
> >
> >
> > Ah, but it gets hairy. You then have to decide which of
> > the two different generator mappings to use for each prime.
> > The point of insisting on consistent ETs is that it removes
> > ambiguity.
>
> You mean like mapping 7/1 as +2 or -10 in 12-ET? Just take the
> one with the smallest absolute value. The advantage of specifying
> 5/12 over 5&7 is that 12-ET is 7-limit consistent, but 7-ET isn't.
> I guess you could run into a case where some prime is mapped to +6
> or -6, though, which would result in two different temperaments.

wow, for all its versatility, 12edo is lousy at mapping the
half-octave. you have to go all the way up to the 42nd prime,
181, to find something that maps really close to 600 cents.

(and actually, 181 happens to be almost exactly that ...
~599.8150645 cents.)

the lowest candidate is 23 (~628.2743473 cents), unless you
accept 11 (~551.3179424) as being close enough ... but i
wouldn't if i could use a higher-cardinality eT, since 11 is
so obviously midway between 500 and 600 cents.

anyway, this reply led me to make a nice graphic that should
have been on my "prime" page long ago,

http://tonalsoft.com/enc/prime.htm

showing the mapping of the first 50 primes to logarithmic
acoustical pitch-height space.

-monz