Chains of fifths and notation

Basing notation schemes which encompass successivly higher prime
limits on a Pythagorean chain-of-fifth (or fourths, etc) method is a
necessity, since we start from the 3-limit. To go to the 5-limit, we
need to add a comma of the form 2^a 3^b 5^c where c = +-1. If a monzo
[a,b,c] gives rise to a temperament with octave period, then the
generator mapping will be +-[0, c, -b], and since c=+-1 once again we
are back with 3/2, 4/3 etc as generator--a Pythagorean system. Looking
at the 5-limit commas, we find really only five reasonable choices,
and so five types of notation systems possible for the higher prime
limits as well.

16/15 fourth-third systems

135/128 pelogic systems

81/80 meantone systems

32805/32768 schismic systems

2954312706550833698643/2951479051793528258560
[-69, 45, -1] counterschismic systems

To go to 7-limit systems, we want 7-limit commas for which the
exponent of 7 is +-1, and so forth. Some 7-limit possibilities are:

36/35, 525/512, 64/63, 875/864, 126/125, 225/224, 5120/5103,
65625/65536, 4375/4374

Aside from 33/32, some 11-limit possibilities include:

77/75, 45/44, 55/54, 56/55, 99/98, 100/99, 176/175, 896/891,
385/384, 441/440, 1375/1372, 6250/6237, 540/539, 5632/5625,
131072/130977, 40283203125/40282095616, 6576668672/6576582375,
781258401/781250000, 13841287201/13841203200

Note that while insuring that the count of p-limit commas depends only
on the exponent for p is a nice property, it is hardly essential. I
think there are advantages, for instance, in using in place of
81/80, 64/63, and 33/32, the commas 5120/5103 and 385/384; they are
related by

64/63 = 81/80 5120/5103

33/32 = 81/80 64/63 385/384

In these terms, my example of 77/75 becomes D double flat, up three
81/80 and a 385/384.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Basing notation schemes which encompass successivly higher prime
> limits on a Pythagorean chain-of-fifth (or fourths, etc) method is a
> necessity, since we start from the 3-limit. To go to the 5-limit, we
> need to add a comma of the form 2^a 3^b 5^c where c = +-1. If a
monzo
> [a,b,c] gives rise to a temperament with octave period, then the
> generator mapping will be +-[0, c, -b], and since c=+-1 once again
we
> are back with 3/2, 4/3 etc as generator--a Pythagorean system.
Looking
> at the 5-limit commas, we find really only five reasonable choices,
> and so five types of notation systems possible for the higher prime
> limits as well.
>
> 16/15 fourth-third systems
>
> 135/128 pelogic systems
>
> 81/80 meantone systems
>
> 32805/32768 schismic systems
>
> 2954312706550833698643/2951479051793528258560
> [-69, 45, -1] counterschismic systems
>
> To go to 7-limit systems, we want 7-limit commas for which the
> exponent of 7 is +-1, and so forth. Some 7-limit possibilities are:
>
> 36/35, 525/512, 64/63, 875/864, 126/125, 225/224, 5120/5103,
> 65625/65536, 4375/4374
>
> Aside from 33/32, some 11-limit possibilities include:
>
> 77/75, 45/44, 55/54, 56/55, 99/98, 100/99, 176/175, 896/891,
> 385/384, 441/440, 1375/1372, 6250/6237, 540/539, 5632/5625,
> 131072/130977, 40283203125/40282095616, 6576668672/6576582375,
> 781258401/781250000, 13841287201/13841203200
>
> Note that while insuring that the count of p-limit commas depends
only
> on the exponent for p is a nice property, it is hardly essential. I
> think there are advantages, for instance, in using in place of
> 81/80, 64/63, and 33/32, the commas 5120/5103 and 385/384; they are
> related by
>
> 64/63 = 81/80 5120/5103
>
> 33/32 = 81/80 64/63 385/384
>
> In these terms, my example of 77/75 becomes D double flat, up three
> 81/80 and a 385/384.^

Gene, FYI Dave and I have the following 11-limit commas notated
*exactly* with single flags in sagittal:

'| for 32768:32805 (5-schisma)
|( for 5103:5120
/| for 80:81
|) for 63:64
|\ for 54:55
(| for 45056:45927

as well as the following notated *exactly* with two flags:

)|( for 16384:16473
~|) for 48:49
(|( for 44:45
//| for 6400:6561
/|) for 35:36
/|\ for 32:33
(|) for 704:729
(|\ for 8192:8505^

and a bunch of others notated *exactly* in high-precision sagittal
(using the 5-schisma) such as:

./| for 2025:2048
.//| for 125:128
'(|\ for 27:28

--George

{{16/15 fourth-third systems

135/128 pelogic systems

81/80 meantone systems

32805/32768 schismic systems

2954312706550833698643/2951479051793528258560
[-69, 45, -1] counterschismic systems}}

We can extend this classification scheme by using planar temperaments
for higher prime limits as well. The wedgie for <81/80, 64/63> is [1,
4, -2, 4, -6, -16], so this goes under the "Dominant Seventh" rubric.
However, <81/80, 36/35> and <81/80, 5120/5103> are equally D7 systems,
giving the same wedgie, and very closely related to using 64/63 to
notate. We pass to the 11-limit by adding 33/32; however the same
wedgie is obtainable by adding 55/54 or 385/384, so any of the nine
systems <81/80, {36/35,64/63,5120/5103}, {33/32,55/54,385/384}> is
equally well an 11-limit D7 notation scheme.

Another very logical system falling under the general Meantone heading
is Septimal Meantone; here to 81/80 we add 126/125, 225/224 or
3136/3125. If now for the 11-limit we add 99/98, 176/175, 441/440,
1375/1372 or 5632/5625, we get 11-limit Meantone; if instead we add
385/384 or 540/549 we get Meanpop systems.

George tells us that each of 36/35, 64/63 and 5120/5103 is sagittally
symbolized, and so is 55/54 (but not 385/384,) so D7 systems are
pretty well covered. So far as Septimal Meantone goes, none of
126/125, 225/224 or 3136/3125 seems to have a symbol, nor do 99/98,
176/174, 441/440, 1375/1372, 5632/5625, 385/384 or 540/539. A symbol
for 385/384 and one for either of 126/125 or 225/224 and either of
99/98 or 176/175 would be nice if other 81/80 systems were to be included.

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

> We can extend this classification scheme by using planar temperaments
> for higher prime limits as well.

If we insist on having commas of the form 2^a 3^b p^{+-1}, we don't
get anything as nice as we have for D7 systems:

11-limit meantone
[81/80, 59049/57344, 387420489/369098752]

meanpop
[81/80, 59049/57344, 17537553/16777216]

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> George tells us that each of 36/35, 64/63 and 5120/5103 is sagittally
> symbolized, and so is 55/54 (but not 385/384,) so D7 systems are
> pretty well covered. So far as Septimal Meantone goes, none of
> 126/125, 225/224 or 3136/3125 seems to have a symbol, nor do 99/98,
> 176/174, 441/440, 1375/1372, 5632/5625, 385/384 or 540/539. A symbol
> for 385/384 and one for either of 126/125 or 225/224 and either of
> 99/98 or 176/175 would be nice if other 81/80 systems were to be
included.

Some of these will be exactly symbolised in the final system, using
accented symbols, it's just that they have not yet been finalised to
the satisfaction of both George and I, and at present we're devoting
our time to the article explaining the basic (unaccented) system.

It's highly probable that each of the following will be the primary
meaning of some accented sagittal symbol.

385/384
126/125
225/224
99/98

But the following are very unlikely to be primary symbol
interpretations, because there are other more popular (usually less
complex) kommas which are very close to them (typically within 0.4 c).
In that case you can simply use the symbol for that nearby komma and
explain somewhere that that's what you are doing (assuming you don't
also need that symbol for its primary purpose, which is fairly
unlikely). We refer to this as using a symbol in a secondary role.

176/175
3136/3125
441/440
1375/1372
5632/5625
540/539

Regards,
-- Dave Keenan

This is a long overdue reply to Gene in "Re: Polyphonic notation" on
the tuning list
/tuning/topicId_46826.html#47497

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > We have also used a temperament to help decide on the actual symbols
> > to be used for the comma ratios in the superset. This is an
> > 8-dimensional temperament with a maximum error of 0.39 cents. The 8
> > dimensions relate to the 9 flags (including the accent mark) that
> make
> > up the symbols, less one degree of freedom because a certain
> > combination is set equal to the apotome.
>
> What commas are being tempered out?

I'm afraid I don't know. I just specified certain combinations of
generators to approximate certain 23-limit ratios (which were
themselves commas, but were not being tempered out) and solved
numerically for the generators. If you're still interested, maybe
there's some other information I could give you if you wanted to work
them out. But it may have to wait some time.

> > Well I think the fact that we have Graham proposing MIRACLE
> > temperament with 10 nominals and Gene proposing ennealimmal
> > temperament with 9 nominals should make it clear that there is
> > unlikely to ever be agreement on which is the ultimate temperament
> for
> > notating everything else including ratios.
>
> My proposal is simply intended to notate effective 7-limit JI
> (extendible to 11-limit) not everything.

Fine.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> This is a long overdue reply to Gene in "Re: Polyphonic notation" on
> the tuning list
> /tuning/topicId_46826.html#47497
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> > --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> >
> > > We have also used a temperament to help decide on the actual
symbols
> > > to be used for the comma ratios in the superset. This is an
> > > 8-dimensional temperament with a maximum error of 0.39 cents.
The 8
> > > dimensions relate to the 9 flags (including the accent mark)
that make
> > > up the symbols, less one degree of freedom because a certain
> > > combination is set equal to the apotome.
> >
> > What commas are being tempered out?
>
> I'm afraid I don't know. I just specified certain combinations of
> generators to approximate certain 23-limit ratios (which were
> themselves commas, but were not being tempered out) and solved
> numerically for the generators. If you're still interested, maybe
> there's some other information I could give you if you wanted to
work
> them out. But it may have to wait some time.

I can give Gene some information, since I figured this out some time
ago.

Gene, what Dave was describing above applies to the extreme-precision
version of the notation (i.e., extremely high precision), which I
don't claim to understand completely.

I do have some data that applies to the medium-precision and high-
precision versions (areas in which I have been working), which for
the 11 limit probably also applies to the extreme-precision version.
In these, the symbol for the 7th harmonic [0, 0, 1] is also used for
[7, -4]; the resulting schismina is 4374:4375 (2*3^7:5^4*7, ~0.396
cents). The symbol for the 11th harmonic [0, 0, 0, 1] is also used
for [10, 5]; the resulting schismina is 184528125:184549376
(3^10*5^5:2^24*11, ~0.199 cents).

The schismina for the 13th harmonic in medium and high-precision
sagittal (which does *not* vanish in extreme-precision sagittal) is
4095:4096 (3^2*5*7*13:2^12, ~0.423) cents, such that [0, 0, 0, 0, 1]
is equivalent to [-9, 4].

I hope this will be of some help.

--George