Common commas and 768

Tonalsoft tells us that 768 tempers out all the "common" commas.
However, I would count 65625/65536, 240047/250000 and 9801/9800 as
common commas, or at least well-known ones, so I don't think this is
true. The temperament you get from tempering out 65625/65536 and
240047/240000 is actually an interesting microtemperament I've
discussed before. It can be called 84&87 and has a period of 1/3
octave, with an approximate 126/125 as generator, and 84, 87 and 171
as MOS.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> Tonalsoft tells us that 768 tempers out all the "common" commas.
> However, I would count 65625/65536, 240047/250000 and 9801/9800 as
> common commas, or at least well-known ones, so I don't think this is
> true. The temperament you get from tempering out 65625/65536 and
> 240047/240000 is actually an interesting microtemperament I've
> discussed before. It can be called 84&87 and has a period of 1/3
> octave, with an approximate 126/125 as generator, and 84, 87 and 171
> as MOS.

Here's a previous posting on "mutt":

/tuning/topicId_53634.html#53850

hi Gene,

--- In tuning@yahoogroups.com,
"Gene Ward Smith" <gwsmith@s...>
wrote:
>
> Tonalsoft tells us that 768
> tempers out all the "common"
> commas.
> However, I would count
> 65625/65536,

2,3,5,7-monzo [-16 1, 5 1 >

... and that would be me, telling
you all those things.

> 240047/250000

are you sure that's right?
i got 2^-4*5^-6 for 1/250000,
but 240047 either has a
divisor higher than 5000,
or it is prime.

> and 9801/9800

2,3,5,7,11-monzo [-3 4, -2 -2 2 >

> as common commas, or at least well-known ones, so I don't
> think this is true.

OK ... i'll put that onto either the bingo page or the
6mu (hexamu, 768-edo) page, or probably both. thanks.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> > 240047/250000
>
> are you sure that's right?
> i got 2^-4*5^-6 for 1/250000,
> but 240047 either has a
> divisor higher than 5000,
> or it is prime.

Typo for 250047/250000, the "three layer" comma. Can anyone think of a
good name deriving from three layer?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
>
> Typo for 250047/250000, the "three layer" comma. Can anyone think of
a
> good name deriving from three layer?

Hum... my best shot: latin "tricortex" (having three skins or films)

I wonder if that makes sense with the exact meaning of three layers in
this context, meaning which I don't happen to know.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> Here's a previous posting on "mutt":
>
> /tuning/topicId_53634.html#53850

it's already in the Tonalsoft Encyclopedia, at the bottom
of the "hexamu" page.

http://tonalsoft.com/enc/hexamu.htm#mutt

-monz
http://tonalsoft.com
microtonal music software

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > > 240047/250000
> >
> > are you sure that's right?
> > i got 2^-4*5^-6 for 1/250000,
> > but 240047 either has a
> > divisor higher than 5000,
> > or it is prime.
>
> Typo for 250047/250000, the "three layer" comma.
> Can anyone think of a good name deriving from three layer?

OK, well, my whole point was to provide the list of
monzos for those commas. here they are:

[-16 1, 5 1 0 > = 65625/65536,
[ -3 6, -5 3 0 > = 250047/250000
[ -3 4, -2 -2 2 > = 9801/9800

-monz

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
<giordanobruno76@y...> wrote:
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> >
> >
> > Typo for 250047/250000, the "three layer" comma. Can anyone think of
> a
> > good name deriving from three layer?
>
> Hum... my best shot: latin "tricortex" (having three skins or films)
>
> I wonder if that makes sense with the exact meaning of three layers in
> this context, meaning which I don't happen to know.

Hmmm...should it be "tricortex" or "tricortical"? Anyway, the meaning
is that it allows one to do the whole 7-limit in three layers, since
anything in the 7-limit can be written |a b c d> where d is -1, 0, or
1. Since the resulting temperament is extremely precise, this is
effectively reducing the JI 7-limit lattice to these three layers.

The TOP octave is 0.00907 cents sharp; calling that c, the TOP errors
for 2, 3, 5, and 7 are c, -log2(3)*c, log2(5)*c, -log2(7)*c, so they
are all indistinguishable from JI. Tricortex tempering is supported by
1578 and 1848, among others, again showing how small the difference
from JI is.

hi Gene,

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

> Hmmm...should it be "tricortex"
> or "tricortical"? Anyway, the
> meaning is that it allows one
> to do the whole 7-limit in
> three layers, since anything
> in the 7-limit can be written
> |a b c d> where d is -1, 0,
> or 1. Since the resulting
> temperament is extremely
> precise, this is effectively
> reducing the JI 7-limit
> lattice to these three layers.

is each of the 5-limit layers still theoretically infinite
in this conceptualization? or does your formulation here
already have 5-limit unison-vectors making each layer finite?

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> is each of the 5-limit layers still theoretically infinite
> in this conceptualization? or does your formulation here
> already have 5-limit unison-vectors making each layer finite?

No, there's just the one comma, so there are three infinite layers.
One is the 5-limit, one is everything connected to 7 by a 5-limit
interval, and one is everything connected to 1/7 by a 5-limit
interval. You can tell what layer any 7-limit interval belongs to by
the exponent of 7 modulo 3.

However, you can take any 5-limit scale, icluding Fokker blocks, and
stellate it by adding the 7/4 to any major triad and the 12/7 to any
minor triad; this will lead to a 7-limit scale which it could be
advantageous to temper tricortically. You can also take any 7-limit
scale, squash it if necessary into the three layers, and then split it
into the +1, 0, and -1 parts, each of which can be drawn on a 5-limit
lattice.

Gene,

You wrote:

> ... 250047/250000, the "three layer" comma. Can anyone think of a
> good name deriving from three layer?

There is a "three layer" theory, and associated techniques,
in landscape painting. The theory is that a landscape will
be convincing if it represents many objects arranged
throughout continuous 3-d space by just three layers -
the foreground, midground and background. This may
give quite a convincing illusion of depth. A few years ago,
I jokingly applied this theory to a piece of music, entitled -
naturally enough - "Landscape Music". Basically it consisted
of just three distinct parts, corresponding to the three layers
of this painting theory.

Accordingly, I propose calling the "three layer" comma,
the Landscape comma!

Regards,
Yahya

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--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> Accordingly, I propose calling the "three layer" comma,
> the Landscape comma!

Very cool!

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...>
wrote:
>
> > Accordingly, I propose calling the "three layer" comma,
> > the Landscape comma!
>
> Very cool!

yes, i really like it too!

-monz