98 out of 99

If you take a 5x5x5 chord cube in the 9-limit lattice of quintads, you
get 153 notes to an octave. Reducing this by 99-et leads to 98 out of
the 99 possible notes, the odd note out being 40, which represents
250/189 (its TM reduced representative.) We may harmonize this by
adding the chord [1 -3 1], which is 25/21-250/189-125/84-250/147-125/63
This is the 1-6/5-4/3-3/2-12/7 utonal quintad over 125/126.

All quintads [i j k] with absolute values less of i,j, and k less than
3, with the addition of [1 -3 1], is therefore one way to harmonize
everything in 99-et. This is a set of 126=5^3+1 quintads, 63 otonal
and 63 utonal, each of which is distinct in 99-et. I may try this for
my next piece.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> If you take a 5x5x5 chord cube in the 9-limit lattice of quintads,
you
> get 153 notes to an octave. Reducing this by 99-et leads to 98 out
of
> the 99 possible notes, the odd note out being 40, which represents
> 250/189 (its TM reduced representative.) We may harmonize this by
> adding the chord [1 -3 1], which is 25/21-250/189-125/84-250/147-
125/63
> This is the 1-6/5-4/3-3/2-12/7 utonal quintad over 125/126.
>
> All quintads [i j k] with absolute values less of i,j, and k less
than
> 3, with the addition of [1 -3 1], is therefore one way to harmonize
> everything in 99-et. This is a set of 126=5^3+1 quintads, 63 otonal
> and 63 utonal, each of which is distinct in 99-et. I may try this
for
> my next piece.

Could you remind me how [a b c] represents a quintad in a 9-limit
lattice? (I know, I should look in the archives...)

>>If you take a 5x5x5 chord cube in the 9-limit lattice of quintads,
>>you get 153 notes to an octave. Reducing this by 99-et leads to
>>98 out of the 99 possible notes, the odd note out being 40, which
>>represents 250/189 (its TM reduced representative.) We may harmonize
>>this by adding the chord [1 -3 1], which is
>>25/21-250/189-125/84-250/147-125/63. This is the 1-6/5-4/3-3/2-12/7
>>utonal quintad over 125/126.
>>
>>All quintads [i j k] with absolute values less of i,j, and k less
>>than 3, with the addition of [1 -3 1], is therefore one way to
>>harmonize everything in 99-et. This is a set of 126=5^3+1 quintads,
>>63 otonal and 63 utonal, each of which is distinct in 99-et. I may
>>try this for my next piece.
>
>Could you remind me how [a b c] represents a quintad in a 9-limit
>lattice? (I know, I should look in the archives...)

Ok, I'll take a stab at this, and maybe Gene can step in later.
This is the lattice *of* quintads -- Z3, I believe. Gene has usually
used done this with the dual to A3 in the 7-limit. I think there may
have been a post at some point about how to get it to work in the
9-limit.... one can extend the chords indefinitely and still use Z3,
but not usually without leaving out certain modulations. For example,
I'm not sure how Z3 can hold modulations by 9:5, 9:7, etc...

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Ok, I'll take a stab at this, and maybe Gene can step in later.
> This is the lattice *of* quintads -- Z3, I believe. Gene has
usually
> used done this with the dual to A3 in the 7-limit. I think there
may
> have been a post at some point about how to get it to work in the
> 9-limit.... one can extend the chords indefinitely and still use Z3,
> but not usually without leaving out certain modulations. For
example,
> I'm not sure how Z3 can hold modulations by 9:5, 9:7, etc...

9/5 and 9/7 are simply 3*3/5 and 3*3/7; in other words I'm not
treating 9 any differently for this purpose, only using it when
constructing chords. This works fine for the 9-limit but obviously
not beyond.

>9/5 and 9/7 are simply 3*3/5 and 3*3/7; in other words I'm not
>treating 9 any differently for this purpose, only using it when
>constructing chords. This works fine for the 9-limit but obviously
>not beyond.

I don't follow. I can extend the scheme to the 11-limit and
not have modulations by any ratios of 11, even though the chords
contain 11-identities. Why are ratios of 9 any different?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>If you take a 5x5x5 chord cube in the 9-limit lattice of
quintads,
> >>you get 153 notes to an octave. Reducing this by 99-et leads to
> >>98 out of the 99 possible notes, the odd note out being 40, which
> >>represents 250/189 (its TM reduced representative.) We may
harmonize
> >>this by adding the chord [1 -3 1], which is
> >>25/21-250/189-125/84-250/147-125/63. This is the 1-6/5-4/3-3/2-
12/7
> >>utonal quintad over 125/126.
> >>
> >>All quintads [i j k] with absolute values less of i,j, and k less
> >>than 3, with the addition of [1 -3 1], is therefore one way to
> >>harmonize everything in 99-et. This is a set of 126=5^3+1
quintads,
> >>63 otonal and 63 utonal, each of which is distinct in 99-et. I may
> >>try this for my next piece.
> >
> >Could you remind me how [a b c] represents a quintad in a 9-limit
> >lattice? (I know, I should look in the archives...)
>
> Ok, I'll take a stab at this, and maybe Gene can step in later.
> This is the lattice *of* quintads -- Z3, I believe. Gene has
usually
> used done this with the dual to A3 in the 7-limit. I think there
may
> have been a post at some point about how to get it to work in the
> 9-limit.... one can extend the chords indefinitely and still use Z3,
> but not usually without leaving out certain modulations. For
example,
> I'm not sure how Z3 can hold modulations by 9:5, 9:7, etc...
>
> -Carl

Do you mean the Z3 group? What's A3? Still don't see how the [i j k]
represents a quintad...Thanks

>>>Could you remind me how [a b c] represents a quintad in a 9-limit
>>>lattice? (I know, I should look in the archives...)
>>
>>Ok, I'll take a stab at this, and maybe Gene can step in later.
>>This is the lattice *of* quintads -- Z3, I believe. Gene has
>>usually used done this with the dual to A3 in the 7-limit. I
>>think there may have been a post at some point about how to get
>>it to work in the 9-limit.... one can extend the chords
>>indefinitely and still use Z3, but not usually without leaving
>>out certain modulations. For example, I'm not sure how Z3 can
>>hold modulations by 9:5, 9:7, etc...
>
>Do you mean the Z3 group?

I mean the cubic lattice. Gene once told me it was called Z3.
Maybe it's the lattice that you get when you assume the symmetries
of the Z3 group??

>What's A3?

The FCC (usually 7-limit) lattice.

>Still don't see how the [i j k]
>represents a quintad...Thanks

Actually, if you follow the thread, you'll see I'm still waiting
for Gene to explain how he can get modulations of 9:5, 9:7, etc,
by moving only distance 1 on the lattice. I thought the whole
point of the observation that the dual of the 7-limit lattice is
also a lattice was that you can represent all the possible
modulations as a single step (there are 6 possible modulations
in the 7-limit, and every point in the cubic lattice is connected
to 6 others).

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >Do you mean the Z3 group?
>
> I mean the cubic lattice. Gene once told me it was called Z3.
> Maybe it's the lattice that you get when you assume the symmetries
> of the Z3 group??

The cyclic group of order 3, the cubic lattice, and the ring of 3adic
integers are all called Z3; we just have to live with it.

> Actually, if you follow the thread, you'll see I'm still waiting
> for Gene to explain how he can get modulations of 9:5, 9:7, etc,
> by moving only distance 1 on the lattice.

You can't, but then you can't get any other modulations this way
either. Moving a distance of 1 always exchanges otonal and utonal
tetrads/quintads.

I thought the whole
> point of the observation that the dual of the 7-limit lattice is
> also a lattice was that you can represent all the possible
> modulations as a single step (there are 6 possible modulations
> in the 7-limit, and every point in the cubic lattice is connected
> to 6 others).

The six tetrads you get are the six tetrads sharing an interval. To do
the 9-limit more intrinsically, we would use a packing, not a lattice,
and we would have ten quintads, not six. We could take all points
[a,b,c] with [a,b,c] mod 5 [3,1,1] for major quintads, and [2,4,4] for
minor quintands, and link them when they share an interval. I don't
know what that looks like, but it's three-dimensional.

>> Actually, if you follow the thread, you'll see I'm still waiting
>> for Gene to explain how he can get modulations of 9:5, 9:7, etc,
>> by moving only distance 1 on the lattice.
>
>You can't, but then you can't get any other modulations this way
>either. Moving a distance of 1 always exchanges otonal and utonal
>tetrads/quintads.

I call those modulations.

> I thought the whole
>> point of the observation that the dual of the 7-limit lattice is
>> also a lattice was that you can represent all the possible
>> modulations as a single step (there are 6 possible modulations
>> in the 7-limit, and every point in the cubic lattice is connected
>> to 6 others).
>
>The six tetrads you get are the six tetrads sharing an interval.

Yes, that's what I meant.

>To do
>the 9-limit more intrinsically, we would use a packing, not a lattice,
>and we would have ten quintads, not six. We could take all points
>[a,b,c] with [a,b,c] mod 5 [3,1,1] for major quintads, and [2,4,4] for
>minor quintands, and link them when they share an interval. I don't
>know what that looks like, but it's three-dimensional.

Fascinating.

-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >To do
> >the 9-limit more intrinsically, we would use a packing, not a lattice,
> >and we would have ten quintads, not six. We could take all points
> >[a,b,c] with [a,b,c] mod 5 [3,1,1] for major quintads, and [2,4,4] for
> >minor quintands, and link them when they share an interval. I don't
> >know what that looks like, but it's three-dimensional.
>
> Fascinating.

There's actually no reason not to simply add extra links to the
lattice to do this, and you get three extra links, not four. To the
six links [+-1,0,0], [0,+-1,0], [0,0,+-1] we add, if the quintad is
otonal, links to quintads [0,2,1], [-1,2,2] and [0,1,2] away; and if
the quintad is utonal, [0,-2,-1], [1,-2,-2] and [0,-1,-2] away.

If we want to make a lattice for this, instead of a packing, we could
link to [0,+-2,+-1], [-+1,+-2,+-2], and [0,+-1,+-2] around every
point; now some of the links no longer involve a shared interval or
even a shared note, however. For a Euclidean lattice, we would use
[[1,-1,-1], [-1,5,4], [-1,4,5]] as the symmetric matrix of the
bilinear form (inner product), and a^2+5b^2+5c^2-2ac-2ac+8bc as the
corresponding quadratric form.