Some 11-limit TM reduced et bases

This gives the basis for the corresponding standard val, and then the
characteristic temperament (the linear temperament obtained from the
first three of the four basis commas.) The name "Subminortone" I
tacked onto the characteristic temperament for 41 is new; now would be
the time to comment, complain, claim, or suggest changes.

12: [36/35, 45/44, 50/49, 56/55]
[4, 4, 4, 12, -3, -5, 5, -2, 14, 20]
[[4, 6, 9, 11, 13], [0, 1, 1, 1, 3]]
"Diminished"

22: [50/49, 55/54, 64/63, 99/98]
[2, -4, -4, 10, -11, -12, 9, 2, 37, 42]
[[2, 3, 5, 6, 6], [0, 1, -2, -2, 5]]
"Pajara"

31: [81/80, 99/98, 121/120, 126/125]
[4, 16, 9, 10, 16, 3, 2, -24, -32, -3]
[[1, 3, 8, 6, 7], [0, -4, -16, -9, -10]]
"Squares"

41: [100/99, 225/224, 243/242, 245/242]
[4, 9, 26, 10, 5, 30, 2, 35, -8, -62]
[[1, 1, 1, -1, 2], [0, 4, 9, 26, 10]]
"Subminortone"

46: [121/120, 126/125, 176/175, 245/243]
[9, 5, -3, 7, -13, -30, -20, -21, -1, 30]
[[1, 1, 2, 3, 3], [0, 9, 5, -3, 7]]
"Alpha"

58: [126/125, 176/175, 243/242, 896/891]
[10, 9, 7, 25, -9, -17, 5, -9, 27, 46]
[[1, -1, 0, 1, -3], [0, 10, 9, 7, 25]]
"Nonkleismic"

72: [225/224, 243/242, 385/384, 4000/3993]
[6, -7, -2, 15, -25, -20, 3, 15, 59, 49]
[[1, 1, 3, 3, 2], [0, 6, -7, -2, 15]]
"Miracle"

118: [385/384, 441/440, 3136/3125, 4375/4374]
[15, -2, -5, 22, -38, -50, -17, -6, 58, 79]
[[1, 4, 2, 2, 7], [0, -15, 2, 5, -22]]
"Hemithird"

152: [540/539, 1375/1372, 4375/4374, 5120/5103]
[24, 32, 40, 24, -5, -4, -45, 3, -55, -71]
[[8, 13, 19, 23, 28], [0, -3, -4, -5, -3]]
"Octoid"

>This gives the basis for the corresponding standard val, and then the
>characteristic temperament (the linear temperament obtained from the
>first three of the four basis commas.)

It seems you order these largest-to-smallest. Why do we want to leave
out the smallest comma in the basis -- my guess was we'd want to omit
the largest.

-Carl

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

> It seems you order these largest-to-smallest. Why do we want to
leave
> out the smallest comma in the basis -- my guess was we'd want to
omit
> the largest.

No, I ordered them by Tenney height, but why in the world would we
ditch the largest comma?

>> It seems you order these largest-to-smallest. Why do we want to
>>leave out the smallest comma in the basis -- my guess was we'd want
>>to omit the largest.
>
>No, I ordered them by Tenney height, but why in the world would we
>ditch the largest comma?

Why in the world would we ditch the highest comma? Once again,
isn't it a combination of these two factors that decides a comma's
'goodness'?

Then again, I'm not sure of the relative benefit of ditching vs.
keeping the best commas.

-Carl

hi Gene,

i tried deriving a periodicity-block for
24-ET from a <3,5,11>-prime-space by using
the following unison-vectors:

[2, 3, 5, 11]-monzo ratio

[ -4, 4, -1, 0] 81:80

[ 7, 0, -3, 0] 128:125

[-17, 2, 0, 4] 131769:131072

but instead of getting a 24-tone periodicity-block,
i got a 48-tone torsional-block.

24-ET represents ratios-of-11 so well that there
has to be a periodicity-block hiding in here somewhere.
can you help?

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Gene,
>
>
> i tried deriving a periodicity-block for
> 24-ET from a <3,5,11>-prime-space by using
> the following unison-vectors:
>
>
> [2, 3, 5, 11]-monzo ratio
>
>
> [ -4, 4, -1, 0] 81:80
>
> [ 7, 0, -3, 0] 128:125
>
> [-17, 2, 0, 4] 131769:131072
>
>
>
> but instead of getting a 24-tone periodicity-block,
> i got a 48-tone torsional-block.
>
>
> 24-ET represents ratios-of-11 so well that there
> has to be a periodicity-block hiding in here somewhere.
> can you help?

i tried using [14 -3 1 2] = 16384:16335 for the third
unison-vector, along with 81:80 and 125:128, and it
worked beautifully.

i got one 12-tone PB in the 11^0 [3,5]-plane, and
another very much like it in the 11^1 plane.

-monz

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >This gives the basis for the corresponding standard val, and then
the
> >characteristic temperament (the linear temperament obtained from
the
> >first three of the four basis commas.)
>
> It seems you order these largest-to-smallest.

no, simplest to most complex.

> Why do we want to leave
> out the smallest comma in the basis -- my guess was we'd want to
omit
> the largest.

he leaves out the most complex, which is intuitive. the simplest will
have the most effect on harmonic progressions in the tuning.

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Gene,
>
>
> i tried deriving a periodicity-block for
> 24-ET from a <3,5,11>-prime-space by using
> the following unison-vectors:
>
>
> [2, 3, 5, 11]-monzo ratio
>
>
> [ -4, 4, -1, 0] 81:80
>
> [ 7, 0, -3, 0] 128:125
>
> [-17, 2, 0, 4] 131769:131072
>
>
>
> but instead of getting a 24-tone periodicity-block,
> i got a 48-tone torsional-block.
>
>
> 24-ET represents ratios-of-11 so well that there
> has to be a periodicity-block hiding in here somewhere.
> can you help?
>
>
>
> -monz

i'm not sure what you're asking gene. you'd like to remove the
torsion? simple -- note that the sum of the three rows in the matrix
above is

[-14 6 -4 4]

which is the square of

[-7 3 -2 2] 3267:3200

using this for the third row of the matrix, you get

[ -4, 4, -1, 0] 81:80

[ 7, 0, -3, 0] 128:125

[-7, 3, -2, 2] 3267:3200

and the torsion is gone.

>he leaves out the most complex, which is intuitive. the simplest will
>have the most effect on harmonic progressions in the tuning.

But isn't this also true for chromatic vectors?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >he leaves out the most complex, which is intuitive. the simplest
will
> >have the most effect on harmonic progressions in the tuning.
>
> But isn't this also true for chromatic vectors?
>
> -Carl

not really, chromatic vectors only determine how far the temperament
is carried out to form a scale, and can look the same for unrelated
temperaments and scales, but the temperament itself is characterized
by the commatic vectors. if you take one of the simplest commas which
vanishes in the equal temperament and re-interpret it as a chromatic
vector, you'll end up with a system that differs more strongly from
the 'native harmony' of the equal temperament than when you do this
with a more complex comma. for example, 31 in the 5-limit is [81/80,
393216/390625], and making the 393216/390625 chromatic maintains the
meantone character that dominates 31-equal's 5-limit behavior, while
making 81/80 chromatic yields the more tenuous würschmidt system . . .

moreover, when there are three or more commatic vectors, the
reduction definition is more arbitrary -- the simplest (or shortest
in the lattice) comma is uniquely and unambiguously defined, but the
rest depend on the precise reduction definition -- for example
minkowski reduction may lead to a very simple second comma and a more
complex third comma, while another basis may sacrifice the simplicity
of the second comma so that the third comma comes out less
complex . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >he leaves out the most complex, which is intuitive. the simplest
> will
> > >have the most effect on harmonic progressions in the tuning.
> >
> > But isn't this also true for chromatic vectors?
> >
> > -Carl
>
> not really, chromatic vectors only determine how far the
temperament
> is carried out to form a scale, and can look the same for unrelated
> temperaments and scales, but the temperament itself is
characterized
> by the commatic vectors. if you take one of the simplest commas
which
> vanishes in the equal temperament and re-interpret it as a
chromatic
> vector, you'll end up with a system that differs more strongly from
> the 'native harmony' of the equal temperament than when you do this
> with a more complex comma. for example, 31 in the 5-limit is
[81/80,
> 393216/390625], and making the 393216/390625 chromatic maintains
the
> meantone character that dominates 31-equal's 5-limit behavior,
while
> making 81/80 chromatic yields the more tenuous würschmidt
system . . .

>>>he leaves out the most complex, which is intuitive. the simplest
>>>will have the most effect on harmonic progressions in the tuning.
>>
>> But isn't this also true for chromatic vectors?
>>
>> -Carl
>
>not really, chromatic vectors only determine how far the temperament
>is carried out to form a scale, and can look the same for unrelated
>temperaments and scales,

But still seems important, in light of the current "hey paul" thread,
and Gene's T[n] thread.

>but the temperament itself is characterized
>by the commatic vectors. if you take one of the simplest commas which
>vanishes in the equal temperament and re-interpret it as a chromatic
>vector, you'll end up with a system that differs more strongly from
>the 'native harmony' of the equal temperament than when you do this
>with a more complex comma. for example, 31 in the 5-limit is [81/80,
>393216/390625], and making the 393216/390625 chromatic maintains the
>meantone character that dominates 31-equal's 5-limit behavior, while
>making 81/80 chromatic yields the more tenuous würschmidt system . . .

Ok, I'll buy that.

-Carl

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> he leaves out the most complex, which is intuitive. the simplest
will
> have the most effect on harmonic progressions in the tuning.

Of course in the case of 41 that came down to choosing 243/242 over
245/242.

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

> but instead of getting a 24-tone periodicity-block,
> i got a 48-tone torsional-block.

You've caught the dreaded torsion disease.

> 24-ET represents ratios-of-11 so well that there
> has to be a periodicity-block hiding in here somewhere.
> can you help?

The TM basis for 3-5-11 24-et is a good starting place:

<81/80, 121/120, 128/125>

By definition, this has no torsion, and your software should now give
you a 24 block.

The three commas can be considered the basis for an 11-limit linear
temperament, which means 7 is now included. The wedgie is

[0, 0, 24, 0, 0, 38, 0, 56, 0, -83]

the prime mapping is

[[24, 38, 56, 67, 83], [0, 0, 0, 1, 0]]

and the rms generators are

[50.00000000, 20.12657339]

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

> The TM basis for 3-5-11 24-et is a good starting place:
>
> <81/80, 121/120, 128/125>

If you add 49/48, you get the TM basis for the 11-limit standard val
h24: <49/48, 81/80, 121/120, 128/125>. The characteristic temperament
for this has basis <49/48, 81/80, 121/120>, the wedgie is

[4, 6, 2, 10, 16, -8, 2, -40, -32, 21]

and the mapping

[[2,4,8,6,9], [0,-2,-8,-1,-5]]

It is Hemifourth with the period reduced to a half-octave, so I
suggest Bihemifourth as a name. Now is a good time to object.

If we toss 121/120 from the mix, we get the TM basis for the 7-limit
standard 24-et val, namely <49/48, 81/80, 128/125>. The first two
commas of this, <49/48, 81/80> are the TM basis for the characteristic
temperament, in this case Hemifourth. Tossing out 49/48 now gives us
the TM basis for 5-limit 12-equal, with characteristic temperament
Meantone.