The diaschismic family

Here is a new analysis of the family in the 7-limit, and old stuff
showing what I've called "diaschismic" in the 11 and 13 limits. I'm
afraid the 13 limit badness is not geometric badness; however both
diaschismic and hemififths come out with low badness figures in the 13
limit and are important 13-limit systems.

7 limit

Pajara [50/49, 64/63]
[2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]

bad 1550.521632 comp 11.92510946 rms 10.90317748

Shrutar [245/243, 2048/2025]
[4, -8, 14, -22, 11, 55] [[2, 3, 5, 5], [0, 2, -4, 7]]

bad 2740.920108 comp 34.89878326 rms 2.250483358

Diaschismic [126/125, 2048/2025]
[2, -4, -16, -11, -31, -26] [[2, 3, 5, 7], [0, 1, -2, -8]]

bad 2780.393711 comp 26.97297092 rms 3.821630806

Ragar [2048/2025, 3136/3125]
[4, -8, -20, -22, -43, -24] [[4, 6, 10, 13], [0, 1, -2, -5]]

bad 2979.183549 comp 36.62986166 rms 2.220376920

Diaquatra [2048/2025, 2401/2400]
[8, -16, -6, -44, -32, 31] [[2, 4, 3, 5], [0, -4, 8, 3]]

bad 4109.453916 comp 43.88939274 rms 2.133364042

11 limit

Diaschismic

[2, -4, -16, -24, -11, -31, -45, -26, -42, -12]

[[2, 3, 5, 7, 9], [0, 1, -2, -8, -12]]

[600.000000000000, 103.783535998448]

bad 4368.478167 comp 76.308394 rms 3.182069

13 limit

Diaschismic

[2, -4, -16, -24, -30, -11, -31, -45, -55, -26, -42, -55, -12, -25, -15]

[[2, 3, 5, 7, 9, 10], [0, 1, -2, -8, -12, -15]]

[600., 103.786589235317]

[46, 58]

unweighted badness 251.546815 unweighted complexity 19.682480 rms
error 2.880707

diaschismic 2048/2025 comma
[599.56, 104.70]

7-limit diaschismic family

0: <<2 -4 -16 -11 -31 -26|| diaschismic {126/125, 2048/2025}
[599.37, 103.79]

12: <<2 -4 -4 -11 -12 2|| pajara {50/49, 64/63}
[598.45, 106.57]

22: <<2 -4 6 -11 4 25|| {28/27, 525/512}
[599.74, 115.71]

24: <<2 -4 8 -11 7 30|| {36/35, 2048/2025}
[598.71, 97.51]

34: <<2 -4 18 -11 23 53|| {875/864, 2048/2025}
[599.27, 107.35]

11-limit diaschismic family

0: <<2 -4 -16 -24 -11 -31 -45 -26 -42 -12|| diaschismic {126/125,
176/175, 896/891}
[599.37, 103.79]

12: <<2 -4 -16 -12 -11 -31 -26 -26 -14 22|| {56/55, 100/99, 2048/2025}
[598.18, 103.58]

46: <<2 -4 -16 22 -11 -31 28 -26 65 117|| {126/125, 385/384, 1232/1215}
[599.74, 104.26]

58: <<2 -4 -16 34 -11 -31 47 -26 93 151|| {126/125, 540/539, 2048/2025}
[599.30, 103.47]

11-limit pajara family

0: <<2 -4 -4 -12 -11 -12 -26 2 -14 -20|| pajara {50/49, 64/63, 99/98}
[598.45, 106.57]

12: <<2 -4 -4 0 -11 -12 -7 2 14 14|| {45/44, 50/49, 56/55}
[596.50, 101.36]

22: <<2 -4 -4 10 -11 -12 9 2 37 42|| {50/49, 55/54, 64/63}
[599.27, 108.86]

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> diaschismic 2048/2025 comma
> [599.56, 104.70]
>
> 7-limit diaschismic family
>
> 0: <<2 -4 -16 -11 -31 -26|| diaschismic {126/125, 2048/2025}
> [599.37, 103.79]
>
> 12: <<2 -4 -4 -11 -12 2|| pajara {50/49, 64/63}
> [598.45, 106.57]
>
> 22: <<2 -4 6 -11 4 25|| {28/27, 525/512}
> [599.74, 115.71]
>
> 24: <<2 -4 8 -11 7 30|| {36/35, 2048/2025}
> [598.71, 97.51]
>
> 34: <<2 -4 18 -11 23 53|| {875/864, 2048/2025}
> [599.27, 107.35]
>
> 11-limit diaschismic family
>
> 0: <<2 -4 -16 -24 -11 -31 -45 -26 -42 -12|| diaschismic {126/125,
> 176/175, 896/891}
> [599.37, 103.79]

Shrutar?

> 12: <<2 -4 -16 -12 -11 -31 -26 -26 -14 22|| {56/55, 100/99,
2048/2025}
> [598.18, 103.58]
>
> 46: <<2 -4 -16 22 -11 -31 28 -26 65 117|| {126/125, 385/384,
1232/1215}
> [599.74, 104.26]
>
> 58: <<2 -4 -16 34 -11 -31 47 -26 93 151|| {126/125, 540/539,
2048/2025}
> [599.30, 103.47]
>
> 11-limit pajara family
>
> 0: <<2 -4 -4 -12 -11 -12 -26 2 -14 -20|| pajara {50/49, 64/63,
99/98}
> [598.45, 106.57]
>
> 12: <<2 -4 -4 0 -11 -12 -7 2 14 14|| {45/44, 50/49, 56/55}
> [596.50, 101.36]
>
> 22: <<2 -4 -4 10 -11 -12 9 2 37 42|| {50/49, 55/54, 64/63}
> [599.27, 108.86]

Paul Erlich wrote:

> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> > wrote:
>>11-limit diaschismic family
>>
>>0: <<2 -4 -16 -24 -11 -31 -45 -26 -42 -12|| diaschismic {126/125,
>>176/175, 896/891}
>>[599.37, 103.79]
> > > Shrutar?

If 7-limit Shrutar is <<4, -8, 14, -22, 11, 55]], wouldn't 11-limit Shrutar be something starting with <<4, -8, 14...?

<<4, -8, 14, -2, -22, 11, -17, 55, 23, -54]] would work.

Map: [[2, 3, 5, 5, 7], [0, 2, -4, 7, -1]]; it matches the description of Shrutar on Graham Breed's catalog of linear temperaments.

It also shows up on my collection of 11-limit temperaments that have consistent mappings of the intervals between 1/1 and 2/1 with numerators of 11 or less (which I found by a brute force approach of running a program to check all possible periods and generators to an accuracy of one cent, up to 31 steps per octave).

<<6, 10, 3, 8, 2, -12, -8, -21, -16, 12]] #54
<<2, 8, 8, 12, 8, 7, 12, -4, 0, 6]] Injera
<<3, 5, -6, 4, 1, -18, -4, -28, -8, 32]] Porcupine
<<1, -13, -2, -6, -23, -6, -13, 32, 31, -10]] #109
<<4, 16, 9, 10, 16, 3, 2, -24, -32, -3]] Squares
<<8, 6, 6, -4, -9, -13, -34, -3, -30, -32]] #92
<<3, 5, -13, 4, 1, -29, -4, -44, -8, 56]] unnamed?
<<3, 5, 16, 4, 1, 17, -4, 23, -8, -44]] unnamed?
<<7, -3, 8, 2, -21, -7, -21, 27, 15, -22]] Orwell
<<5, 1, 12, -8, -10, 5, -30, 25, -22, -64]] Magic
<<4, -8, 14, -2, -22, 11, -17, 55, 23, -54]] Shrutar
<<7, 2, -11, -10, -13, -37, -40, -31, -30, 10]] unnamed?
<<5, -11, -12, -3, -29, -33, -22, 3, 31, 33]] Tritonic
<<10, 9, 7, 25, -9, -17, 5, -9, 27, 46]] Nonkleismic
<<1, 21, 15, 11, 31, 21, 14, -24, -47, -21]] unnamed?
<<6, -7, -2, 15, -25, -20, 3, 15, 59, 49]] Miracle
<<9, 5, -3, 7, -13, -30, -20, -21, -1, 30]] Quartaminorthirds (Valentine)

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> <<3, 5, -13, 4, 1, -29, -4, -44, -8, 56]] unnamed?
> <<3, 5, 16, 4, 1, 17, -4, 23, -8, -44]] unnamed?

The 7-limit reductions of these are, in terms of nexials,
porcupine-7 and porcupine+22 respectively.

> <<7, 2, -11, -10, -13, -37, -40, -31, -30, 10]] unnamed?

TM basis {100/99, 245/242, 525/512}; 26&29.

> <<1, 21, 15, 11, 31, 21, 14, -24, -47, -21]] unnamed?

TM basis {121/120, 441/440, 686/675}. The 7-limit TM basis is
{686/675, 5120/5103}, and the 5-limit comma |31 -21 1>. This is the
29&46 system.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> Paul Erlich wrote:
>
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> >>11-limit diaschismic family
> >>
> >>0: <<2 -4 -16 -24 -11 -31 -45 -26 -42 -12|| diaschismic {126/125,
> >>176/175, 896/891}
> >>[599.37, 103.79]
> >
> >
> > Shrutar?
>
> If 7-limit Shrutar is <<4, -8, 14, -22, 11, 55]],

Hmm? There's no such thing as 7-limit Shrutar. Where are you getting
this wedgie from?

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

> > If 7-limit Shrutar is <<4, -8, 14, -22, 11, 55]],
>
> Hmm? There's no such thing as 7-limit Shrutar. Where are you getting
> this wedgie from?

No such thing as 7-limit Shrutar?? That's news to me.

If <<a1 a2 a3 a4 a5 a6 a7 a8 a9 a10|| is an 11-limit wedgie, then
<<a1 a2 a3 a5 a6 a8||, so long as gcd(a1,a2,a3,a5,a6,a8)=1, is a
7-limit wedgie. While we can debate the merits of names going uplimit,
obviously if no other name takes precedence this deserves to be called
by the 11-limit name. Hence the wedgie above is 7-limit Shrutar, as it
has been called by some of us for some time now.

TM basis: {245/243, 2048/2025}
Mapping: [<2 3 5 5|, <0 2 -4 7|]
TOP generators: [599.46, 52.64]
TOP error: 1.079
L1 TOP complexity 8.438
L1 TOP badness: 76.826

There's not much of an 11-limit family tree, shrutar-22 and shrutar+46
are marginal and the rest worse. The TOP generators for 11-limit
shrutar are [599.82, 52.37], pretty close. This is a pretty clear-cut
case, it seems to me.

Naturally, this still works for higher limits; to project a 13-limit
wedgie to the 11-limit, we take

<<a1 a2 a3 a4 a6 a7 a8 a10 a11 a13||

We are simply ignoring the basis elements associated with the 13adic
valuation.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > > If 7-limit Shrutar is <<4, -8, 14, -22, 11, 55]],
> >
> > Hmm? There's no such thing as 7-limit Shrutar. Where are you
getting
> > this wedgie from?
>
> No such thing as 7-limit Shrutar?? That's news to me.
>
> If <<a1 a2 a3 a4 a5 a6 a7 a8 a9 a10|| is an 11-limit wedgie, then
> <<a1 a2 a3 a5 a6 a8||, so long as gcd(a1,a2,a3,a5,a6,a8)=1, is a
> 7-limit wedgie. While we can debate the merits of names going
uplimit,
> obviously if no other name takes precedence this deserves to be
called
> by the 11-limit name. Hence the wedgie above is 7-limit Shrutar, as
it
> has been called by some of us for some time now.

Well enough, but please take a step back, Gene. Herman was starting
with 7-limit shrutar, and trying to construct 11-limit shrutar from
it. But the tuning was originally defined as an 11-limit one, and
that's what I was asking about. Instead of an answer, we're going
around in circles. All I remember for sure is that 896/891 and
2048/2025 vanish.

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

> Well enough, but please take a step back, Gene. Herman was starting
> with 7-limit shrutar, and trying to construct 11-limit shrutar from
> it. But the tuning was originally defined as an 11-limit one, and
> that's what I was asking about. Instead of an answer, we're going
> around in circles. All I remember for sure is that 896/891 and
> 2048/2025 vanish.

It's h46/\h22 =<<4 -8 14 -2 -22 11 -17 55 23 -54||, and 896/891 and
2048/2025 do indeed vanish, though the TM basis is actually
{121/120, 176/175, 245/243}.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > Well enough, but please take a step back, Gene. Herman was
starting
> > with 7-limit shrutar, and trying to construct 11-limit shrutar
from
> > it. But the tuning was originally defined as an 11-limit one, and
> > that's what I was asking about. Instead of an answer, we're going
> > around in circles. All I remember for sure is that 896/891 and
> > 2048/2025 vanish.
>
> It's h46/\h22 =<<4 -8 14 -2 -22 11 -17 55 23 -54||, and 896/891 and
> 2048/2025 do indeed vanish, though the TM basis is actually
> {121/120, 176/175, 245/243}.

Oh, so it's not considered part of the diaschismic family because its
generator is only half the diaschismic one (yes?) . . .

Paul Erlich wrote:

> Well enough, but please take a step back, Gene. Herman was starting > with 7-limit shrutar, and trying to construct 11-limit shrutar from > it. But the tuning was originally defined as an 11-limit one, and > that's what I was asking about. Instead of an answer, we're going > around in circles. All I remember for sure is that 896/891 and > 2048/2025 vanish.

I was just going from Gene's Jan. 21 list of 114 7-limit temperaments, which identified <<4, -8, 14, -22, 11, 55]] as "Shrutar". The map of [[2, 3, 5, 5, 7], [0, 2, -4, 7, -1]] is consistent with 896/891 and 2048/2025 vanishing; this temperament has a 52 cent generator and half-octave period, consistent with the description of (the original 11-limit) Shrutar at http://x31eq.com/catalog.htm.

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

> > It's h46/\h22 =<<4 -8 14 -2 -22 11 -17 55 23 -54||, and 896/891 and
> > 2048/2025 do indeed vanish, though the TM basis is actually
> > {121/120, 176/175, 245/243}.
>
> Oh, so it's not considered part of the diaschismic family because its
> generator is only half the diaschismic one (yes?) . . .

In the strict sense I was using for "family", but there is a broader
sense based on the 5-limit--7-limit--11-limit etc comma family tree
structure. It still has 2048/2025 as its 5-limit comma.