Fwd: [tuning-math] gene's lists, monzo's lines

Do we now have names for all the temperaments that weren't
on monz's chart?

-Carl

>Date: Wed, 06 Mar 2002 01:13:08 -0800
>Subject: [tuning-math] gene's lists, monzo's lines
>Reply-To: tuning-math@yahoogroups.com
>
>5-limit
>
>>135/128
>>
>>Map:
>>
>>[ 0 1]
>>[-1 2]
>>[ 3 1]
>>
>>Generators: a = 10.0215/23; b = 1
>>
>>badness: 46.1
>>rms: 18.1
>>g: 2.94
>>errors: [-24.8, -17.7, 7.1]
>
>Not on monz's chart. What's "g"?
>
>>648/625
>>
>>Map:
>>
>>[ 0 4]
>>[ 1 5]
>>[ 1 8]
>>
>>Generators: a = 21.0205/64; b = 1/4
>>
>>badness: 385
>>rms: 11.06
>>g: 3.266
>>errors: [-7.82, 7.82, 15.64]
>>
>>64-et, anyone? It could also be used to temper the 12-et.
>
>diminished.
>
>>250/243
>>
>>Map:
>>
>>[ 0 1]
>>[-3 2]
>>[-5 3]
>>
>>Generators: a = 2.9883/22; b = 1
>>
>>badness: 360
>>rms: 7.98
>>g: 3.559
>>errors: [9.06, -1.29, -10.35]
>>
>>One way to cure those 22-et major thirds of what ails them.
>
>porcupine.
>
>>128/125
>>
>>Map:
>>
>>[ 0 3]
>>[-1 6]
>>[ 0 7]
>>
>>Generators: a = 11.052/27 (~4/3); b = 1/3
>>
>>badness: 142
>>rms: 9.68
>>g: 2.449
>>errors: [6.84, 13.69, 6.84]
>
>augmented
>
>>3125/3072
>>
>>Map:
>>
>>[ 0 1]
>>[ 5 0]
>>[ 1 2]
>>
>>Generators: a = 12.9822/41 (=6.016/19); b = 1
>>
>>badness: 239
>>rms: 4.57
>>g: 3.74
>>errors: [-2.115, -6.346, -4.231]
>>
>>Graham has named this one: Magic.
>
>>81/80
>>
>>Map:
>>
>>[ 0 1]
>>[-1 2]
>>[-4 4]
>>
>>Generators: a = 20.9931/50; b = 1
>>
>>badness: 108
>>rms: 4.22
>>g: 2.944
>>errors: [-5.79, -1.65, 4.14]
>>
>>Nothing left to say about this one. :)
>
>>2048/2025
>>
>>Map:
>>
>>[ 0 2]
>>[-1 4]
>>[ 2 3]
>>
>>Generators: 14.0123/34 (~4/3); b = 1/2
>>
>>badness: 211
>>rms: 2.613
>>g: 4.32
>>errors: [3.49, 2.79, -.70]
>>
>>A good way to take advantage of the 34-ets excellent 5-limit
>>harmonies is two gothish 17-et chains of fifths a sqrt(2)
>>apart.
>
>diaschismic
>
>>78732/78125 = 2^2 3^9 5^-7
>>
>>Map:
>>
>>[ 0 1]
>>[ 7 -1]
>>[ 9 -1]
>>
>>Generators: 23.9947/65 (~9/7); b = 1
>>
>>badness: 346
>>rms: 1.157
>>g: 6.68
>>errors: [-1.1, 0.5, 1.6]
>
>un-named on monz's chart!
>
>>393216/390625 = 2^17 3 5^-8
>>
>>Map:
>>
>>[ 0 1]
>>[ 8 -1]
>>[ 1 2]
>>
>>Generators: a = 31.9951/99 (~5/4); b = 1
>>Works with 31,34,65,99,164
>>
>>badness: 251
>>rms: 1.072
>>g: 6.16
>>error: [.602, 1.506, .904]
>
>wuerschmidt
>
>>2109375/2097152 = 2^-21 3^3 5^7 Orwell
>>
>>Map:
>>
>>[ 0 1]
>>[ 7 0]
>>[-3 3]
>>
>>Generators: a = 19.01127197/84; b = 1
>>
>>badness: 305.93
>>rms: .8004
>>g: 7.257
>>errors: [-.828, -1.082, -.255]
>>
>>ets: 22,31,53,84
>
>>15625/15552 = 2^-6 36-5 5^6 Kleismic
>>
>>Map:
>>
>>[ 0 1]
>>[ 6 0]
>>[ 5 1]
>>
>>Generators: a = 14.00435233/53 (~6/5); b = 1
>>
>>badness: 97
>>rms: 1.030
>>g: 4.546
>>errors: [.523, -.915, -1.438]
>>
>>ets: 19,34,53,68,72,87,140
>
>>1600000/1594323 = 2^9 3^-13 5^-2 Acute Minor Third system
>>
>>Map:
>>
>>[ 0 1]
>>[-5 3]
>>[-13 6]
>>
>>Generators: a = 28.00947813/99 (~243/200); b = 1
>>
>>badness: 305.53
>>rms: .3831
>>g: 9.273
>>error: [-.5009, .0716, -.4293]
>
>not on monz's chart.
>
>>6115295232/6103515625 = 2^23 3^6 5^-15 Semisuper
>>
>>Map:
>>
>>[ 0 2]
>>[ 7 -3]
>>[ 3 2]
>>
>>Generators: a = 52.00397043/118 (~3125/2304); b = 1/2
>/.../
>>badness: 190
>>rms: .1940
>>g: 9.933
>>errors: [.0226, .2081, .2255]
>
>not on monz's chart.
>
>>32805/32768 Shismic
>>
>>Map:
>>
>>[ 0 1]
>>[-1 2]
>>[ 8 1]
>>
>>Generators: a = 120.000624/289 (~4/3); b = 1
>>
>>badness: 55
>>rms: .1617
>>g: 6.976
>>errors: [-.2275, -.1338, .0937]
>
>7-limit
>
>//augmented
>
>>When extended to the 7-limit, this becomes the
>>
>>[ 0 3]
>>[-1 6]
>>[ 0 7]
>>[ 2 6]
>>
>>system I've already mentioned in several contexts, such as
>>the 15+12 system of the 27-et. Both as a 5-limit and a
>>7-limit system, it is good enough to deserve a name of its
>>own.
>
>Jeez- I just realized that the wholetone scale contains
>4:5:7 chords. Here's the 4:5:6:7 in augmented in 27-et:
>
>27 1200
>0 0
>9 400
>16 711
>22 978
>
>This nonatonic looks interesting: 0 2 4 9 11 13 18 20 22 (27)
>
>>(1) [6,10,10,-5,1,2] ets: 22
>>
>>[0 2]
>>[3 1]
>>[5 1]
>>[5 2]
>>
>>a = 7.98567775 / 22 (~9/7) ; b = 1/2
>>measure 3165
>
>What is this? What's "measure"?
>
>>(4) [10,14,14,-7,6,-1] ets: 26
>>
>>[0 2]
>>[5 2]
>>[7 3]
>>[7 4]
>>
>>a = 3.026421762 / 26; b = 1/2
>>measure 8510
>
>This and the above look suspiciously like
>the decatonic and double-diatonic systems.
>But they're not, are they?
>
>-Carl

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

> Do we now have names for all the temperaments that weren't
> on monz's chart?

Which chart?

The first one corresponds with what Erv Wilson called Mavila, in that
Meta-Mavila is constructed analogously to Meta-Meantone in terms of
how the beating rates are related to one another, but the unison
vector is 135/128 instead of 81/80. We used to call it Pelogic but it
doesn't seem much like authentic Pelogs. The paper I sent you has
this and most others below . . . if you need me to send you another
copy, let me know.

> -Carl
>
> >Date: Wed, 06 Mar 2002 01:13:08 -0800
> >Subject: [tuning-math] gene's lists, monzo's lines
> >Reply-To: tuning-math@yahoogroups.com
> >
> >5-limit
> >
> >>135/128
> >>
> >>Map:
> >>
> >>[ 0 1]
> >>[-1 2]
> >>[ 3 1]
> >>
> >>Generators: a = 10.0215/23; b = 1
> >>
> >>badness: 46.1
> >>rms: 18.1
> >>g: 2.94
> >>errors: [-24.8, -17.7, 7.1]
> >
> >Not on monz's chart. What's "g"?

Probably geometric complexity, using a Hahn lattice or something
similar . . .

> >>78732/78125 = 2^2 3^9 5^-7
> >>
> >>Map:
> >>
> >>[ 0 1]
> >>[ 7 -1]
> >>[ 9 -1]
> >>
> >>Generators: 23.9947/65 (~9/7); b = 1
> >>
> >>badness: 346
> >>rms: 1.157
> >>g: 6.68
> >>errors: [-1.1, 0.5, 1.6]
> >
> >un-named on monz's chart!

Semisixths -- I call this 5-limit version "Sensipent" and an awesome
7-limit extension "Sensisept".

> >>1600000/1594323 = 2^9 3^-13 5^-2 Acute Minor Third system
> >>
> >>Map:
> >>
> >>[ 0 1]
> >>[-5 3]
> >>[-13 6]
> >>
> >>Generators: a = 28.00947813/99 (~243/200); b = 1
> >>
> >>badness: 305.53
> >>rms: .3831
> >>g: 9.273
> >>error: [-.5009, .0716, -.4293]
> >
> >not on monz's chart.

"Acute Minor Thirds" became "AMT" which became "Amity".

> >>6115295232/6103515625 = 2^23 3^6 5^-15 Semisuper
> >>
> >>Map:
> >>
> >>[ 0 2]
> >>[ 7 -3]
> >>[ 3 2]
> >>
> >>Generators: a = 52.00397043/118 (~3125/2304); b = 1/2
> >/.../
> >>badness: 190
> >>rms: .1940
> >>g: 9.933
> >>errors: [.0226, .2081, .2255]
> >
> >not on monz's chart.

Vishnu.

> >This nonatonic looks interesting: 0 2 4 9 11 13 18 20 22 (27)

Compare with the 9-note ring in the "TOP Augene" horagram, which also
has a 27-note ring.

> >>(1) [6,10,10,-5,1,2] ets: 22
> >>
> >>[0 2]
> >>[3 1]
> >>[5 1]
> >>[5 2]
> >>
> >>a = 7.98567775 / 22 (~9/7) ; b = 1/2
> >>measure 3165
> >
> >What is this?

"Hedgehog" in my paper, previously known as "Biporky" and
even "Erethrozoic" or something.

> What's "measure"?

Probably one of badness.

> >>(4) [10,14,14,-7,6,-1] ets: 26
> >>
> >>[0 2]
> >>[5 2]
> >>[7 3]
> >>[7 4]
> >>
> >>a = 3.026421762 / 26; b = 1/2
> >>measure 8510

This one isn't on the list of 114:

/tuning-math/message/8809

> >This and the above look suspiciously like
> >the decatonic and double-diatonic systems.
> >But they're not, are they?

Nope.

>> Do we now have names for all the temperaments that weren't
>> on monz's chart?
>
>Which chart?

I believe it was the one that eventually wound up on his ET
page -- your favorite page.

>The first one corresponds with what Erv Wilson called Mavila,

Ah, good. Just wondering if any of these oldies fell through
the cracks.

>if you need me to send you another copy, let me know.

I'm hoping it'll turn up (unless I left in on the plane...).

>> >Date: Wed, 06 Mar 2002 01:13:08 -0800
>> >Subject: [tuning-math] gene's lists, monzo's lines
>> >Reply-To: tuning-math@yahoogroups.com
>> >
>> >5-limit
>> >
>> >>135/128
>> >>
>> >>Map:
>> >>
>> >>[ 0 1]
>> >>[-1 2]
>> >>[ 3 1]
>> >>
>> >>Generators: a = 10.0215/23; b = 1
>> >>
>> >>badness: 46.1
>> >>rms: 18.1
>> >>g: 2.94
>> >>errors: [-24.8, -17.7, 7.1]
>> >
>> >Not on monz's chart. What's "g"?
>
>Probably geometric complexity, using a Hahn lattice or something
>similar . . .

In that thread, it was the 'average number of generators needed
to get consonances' according to Gene, and you said you thought
he used rms average.

>> >>78732/78125 = 2^2 3^9 5^-7
>> >>
>> >>Map:
>> >>
>> >>[ 0 1]
>> >>[ 7 -1]
>> >>[ 9 -1]
>> >>
>> >>Generators: 23.9947/65 (~9/7); b = 1
>> >>
>> >>badness: 346
>> >>rms: 1.157
>> >>g: 6.68
>> >>errors: [-1.1, 0.5, 1.6]
>> >
>> >un-named on monz's chart!
>
>Semisixths -- I call this 5-limit version "Sensipent" and an awesome
>7-limit extension "Sensisept".

Cool.

>> >>1600000/1594323 = 2^9 3^-13 5^-2 Acute Minor Third system
>> >>
>> >>Map:
>> >>
>> >>[ 0 1]
>> >>[-5 3]
>> >>[-13 6]
>> >>
>> >>Generators: a = 28.00947813/99 (~243/200); b = 1
>> >>
>> >>badness: 305.53
>> >>rms: .3831
>> >>g: 9.273
>> >>error: [-.5009, .0716, -.4293]
>> >
>> >not on monz's chart.
>
>"Acute Minor Thirds" became "AMT" which became "Amity".

Excellent.

>> >>6115295232/6103515625 = 2^23 3^6 5^-15 Semisuper
>> >>
>> >>Map:
>> >>
>> >>[ 0 2]
>> >>[ 7 -3]
>> >>[ 3 2]
>> >>
>> >>Generators: a = 52.00397043/118 (~3125/2304); b = 1/2
>> >/.../
>> >>badness: 190
>> >>rms: .1940
>> >>g: 9.933
>> >>errors: [.0226, .2081, .2255]
>> >
>> >not on monz's chart.
>
>Vishnu.

Thanks -- I'd seen Vishnu around but wasn't familiar with him.

>> >This nonatonic looks interesting: 0 2 4 9 11 13 18 20 22 (27)
>
>Compare with the 9-note ring in the "TOP Augene" horagram, which also
>has a 27-note ring.

There are some nice triad and tetrad patterns in this scale, which
I was just playing with Scala. I suppose I'll compare to Augene
when I find the paper. But couldn't you post gifs as you did for
the 5-limit horograms? We could make a permanent horogram spot at
lumma.org/tuning/erlich.

>> >>(1) [6,10,10,-5,1,2] ets: 22
>> >>
>> >>[0 2]
>> >>[3 1]
>> >>[5 1]
>> >>[5 2]
>> >>
>> >>a = 7.98567775 / 22 (~9/7) ; b = 1/2
>> >>measure 3165
>> >
>> >What is this?
>
>"Hedgehog" in my paper, previously known as "Biporky" and
>even "Erethrozoic" or something.

Good to know...

>> What's "measure"?
>
>Probably one of badness.
>
>> >>(4) [10,14,14,-7,6,-1] ets: 26
>> >>
>> >>[0 2]
>> >>[5 2]
>> >>[7 3]
>> >>[7 4]
>> >>
>> >>a = 3.026421762 / 26; b = 1/2
>> >>measure 8510
>
>This one isn't on the list of 114:
>
>/tuning-math/message/8809

Hrm.

>> >This and the above look suspiciously like
>> >the decatonic and double-diatonic systems.
>> >But they're not, are they?
>
>Nope.

Hrm.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Do we now have names for all the temperaments that weren't
> >> on monz's chart?
> >
> >Which chart?
>
> I believe it was the one that eventually wound up on his ET
> page -- your favorite page.

Well, the page must have looked different then, because things you
say "not on monz's chart" about are there now, at tonalsoft.com.

> >The first one corresponds with what Erv Wilson called Mavila,
>
> Ah, good. Just wondering if any of these oldies fell through
> the cracks.

Apparently the last one did:

> >> >>(4) [10,14,14,-7,6,-1] ets: 26
> >> >>
> >> >>[0 2]
> >> >>[5 2]
> >> >>[7 3]
> >> >>[7 4]
> >> >>
> >> >>a = 3.026421762 / 26; b = 1/2
> >> >>measure 8510
> >
> >This one isn't on the list of 114:
> >
> >/tuning-math/message/8809
>
> Hrm.

> >> >This nonatonic looks interesting: 0 2 4 9 11 13 18 20 22 (27)
> >
> >Compare with the 9-note ring in the "TOP Augene" horagram, which
also
> >has a 27-note ring.
>
> There are some nice triad and tetrad patterns in this scale, which
> I was just playing with Scala. I suppose I'll compare to Augene
> when I find the paper. But couldn't you post gifs as you did for
> the 5-limit horograms?

I did that a while ago: go to
/tuning-math/files/Erlich/sevenlimit.zip
and open up tripletone.bmp . . . Gene came up with the name
tripletone but then retired it, as I recall, so I used "Augene" in my
paper.

> >> >This and the above look suspiciously like
> >> >the decatonic and double-diatonic systems.
> >> >But they're not, are they?
> >
> >Nope.
>
> Hrm.

The decatonic and double-diatonic scales are subsets of what are
called "Pajara" and "Injera" in my paper -- specifically, they
correspond to the 10-note ring of the Pajara horagram and the 14-note
ring of the Injera horagram, respectively. These systems are ones I
thought I had invented early on, so I did take the liberty of naming
them.

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

> >> >This nonatonic looks interesting: 0 2 4 9 11 13 18 20 22 (27)
> >
> >Compare with the 9-note ring in the "TOP Augene" horagram, which
also
> >has a 27-note ring.
>
> There are some nice triad and tetrad patterns in this scale,

What tetrads? It seems you need to go outside the 9-note scale to get
any 7-limit tetrads in this system.

>> >Which chart?
>>
>> I believe it was the one that eventually wound up on his ET
>> page -- your favorite page.
>
>Well, the page must have looked different then,

No doubt.

>> >The first one corresponds with what Erv Wilson called Mavila,
>>
>> Ah, good. Just wondering if any of these oldies fell through
>> the cracks.
>
>Apparently the last one did:
>
>> >> >>(4) [10,14,14,-7,6,-1] ets: 26
>> >> >>
>> >> >>[0 2]
>> >> >>[5 2]
>> >> >>[7 3]
>> >> >>[7 4]
>> >> >>
>> >> >>a = 3.026421762 / 26; b = 1/2
>> >> >>measure 8510
>> >
>> >This one isn't on the list of 114:
>> >
>> >/tuning-math/message/8809
>>
>> Hrm.

Hmm.

>> There are some nice triad and tetrad patterns in this scale, which
>> I was just playing with Scala. I suppose I'll compare to Augene
>> when I find the paper. But couldn't you post gifs as you did for
>> the 5-limit horograms?
>
>I did that a while ago: go to
>/tuning-math/files/Erlich/sevenlimit.zip
>and open up tripletone.bmp . . . Gene came up with the name
>tripletone but then retired it, as I recall, so I used "Augene" in my
>paper.

Awesome!

Strangely, I think I had these downloaded, meant to bulk convert them
to png or gif, and somehow dropped the ball and deleted them. Durn
gremlins...

>> >> >This and the above look suspiciously like
>> >> >the decatonic and double-diatonic systems.
>> >> >But they're not, are they?
>> >
>> >Nope.
>>
>> Hrm.
>
>The decatonic and double-diatonic scales are subsets of what are
>called "Pajara" and "Injera" in my paper -- specifically, they
>correspond to the 10-note ring of the Pajara horagram and the 14-note
>ring of the Injera horagram, respectively.

Right, I knew that.

-Carl

>> >> >This nonatonic looks interesting: 0 2 4 9 11 13 18 20 22 (27)
>> >
>> >Compare with the 9-note ring in the "TOP Augene" horagram, which
>> >also has a 27-note ring.
>>
>> There are some nice triad and tetrad patterns in this scale,
>
>What tetrads? It seems you need to go outside the 9-note scale to get
>any 7-limit tetrads in this system.

There's 4:5:6:15, interspersed with a nice 7b5 chord on 1-4-6-9.

There are actually three 4:5:6:7 approximations in the scale (though
not respecting the mapping) on 1-4-6-8.

And lots of nice triads.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> >This nonatonic looks interesting: 0 2 4 9 11 13 18 20 22 (27)
> >> >
> >> >Compare with the 9-note ring in the "TOP Augene" horagram,
which
> >> >also has a 27-note ring.
> >>
> >> There are some nice triad and tetrad patterns in this scale,
> >
> >What tetrads? It seems you need to go outside the 9-note scale to
get
> >any 7-limit tetrads in this system.
>
> There's 4:5:6:15, interspersed with a nice 7b5 chord on 1-4-6-9.
>
> There are actually three 4:5:6:7 approximations in the scale (though
> not respecting the mapping) on 1-4-6-8.

In order to get the approximation without respecting the mapping,
you'd have to exploit the "wrapping around" of the ET. But since your
chain is only two generators long, there's a gap of seven generators
to get around, and the tetrad mapping only spans three generators.
Something's not checking out. The template 1-4-6-8 gives me:

0-9-13-20
2-11-18-22
4-13-20-27

and then of course the pattern repeats, transposed up 1/3 octave.
None of these are 4:5:6:7 approximations.

>>> There are some nice triad and tetrad patterns in this scale, which
>>> I was just playing with Scala. I suppose I'll compare to Augene
>>> when I find the paper. But couldn't you post gifs as you did for
>>> the 5-limit horograms?
>>
>>I did that a while ago: go to
>>/tuning-math/files/Erlich/sevenlimit.zip
>>and open up tripletone.bmp . . . Gene came up with the name
>>tripletone but then retired it, as I recall, so I used "Augene" in my
>>paper.
>
>Awesome!
>
>Strangely, I think I had these downloaded, meant to bulk convert them
>to png or gif,

Done!

It's missing Hedgehog. And, if they're mentioned in your paper,
Amity and Vishnu.

Also, Pelogic is still called Pelogic. Should it be called
Mavilla (in the 5- and 7-limit?)?

-Carl

>> There are actually three 4:5:6:7 approximations in the scale (though
>> not respecting the mapping) on 1-4-6-8.
>
>In order to get the approximation without respecting the mapping,
>you'd have to exploit the "wrapping around" of the ET. But since your
>chain is only two generators long, there's a gap of seven generators
>to get around, and the tetrad mapping only spans three generators.
>Something's not checking out. The template 1-4-6-8 gives me:
>
>0-9-13-20
>2-11-18-22
>4-13-20-27
>
>and then of course the pattern repeats, transposed up 1/3 octave.
>None of these are 4:5:6:7 approximations.

That's...

0-9-13-20
0-9-16-20
0-9-16-23

...on the same root. The bottom ones sounds like 4:5:6:7 to me.
Maybe it's only so with all the distortion in Scala's output on
my laptop speakers?

-Carl

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

> It's missing Hedgehog. And, if they're mentioned in your paper,

Yes.

> Amity and Vishnu.

Among others.

> Also, Pelogic is still called Pelogic. Should it be called
> Mavilla (in the 5- and 7-limit?)?

Mavila (one l) in 5-limit. My paper doesn't have any 7-limit
extensions of it. Did you find your copy yet?

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> There are actually three 4:5:6:7 approximations in the scale
(though
> >> not respecting the mapping) on 1-4-6-8.
> >
> >In order to get the approximation without respecting the mapping,
> >you'd have to exploit the "wrapping around" of the ET. But since
your
> >chain is only two generators long, there's a gap of seven
generators
> >to get around, and the tetrad mapping only spans three generators.
> >Something's not checking out. The template 1-4-6-8 gives me:
> >
> >0-9-13-20
> >2-11-18-22
> >4-13-20-27
> >
> >and then of course the pattern repeats, transposed up 1/3 octave.
> >None of these are 4:5:6:7 approximations.
>
> That's...
>
> 0-9-13-20
> 0-9-16-20
> 0-9-16-23
>
> ...on the same root. The bottom ones sounds like 4:5:6:7 to me.

But you wrote that it should be 0-9-16-22 in a recent post here, so
that's what I was looking for. I guess we had two very different
meanings of "not respecting the mapping" in mind!

> Maybe it's only so with all the distortion in Scala's output on
> my laptop speakers?

Well, if that's the chord that you want to approximate 4:5:6:7 with,
you're better off ditching Augene (formerly Tripletone) and going for
August instead. In TOP August, the nonatonic you mentioned (known as
the Tcherepnin scale in 12-equal) would have step sizes of 107.31
cents and 185.37 cents, and would contain three approximate 4:5:6:7
chords.

>> It's missing Hedgehog. And, if they're mentioned in your paper,
>
>Yes.
>
>> Amity and Vishnu.
>
>Among others.
>
>> Also, Pelogic is still called Pelogic. Should it be called
>> Mavilla (in the 5- and 7-limit?)?
>
>Mavila (one l) in 5-limit.

Pelogic in the 7-limit, then?

>My paper doesn't have any 7-limit
>extensions of it. Did you find your copy yet?

Nope. :(

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> It's missing Hedgehog. And, if they're mentioned in your paper,
> >
> >Yes.
> >
> >> Amity and Vishnu.
> >
> >Among others.
> >
> >> Also, Pelogic is still called Pelogic. Should it be called
> >> Mavilla (in the 5- and 7-limit?)?
> >
> >Mavila (one l) in 5-limit.
>
> Pelogic in the 7-limit, then?

I advise against that.

> >My paper doesn't have any 7-limit
> >extensions of it.

Which is why I didn't come up with any names for such.

> Did you find your copy yet?
>
> Nope. :(

Well, if you're sure you lost it, it only costs me $1.06 to send you
another one.

>> >> It's missing Hedgehog. And, if they're mentioned in your paper,
>> >
>> >Yes.
>> >
>> >> Amity and Vishnu.
>> >
>> >Among others.
>> >
>> >> Also, Pelogic is still called Pelogic. Should it be called
>> >> Mavilla (in the 5- and 7-limit?)?
>> >
>> >Mavila (one l) in 5-limit.
>>
>> Pelogic in the 7-limit, then?
>
>I advise against that.

What should I call it? The 7-limit horagram you just linked
to calls it Pelogic.

>> >My paper doesn't have any 7-limit extensions of it.
>
>Which is why I didn't come up with any names for such.

Ah.

>> Did you find your copy yet?
>>
>> Nope. :(
>
>Well, if you're sure you lost it, it only costs me $1.06 to send you
>another one.

Still have my address?

But: Have you considered scanning it into a pdf?

-Carl

>> >> There are actually three 4:5:6:7 approximations in the scale
>> >> (though not respecting the mapping) on 1-4-6-8.
>> >
>> >In order to get the approximation without respecting the mapping,
>> >you'd have to exploit the "wrapping around" of the ET. But since
>> >your chain is only two generators long, there's a gap of seven
>> >generators to get around, and the tetrad mapping only spans three
>> >generators.
>> >
>> >Something's not checking out. The template 1-4-6-8 gives me:
>> >
>> >0-9-13-20
>> >2-11-18-22
>> >4-13-20-27
>> >
>> >and then of course the pattern repeats, transposed up 1/3 octave.
>> >None of these are 4:5:6:7 approximations.
>>
>> That's...
>>
>> 0-9-13-20
>> 0-9-16-20
>> 0-9-16-23
>>
>> ...on the same root. The bottom ones sounds like 4:5:6:7 to me.
>
>But you wrote that it should be 0-9-16-22 in a recent post here,

I did? I don't find that string in my archive (or with commas,
or spaces). All I remember saying is 1-4-6-8, and that's right.

>> Maybe it's only so with all the distortion in Scala's output on
>> my laptop speakers?
>
>Well, if that's the chord that you want to approximate 4:5:6:7 with,
>you're better off ditching Augene (formerly Tripletone) and going for
>August instead. In TOP August, the nonatonic you mentioned (known as
>the Tcherepnin scale in 12-equal) would have step sizes of 107.31
>cents and 185.37 cents, and would contain three approximate 4:5:6:7
>chords.

Thanks!

-Carl

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

> I did?

Yup -- in the first post of the thread, msg. #11605.

>I don't find that string in my archive (or with commas,
> or spaces).

Here's what you wrote:

"Here's the 4:5:6:7 in augmented in 27-et:

>27 1200
>0 0
>9 400
>16 711
>22 978"

Given that Augene is compatible with 27-equal but August is not, the
smallest distributionally even scale from "augmented" that yields
this chord has 12 notes. Try August instead if you want decent
4:5:6:7s from the 9-note "augmented" scale.

>> I did?
>
>Yup -- in the first post of the thread, msg. #11605.

Oh, that was actually from the old part of the message.

>>I don't find that string in my archive (or with commas,
>> or spaces).
>
>Here's what you wrote:
>
>"Here's the 4:5:6:7 in augmented in 27-et:
>
>>27 1200
>>0 0
>>9 400
>>16 711
>>22 978"
>
>Given that Augene is compatible with 27-equal but August is not, the
>smallest distributionally even scale from "augmented" that yields
>this chord has 12 notes. Try Augene instead if you want decent
>4:5:6:7s from the 9-note "augmented" scale.

Will do!

-Carl

You replied to a deleted version of the message! I changed "Try
Augene instead" to "Try August instead" in the version that's
currently up on the website . . .

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> I did?
> >
> >Yup -- in the first post of the thread, msg. #11605.
>
> Oh, that was actually from the old part of the message.
>
> >>I don't find that string in my archive (or with commas,
> >> or spaces).
> >
> >Here's what you wrote:
> >
> >"Here's the 4:5:6:7 in augmented in 27-et:
> >
> >>27 1200
> >>0 0
> >>9 400
> >>16 711
> >>22 978"
> >
> >Given that Augene is compatible with 27-equal but August is not,
the
> >smallest distributionally even scale from "augmented" that yields
> >this chord has 12 notes. Try Augene instead if you want decent
> >4:5:6:7s from the 9-note "augmented" scale.
>
> Will do!
>
> -Carl

>You replied to a deleted version of the message! I changed "Try
>Augene instead" to "Try August instead" in the version that's
>currently up on the website . . .

I was just composing a message asking you about that!

(I get the list by e-mail, so deletions/updates haf na' ahfect!)

-Carl

>> >> I did?
>> >
>> >Yup -- in the first post of the thread, msg. #11605.
>>
>> Oh, that was actually from the old part of the message.
>>
>> >>I don't find that string in my archive (or with commas,
>> >> or spaces).
>> >
>> >Here's what you wrote:
>> >
>> >"Here's the 4:5:6:7 in augmented in 27-et:
>> >
>> >>27 1200
>> >>0 0
>> >>9 400
>> >>16 711
>> >>22 978"
>> >
>> >Given that Augene is compatible with 27-equal but August is not,
>> >the smallest distributionally even scale from "augmented" that
>> >yields this chord has 12 notes. Try Augene instead if you want
>> >decent 4:5:6:7s from the 9-note "augmented" scale.
>>
>> Will do!
>>
>> -Carl

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

> >> >>(4) [10,14,14,-7,6,-1] ets: 26
> >> >>
> >> >>[0 2]
> >> >>[5 2]
> >> >>[7 3]
> >> >>[7 4]
> >> >>
> >> >>a = 3.026421762 / 26; b = 1/2
> >> >>measure 8510
> >
> >This one isn't on the list of 114:
> >
> >/tuning-math/message/8809
>
> Hrm.

I think the ordering conventions for wedgies changed since then. Ugh!
Gene's giant list (which he e-mailed me) has this:

[10, 14, 14, -1, -6, -7] 3.106578 20.019718

and since 3.106578/10 + 20.019718/24 = 1.1448127 > 1, this didn't
make it into my paper. Surprised it wasn't in this list of 126, from
which the moats, and thus indirectly the paper's cutoff, were derived:

/tuning-math/message/9268

But I guess it's actually worse than the last entry there on both
damage and complexity (whose orders are reversed). That's why it
doesn't appear on this graph:

/tuning-math/files/Erlich/7lin23.gif

where, if it did appear, would be at coordinates (20.019718,
3.106578).

Bless you, Paul, for doing this research.

-Carl

>> >> >>(4) [10,14,14,-7,6,-1] ets: 26
>> >> >>
>> >> >>[0 2]
>> >> >>[5 2]
>> >> >>[7 3]
>> >> >>[7 4]
>> >> >>
>> >> >>a = 3.026421762 / 26; b = 1/2
>> >> >>measure 8510
>> >
>> >This one isn't on the list of 114:
>> >
>> >/tuning-math/message/8809
>>
>> Hrm.
>
>I think the ordering conventions for wedgies changed since then. Ugh!
>Gene's giant list (which he e-mailed me) has this:
>
>[10, 14, 14, -1, -6, -7] 3.106578 20.019718
>
>and since 3.106578/10 + 20.019718/24 = 1.1448127 > 1, this didn't
>make it into my paper. Surprised it wasn't in this list of 126, from
>which the moats, and thus indirectly the paper's cutoff, were derived:
>
>/tuning-math/message/9268
>
>But I guess it's actually worse than the last entry there on both
>damage and complexity (whose orders are reversed). That's why it
>doesn't appear on this graph:
>
>/tuning-math/files/Erlich/7lin23.gif
>
>where, if it did appear, would be at coordinates (20.019718,
>3.106578).