Yahoo Tuning Groups Ultimate Backup tuning Semimarvelous Blue Drawf

will this do?

http://www.traxinspace.com/pub/123320/46615_l.jpg

:-)

I didn't think it was appropriate for my "serious" site but I've been using
it on the tuning muggles sites I posted it to.

Chris

On Wed, Nov 10, 2010 at 1:40 PM, genewardsmith
<genewardsmith@...>wrote:

>
>
>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "christopherv"
> <chrisvaisvil@...> wrote:
> >
> >
> > Online play and tuning
> >
> > http://chrisvaisvil.com/?p=314
> >
> > download
> >
> > http://micro.soonlabel.com/Semimarvelous_dwarf/semimarvelous-piano.mp3
>
> Very cool! Maybe I'm deluding myself, but I think the equal beating is
> making a difference. Whatever, you are having fun with the sonorities.
>
> You wouldn't happen to have an image file of a Blue Drawf, would you? I
> keep thinking Smurf.
>
>
>

🔗genewardsmith <genewardsmith@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Brofessor <kraiggrady@...>

Gene~
is there a translation of this type of notation matrix somewhere. i have not been able to crack the code . or could it possibly also be spelled out in a different way too.
Usually an example goes a long way.

To say that such and such comma is tuned out only tells me what has been omitted , not specifically what is in place.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@> wrote:
> >
> > will this do?
> >
> > http://www.traxinspace.com/pub/123320/46615_l.jpg
> >
> > :-)
> >
> >
> > I didn't think it was appropriate for my "serious" site but I've been using
> > it on the tuning muggles sites I posted it to.
>
> Perfect!
>
> Would it be OK to link your mp3 to pages on the Xenwiki it would go with? Or what about that question in general?
>

🔗genewardsmith <genewardsmith@...>

🔗Brofessor <kraiggrady@...>

here is an example from wikispaces i couldn't figure out.
on the rank 2 temperments i have this
One period per octave
father [<1, 2, 2], <0, -1, 1]>

* mother [<1, 2, 2, 2], <0, -1, 1, 2]>
* father [<1, 2, 2, 4], <0, -1, 1, -3]>
what do the >< and bracket signs mean and why are there two sets of numbers and why if there are two generators are there 4 numbers?

--- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@...> wrote:
>
> how doe one read these matrixes and what are they telling one?
> octave, fifths thirds. i understand but what is that telling me.
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@> wrote:
> > >
> > > Gene~
> > > is there a translation of this type of notation matrix somewhere. i have not been able to crack the code . or could it possibly also be spelled out in a different way too.
> >
> > Could you explain what you mean by "this type of notation matrix"?
> >
>

🔗Carl Lumma <carl@...>

Hi Kraig

(hit "use fixed width font" on the right if reading
on Yahoo's website)

> here is an example from wikispaces i couldn't figure out.
> on the rank 2 temperments i have this
> One period per octave
> father [<1, 2, 2], <0, -1, 1]>

I wish Gene would write them this way

father
< 1 2 2 |
< 0 -1 1 |

The commas hurt my eyes, and placing one over also makes
it easier to read. And this should help explain it

2 3 5
< 1 2 2 | period
< 0 -1 1 | generator

Better?

The brackets are due to Dirac
http://en.wikipedia.org/wiki/Dirac_notation

I wouldn't worry about it.

-Carl

🔗Carl Lumma <carl@...>

🔗Herman Miller <hmiller@...>

🔗Brofessor <kraiggrady@...>

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@...> wrote:

> > 2 3 5
> > < 1 2 2 | period
> > < 0 -1 1 | generator
> >
> > Better?
> >
>
> no that doesn't tell me much that i understand you have
> a 225/1 generator divided by a -16/15?
>

The 2:1 is mapped by 1 period and 0 generators.
That means the period is an octave.
From the 3:1 mapping you can see the generator is some
kind of fourth.
From the 5:1 mapping, it also seems the generator is
some kind of major third.
This means it serves as both 4:3 and 5:4, and the
difference between them (16:15) is tempered out.
This means father isn't a very accurate temperament.
However, various detemperings are possible. Including
all the way to JI, where it turns into a constant structure.

-Carl

🔗Brofessor <kraiggrady@...>

what 3:1 i am sorry i don't see it
but the bottom set of numbers are directly below the ones above

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@> wrote:
>
> > > 2 3 5
> > > < 1 2 2 | period
> > > < 0 -1 1 | generator
> > >
> > > Better?
> > >
> >
> > no that doesn't tell me much that i understand you have
> > a 225/1 generator divided by a -16/15?
> >
>
> The 2:1 is mapped by 1 period and 0 generators.
> That means the period is an octave.
> From the 3:1 mapping you can see the generator is some
> kind of fourth.
> From the 5:1 mapping, it also seems the generator is
> some kind of major third.
> This means it serves as both 4:3 and 5:4, and the
> difference between them (16:15) is tempered out.
> This means father isn't a very accurate temperament.
> However, various detemperings are possible. Including
> all the way to JI, where it turns into a constant structure.
>
> -Carl
>

🔗Brofessor <kraiggrady@...>

i see this as going two fifths up say from C would be D above going down a fifth, this is F

🔗Brofessor <kraiggrady@...>

Since the scale was given in cents, i couldn't tell in which sense it is equal beating?
Is it the basic triad or is it like Silver's, the whole scale.
Since the Meru scales all produce proportional triads, i have not understood why the inversions work so well or the whole scale, but they do.
In fact if i play all the members of the scale i usually cannot explain the resultant beat i ma hearing by looking at the difference tones.

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> On 11/10/2010 1:40 PM, genewardsmith wrote:
> >
> >
> > --- In tuning@yahoogroups.com, "christopherv"<chrisvaisvil@>
> > wrote:
> >>
> >>
> >> Online play and tuning
> >>
> >> http://chrisvaisvil.com/?p=314
> >>
> >> download
> >>
> >> http://micro.soonlabel.com/Semimarvelous_dwarf/semimarvelous-piano.mp3
>
> I love the range of different flavors and textures that you're getting
> from this.
>
> >>
> > Very cool! Maybe I'm deluding myself, but I think the equal beating
> > is making a difference. Whatever, you are having fun with the
> > sonorities.
>
> I agree about the equal beating. I can never be sure if I'm imagining
> things, as the effect is subtle, but I find that I like the effect of
> these equal-beating scales. On the other hand, anything sounds great in
> Pianoteq.
>

🔗Carl Lumma <carl@...>

The columns are the primes. 3:1 is the middle column.
It is composed of 2 periods and -1 generators. We know
the period is an octave, 2 of them takes us to 4:1.
Then we must come down a fourth to get to 3:1. That
is the -1. The generator is a fourth.
Similarly in the 5 column, we must go up a major third
to get to 5:1. The 1 in the generator row means the
generator is a major third.

So it must function both as a fourth and a major third.
That means 16:15 vanishes. Or, in JI, that 16:15 is a
unison vector of the periodicity block / constant structure.

-Carl

--- In tuning@...m, "Brofessor" <kraiggrady@...> wrote:
>
> what 3:1 i am sorry i don't see it
> but the bottom set of numbers are directly below the ones above
>
> > > > 2 3 5
> > > > < 1 2 2 | period
> > > > < 0 -1 1 | generator
> > > >

🔗genewardsmith <genewardsmith@...>

🔗Brofessor <kraiggrady@...>

Thanks Carl- i am not trying to be difficult and i appreciate you taking the time.

Before i get to the last column i don't understand why the second column is not a 4/1 minus a 3/2. hence a 8/3
are negative numbers also treated as the inversions of the positive?

In the last column it is the 4/1 period plus the 5/4 which is 5/1

these you octave reduce?

why couldn't in the last column it be a 1 period for the 5 column instead of a 2 then you would have a 5/2 which would put the 8/3 and 5/2 a 16/15 apart?

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> The columns are the primes. 3:1 is the middle column.
> It is composed of 2 periods and -1 generators. We know
> the period is an octave, 2 of them takes us to 4:1.
> Then we must come down a fourth to get to 3:1. That
> is the -1. The generator is a fourth.
> Similarly in the 5 column, we must go up a major third
> to get to 5:1. The 1 in the generator row means the
> generator is a major third.
>
> So it must function both as a fourth and a major third.
> That means 16:15 vanishes. Or, in JI, that 16:15 is a
> unison vector of the periodicity block / constant structure.
>
> -Carl
>
> --- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@> wrote:
> >
> > what 3:1 i am sorry i don't see it
> > but the bottom set of numbers are directly below the ones above
> >
> > > > > 2 3 5
> > > > > < 1 2 2 | period
> > > > > < 0 -1 1 | generator
> > > > >
>

🔗Brofessor <kraiggrady@...>

do you end up with a chain of tetrads then taht are linked in someway.
I am trying to visualize first. I don't understand the logic behind the math why e would be subtracted from 3 and 5 but added to 7.
spell out one tetrad and see if i can understand that in VPS.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@> wrote:
> >
> > Since the scale was given in cents, i couldn't tell in which sense it is equal beating?
> > Is it the basic triad or is it like Silver's, the whole scale.
>
> The otonal tetrads, not merely the triads, are equal beating. The approximations for 3, 5, and 7 are 3 - e, 5 - e, and 7 + e, where e is a small positive quantity. Since 225/224 is tempered out, this quantity must satisfy (3-e)^2(5-e)^2/(32(7+e)) = 1, which means e is the smallest real root of e^4 - 16e^3 + 94e^2 - 272e + 1, an algebraic integer (in fact, an algebraic unit.)
>

🔗genewardsmith <genewardsmith@...>

🔗Carl Lumma <carl@...>

Hi Kraig!

> Thanks Carl- i am not trying to be difficult and i appreciate
> you taking the time.

Sure!

> Before i get to the last column i don't understand why the
> second column is not a 4/1 minus a 3/2. hence a 8/3 are
> negative numbers also treated as the inversions of the positive?

Let's repaste the original here

> > > > > > 2 3 5
> > > > > > < 1 2 2 | period
> > > > > > < 0 -1 1 | generator
> > > > > >

The middle column's target is 3/1, and 2 periods gets us
to 4/1. So 4/1 + x = 3/1. Solving for x here gives -4/3.
The negative sign goes in there so we know it's downward
one generator to get from 4/1 to 3/1.

> In the last column it is the 4/1 period plus the 5/4 which is 5/1

Yep.

> these you octave reduce?

Not really. It might help to look at the mapping for 5-limit JI

< 1 0 0 | period
< 0 1 0 | generator 1
< 0 0 1 | generator 2

The "rank" of the thing the mapping represents is the number of
rows. 5-limit JI being "rank 3".

> why couldn't in the last column it be a 1 period for the
> 5 column instead of a 2 then you would have a 5/2 which
> would put the 8/3 and 5/2 a 16/15 apart?

Hm, not sure, but there will always been different ways to
write the mapping for the same comma. Like, this could be
rewritten for octave and fifth

> > > > > > 2 3 5
> > > > > > < 1 1 3 | period
> > > > > > < 0 1 -1 | generator

Now the generator's a fifth, so 3/1 is 1 octave + 1 fifth.
And 5/1 is 3 octaves - 1 fifth. Keep in mind this is a
very sharp fifth, just like before it was a very flat fourth.

So yeah, you can always rewrite a mapping in many ways.
That's why Gene and others have devised a standard way of
choosing one. They choose the mapping where the period is
an octave, or simple fraction of the octave (half, third,
quarter, etc) and the generator such that it is less than
half the period. (It's possible to prove such a mapping
always exists.) So for father, the period is an octave, the
generator could be a fifth or a fourth, but a fifth would
be > than half an octave so we choose the fourth. It's just
a standard that makes it easier to recognize these mappings.

-Carl

🔗Ozan Yarman <ozanyarman@...>

Chris, this piece is superb. I like the Bluesy colour of the piece with 7-limitish chords. Bravo!

Oz.

✩ ✩ ✩
www.ozanyarman.com

Chris Vaisvil wrote:
> This is a solo piano piece performed on a M-Audio 88es driving pianoteq
> which was re-tuned to Gene Ward Smith’s 17 per octave equal beating
> dwarf(<17 27 40|). The piece was originally recorded in pianoteq using
> the standalone mode and then brought into Sonar 8.5 to edit some stray
> notes, change a chord, and correct some timing issues (some of those
> still exist).
>
> http://micro.soonlabel.com/Semimarvelous_dwarf/semimarvelous-piano.mp3
>
> online play
> http://notonlymusic.com/board/viewtopic.php?f=23&t=679&start=0
> <http://notonlymusic.com/board/viewtopic.php?f=23&t=679&start=0>
>
> I am finding this tuning to be really rich with extended chords and I’m
> enjoying the time I have been spending with it. The scala format for
> Gene’s tuning follows:
>
> ! dwarf17marveq.scl
> Semimarvelous dwarf: equal beating dwarf(<17 27 40|)
> 17
> !
> 70.247930173690388400
> 115.13195688812420070
> 185.37988706181458910
> 269.90670087373119520
> 314.79072758816500750
> 385.03865776185539590
> 500.17061464997959660
> 570.41854482366998500
> 615.30257153810379730
> 699.82938535002040340
> 770.07731552371079180
> 814.96134223814460410
> 885.20927241183499250
> 955.45720258552538090
> 1000.3412292999591932
> 1084.8680431118757993
> 1200.0000000000000000
> ! eight tetrads/pentads, representible by [[0, -1, 0], [0, -1, 1],
> ! [1, -1, 1], [1, -1, 2], [0, 0, 2], [0, -1, -2], [0, 0, 1], [0, -1, -1]]
>

🔗Brofessor <kraiggrady@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi Kraig!

I suggest you start over and explain every single number in every column because it does not work fro me logically

>
> Let's repaste the original here
>
> > > > > > > 2 3 5
> > > > > > > < 1 2 2 | period
> > > > > > > < 0 -1 1 | generator
> > > > > > >

>
> The middle column's target is 3/1,
because it is in the 3 column?
if so what does the 2 in the period column mean

and 2 periods gets us
> to 4/1.
if this is the 3 column why isn't this 6/1?

So 4/1 + x = 3/1. Solving for x here gives -4/3.

which number is x , where does this come from?

> The negative sign goes in there so we know it's downward
> one generator to get from 4/1 to 3/1.

if we are in a column that is designated as 3 a negative to me means moving in the chain in the negative direction.

>
> > In the last column it is the 4/1 period plus the 5/4 which is 5/1
>
> Yep.
>
> > these you octave reduce?
>
> Not really. It might help to look at the mapping for 5-limit JI
>

> < 1 0 0 | period
> < 0 1 0 | generator 1
> < 0 0 1 | generator 2
>
> The "rank" of the thing the mapping represents is the number of
> rows. 5-limit JI being "rank 3".
>
> > why couldn't in the last column it be a 1 period for the
> > 5 column instead of a 2 then you would have a 5/2 which
> > would put the 8/3 and 5/2 a 16/15 apart?
>
> Hm, not sure, but there will always been different ways to
> write the mapping for the same comma. Like, this could be
> rewritten for octave and fifth
>
> > > > > > > 2 3 5
> > > > > > > < 1 1 3 | period
> > > > > > > < 0 1 -1 | generator
>
> Now the generator's a fifth, so 3/1 is 1 octave + 1 fifth.
> And 5/1 is 3 octaves - 1 fifth. Keep in mind this is a
> very sharp fifth, just like before it was a very flat fourth.
>
> So yeah, you can always rewrite a mapping in many ways.
> That's why Gene and others have devised a standard way of
> choosing one. They choose the mapping where the period is
> an octave, or simple fraction of the octave (half, third,
> quarter, etc) and the generator such that it is less than
> half the period. (It's possible to prove such a mapping
> always exists.) So for father, the period is an octave, the
> generator could be a fifth or a fourth, but a fifth would
> be > than half an octave so we choose the fourth. It's just
> a standard that makes it easier to recognize these mappings.
>
> -Carl
>

🔗Carl Lumma <carl@...>

Kraig wrote:

> I suggest you start over and explain every single number
> in every column because it does not work fro me logically

These are mappings from the temperament to the primes.
The temperament is generated by two intervals (in the
case of a rank 2 or linear temperament): the period and
the generator. All we are doing is counting how many
periods and generators we must stack to reach each prime
number.

> > Let's repaste the original here
> >
> > > > > > > > 2 3 5
> > > > > > > > < 1 2 2 | period
> > > > > > > > < 0 -1 1 | generator
> > > > > > > >
> >
> > The middle column's target is 3/1,
>
> because it is in the 3 column?

Yes.

> if so what does the 2 in the period column mean

The period row?

Read it by columns. To reach 3, stack 2 periods
and -1 generators. etc.

> > and 2 periods gets us to 4/1.
>
> if this is the 3 column why isn't this 6/1?

The columns are the targets, the rows are the things
you're stacking.

> So 4/1 + x = 3/1. Solving for x here gives -4/3.
>
> which number is x , where does this come from?

We are trying to find the sizes of the period and
generator knowing only the mapping. In other words,
we are finding a tuning for this temperament.

This is not always easy, but in this case (father)
it is easy. Because the 2:1 is mapped by 1 period and
zero generators, that means the period is a 2:1.
Then you can put that into the second column to deduce
the size of the generator, using the equation above.

> > The negative sign goes in there so we know it's downward
> > one generator to get from 4/1 to 3/1.
>
> if we are in a column that is designated as 3 a negative
> to me means moving in the chain in the negative direction.

That's exactly right, but the chain is a chain of things
described by the _row_ you are on.

> >It might help to look at the mapping for 5-limit JI
> >
> > < 1 0 0 | period
> > < 0 1 0 | generator 1
> > < 0 0 1 | generator 2

Maybe this makes sense now. 5-limit JI is the 'temperament'
with an octave period, and generators of 3:1 and 5:1.

-Carl

🔗Brofessor <kraiggrady@...>

much closer and thanks for you patience

So i think my trouble was not i understanding was that the minus sign means to subtract from the above as opposed to moving in the opposite direction.
It is not associative then. it is more that we are doing -(+1)

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Kraig wrote:

>
> Read it by columns. To reach 3, stack 2 periods
> and -1 generators. etc.

so how do i read the last columm
to reach 5 stack 2 periods (4/1)
and then add 5/1?
why would this be notated with a negative 2
or do we always assume we are looking for the difference of the upper number to the lower?
Just so i understand,
the number in the bottom plower column will always be O?

> -Carl
>

🔗Carl Lumma <carl@...>

> > Read it by columns. To reach 3, stack 2 periods and -1
> > generators. etc.
>
> so how do i read the last columm
> to reach 5 stack 2 periods (4/1)
> and then add 5/1?

2 periods then add 1 generator (x). So the two equations are

4/1 - x = 3/1
4/1 + x = 5/1
---------------
16/1 + 0 = 15/1

16/15 = 0

Which is the comma that disappears. Hopefully this helps
more than hurts!

> why would this be notated with a negative 2

There's no negative 2. Maybe this formatting will help:

2___3___5
1...2...2 period
0..-1...1 generator

> or do we always assume we are looking for the difference
> of the upper number to the lower?

We add the rows in each column to get to the prime for
that column. If we find a negative number, we're adding
a negative, which is the same as subtracting.

> Just so i understand,
> the number in the bottom plower column will always be O?

The column for the 2:1 will always be all zeros except for
the period row. So you can always find the size of the
period by dividing the octave by that number. Usually it
is 1 and the period is the octave itself.
A familiar counterexample is the diminished temperament
which has a 4 there, indicating the period is 1/4-octave
(300 cents). The Messiaen octatonic scale is associated
with this tempearment, and indeed, repeats every minor 3rd.
Another one playable in 12-ET is the augmented temperament,
which has a 3 there, indicating the period is 1/3-octave.
And indeed, the associated hexatonic scale 0-1-4-5-8-9
repeats every major 3rd.

Hope that helps. Sorry I'm not writing more clearly.

-Carl

🔗Brofessor <kraiggrady@...>

Thanks Carl.
I think you did a good job and were clear considering it was a place where my intuition was at quite a bit of odds to this.
I guess my question now is once you know that one is tempering out the 16/15 how are people proceeding in doing this. I am assuming one tempers toward the mediant, and if so which one?

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > > Read it by columns. To reach 3, stack 2 periods and -1
> > > generators. etc.
> >
> > so how do i read the last columm
> > to reach 5 stack 2 periods (4/1)
> > and then add 5/1?
>
> 2 periods then add 1 generator (x). So the two equations are
>
> 4/1 - x = 3/1
> 4/1 + x = 5/1
> ---------------
> 16/1 + 0 = 15/1
>
> 16/15 = 0
>
> Which is the comma that disappears. Hopefully this helps
> more than hurts!
>
> > why would this be notated with a negative 2
>
> There's no negative 2. Maybe this formatting will help:
>
> 2___3___5
> 1...2...2 period
> 0..-1...1 generator
>
> > or do we always assume we are looking for the difference
> > of the upper number to the lower?
>
> We add the rows in each column to get to the prime for
> that column. If we find a negative number, we're adding
> a negative, which is the same as subtracting.
>
> > Just so i understand,
> > the number in the bottom plower column will always be O?
>
> The column for the 2:1 will always be all zeros except for
> the period row. So you can always find the size of the
> period by dividing the octave by that number. Usually it
> is 1 and the period is the octave itself.
> A familiar counterexample is the diminished temperament
> which has a 4 there, indicating the period is 1/4-octave
> (300 cents). The Messiaen octatonic scale is associated
> with this tempearment, and indeed, repeats every minor 3rd.
> Another one playable in 12-ET is the augmented temperament,
> which has a 3 there, indicating the period is 1/3-octave.
> And indeed, the associated hexatonic scale 0-1-4-5-8-9
> repeats every major 3rd.
>
> Hope that helps. Sorry I'm not writing more clearly.
>
> -Carl
>

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@...> wrote:

> I guess my question now is once you know that one is tempering
> out the 16/15 how are people proceeding in doing this. I am
> assuming one tempers toward the mediant, and if so which one?

The simplest way is to use a generator half way in size
between a third and a fourth. 16/15 is 112 cents, half
of that is 56 cents. 498 - 56 or 386 + 56 = 442 cents.
So that's one choice of generator. So your 4:5:6:8 chord
would be

0 442 758 1200

Pretty crazy, but this is a temperament for the bold.
Paul Erlich devised a way of tempering the octave a bit
too, so it would come out

0 447 739 1186

Herman Miller has a MIDI demo on this page:

http://www.io.com/~hmiller/music/zireen-music.html

I should say again that it is also possible to detemper
it all the way to JI, when one is left with a periodicity
block where 16/15 is a unison vector.

-Carl

🔗Brofessor <kraiggrady@...>

i would have assume yes the geometric mean being the most common.
I see what you wrote uses the practice of mirroring the generator back and forth around th a starting point.
I find it much harder to see and follow and count the steps in a MOS.
with my use of Horogram rhythms i think it would really be difficult to see and work with regardless........
By coincidence one of the recurrent sequences i found independently is
Hn = Hn-5+ 2Hn-4 or (1+2G^1)^(1/5) this gives one a generator of 441.716
So one could use this converged figure or a numerical series that comverges on this. it shows where a proportional triad would be

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@> wrote:
>
> > I guess my question now is once you know that one is tempering
> > out the 16/15 how are people proceeding in doing this. I am
> > assuming one tempers toward the mediant, and if so which one?
>
> The simplest way is to use a generator half way in size
> between a third and a fourth. 16/15 is 112 cents, half
> of that is 56 cents. 498 - 56 or 386 + 56 = 442 cents.
> So that's one choice of generator. So your 4:5:6:8 chord
> would be
>
> 0 442 758 1200
>
> Pretty crazy, but this is a temperament for the bold.
> Paul Erlich devised a way of tempering the octave a bit
> too, so it would come out
>
> 0 447 739 1186
>
> Herman Miller has a MIDI demo on this page:
>
> http://www.io.com/~hmiller/music/zireen-music.html
>
> I should say again that it is also possible to detemper
> it all the way to JI, when one is left with a periodicity
> block where 16/15 is a unison vector.
>
> -Carl
>

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> The simplest way is to use a generator half way in size
> between a third and a fourth. 16/15 is 112 cents, half
> of that is 56 cents. 498 - 56 or 386 + 56 = 442 cents.
> So that's one choice of generator. So your 4:5:6:8 chord
> would be
>
> 0 442 758 1200

Minimax tuning. Minor thirds/major sixths are "eigenmonzos", meaning pure just intervals.

> Pretty crazy, but this is a temperament for the bold.

Or the deranged.

> Paul Erlich devised a way of tempering the octave a bit
> too, so it would come out
>
> 0 447 739 1186

TOP tuning.

> Herman Miller has a MIDI demo on this page:
>
> http://www.io.com/~hmiller/music/zireen-music.html

The master of low-complexity temperament strikes again!

🔗Brofessor <kraiggrady@...>

It is a nice piece. i like it.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
>
> > The simplest way is to use a generator half way in size
> > between a third and a fourth. 16/15 is 112 cents, half
> > of that is 56 cents. 498 - 56 or 386 + 56 = 442 cents.
> > So that's one choice of generator. So your 4:5:6:8 chord
> > would be
> >
> > 0 442 758 1200
>
> Minimax tuning. Minor thirds/major sixths are "eigenmonzos", meaning pure just intervals.
>
> > Pretty crazy, but this is a temperament for the bold.
>
> Or the deranged.
>
> > Paul Erlich devised a way of tempering the octave a bit
> > too, so it would come out
> >
> > 0 447 739 1186
>
> TOP tuning.
>
> > Herman Miller has a MIDI demo on this page:
> >
> > http://www.io.com/~hmiller/music/zireen-music.html
>
> The master of low-complexity temperament strikes again!
>

🔗Brofessor <kraiggrady@...>

The proportional triad doesn't occur in the 5 tone scale unfortunately. One has to get to the 6th tone where one gets something like a stretched out 1-7-10. The MOS of the 441.715778 generator coming from the recurrent series i posted
leads to scales of
1,2,3,5,8,11,19,30,49 tones etc.
30 being one interesting spot before one has to go quite a bit higher to get one better.

The 5 note stills sounds in the Pelog ballpark or possibly just the closest familiar scale that comes to mind.

--- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@...> wrote:
>
> It is a nice piece. i like it.
>
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > > The simplest way is to use a generator half way in size
> > > between a third and a fourth. 16/15 is 112 cents, half
> > > of that is 56 cents. 498 - 56 or 386 + 56 = 442 cents.
> > > So that's one choice of generator. So your 4:5:6:8 chord
> > > would be
> > >
> > > 0 442 758 1200
> >
> > Minimax tuning. Minor thirds/major sixths are "eigenmonzos", meaning pure just intervals.
> >
> > > Pretty crazy, but this is a temperament for the bold.
> >
> > Or the deranged.
> >
> > > Paul Erlich devised a way of tempering the octave a bit
> > > too, so it would come out
> > >
> > > 0 447 739 1186
> >
> > TOP tuning.
> >
> > > Herman Miller has a MIDI demo on this page:
> > >
> > > http://www.io.com/~hmiller/music/zireen-music.html
> >
> > The master of low-complexity temperament strikes again!
> >
>

🔗Jacques Dudon <fotosonix@...>

Kraig wrote :

> I see what you wrote uses the practice of mirroring the generator > back and forth around th a starting point.
> I find it much harder to see and follow and count the steps in a MOS.

I agree with you and for other reasons. I think having the generator going one direction rather than symmetrical lets you encompass at once two times more harmonic possibilities of the temperament, if any.

> By coincidence one of the recurrent sequences i found independently is
> Hn = Hn-5+ 2Hn-4 or (1+2G^1)^(1/5) this gives one a generator of > 441.716
> So one could use this converged figure or a numerical series that > comverges on this. it shows where a proportional triad would be

like 56 72 93 120 155 200 258 333 430 555 716 924 1193 1540 1987 2564 3310 4273 5514 7115 9184...

You might be interested to know it is in fact the solution of a 4th degree polynomial which I named in your honor "Poussah-Grady" :

x^4 = x^3 - x^2 + x + 1
(155 = 120 -93 +72 +56)...

of which x^5 = 2x + 1 is definitively a superb -c 5th degree property.
(200 = 2*72 +56)...

It means that only 4 notes are necessary to start these series.

It could also apply more precisely to "Clyde" (~1.29036), if I refer to the list of linear temperaments compiled by Graham you may find in my TL files folder (linear-versus fractals), with other choices as well.

Same folder is a photosonic disk I made of it :
http://f1.grp.yahoofs.com/v1/sG_eTC9D07R82Q56YgFBl-wPkH8zU7rg-Y_GeRvxwW_MMpAb2ZnGbvRV8A9SGOridNb6RG01XaliwcuMRpZ-rAzwwKij4sRdFA/JacquesDudon/Poussah-Grady_1236.pdf

.../...
> The proportional triad doesn't occur in the 5 tone scale > unfortunately. One has to get to the 6th tone where one gets > something like a stretched out 1-7-10. The MOS of the 441.715778 > generator coming from the recurrent series i posted
> leads to scales of
> 1,2,3,5,8,11,19,30,49 tones etc.
> 30 being one interesting spot before one has to go quite a bit > higher to get one better.

I find myself 19 much more balanced than 30 or 49 (see for example 9184/72, that reduces to 288/287 in my series)

> The 5 note stills sounds in the Pelog ballpark or possibly just the > closest familiar scale that comes to mind.

Curiously Herman's piece (is it what he calls Nai Yulung ?) :
http://www.io.com/~hmiller/music/zireen-music.html
seems developped around a structure similar to the pentatonic raga Chandrakaus :
C : Eb : F# : G : Bb : C (Madhyama raga started here in G in North indian music)
I don't how he got there really, unless from higher numbers of generations, or if it is "detempered all the way to JI,
when one is left with a periodicity block where 16/15 is a unison vector" as Carl says.

.../...
> It is a nice piece. i like it.

Me too, very much. Qualifies for the "Ethno extra" series, Asian-jazz category ... ;-)
- - - - - - -
Jacques

🔗genewardsmith <genewardsmith@...>

🔗Brofessor <kraiggrady@...>

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> > By coincidence one of the recurrent sequences i found independently is
> > Hn = Hn-5+ 2Hn-4 or (1+2G^1)^(1/5) this gives one a generator of
> > 441.716
> > So one could use this converged figure or a numerical series that
> > comverges on this. it shows where a proportional triad would be
>
> like 56 72 93 120 155 200 258 333 430 555 716 924 1193 1540 1987 2564
> 3310 4273 5514 7115 9184...

a fine seed this is
>
> You might be interested to know it is in fact the solution of a 4th
> degree polynomial which I named in your honor "Poussah-Grady" :
>
> x^4 = x^3 - x^2 + x + 1
> (155 = 120 -93 +72 +56)...

In order for me to use adding and subtracting, i would have to conceive it in terms of 2nd order difference tones, but i recognize that such properties add to the overall use of the tuning.
>
> of which x^5 = 2x + 1 is definitively a superb -c 5th degree property.
> (200 = 2*72 +56)...
>
> It means that only 4 notes are necessary to start these series.
>
> It could also apply more precisely to "Clyde" (~1.29036), if I refer
> to the list of linear temperaments compiled by Graham you may find in
> my TL files folder (linear-versus fractals), with other choices as well.

I will look at both of these . thanks.
you might enjoy the following

http://anaphoria.com/MERUcream.PDF

which i think is the better part of
http://anaphoria.com/MERU2nds.pdf
>
> Same folder is a photosonic disk I made of it :
> http://f1.grp.yahoofs.com/v1/sG_eTC9D07R82Q56YgFBl-wPkH8zU7rg-
> Y_GeRvxwW_MMpAb2ZnGbvRV8A9SGOridNb6RG01XaliwcuMRpZ-rAzwwKij4sRdFA/
> JacquesDudon/Poussah-Grady_1236.pdf

>
> .../...
> > The proportional triad doesn't occur in the 5 tone scale
> > unfortunately. One has to get to the 6th tone where one gets
> > something like a stretched out 1-7-10. The MOS of the 441.715778
> > generator coming from the recurrent series i posted
> > leads to scales of
> > 1,2,3,5,8,11,19,30,49 tones etc.
> > 30 being one interesting spot before one has to go quite a bit
> > higher to get one better.
>
> I find myself 19 much more balanced than 30 or 49 (see for example
> 9184/72, that reduces to 288/287 in my series)

I agree. a mistake on my part
>
> > The 5 note stills sounds in the Pelog ballpark or possibly just the
> > closest familiar scale that comes to mind.
>
> Curiously Herman's piece (is it what he calls Nai Yulung ?) :
> http://www.io.com/~hmiller/music/zireen-music.html
> seems developped around a structure similar to the pentatonic raga
> Chandrakaus :
> C : Eb : F# : G : Bb : C (Madhyama raga started here in G in North
> indian music)
> I don't how he got there really, unless from higher numbers of
> generations, or if it is "detempered all the way to JI,
> when one is left with a periodicity block where 16/15 is a unison
> vector" as Carl says.

yes in this ballbark and by Pelog i wasn't meaning the typical m2 M2 M3 m2 M3 version but some of the more unusal one like this one here. Lou harrisno would pull ones like this out
Being so non just made me not thing of India
>
> .../...
> > It is a nice piece. i like it.
>
>
> Me too, very much. Qualifies for the "Ethno extra" series, Asian-jazz
> category ... ;-)

> - - - - - - -
> Jacques
>

🔗genewardsmith <genewardsmith@...>

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

> The minimax generator is 1.29029934 and the pure-octaves TE generator is 1.29036478; in between is the Poussami generator, the real root of 5x^3 - 6x - 3, which comes in at 1.29034778. This is an excellent generator for Clyde, therefore.
>

I've added this and some other recommended algebraic generators to the appropriate pages on the Xenwiki. There seem to be a lot of things on Jacques' list which make for recommendable tunings, but are still three-term polynomials of small degree and coefficients not too large.

I don't have a good name for these, as while they are algebraic they are no more so than minimax or least squares tunings. I don't want to call them fractals, because they aren't fractals, or -c, since that isn't a name, or differentially coherent, since that is too much name and anyway someone would need to write a web page defining it.

The choices I made may be familiar to Jacques but they weren't to me. On the Meantone family page, I recommended Cybozem for septimal meantone, Traverse for unidecimal meantone/huyghens, Cybozem or Radieubiz for meanpop, Squarto for flattone, Mohabis for mohajira, Rabrindanath for mothra, Sceptre2 for squares, and Radix for liese. These are not terms I can recall being bandied about.

🔗Jacques Dudon <fotosonix@...>

On the 14th of november 2010 Gene wrote :

> > The minimax generator is 1.29029934 and the pure-octaves TE > generator is 1.29036478; in between is the Poussami generator, the > real root of 5x^3 - 6x - 3, which comes in at 1.29034778. This is > an excellent generator for Clyde, therefore.

By the way, 40/31 = 1.2903225806452 makes it also for Cyde then !

What does TE means ? and what's the difference with minimax ? (or where do I find that ?)
What is specific of the Clyde temperament exactly and in what it differs from his neighbours Sensipent, Sensisept or Sensi ?

All this zone (lets say 1.290 - 1.293) has an incredible diversity of recurrent sequences with various acoustic qualities and I would like to understand better the temperaments it may apply for.
I mentionned Poussami and other Poussahs, but I am finding a huge number of -c solutions with slightly higher ratios in zones around for example :

Pierre :
x^9 = x^7 + 4
x = 1.29141092246253

Retrixring (or its complement Tierxring) :
x^2 = 12 - 8x
x = 1.2915026221292

Carene, that comes with a superfine 3rd harmonic approximation (x^7 = 5.999474932) :
4x^8 = 4x^7 + 7
x = 1.2916921930394356

or even with slightly extended fifths such as Calice :
2x^6 = x + 8
x = 1.291748324976139

or Calista :
x^7 = 3x^2 + 1
x = 1.2919315501289

or Irina :
2x^4 = 2x + 3
x = 1.29272801434

or Radevati :
x^6 = x^2 + 3
x = 1.292942335

Just to show a few solid guys. Would they refer to Sensi/Sensipent/Sensisept temperaments ?

> I've added this and some other recommended algebraic generators to > the appropriate pages on the Xenwiki. There seem to be a lot of > things on Jacques' list which make for recommendable tunings, but > are still three-term polynomials of small degree and coefficients > not too large.

You mean "and" are still 3 terms polynomials with not too large coefficients, or are they still too high ? :-)
I am honored of this historical entry really, thank you very much. Wow, I have no excuse to not learn more about linear temperaments now ! ;-)

> I don't have a good name for these, as while they are algebraic > they are no more so than minimax or least squares tunings. I don't > want to call them fractals, because they aren't fractals, or -c, > since that isn't a name, or differentially coherent, since that is > too much name and anyway someone would need to write a web page > defining it.

It's not because it's not in familiar use yet that the word "fractals" would be incorrect ; after asking mathematicians who saw and heard my photosonic disks of those, the term "linear fractals" they said was correct - but I will enquire more precisely about that soon with eminent mathematicians. If I find it's not precisely and mathematically correct, I would give up this name.
One important distinction I make between these is based on their geometrical resolution as fractal waveforms, whether I could have it resolved or not (by "resolved" I mean forever self-generating with no accident). If yes, there is no doubt these are fractals (all Pisot-Vijayaraghavan numbers used as generators for example) ; if not, but if they have valid -c algorithms, these can be made audible and are anyway potentially fractals as well.

> The choices I made may be familiar to Jacques but they weren't to > me. On the Meantone family page, I recommended Cybozem for septimal > meantone, Traverse for unidecimal meantone/huyghens, Cybozem or > Radieubiz for meanpop, Squarto for flattone, Mohabis for mohajira, > Rabrindanath for mothra, Sceptre2 for squares, and Radix for liese. > These are not terms I can recall being bandied about.

New word, or expression, for me, can't find it in my dictionary. It means what exactly and in this context ?

- - - - - -
Jacques

🔗genewardsmith <genewardsmith@...>

🔗Brofessor <kraiggrady@...>

🔗shaahin <acousticsoftombak@...>

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> What does TE means ? and what's the difference with minimax ? (or
> where do I find that ?)

TE tuning is another (new, replacement) name for TOP-RMS:

http://xenharmonic.wikispaces.com/Tenney-Euclidean+Tuning

> What is specific of the Clyde temperament exactly and in what it
> differs from his neighbours Sensipent, Sensisept or Sensi ?

http://xenharmonic.wikispaces.com/Kleismic+family

> All this zone (lets say 1.290 - 1.293) has an incredible diversity of
> recurrent sequences with various acoustic qualities and I would like
> to understand better the temperaments it may apply for.

Sounds worthy of more attention.

> or Calista :
> x^7 = 3x^2 + 1
> x = 1.2919315501289

Calista I've noted as a sensi tuning.

> It's not because it's not in familiar use yet that the word
> "fractals" would be incorrect ; after asking mathematicians who saw
> and heard my photosonic disks of those, the term "linear fractals"
> they said was correct -

What, precisely, is the linear fractal in question? What is self-similar, and how do you equate that to a real algebraic number?

If yes, there is no doubt these are fractals (all Pisot-
> Vijayaraghavan numbers used as generators for example)

A PV number gives a recurrence which is an exponential with damped oscillations dying away exponentially. Why is that a fractal?

> > These are not terms I can recall being bandied about.
>
> New word, or expression, for me, can't find it in my dictionary. It
> means what exactly and in this context ?

bandy - transitive verb
To give and receive (words, for example); exchange: The old friends bandied compliments when they met.

🔗Jacques Dudon <fotosonix@...>

Kraig wrote :
> --- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
>> By the way, 40/31 = 1.2903225806452 makes it also for Clyde then !
>
>>
> 40/31 was the first two numbers i was using for my seed as this is > what appears in the scale tree above this proportion . Always the > first place i look .

Considering that 31/24 is right in the middle of Clyde's close neighbour, the Sensi temperament, this suggests an easy way to draw scales for these temperaments on a lattice :
Take a 5-limit lattice and define a paralelogram (or rectangle, or square) with 1/1, 5/4, 15/8 and 3/2 as sommets.
then find the centre of the parallelogram and draw a point : that's where the "31" hides, and you may be able to view your favorite scales from a new point of view now, using the Sensi mapping :
<0, 7, 9, 13, -15, 10]
or alternatively Clyde :
<0, -12, -10, -25]
The generator vector 31/24 is oblique and goes from 3/2 to 31/16.
Since 40/31 is a translation of the same, you also assume you dissolved 961/960.

In a even more microtonal vision I imagine 40/31 and 31/24 could be the two generators in the x and y directions of a 2D lattice where you would find 3/2 after 7 or -12 of those, 5/4 after 9 or -10, 7/4 after 13 or -25, using each time specific numbers of both generators, along zig-zaging stairways of calculated inclinations (and where ten 40/31 + nine 31/24 would reproduce the paving with a minimal schisma). With many 5/3 as well, and plenty of 961/960 commas this time, in the other oblique direction.
Wouldn't it be a great planar temperament ?
It's like if I hear the sound of these commas descending now...

Sweet dreams everybody,
- - - - - - -
Jacques

🔗Jacques Dudon <fotosonix@...>

Gene wrote :
> Sometimes I couldn't find one. For instance for 7-limit Rodan, I've > now listed
> 20x^2-36x+15 and for 11-limit Rodan, x^2+16x-31, and I hope Jacques > will give me
> an opinion on these.

This might be a case of a more difficult temperament to reach from equations, and if we look at the mapping it explains why :
[<1, 1, -1, 3, 6, 8], <0, 3, 17, -1, -13, -22]>
The generator is very close to a simple ratio (8/7 or 7/4) but must also be really tuned specifically in order to reach h.5 and h.11 or even h.13 after high numbers of reiterations.
Of these sequences you find, the 1st one, with 20 for the first coefficient, will make it difficult to create series.
The second one looks better, but is not converging and does not allows series either.
Anyway it will be always difficult to make small numbers series with generators so close to a simple ratio.
The acoustic qualities of the proposition, and the precision needed then remain the only criterias.
Here are two suggestions :

Rasotpa : x^6 = 8x^2 + 4
x = 1.7468183196
(965.67547973352 c.)
(slightly more away, but with a pertinent -c in the context, of a 64 - 55 = 9 type) ;

Perhaps a more precise Rodan temperament would be "Gatetone" :
12x^3 = 7x + 10
x = 1.145037725330437
(234.47415755052 c.)

I am sure there must be others, but this temperament has some character... quite interesting I think, I like to hear !

> (Jacques) :
> > What is specific of the Clyde temperament exactly and in what it
> > differs from his neighbours Sensipent, Sensisept or Sensi ?
>
> http://xenharmonic.wikispaces.com/Kleismic+family

I had not seen this page. I found the mappings, thanks.

> > All this zone (lets say 1.290 - 1.293) has an incredible > diversity of
> > recurrent sequences with various acoustic qualities and I would like
> > to understand better the temperaments it may apply for.
>
> Sounds worthy of more attention.
>
> > or Calista :
> > x^7 = 3x^2 + 1
> > x = 1.2919315501289
>
> Calista I've noted as a sensi tuning.
>
> > It's not because it's not in familiar use yet that the word
> > "fractals" would be incorrect ; after asking mathematicians who saw
> > and heard my photosonic disks of those, the term "linear fractals"
> > they said was correct -
>
> What, precisely, is the linear fractal in question? What is self-> similar, and
> how do you equate that to a real algebraic number?

For the resolved ones, self-similar are the semi-periodic patterns of the fractal waveforms at all degrees of deployment and therefore the frequencies generated, in a completely audible and analysable manner.
I usually start from algebraic number to make those, but the other way would not be impossible, that's actually what we can do when looking at the phyllotaxies of many plants such as a sunflower heart etc., when we find these are based on Phi.
That's also what I do when I use specific fractal waveforms as horagrams for Meta-temperaments (like I did for example for Miracle).
I will upload visual examples of such fractals some day and we can discuss it then.

> If yes, there is no doubt these are fractals (all Pisot-
> Vijayaraghavan numbers used as generators for example)
>
> A PV number gives a recurrence which is an exponential with damped > oscillations
> dying away exponentially. Why is that a fractal ?

There may be several definitions of a fractal. What is certain, they can be resolved geometrically as fractal waveforms, that means auto-generating their own patterns forever with no accident, and only certain ratios have this property.
Why P-V numbers act that way is certainly an interesting mathematical question. From my observation it is related with the property of those numbers to have their powers getting closer and closer to whole numbers when increasing, as with Phi^n.

- - - - - - - -
Jacques

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> Considering that 31/24 is right in the middle of Clyde's close
> neighbour, the Sensi temperament, this suggests an easy way to draw
> scales for these temperaments on a lattice :

31/24 is very close to the 24/65 generator of sensi, making it an excellent 5-limit generator but not so good in the 7-limit.

With many 5/3 as well, and plenty of 961/960 commas this
> time, in the other oblique direction.
> Wouldn't it be a great planar temperament ?

It would be fine, except it raises the question of why you went all the way to the 31-limit in the first place, and it wouldn't be planar unless you tossed in more commas. Of course we could start with commas defining sensi, such as 126/125 and 245/243, and since 961/960 uses only 2, 3, 5, and 31 in its factorization, you could stop with a {2,3,5,7,31} system. Or even more accurately, {2,3,5,31}, with 78732/78125 and 961/960 as commas. Of course, that still isn't planar.

We have a mapping

[<1 6 8 0 0 0 0 0 0 0 10|, <0 -7 -9 0 0 0 0 0 0 0 -8|]

in case anyone is interested. Lots of opportunity to slip 31 in among the 5-limit in case for some reason that is what you wanted.

🔗Jacques Dudon <fotosonix@...>

Jacques wrote :

> Perhaps a more precise Rodan temperament would be "Gatetone" :
> 12x^3 = 7x + 10
> x = 1.145037725330437
> (234.47415755052 c.)

Last thought of the night and perhaps now the ultimate solution,
now based on a special -c property of the 9/8 tone and equal-beating with the 8/7 :
4x^6 = 7x + 1
x = 1.1450375644954844
234.47391437688 c.
This is only 2/10000 of a cent less than the precedent, as a consequence if necessary the precedent (12x^3 = 7x + 10) will help to construct the seed for this one, of higher degree but who generates simpler series. Both are convergent.
I suggest the name "Gatetone" for 4x^6 - 7x - 1 and changing to "Gatedone" for the other.
Here is a correct start and the 26 first notes :

151740
173748
198948
227803 (~h.3)
260843
298675
341944
391596
448392,25
513426
587892
673158,25
770791,51
882585,44
1010593,50
1157167,51
1325000
1517174,69 (~h.5)
1737222,39
1989185,09
2277691,52
2608041,88
2986305,70
3419432,86
3915379,51
4483256,43

For the dessert now this is another interesting Rodan, exploring the 11/10 -c :
64 - x^30 = x^13
x = 1.1450551129838795
234.5004465264 c.
The relatively correct solution suggests that, for temperaments having a minimum of 31 notes, Rodan proposes some 11/10 intervals with potential differentially-coherent qualities, which is a surprise for a temperament based on a 8/7 generator.
Tested on "Gatetone," 5.8443337236735 / 5.8164305762132 =
the difference tone of its 11/10s is only 8 cents off with 1/1 - not bad at all.
- - - - - - -
Jacques

🔗genewardsmith <genewardsmith@...>

🔗Jacques Dudon <fotosonix@...>

Gene wrote :

(Jacques) :
> > Last thought of the night and perhaps now the ultimate solution,
> > now based on a special -c property of the 9/8 tone and equal-beating
> > with the 8/7 :
> > 4x^6 = 7x + 1
> > x = 1.1450375644954844
> > 234.47391437688 c.
>
> Thanks; this makes a good 13-limit tuning. It might be noted that > the range from 1.14496 to 1.145 is of interest.

By hazard I have a less exciting 14x^3 = 7x + 13 >
x = 1.14498476578144,
but to find better I would need to know why do you say this zone is of interest, as it would make it worse for 5 and 11-limit than the above.
More generally, I will be happy to improve my list by finding algebraic solutions for any temperaments at my leasure times, if I can be provided with the specific mappings (and limits) of the temperament and approximative ratio ranges.
- - - - - - -
Jacques

(Jacques) :
> > Considering that 31/24 is right in the middle of Clyde's close
> > neighbour, the Sensi temperament, this suggests an easy way to draw
> > scales for these temperaments on a lattice :
>
> 31/24 is very close to the 24/65 generator of sensi, making it an > excellent 5-limit generator but not so good in the 7-limit.

Too bad, since 31 - 24 = 7 :-(
at least 40 - 31 = 9.

> With many 5/3 as well, and plenty of 961/960 commas this
> > time, in the other oblique direction.
> > Wouldn't it be a great planar temperament ?
>
> It would be fine, except it raises the question of why you went all > the way to the 31-limit in the first place, and it wouldn't be > planar unless you tossed in more commas. Of course we could start > with commas defining sensi, such as 126/125 and 245/243, and since > 961/960 uses only 2, 3, 5, and 31 in its factorization, you could > stop with a {2,3,5,7,31} system. Or even more accurately, > {2,3,5,31}, with 78732/78125 and 961/960 as commas. Of course, that > still isn't planar.
>
> We have a mapping
>
> [<1 6 8 0 0 0 0 0 0 0 10|, <0 -7 -9 0 0 0 0 0 0 0 -8|]
>
> in case anyone is interested. Lots of opportunity to slip 31 in > among the 5-limit in case for some reason that is what you wanted.

To draw superb lattices of such scales on squared paper, definitively !
But the worse thing, you will never guess, is the number of 961/960 that fit in a octave = 666 !!!!!
and I certainly don't want to be bewitched by the numerologists ! :D

(unless I temper it to 960/959 may be, since 665 of this one fits in the octave, and with 389 the best approximation of 3/2 ever offered by a ET)

🔗Chris Vaisvil <chrisvaisvil@...>

The vector notation is still opaque to me.

Does anyone know if I'd find a useful interpretation by doing in the following:
"1.14496 to 1.145 is of interest."

12 steps of 1.14496
17 steps of 1.14496

and so on?

Thanks,

Chris

On Wed, Nov 17, 2010 at 7:28 AM, Jacques Dudon <fotosonix@...> wrote:
>
>
>
> Gene wrote :
> (Jacques) :
>
> > Last thought of the night and perhaps now the ultimate solution,
> > now based on a special -c property of the 9/8 tone and equal-beating
> > with the 8/7 :
> > 4x^6 = 7x + 1
> > x = 1.1450375644954844
> > 234.47391437688 c.
>
> Thanks; this makes a good 13-limit tuning. It might be noted that the range from 1.14496 to 1.145 is of interest.
>

🔗jacques.dudon <fotosonix@...>

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> The vector notation is still opaque to me.
>
> Does anyone know if I'd find a useful interpretation by doing in the following:
> "1.14496 to 1.145 is of interest."
>
> 12 steps of 1.14496
> 17 steps of 1.14496
>
> and so on?
>
> Thanks,
>
> Chris

I'm not certain to understand the question, but yes you can interpret those as frequency ratios "steps". 17 of these quasi-septimal tones (which would require 18 notes) as you can check would lead you here to the major third of the temperament. This is why I said a slightly higher generator (like around 1,14505) would be more pertinent in 5-limit, as well as in 11-limit. But may be Gene wanted a lower range to optimise a 7-limit model.

>
> On Wed, Nov 17, 2010 at 7:28 AM, Jacques Dudon <fotosonix@...> wrote:
> >
> >
> >
> > Gene wrote :
> > (Jacques) :
> >
> > > Last thought of the night and perhaps now the ultimate solution,
> > > now based on a special -c property of the 9/8 tone and equal-beating
> > > with the 8/7 :
> > > 4x^6 = 7x + 1
> > > x = 1.1450375644954844
> > > 234.47391437688 c.
> >
> > Thanks; this makes a good 13-limit tuning. It might be noted that the range from 1.14496 to 1.145 is of interest.

🔗genewardsmith <genewardsmith@...>

🔗Brofessor <kraiggrady@...>

With any generator you can find all the MOS by using
the 1/f formula which you can find
http://anaphoria.com/wilsonintroMOS.html
under the section toward the end
FURTHER DEVELOPMENTS BY WILSON BEYOND THIS LETTER.
which gives me MOS at ..5,6,11,16,21,26,31 etc.
it seems to be between a few recurrent sequences above and below though but nothing right in their this exact

--- In tuning@yahoogroups.com, "jacques.dudon" <fotosonix@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@> wrote:
> >
> > The vector notation is still opaque to me.
> >
> > Does anyone know if I'd find a useful interpretation by doing in the following:
> > "1.14496 to 1.145 is of interest."
> >
> > 12 steps of 1.14496
> > 17 steps of 1.14496
> >
> > and so on?
> >
> > Thanks,
> >
> > Chris
>
> I'm not certain to understand the question, but yes you can interpret those as frequency ratios "steps". 17 of these quasi-septimal tones (which would require 18 notes) as you can check would lead you here to the major third of the temperament. This is why I said a slightly higher generator (like around 1,14505) would be more pertinent in 5-limit, as well as in 11-limit. But may be Gene wanted a lower range to optimise a 7-limit model.
>
> >
> > On Wed, Nov 17, 2010 at 7:28 AM, Jacques Dudon <fotosonix@> wrote:
> > >
> > >
> > >
> > > Gene wrote :
> > > (Jacques) :
> > >
> > > > Last thought of the night and perhaps now the ultimate solution,
> > > > now based on a special -c property of the 9/8 tone and equal-beating
> > > > with the 8/7 :
> > > > 4x^6 = 7x + 1
> > > > x = 1.1450375644954844
> > > > 234.47391437688 c.
> > >
> > > Thanks; this makes a good 13-limit tuning. It might be noted that the range from 1.14496 to 1.145 is of interest.
>

🔗genewardsmith <genewardsmith@...>

You have the right idea--check the tunings. The basis of my statement was the following data:

7 and 9 limit minimax 1.1449257694804816458

7 limit TE 1.1449997654313920128

7 limit least squares 1.1449955625235751975

9 limit least squares 1.1449872685843582319

11 limit minimax 1.1449535937083555327

11 limit TE 1.1450275218156028366

11 limit least squares 1.1450227861877296014

🔗Brofessor <kraiggrady@...>

this is close as far as MOS but diverges with your at 31 as your
continues to 36 and beyond
http://anaphoria.com/chain7.PDF <http://anaphoria.com/chain7.PDF>
--- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@...> wrote:
>
> With any generator you can find all the MOS by using
> the 1/f formula which you can find
> http://anaphoria.com/wilsonintroMOS.html
> under the section toward the end
> FURTHER DEVELOPMENTS BY WILSON BEYOND THIS LETTER.
> which gives me MOS at ..5,6,11,16,21,26,31 etc.
> it seems to be between a few recurrent sequences above and below
though but nothing right in their this exact
>
> --- In tuning@yahoogroups.com, "jacques.dudon" fotosonix@ wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@> wrote:
> > >
> > > The vector notation is still opaque to me.
> > >
> > > Does anyone know if I'd find a useful interpretation by doing in
the following:
> > > "1.14496 to 1.145 is of interest."
> > >
> > > 12 steps of 1.14496
> > > 17 steps of 1.14496
> > >
> > > and so on?
> > >
> > > Thanks,
> > >
> > > Chris
> >
> > I'm not certain to understand the question, but yes you can
interpret those as frequency ratios "steps". 17 of these quasi-septimal
tones (which would require 18 notes) as you can check would lead you
here to the major third of the temperament. This is why I said a
slightly higher generator (like around 1,14505) would be more pertinent
in 5-limit, as well as in 11-limit. But may be Gene wanted a lower range
to optimise a 7-limit model.
> >
> > >
> > > On Wed, Nov 17, 2010 at 7:28 AM, Jacques Dudon <fotosonix@> wrote:
> > > >
> > > >
> > > >
> > > > Gene wrote :
> > > > (Jacques) :
> > > >
> > > > > Last thought of the night and perhaps now the ultimate
solution,
> > > > > now based on a special -c property of the 9/8 tone and
equal-beating
> > > > > with the 8/7 :
> > > > > 4x^6 = 7x + 1
> > > > > x = 1.1450375644954844
> > > > > 234.47391437688 c.
> > > >
> > > > Thanks; this makes a good 13-limit tuning. It might be noted
that the range from 1.14496 to 1.145 is of interest.
> >
>

🔗jacques.dudon <fotosonix@...>

Really, this Rodan temperament is full of surprises ! found again two very close solutions (7/10000 of a cent) for what I think would be ideal solutions this time for the 7-limit version (making abstraction of 5 here) :
7x^4 = 14x - 4 x = 1.1449462741781
16x^3 = 7x + 16 x = 1.1449467513243
(234.336 c.)

Perhaps slightly passed the lower edge of what you suggested, but I think worth the trip considering specifically 3 and 7 acoustical interactions, equal-beating and tutti quanti.
Again to have two sequences with so close solutions is really a plus as they can potentially share series and qualities.

10509
12032
13776
15773
18059
etc.

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> Gene wrote :
>
> (Jacques) :
> > > Last thought of the night and perhaps now the ultimate solution,
> > > now based on a special -c property of the 9/8 tone and equal-beating
> > > with the 8/7 :
> > > 4x^6 = 7x + 1
> > > x = 1.1450375644954844
> > > 234.47391437688 c.
> >
> > Thanks; this makes a good 13-limit tuning. It might be noted that
> > the range from 1.14496 to 1.145 is of interest.
>

🔗jacques.dudon <fotosonix@...>

🔗genewardsmith <genewardsmith@...>

🔗jacques.dudon <fotosonix@...>

It is interesting that Erv uses here 384 as a reference Pitch. I found 6, 12 and octaves to be the most common pitch of my photosonic disks. It's funny he mentions Phi, Pi, 22/7 in here, and this article of Arnold Keyserling, whom I met in Paris in the seventies where he demonstrated his "Chakraphone" to me. I thought he had tuned it to 5-ET, but I may be wrong, anyway it was going chakra after chakra by octaves up. Now I think his use of the A E I O U code is probably inspired by the Kototama.

--- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@...> wrote:
>
> this is close as far as MOS but diverges with your at 31 as your
> continues to 36 and beyond
> http://anaphoria.com/chain7.PDF <http://anaphoria.com/chain7.PDF>
> --- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@> wrote:
> >
> > With any generator you can find all the MOS by using
> > the 1/f formula which you can find
> > http://anaphoria.com/wilsonintroMOS.html
> > under the section toward the end
> > FURTHER DEVELOPMENTS BY WILSON BEYOND THIS LETTER.
> > which gives me MOS at ..5,6,11,16,21,26,31 etc.
> > it seems to be between a few recurrent sequences above and below
> though but nothing right in their this exact
> >
> > --- In tuning@yahoogroups.com, "jacques.dudon" fotosonix@ wrote:
> > >
> > >
> > >
> > > --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@> wrote:
> > > >
> > > > The vector notation is still opaque to me.
> > > >
> > > > Does anyone know if I'd find a useful interpretation by doing in
> the following:
> > > > "1.14496 to 1.145 is of interest."
> > > >
> > > > 12 steps of 1.14496
> > > > 17 steps of 1.14496
> > > >
> > > > and so on?
> > > >
> > > > Thanks,
> > > >
> > > > Chris
> > >
> > > I'm not certain to understand the question, but yes you can
> interpret those as frequency ratios "steps". 17 of these quasi-septimal
> tones (which would require 18 notes) as you can check would lead you
> here to the major third of the temperament. This is why I said a
> slightly higher generator (like around 1,14505) would be more pertinent
> in 5-limit, as well as in 11-limit. But may be Gene wanted a lower range
> to optimise a 7-limit model.
> > >
> > > >
> > > > On Wed, Nov 17, 2010 at 7:28 AM, Jacques Dudon <fotosonix@> wrote:
> > > > >
> > > > >
> > > > >
> > > > > Gene wrote :
> > > > > (Jacques) :
> > > > >
> > > > > > Last thought of the night and perhaps now the ultimate
> solution,
> > > > > > now based on a special -c property of the 9/8 tone and
> equal-beating
> > > > > > with the 8/7 :
> > > > > > 4x^6 = 7x + 1
> > > > > > x = 1.1450375644954844
> > > > > > 234.47391437688 c.
> > > > >
> > > > > Thanks; this makes a good 13-limit tuning. It might be noted
> that the range from 1.14496 to 1.145 is of interest.
> > >
> >
>

Semimarvelous Blue Drawf

🔗christopherv <chrisvaisvil@...>

🔗genewardsmith <genewardsmith@...>

🔗cameron <misterbobro@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗genewardsmith <genewardsmith@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Brofessor <kraiggrady@...>

🔗genewardsmith <genewardsmith@...>

🔗Brofessor <kraiggrady@...>

🔗Brofessor <kraiggrady@...>

🔗Carl Lumma <carl@...>

🔗Carl Lumma <carl@...>

🔗Herman Miller <hmiller@...>

🔗Brofessor <kraiggrady@...>

🔗Carl Lumma <carl@...>

🔗Brofessor <kraiggrady@...>

🔗Brofessor <kraiggrady@...>

🔗Brofessor <kraiggrady@...>

🔗Carl Lumma <carl@...>

🔗genewardsmith <genewardsmith@...>

🔗Brofessor <kraiggrady@...>

🔗Brofessor <kraiggrady@...>

🔗genewardsmith <genewardsmith@...>

🔗Carl Lumma <carl@...>

🔗Ozan Yarman <ozanyarman@...>

🔗Brofessor <kraiggrady@...>

🔗Carl Lumma <carl@...>

🔗Brofessor <kraiggrady@...>

🔗Carl Lumma <carl@...>

🔗Brofessor <kraiggrady@...>

🔗Carl Lumma <carl@...>

🔗Brofessor <kraiggrady@...>

🔗genewardsmith <genewardsmith@...>

🔗Brofessor <kraiggrady@...>

🔗Brofessor <kraiggrady@...>

🔗Jacques Dudon <fotosonix@...>

🔗genewardsmith <genewardsmith@...>

🔗genewardsmith <genewardsmith@...>

🔗Brofessor <kraiggrady@...>

🔗genewardsmith <genewardsmith@...>

🔗Jacques Dudon <fotosonix@...>

🔗genewardsmith <genewardsmith@...>

🔗Brofessor <kraiggrady@...>

🔗Brofessor <kraiggrady@...>

🔗shaahin <acousticsoftombak@...>

🔗genewardsmith <genewardsmith@...>

🔗Jacques Dudon <fotosonix@...>

🔗Jacques Dudon <fotosonix@...>

🔗genewardsmith <genewardsmith@...>

🔗Jacques Dudon <fotosonix@...>

🔗genewardsmith <genewardsmith@...>

🔗Jacques Dudon <fotosonix@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗jacques.dudon <fotosonix@...>

🔗genewardsmith <genewardsmith@...>

🔗Brofessor <kraiggrady@...>

🔗genewardsmith <genewardsmith@...>

🔗Brofessor <kraiggrady@...>

🔗jacques.dudon <fotosonix@...>

🔗jacques.dudon <fotosonix@...>

🔗genewardsmith <genewardsmith@...>

🔗jacques.dudon <fotosonix@...>