Yahoo Tuning Groups Ultimate Backup tuning a rational decision

--- In tuning@yahoogroups.com, "David Beardsley" <db@b...> wrote:
> ----- Original Message -----
> From: "Joseph Pehrson" <jpehrson@r...>
>
> > I decided that the _Blacklight_ description should mention
Blackjack
> > as a 21-note *NEAR* Just Intonation scale...
>
> So if Blackjack is near Just, what JI intervals
> is it close to?

intervals:

the 21-tone blackjack scale has 41 different interval sizes, 27 of
which are within 4 cents of the partchian 11-limit consonances (there
are 29 of the latter, but 10:11 and 11:20 are not found in blackjack).

i assigned ratios and showed cents deviations accordingly for these
27 interval types, and more arbitrarily for the other 14 types, in
the following chart of all intervals in blackjack:

/tuning/files/monz/blackjackintmat3.xls

this categorizes all intervals by color according to the partchian
consonance scale -- and note how all the colored intervals deviate by
a cent, two cents, or at the very most four cents (in the case of 8:9
and 9:16) from the simple ratios they represent.

chords:

the potential of the scale to emulate ji sounds goes beyond intervals
and extends quite well to chords of 3, 4, or more notes. for example
there are eight near-4:5:6:7 chords, and eight near-1/(4:5:6:7)
chords. all six intervals in each of these chords are within 3 cents
of just intonation.

joseph and i have been discussing several of the other kinds of
chords in blackjack here recently, so you could look over those
posts . . . and of course there were a large number of posts about
the chords, hexanies, etc. when blackjack was first
discovered/invented.

pitches:

if one were to arbitrarily "break" many of the consonant chords
(actually, the errors only go up to 7.7 cents), one could render the
rest in true just intonation, and thus express the entire set of 21
*pitches* as a set of ratios. i've created lattices (with pitch
ratios shown) for various possibilities, showing the "broken"
intervals in gray, and the pure 7-limit consonances in colors:

/tuning/files/perlich/scales/blackjust1.g
if

/tuning/files/perlich/scales/blackjust2.g
if

/tuning/files/perlich/scales/blackjust3.g
if

/tuning/files/perlich/scales/blackjust4.g
if -- this one shows exactly what interval each of the broken
consonances is detuned by: 224:225 is 7.7 cents, 2400:2401 is 0.7
cents, and 16807:16875 is 7.0 cents . . .

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Afmmjr@aol.com

🔗David Beardsley <db@biink.com>

----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>

> pitches:
>
> if one were to arbitrarily "break" many of the consonant chords
> (actually, the errors only go up to 7.7 cents), one could render the
> rest in true just intonation, and thus express the entire set of 21
> *pitches* as a set of ratios. i've created lattices (with pitch
> ratios shown) for various possibilities, showing the "broken"
> intervals in gray, and the pure 7-limit consonances in colors:
>
> /tuning/files/perlich/scales/blackjust1.g
> if

/tuning/files/perlich/scales/blackjust1.gif

> /tuning/files/perlich/scales/blackjust2.g
> if

/tuning/files/perlich/scales/blackjust2.gif

> /tuning/files/perlich/scales/blackjust3.g
> if

/tuning/files/perlich/scales/blackjust3.gif

> /tuning/files/perlich/scales/blackjust4.g
> if -- this one shows exactly what interval each of the broken
> consonances is detuned by: 224:225 is 7.7 cents, 2400:2401 is 0.7
> cents, and 16807:16875 is 7.0 cents . . .

/tuning/files/perlich/scales/blackjust4.gif

Sometimes it helps to enclose the link with < and >, and the link
will usually survive the wraparound. I hope these work!

Thanks for the links. The lattices are a bit too much for me.
I think what really interests me is pitch. What are the near Just ratios?
Two years ago I set up a spreadsheet to investigate this and then
lost interest. I think I was offered a gig and decided to
do my first show with the JI guitar - I need to get my act together!

On the spread sheet I had started to look for near Just ratios.
I see I got as far as 3/2, and then there are no entries above 3/2.
It was an analysis of 72-tet, Blackjack was part of it.
I'm curious to see what others came up with, I think my
analysis is flawed, trying to shoe horn the 13th harmonic into 72tet,

Any thoughts?

* David Beardsley
* microtonal guitar
* http://biink.com/db

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "David Beardsley" <db@b...> wrote:
> ----- Original Message -----
> From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
>
>
> > pitches:
> >
> > if one were to arbitrarily "break" many of the consonant chords
> > (actually, the errors only go up to 7.7 cents), one could render
the
> > rest in true just intonation, and thus express the entire set of
21
> > *pitches* as a set of ratios. i've created lattices (with pitch
> > ratios shown) for various possibilities, showing the "broken"
> > intervals in gray, and the pure 7-limit consonances in colors:
> >
> >
/tuning/files/perlich/scales/blackjust1.
g
> > if
>
>
/tuning/files/perlich/scales/blackjust1.
gif
>
>
> >
/tuning/files/perlich/scales/blackjust2.
g
> > if
>
>
/tuning/files/perlich/scales/blackjust2.
gif
>
>
> >
/tuning/files/perlich/scales/blackjust3.
g
> > if
>
>
/tuning/files/perlich/scales/blackjust3.
gif
>
> >
/tuning/files/perlich/scales/blackjust4.
g
> > if -- this one shows exactly what interval each of the broken
> > consonances is detuned by: 224:225 is 7.7 cents, 2400:2401 is 0.7
> > cents, and 16807:16875 is 7.0 cents . . .
>
>
/tuning/files/perlich/scales/blackjust4.
gif
>
> Sometimes it helps to enclose the link with < and >, and the link
> will usually survive the wraparound. I hope these work!
>
> Thanks for the links. The lattices are a bit too much for me.
> I think what really interests me is pitch. What are the near Just
> ratios?

if it's too much to read them off the lattices above, i'll tabulate
them for you below. note that the choice of which pitch to treat
as 1/1 is somewhat arbitrary. in the last column i read the intervals
from C in the interval matrix (no particular reason) . . .

pitch blackjust1 blackjust2 blackjust3 blackjust4 intmatfrC
B[ ... 128/105 ... 80/49 ..... 3/2 ...... 1/1 ..... 11/6
Bv ..... 5/4 ...... 5/3 ..... 75/49 .... 50/49 .... 15/8 or 28/15
C< .... 21/16 ..... 7/4 ...... 8/5 ..... 15/14 .... 49/25 or 96/49
C ...... 4/3 ..... 25/14 .... 49/30 .... 49/45 ..... 1/1
C#v .... 7/5 ..... 15/8 ..... 12/7 ...... 8/7 ..... 21/20 or 22/21
Db^ ... 10/7 ..... 40/21 ..... 7/4 ...... 7/6 ..... 15/14 or 16/15
D ...... 3/2 ...... 1/1 ..... 90/49 .... 60/49 ..... 9/8
D> .... 32/21 .... 50/49 .... 28/15 ..... 5/4 ...... 8/7
Eb^ .... 8/5 ..... 15/14 .... 49/25 .... 98/75 ..... 6/5
E[ ... 105/64 ... 160/147 .... 1/1 ...... 4/3 ..... 11/9
E> .... 12/7 ...... 8/7 ..... 21/20 ..... 7/5 ...... 9/7
F< ..... 7/4 ...... 7/6 ..... 15/14 .... 10/7 ..... 55/42 or 21/16
F] .... 64/35 .... 60/49 .... 28/25 ..... 3/2 ..... 11/8
F#v ... 15/8 ...... 5/4 ...... 8/7 ..... 75/49 ..... 7/5
G< .... 63/32 .... 21/16 ..... 6/5 ...... 8/5 ..... 22/15 or 72/49
G ...... 1/1 ...... 4/3 ..... 60/49 .... 49/30 ..... 3/2
G#v ... 21/20 ..... 7/5 ...... 9/7 ..... 12/7 ..... 11/7
Ab^ ... 15/14 .... 10/7 ..... 98/75 ..... 7/4 ...... 8/5
A ...... 9/8 ...... 3/2 ..... 48/35 .... 90/49 .... 27/16 or 42/25
A> ..... 8/7 ..... 32/21 ..... 7/5 ..... 28/15 .... 12/7
Bb^ .... 6/5 ...... 8/5 ..... 72/49 .... 49/25 ..... 9/5

🔗Graham Breed <graham@microtonal.co.uk>

David Beardsley wrote:

> Sometimes it helps to enclose the link with < and >, and the link
> will usually survive the wraparound. I hope these work!

I've found it doesn't help with wraparound, and in some cases the trailing > gets included in the URL, so it's better not to bother.

> Thanks for the links. The lattices are a bit too much for me.
> I think what really interests me is pitch. What are the near Just ratios?
> Two years ago I set up a spreadsheet to investigate this and then
> lost interest. I think I was offered a gig and decided to
> do my first show with the JI guitar - I need to get my act together!

It's sounds like you had your priorities straight!

> On the spread sheet I had started to look for near Just ratios.
> I see I got as far as 3/2, and then there are no entries above 3/2.
> It was an analysis of 72-tet, Blackjack was part of it.
> I'm curious to see what others came up with, I think my
> analysis is flawed, trying to shoe horn the 13th harmonic into 72tet,

To find linear temperaments (and equal temperaments as a first step) checking closest approximations to a set of consonances, typically odd limits, see:

http://x31eq.com/temper.html

and you can run the scripts online

http://x31eq.com/temper

where you can also find an explanation of the algorithm. I need to clean up the output, and it may be possible to speed it up (anybody using gcc as a cross compiler?)

All this may or may not relate to what you're interested in ;-)

Graham

🔗David Beardsley <db@biink.com>

----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>

> --- In tuning@yahoogroups.com, "David Beardsley" <db@b...> wrote:
> > ----- Original Message -----

> > The lattices are a bit too much for me.
> > I think what really interests me is pitch. What are the near Just
> > ratios?
>
> if it's too much to read them off the lattices above, i'll tabulate
> them for you below. note that the choice of which pitch to treat
> as 1/1 is somewhat arbitrary. in the last column i read the intervals
> from C in the interval matrix (no particular reason) . . .

Thanks. Now I see what you're getting at with the other charts:
Use the Blackjack scale but modulate around the 72tet keys?

This chart that I just snipped looks like plans for using Blackjack
on a guitar, checking out what the possibilities are for tuning
each string....maybe I'll try tie-on-frets on the fretless
to try it out...or maybe I'll buy a graphite fretless neck and stick
it on the Mexican/Roland strat. I don't think I want to give
up the G&L fretless for this. Hmmm.....

I'm more interested in a JI guitar with the frets going straight
across the neck these days. The Catler guitar, no matter how much
I enjoy exploring it, is a closed system. Maybe I can lower the
3/2 strings to 4/3, but 4/3 is part of the original tuning anyway.

* David Beardsley
* microtonal guitar
* http://biink.com/db

🔗monz <monz@attglobal.net>

hi Dave,

> From: "David Beardsley" <db@biink.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, July 04, 2003 5:09 PM
> Subject: Re: [tuning] Re: a rational decision
>
>
> ----- Original Message -----
> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
>
> > --- In tuning@yahoogroups.com, "David Beardsley" <db@b...> wrote:
> > > ----- Original Message -----
>
> > > The lattices are a bit too much for me.
> > > I think what really interests me is pitch. What are the near Just
> > > ratios?
> >
> > if it's too much to read them off the lattices above, i'll tabulate
> > them for you below. note that the choice of which pitch to treat
> > as 1/1 is somewhat arbitrary. in the last column i read the intervals
> > from C in the interval matrix (no particular reason) . . .
>
> <chart snip>
>
> Thanks. Now I see what you're getting at with the other charts:
> Use the Blackjack scale but modulate around the 72tet keys?
>
> This chart that I just snipped looks like plans for using Blackjack
> on a guitar, checking out what the possibilities are for tuning
> each string....maybe I'll try tie-on-frets on the fretless
> to try it out...or maybe I'll buy a graphite fretless neck and stick
> it on the Mexican/Roland strat. I don't think I want to give
> up the G&L fretless for this. Hmmm.....
>
> I'm more interested in a JI guitar with the frets going straight
> across the neck these days. The Catler guitar, no matter how much
> I enjoy exploring it, is a closed system. Maybe I can lower the
> 3/2 strings to 4/3, but 4/3 is part of the original tuning anyway.

it sounds like you might like to try something
like Robin Perry's "just-about intonation" guitar.
that had straight frets and gave a great approximation
to lots of JI ratios.

/tuning/topicId_14767.html#14767

-monz

🔗David Beardsley <db@biink.com>

----- Original Message -----
From: "Graham Breed" <graham@microtonal.co.uk>

> David Beardsley wrote:
>
> > Sometimes it helps to enclose the link with < and >, and the link
> > will usually survive the wraparound. I hope these work!
>
> I've found it doesn't help with wraparound, and in some cases the
> trailing > gets included in the URL, so it's better not to bother.

It usually works for me, except on some REALLY long URLs.

> > Thanks for the links. The lattices are a bit too much for me.
> > I think what really interests me is pitch. What are the near Just
ratios?
> > Two years ago I set up a spreadsheet to investigate this and then
> > lost interest. I think I was offered a gig and decided to
> > do my first show with the JI guitar - I need to get my act together!
>
> It's sounds like you had your priorities straight!

I'm still learning Catler's tuning system, but things seem to
have worked out in some ways: the Kyle Gann review,
the LA Times review from playing LA Microfest this
Spring, more gigs - if I could get my act together and record a CD,
I could probably get some paying gigs.

Now I'm planning another JI guitar, so I'm looking at
what I want to do for the next guitar(s). I have two or
three other tunings I'm thinking about.

> > On the spread sheet I had started to look for near Just ratios.
> > I see I got as far as 3/2, and then there are no entries above 3/2.
> > It was an analysis of 72-tet, Blackjack was part of it.
> > I'm curious to see what others came up with, I think my
> > analysis is flawed, trying to shoe horn the 13th harmonic into 72tet,
>
> To find linear temperaments (and equal temperaments as a first step)
> checking closest approximations to a set of consonances, typically odd
> limits, see:
>
> http://x31eq.com/temper.html
>
> and you can run the scripts online
>
> http://x31eq.com/temper
>
> where you can also find an explanation of the algorithm. I need to
> clean up the output, and it may be possible to speed it up (anybody
> using gcc as a cross compiler?)
>
> All this may or may not relate to what you're interested in ;-)

I'm more interested in Just, but I'll take a look at those examples.

Thanks.

* David Beardsley
* microtonal guitar
* http://biink.com/db

🔗David Beardsley <db@biink.com>

🔗monz <monz@attglobal.net>

hi Dave,

> From: "David Beardsley" <db@biink.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, July 04, 2003 5:53 PM
> Subject: Re: [tuning] Re: a rational decision
>
>
> Thanks Joe. I'll look at it closer in a day or two or in the morning.
>
> Right now I'm practicing, well ...playing. I think the blues
> I'm playing is a Coltrane tune, more research for this weekend.
> Bending for that 11/9, it's so sweet...

since you just made another comment about
preferring JI, and you haven't read the link
i sent yet, i thought maybe i should point out
now that Robin's intention on his "just-about"
guitar was essentially to get JI, within the
boundaries of "normal human tuning error",
which is generally assumed to be around
5 cents, or 240edo.

that, and your preference for straight frets,
is why i thought you'd be especially interested.

Graham, Dave, paul, et al: does this make
it a microtemperament? or exactly what
family of temperaments is it?

-monz

🔗Afmmjr@aol.com

🔗monz <monz@attglobal.net>

hi Johnny and Dave,

> From: <Afmmjr@aol.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, July 04, 2003 8:18 PM
> Subject: Re: [tuning] Re: a rational decision
>
>
> In a message dated 7/4/03 11:07:03 PM Eastern Daylight Time,
> monz@attglobal.net writes:
>
>
> > > Bending for that 11/9, it's so sweet...
> >
> > Robin's intention on his "just-about"
> > guitar was essentially to get JI, within the
> > boundaries of "normal human tuning error",
> > which is generally assumed to be around
> > 5 cents, or 240edo.
> >
> > that, and your preference for straight frets,
> > is why i thought you'd be especially interested.
> >
>
> But would a 5-cent flat, or 5-cent sharp 11/9 be
> as sweet as to transfix David?
> I suspect not.

yes, Johnny, actually, you're right.

i'm very familiar with Dave's music, and he
*wouldn't* be transfixed by a near-JI ratio
that's that far off.

in the type of extremely slow-moving music
he creates, an interval like that would beat ...
very slowly, to be sure, but still very perceptible
in the time-frame of *his* music.

but still, for purposes of experimentation, i
thought the "just-about" was a great guitar,
and i still think Dave would enjoy playing
around with it. with its pseudo-12edo fret-space
sizes, it's a lot easier to play than, say,
my 31edo guitar.

-monz

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "David Beardsley" <db@b...> wrote:
> ----- Original Message -----
> From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
>
> > --- In tuning@yahoogroups.com, "David Beardsley" <db@b...> wrote:
> > > ----- Original Message -----
>
> > > The lattices are a bit too much for me.
> > > I think what really interests me is pitch. What are the near
Just
> > > ratios?
> >
> > if it's too much to read them off the lattices above, i'll
tabulate
> > them for you below. note that the choice of which pitch to treat
> > as 1/1 is somewhat arbitrary. in the last column i read the
intervals
> > from C in the interval matrix (no particular reason) . . .
>
> <chart snip>
>
> Thanks. Now I see what you're getting at with the other charts:
> Use the Blackjack scale but modulate around the 72tet keys?

no, it's all just 21 notes. modulating the scale around was never an
option for joseph, for whom the charts were originally created . . .

> This chart that I just snipped looks like plans for using Blackjack
> on a guitar, checking out what the possibilities are for tuning
> each string

it looks a bit that way, but i think maybe that's misleading you into
thinking that there are more than 21 pitches involved somehow? it's
all the same fixed-pitch 21-tone scale. the pitches are named (using
our ascii 72-equal notation) in the leftmost column.

>....maybe I'll try tie-on-frets on the fretless
> to try it out...

dave keenan actually developed a blackjack fretting where the frets go
straight across, but the open string tuning is quite unconventional. i
designed a canasta fretboard (which includes 11 transpositions of
blackjack) where the open string tuning is exactly the conventional
12-equal one. the frets don't go straight across, but it's what i'd
use for blackjack.

> I'm more interested in a JI guitar with the frets going straight
> across the neck these days.

dave keenan and i have done a lot of work in this area as well,
including off the lists.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗David Beardsley <db@biink.com>

----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>

> --- In tuning@yahoogroups.com, "David Beardsley" <db@b...> wrote:

> > Thanks. Now I see what you're getting at with the other charts:
> > Use the Blackjack scale but modulate around the 72tet keys?
>
> no, it's all just 21 notes. modulating the scale around was never an
> option for joseph, for whom the charts were originally created . . .
>
> > This chart that I just snipped looks like plans for using Blackjack
> > on a guitar, checking out what the possibilities are for tuning
> > each string
>
> it looks a bit that way, but i think maybe that's misleading you into
> thinking that there are more than 21 pitches involved somehow? it's
> all the same fixed-pitch 21-tone scale. the pitches are named (using
> our ascii 72-equal notation) in the leftmost column.

Ah...I got it.

> >....maybe I'll try tie-on-frets on the fretless
> > to try it out...
>
> dave keenan actually developed a blackjack fretting where the frets go
> straight across, but the open string tuning is quite unconventional. i

I remember this, strings tuned in thirds.

> designed a canasta fretboard (which includes 11 transpositions of
> blackjack) where the open string tuning is exactly the conventional
> 12-equal one. the frets don't go straight across, but it's what i'd
> use for blackjack.

That would work.

> > I'm more interested in a JI guitar with the frets going straight
> > across the neck these days.
>
> dave keenan and i have done a lot of work in this area as well,
> including off the lists.

Any thoughts on open string tunings for a JI guitar? One solution
I'm looking forward to trying is something like C G C G C D or
C G C G C E. Maybe a sort of a baritone G C G C G C?

* David Beardsley
* microtonal guitar
* http://biink.com/db

🔗David Beardsley <db@biink.com>

----- Original Message -----
From: "monz" <monz@attglobal.net>

> hi Johnny and Dave,
>
> > From: <Afmmjr@aol.com>
> >
> > In a message dated 7/4/03 11:07:03 PM Eastern Daylight Time,
> > monz@attglobal.net writes:
> >
> >
> > > > Bending for that 11/9, it's so sweet...
> > >
> > > Robin's intention on his "just-about"
> > > guitar was essentially to get JI, within the
> > > boundaries of "normal human tuning error",
> > > which is generally assumed to be around
> > > 5 cents, or 240edo.
> > >
> > > that, and your preference for straight frets,
> > > is why i thought you'd be especially interested.
> > >
> >
> > But would a 5-cent flat, or 5-cent sharp 11/9 be
> > as sweet as to transfix David?
> > I suspect not.
>
> yes, Johnny, actually, you're right.

Totally!

> i'm very familiar with Dave's music, and he
> *wouldn't* be transfixed by a near-JI ratio
> that's that far off.
>
> in the type of extremely slow-moving music
> he creates, an interval like that would beat ...
> very slowly, to be sure, but still very perceptible
> in the time-frame of *his* music.

In this case, I was playing a 12tet guitar. I moved on
to the JI guitar later.

> but still, for purposes of experimentation, i
> thought the "just-about" was a great guitar,
> and i still think Dave would enjoy playing
> around with it. with its pseudo-12edo fret-space
> sizes, it's a lot easier to play than, say,
> my 31edo guitar.

I'll check it out, even though I'm not an ET sort of guy.
I did try Jon Catlers 31 tet a few times.Instead of trying
to find "pleasing" near JI intervals, I was checking out the
microtones, trying for microtonal clusters. Lots of fun.

* David Beardsley
* microtonal guitar
* http://biink.com/db

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@yahoogroups.com, Afmmjr@a... wrote:
> In a message dated 7/4/03 11:07:03 PM Eastern Daylight Time,
> monz@a... writes:
>
>
> > > Bending for that 11/9, it's so sweet...
> >
> > Robin's intention on his "just-about"
> > guitar was essentially to get JI, within the
> > boundaries of "normal human tuning error",
> > which is generally assumed to be around
> > 5 cents, or 240edo.
> >
> > that, and your preference for straight frets,
> > is why i thought you'd be especially interested.
> >
>
> But would a 5-cent flat, or 5-cent sharp 11/9 be as sweet as to
> transfix David?
> I suspect not.

I'm certain it wouldn't. I suspect it would need to be within half a
cent. But the point is, he is already bending for it. Surely it would
be more convenient if it was closer to start with.

If you want lots of straight frets with JI, you're gonna hafta be
willing to bend something. And lets face it, even the best guitars
typically have intonation errors of +-3 cents anyway, so you have to
bend any sustained JI harmony to really make it sing, even when the
guitar is fretted for strict JI. But you don't have to put up with 5
cent initial errors in the tuning either.

Folks on this list have discovered a number of linear
microtemperaments suitable for use on a guitar (or Chapman stick),
that have no error greater than 2.8 cents at the chosen harmonic
limit. (Monz, somewhat arbitrarily, we don't call anything with an
error over 2.8 cents a microtemperament). At the 7-limit, Blackjack is
a MOS of such a microtemperament -- the "miracle" microtemperament.
And miracle only goes up to 3.3 cent errors at the 11-limit (with the
minimax generator, not as a subset of 72-ET).

However it does have the drawback that its generator is not a fifth or
fourth so the optimum open tuning (for maximising straight frets) is
very different from standard open tuning. Paul and I have designed a
number of other straight-across frettings using other
microtemperaments that do allow the standard open tuning to be preserved.

It appears that Robin Perry has discovered the power of the secor too.
The secor being the name we've given to the 116.7 cent generator of
the "miracle" temperament, after its discoverer (in 1975) George Secor.

Robin has (understandably) avoided the 33 cent intervals of the strict
miracle MOSs, but notice how close 70, 85 and 115 cents are to 66.7
83.3 and 116.7, all of which occur in Blackjack. But I strongly
suspect it's more closely related to the "Byzantine" planar
microtemperament, which is an excellent choice when you want longer
chains of fifths (more diatonicity/tetrachordality) than are provided
in pure miracle. The two essentially become identical when taken out
to 31 notes (Canasta). But it might be interesting to look at smaller
"Byzantine" subsets on a guitar, such as the 19-note one derived from
Rami Vitale's tuning, or others that don't have 33 cent spacing.

As for open tunings for strict JI, I don't thing there is much doubt
that alternating fourths and fifths like CGCGCG gives the most
straight frets, but of course it has its own problems.

Of course there are always tradeoffs that have to be decided. In
addition to open tuning, intonation error and straight frets, some
others are minimum fret spacing, transposability, closed cycle of
fifths, tetrachordality. However, given someone's personal preferences
for these things we can usually come up with a microtempered fretting
that uses far fewer split frets than a strict JI fretting.

This is really a "Middle way". It is not ET and it is not strict JI,
but it's close enough to JI that it should only require slight
variations in finger pressure to make it JI on sustained harmonies and
you can't tell the difference on faster stuff.

-- Dave Keenan

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Dave Keenan" <D.KEENAN@U...> wrote:
> Folks on this list have discovered a number of linear
> microtemperaments suitable for use on a guitar (or Chapman stick),
> that have no error greater than 2.8 cents at the chosen harmonic
> limit. (Monz, somewhat arbitrarily, we don't call anything with an
> error over 2.8 cents a microtemperament). At the 7-limit, Blackjack
is
> a MOS of such a microtemperament -- the "miracle" microtemperament.

As I was telling Monz when remarking that his definition
of "microtemperament" needed fixing, I wouldn't call anything a micro
unless all errors were less than a cent. I've taken to calling
absurdly accurate temperaments "nanos", and we could call the range
from 1-3 cents "minis" or "millis" or something. However, I think a
better idea would be to steal the magnitude scale from astronomy,
which proceeds by way of the fifth root of 100. I don't see we need
as much range, and we could go for a finer resolution, such as 2:

0 th magnitude .25 to .5 cent max error
1st magnitude .5 to 1 cent max error
2nd magnitude 1 to 2 cent max error
3rd magnitude 2 to 4 cent max error
4th magnitude 4 to 8 cent max error
5th magnitude 8 to 16 cent max error
6th magnitude 16 to 32 cent max error

Of course this demotes miracle to a lowly 3rd magnitude.

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <D.KEENAN@U...> wrote:
> > Folks on this list have discovered a number of linear
> > microtemperaments suitable for use on a guitar (or Chapman stick),
> > that have no error greater than 2.8 cents at the chosen harmonic
> > limit. (Monz, somewhat arbitrarily, we don't call anything with an
> > error over 2.8 cents a microtemperament). At the 7-limit, Blackjack
> is
> > a MOS of such a microtemperament -- the "miracle" microtemperament.
>
> As I was telling Monz when remarking that his definition
> of "microtemperament" needed fixing, I wouldn't call anything a micro
> unless all errors were less than a cent. I've taken to calling
> absurdly accurate temperaments "nanos", and we could call the range
> from 1-3 cents "minis" or "millis" or something. However, I think a
> better idea would be to steal the magnitude scale from astronomy,
> which proceeds by way of the fifth root of 100. I don't see we need
> as much range, and we could go for a finer resolution, such as 2:
>
> 0 th magnitude .25 to .5 cent max error
> 1st magnitude .5 to 1 cent max error
> 2nd magnitude 1 to 2 cent max error
> 3rd magnitude 2 to 4 cent max error
> 4th magnitude 4 to 8 cent max error
> 5th magnitude 8 to 16 cent max error
> 6th magnitude 16 to 32 cent max error
>
> Of course this demotes miracle to a lowly 3rd magnitude.

Hi Gene,

I suppose this is all entirely defensible, but if you need so many
categories it is probably best to just say what you mean in cents, few
people are going to remember such a system.

Initially microtemperament meant "having less tha about half the
maximum error of 1/4-comma meantone" (i.e. 2.7 cents or less), but
later, after you and Graham Breed had unearthed lots of interesting
temperaments, I looked at the clustering of their maximum errors and
found there were several at or just below 2.8 cents having similar
complexity, and so, like the biological taxonomists attempting to
"carve nature at its joints", I thought that just above 2.8 cents
seemed like a good place to draw the line.

You might want to look more systematically at the clustering of the
errors of the best linear temperaments.

-- Dave Keenan

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning@yahoogroups.com, "David Beardsley" <db@b...> wrote:
> ----- Original Message -----
> From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
>
> > --- In tuning@yahoogroups.com, "David Beardsley" <db@b...> wrote:
>
> > > Thanks. Now I see what you're getting at with the other charts:
> > > Use the Blackjack scale but modulate around the 72tet keys?
> >
> > no, it's all just 21 notes. modulating the scale around was never
an
> > option for joseph, for whom the charts were originally created . .
.
> >
> > > This chart that I just snipped looks like plans for using
Blackjack
> > > on a guitar, checking out what the possibilities are for tuning
> > > each string
> >
> > it looks a bit that way, but i think maybe that's misleading you
into
> > thinking that there are more than 21 pitches involved somehow?
it's
> > all the same fixed-pitch 21-tone scale. the pitches are named
(using
> > our ascii 72-equal notation) in the leftmost column.
>
> Ah...I got it.
>
>
> > >....maybe I'll try tie-on-frets on the fretless
> > > to try it out...
> >
> > dave keenan actually developed a blackjack fretting where the
frets go
> > straight across, but the open string tuning is quite
unconventional. i
>
> I remember this, strings tuned in thirds.
>
>
> > designed a canasta fretboard (which includes 11 transpositions of
> > blackjack) where the open string tuning is exactly the
conventional
> > 12-equal one. the frets don't go straight across, but it's what
i'd
> > use for blackjack.
>
> That would work.

would you like to see what it looks like? kinda ugly diagram, but
basically 31 frets per octave, with most frets at one 72-equal
position on some strings and another 72-equal position on others (thus
split or bent frets are required). this is canasta, and is apparently
joe monzo's favorite scale:

http://sonic-arts.org/dict/canasta.htm

> > > I'm more interested in a JI guitar with the frets going straight
> > > across the neck these days.
> >
> > dave keenan and i have done a lot of work in this area as well,
> > including off the lists.
>
> Any thoughts on open string tunings for a JI guitar? One solution
> I'm looking forward to trying is something like C G C G C D or
> C G C G C E. Maybe a sort of a baritone G C G C G C?

dave keenan and i had a huge discussion about the "shrutar", a
22-tone, mostly-just intonation guitar, with the all the open strings
tuned to 1/1s and 3/2s. dave, what is the latest status of this
tuning? my acoustic duo, "IT", with garo papazian on doumbek, is ready
to go into the studio -- i'd like to microtonalize it first!

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning@yahoogroups.com, "David Beardsley" <db@b...> wrote:
> > > > I'm more interested in a JI guitar with the frets going straight
> > > > across the neck these days.
> > >
> > > dave keenan and i have done a lot of work in this area as well,
> > > including off the lists.
> >
> > Any thoughts on open string tunings for a JI guitar? One solution
> > I'm looking forward to trying is something like C G C G C D or
> > C G C G C E. Maybe a sort of a baritone G C G C G C?
>
> dave keenan and i had a huge discussion about the "shrutar", a
> 22-tone, mostly-just intonation guitar, with the all the open strings
> tuned to 1/1s and 3/2s. dave, what is the latest status of this
> tuning? my acoustic duo, "IT", with garo papazian on doumbek, is ready
> to go into the studio -- i'd like to microtonalize it first!

The "shrutar" design has been stable for over two years now. I think
it is far more likely to interest Dave Beardsley than is Blackjack or
Canasta.

Here is the scale.

! shrutar.scl
!
Paul Erlich's Shrutar tuning (from 9th fret) tempered in cooperation
with Dave Keenan
22
!
33/32
101.95500 ! 16/15 / (2048/2025)^(1/2)
12/11
9/8
262.02693 ! 7/6 / (896/891)^(1/2)
6/5
5/4
9/7
4/3
11/8
595.11186 ! 45/32 * (2048/2025)^(1/4)
643.83808 ! 16/11 / (896/891)^(1/2)
3/2
760.07192 ! 14/9 / (896/891)^(1/2)
808.79814 ! 8/5 / (2048/2025)^(1/4)
18/11
27/16
7/4
9/5
15/8
1141.88308 ! 27/14 * (896/891)^(1/2)
2/1

And here is the fretting.

! shrutarfrets.scl
!
Paul Erlich's Shrutar fretting, tempered in cooperation with Dave Keenan
22
!
58.11692355 ! 33/32 * (896/891)^(1/2)
106.8431431 ! 16/15 / (2048/2025)^(1/4)
12/11
9/8
7/6
6/5
5/4
439.9280756 ! 9/7 * (896/891)^(1/2)
4/3
11/8
600.0 ! 45/32 * (2048/2025)^(1/2)
16/11
3/2
760.0719244 ! 14/9 / (896/891)^(1/2)
8/5
5/3
12/7
16/9
11/6
1093.156857 ! 15/8 * (2048/2025)^(1/4)
1141.883076 ! 64/33 / (896/891)^(1/2)
2/1

All frets are straight across. The first scala file above may be read
as how the scale appears on the 3/2 strings, and the second how it
appears on the 1/1 strings. You will notice that most pitches are just
on both strings, some are just on one string and shifted 5 cents on
the other, and only 4 pitches are tempered on both strings. These four
are at the extremities of the JI lattice and allow them to wrap around
to the other side of the lattice and perform double duty without any
consonance having more than a 5 cent error. The central part of the
11-limit lattice is strictly just.

-- Dave Keenan

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning@yahoogroups.com, "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> > --- In tuning@yahoogroups.com, "David Beardsley" <db@b...> wrote:
> > > > > I'm more interested in a JI guitar with the frets going
straight
> > > > > across the neck these days.
> > > >
> > > > dave keenan and i have done a lot of work in this area as
well,
> > > > including off the lists.
> > >
> > > Any thoughts on open string tunings for a JI guitar? One
solution
> > > I'm looking forward to trying is something like C G C G C D or
> > > C G C G C E. Maybe a sort of a baritone G C G C G C?
> >
> > dave keenan and i had a huge discussion about the "shrutar", a
> > 22-tone, mostly-just intonation guitar, with the all the open
strings
> > tuned to 1/1s and 3/2s. dave, what is the latest status of this
> > tuning? my acoustic duo, "IT", with garo papazian on doumbek, is
ready
> > to go into the studio -- i'd like to microtonalize it first!
>
> The "shrutar" design has been stable for over two years now. I think
> it is far more likely to interest Dave Beardsley than is Blackjack
or
> Canasta.

maybe, maybe not. neither of us is inside his head. i think dave
beardsley can decide for himself.

thanks for the info, but i know that when we last left it, there was
still some room for modification, in my mind at least, i don't know
if i had expressed any open questions about it to you or not . . .

now wouldn't it be possible to develop a 31-tone analogue to shrutar?
that is, if you don't care about preserving any characteristics of
the 22-shruti indian scale?

>
> Here is the scale.
>
> ! shrutar.scl
> !
> Paul Erlich's Shrutar tuning (from 9th fret) tempered in cooperation
> with Dave Keenan
> 22
> !
> 33/32
> 101.95500 ! 16/15 / (2048/2025)^(1/2)
> 12/11
> 9/8
> 262.02693 ! 7/6 / (896/891)^(1/2)
> 6/5
> 5/4
> 9/7
> 4/3
> 11/8
> 595.11186 ! 45/32 * (2048/2025)^(1/4)
> 643.83808 ! 16/11 / (896/891)^(1/2)
> 3/2
> 760.07192 ! 14/9 / (896/891)^(1/2)
> 808.79814 ! 8/5 / (2048/2025)^(1/4)
> 18/11
> 27/16
> 7/4
> 9/5
> 15/8
> 1141.88308 ! 27/14 * (896/891)^(1/2)
> 2/1
>
> And here is the fretting.
>
> ! shrutarfrets.scl
> !
> Paul Erlich's Shrutar fretting, tempered in cooperation with Dave
Keenan
> 22
> !
> 58.11692355 ! 33/32 * (896/891)^(1/2)
> 106.8431431 ! 16/15 / (2048/2025)^(1/4)
> 12/11
> 9/8
> 7/6
> 6/5
> 5/4
> 439.9280756 ! 9/7 * (896/891)^(1/2)
> 4/3
> 11/8
> 600.0 ! 45/32 * (2048/2025)^(1/2)
> 16/11
> 3/2
> 760.0719244 ! 14/9 / (896/891)^(1/2)
> 8/5
> 5/3
> 12/7
> 16/9
> 11/6
> 1093.156857 ! 15/8 * (2048/2025)^(1/4)
> 1141.883076 ! 64/33 / (896/891)^(1/2)
> 2/1
>
> All frets are straight across. The first scala file above may be
read
> as how the scale appears on the 3/2 strings, and the second how it
> appears on the 1/1 strings. You will notice that most pitches are
just
> on both strings, some are just on one string and shifted 5 cents on
> the other, and only 4 pitches are tempered on both strings. These
four
> are at the extremities of the JI lattice and allow them to wrap
around
> to the other side of the lattice and perform double duty without any
> consonance having more than a 5 cent error. The central part of the
> 11-limit lattice is strictly just.
>
> -- Dave Keenan

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, Afmmjr@a... wrote:

/tuning/topicId_45243.html#45244

> In a message dated 7/2/03 10:46:54 PM Eastern Daylight Time,
jpehrson@r...
> writes:
>
>
> > I decided that the _Blacklight_ description should mention
Blackjack
> > as a 21-note *NEAR* Just Intonation scale...
> >
> > A rational decision, yes??
> >
> > J. Pehrson
> >
> >
> >
> >
>
> Yes, Joe. I could not in good conscience call Blackjack a "just
intonation
> scale" and we should be able to agree on program notes. No?
>
> best, Johnny

***That makes sense, Johnny... Sorry I haven't gotten back to you via
phone. I was away for the long weekend and am still catching up with
things...

best,

Joseph

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <D.KEENAN@U...> wrote:
> > The "shrutar" design has been stable for over two years now. I think
> > it is far more likely to interest Dave Beardsley than is Blackjack
> or
> > Canasta.
>
> maybe, maybe not. neither of us is inside his head. i think dave
> beardsley can decide for himself.

Well of course he can. I did start the sentence with "I think". Am I
not allowed to express an opinion?

> thanks for the info, but i know that when we last left it, there was
> still some room for modification, in my mind at least, i don't know
> if i had expressed any open questions about it to you or not . . .

As I remember, when we last left it, it had satisfied every
requirement you had ever raised, both harmonic and melodic, with
essentially no room remaining for modification. The final clincher was
minimising the difference between the major and minor whole-tones in
the Indian diatonics.

And we found that it happened to contain some Just Arabic scales too.

> now wouldn't it be possible to develop a 31-tone analogue to shrutar?
> that is, if you don't care about preserving any characteristics of
> the 22-shruti indian scale?

It depends how loose the analogy is. Based on the corresponding linear
temperament, I'm guessing that as soon as you go beyond 22 notes the
minimum fret spacing will nearly halve, and will stay that way until
46 notes. I think the shrutar tuning is just perfect at 22 notes.

Possibly the only thing detracting from its perfection is the fact
that no one has built one yet.

-- Dave Keenan

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning@yahoogroups.com, "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> > --- In tuning@yahoogroups.com, "Dave Keenan" <D.KEENAN@U...>
wrote:
> > > The "shrutar" design has been stable for over two years now. I
think
> > > it is far more likely to interest Dave Beardsley than is
Blackjack
> > or
> > > Canasta.
> >
> > maybe, maybe not. neither of us is inside his head. i think dave
> > beardsley can decide for himself.
>
> Well of course he can. I did start the sentence with "I think". Am
I
> not allowed to express an opinion?
>
> > thanks for the info, but i know that when we last left it, there
was
> > still some room for modification, in my mind at least, i don't
know
> > if i had expressed any open questions about it to you or not . .
.
>
> As I remember, when we last left it, it had satisfied every
> requirement you had ever raised, both harmonic and melodic, with
> essentially no room remaining for modification. The final clincher
was
> minimising the difference between the major and minor whole-tones
in
> the Indian diatonics.
>
> And we found that it happened to contain some Just Arabic scales
too.
>
> > now wouldn't it be possible to develop a 31-tone analogue to
shrutar?
> > that is, if you don't care about preserving any characteristics
of
> > the 22-shruti indian scale?
>
> It depends how loose the analogy is. Based on the corresponding
linear
> temperament, I'm guessing that as soon as you go beyond 22 notes
the
> minimum fret spacing will nearly halve, and will stay that way
until
> 46 notes. I think the shrutar tuning is just perfect at 22 notes.

the linear temperament would be one of the characteristics we don't
care about preserving.

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, "David Beardsley" <db@b...> wrote:

/tuning/topicId_45243.html#45254

> ----- Original Message -----
> From: "Joseph Pehrson" <jpehrson@r...>
>
> > I decided that the _Blacklight_ description should mention
Blackjack as a 21-note *NEAR* Just Intonation scale...
>
> So if Blackjack is near Just, what JI intervals
> is it close to?

***Hi David...

Whew... I just see that Paul Erlich has posted the solution to this
riddle. Thank god for that... I would be floundering in my
description (I'd get *some* of them right, though... :)

>
> BTW: I really enjoyed Blacklight. I only had time to
> listen to it twice, yesterday, but I'll investigate further
> this weekend.
>

***That's fantastic, David, and thank you. For myself, I only had a
chance to listen to it *once...* :)

J. Pehrson