Re:Designing a Canasta guitar

In-Reply-To: <3.0.6.32.20010604202252.00a70570@uq.net.au>
Dave Keenan wrote:

> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > Hey Dave, I think the fretting should be Canasta, since that doesn't
> > introduce any smaller intervals . . .
>
> Yeah. I guess so. And those smallest intervals are no smaller than they
> would be on a 36-EDO guitar. Eek.

I find the large-small pattern on my meantone guitar works very well for
9-limit chords. I pulled the bonus fret out because it was too difficult
to play F major. But perhaps I'm in a minority, because everybody else
seems to be rushing towards these complete middle-number ETs. But I'd
rather use the same kind of fretting, and retune the strings to make
different chords available.

If you're fretting to Canasta, why not make it 31-equal? You won't get
the guitar much more accurate anyway.

> > and there should be at least
> > one Blackjack scale playable in full on any string.
>
> That's easy. It only means that the open string "chord" must not span
> more
> than 10 generators in a chain. However, I think we should try to keep
> it to
> a span of 8 generators.

The tuning I started the thread with uses 9 generators doesn't it? That
was designed to address the practical issue of retuning without
restringing. I was also aiming to make it easy to get to chords, by
starting with a lattice structure that can become an 11-limit chord with
only a few notes having to change. Fretting to Blackjack should also
mean a lot of the notes you want will be there on the fretboard. I don't
see that it's important to be able to play a Blackjack scale. The value
of these scales was always in the intervals they contained.

You mean the same Blackjack should be playable on all strings?

So, with whatever criteria, we have a big list of tunings. I'd rather
they were chosen on the basis of experience with playing Miracle-based
music, and so knowledge of what chords are likely to be wanted. My
tuning sounds okay, I think it's a good place to start.

Graham

--- In tuning@y..., graham@m... wrote:

> If you're fretting to Canasta, why not make it 31-equal? You won't
get
> the guitar much more accurate anyway.

Frankly, that's not true. On steel-string acoustic guitar, the
difference between a hexad with 4 cent errors and a hexad with 11
cent errors is significant. But I'm picking up my 31-equal guitar
from the shop today and I'm not planning on getting a Canasta guitar
anytime soon!
>
> You mean the same Blackjack should be playable on all strings?

That's what I mean . . . so that if a composer writes a Blackjack
piece, they won't have any special difficulties stemming from certain
notes not being available on certain strings.

--- In tuning@y..., graham@m... wrote:
> I find the large-small pattern on my meantone guitar works very well
for
> 9-limit chords. I pulled the bonus fret out because it was too
difficult
> to play F major.

So how many steps to the octave now?

> But perhaps I'm in a minority, because everybody
else
> seems to be rushing towards these complete middle-number ETs. But
I'd
> rather use the same kind of fretting, and retune the strings to make
> different chords available.

I'm with you here. I once refretted a guitar for meantone too, with
standard open string tuning and only 12 frets to the octave. Eb to G#
on every string using split frets. It was destroyed by a child
standing on it. :-(

I'm thinking of making a Miracle replacement. I'd go for continuous
frets now. I think it will be too hard to find my way around with more
than 21 frets to the octave.

> If you're fretting to Canasta, why not make it 31-equal? You won't
get
> the guitar much more accurate anyway.

No way. I disagree totally. Ratios of 11 with 10c errors are not
ratios of 11 at all. Similarly for ratios of 9. However I might lean
towards the narrow side of an optimum 11-limit Miracle generator to
widen those small steps a little. But no smaller than a 116.562 cent
generator.

116.693 cents is an interesting sort of optimum for the Miracle
generator. You might call it the phase-locking avoidance optimum. With
this size of generator, every 11-limit ratio has an error of at least
0.4 cents while none has an error of more than 3.6 c. The maximin
nearest to the minimax. In 72-EDO the minimum error drops to 0.2 cents
(in the 6:7), but as Paul Erlich mentioned, 72-EDO is good enough in
this regard.

There is another of these kind of optima at 116.562 cents. Every
11-limit ratio here has an error of at least 0.3 cents while none has
more than 5.2 cents.

> The tuning I started the thread with uses 9 generators doesn't it?
That
> was designed to address the practical issue of retuning without
> restringing.

8 generators I think (i.e. 9 notes in the chain). Yes. I'll worry
about trying to avoiding restringing last. I'm more concerned with max
playable chords to the 11-limit.

> I was also aiming to make it easy to get to chords, by
> starting with a lattice structure that can become an 11-limit chord
with
> only a few notes having to change.

Yes but there are so many such chords to choose from.

> Fretting to Blackjack should
also
> mean a lot of the notes you want will be there on the fretboard. I
don't
> see that it's important to be able to play a Blackjack scale. The
value
> of these scales was always in the intervals they contained.

I agree.

> So, with whatever criteria, we have a big list of tunings. I'd
rather
> they were chosen on the basis of experience with playing
Miracle-based
> music, and so knowledge of what chords are likely to be wanted. My
> tuning sounds okay, I think it's a good place to start.

A bit of a chicken and egg thing, no? At least in the case of a
guitar.

I need to decide how far down the fretboard the POS
(point-of-symmetry) should go. i.e. what rotation of Blackjack (or
Canasta) to put on the fretboard.

The optimum choice for this depends _totally_ on what the open string
tuning is. If a bunch of likely-looking open string tunings all had
the same optimum rotation, that would be good enough.

There is another useful stopping point on the Miracle chain between
Blackjack and Canasta. It has 26 notes. It is important because it
allows JI diatonics (one major one minor) and a bagpipe scale. It is
also one more note than is needed to allow a Pythagorean pentatonic
(chain of 4 fifths) and the Arabic neutral diatonic (which is _not_
the same as Mohajira, the chain of neutral thirds).

But I'd be happy if these scales could only be played in one position
far down the neck, making use of "parasitic" notes caused by the
continuous frets. It would be nice if the low string could make the
appropriate drone for the bagpipe scale, but this looks difficult.

-- Dave Keenan

In-Reply-To: <9fk8su+hnp1@eGroups.com>
Dave Keenan wrote:

> --- In tuning@y..., graham@m... wrote:
> > I find the large-small pattern on my meantone guitar works very well
> for
> > 9-limit chords. I pulled the bonus fret out because it was too
> difficult
> > to play F major.
>
> So how many steps to the octave now?

19. That's per-string, as the frets are straight.

> > If you're fretting to Canasta, why not make it 31-equal? You won't
> get
> > the guitar much more accurate anyway.
>
> No way. I disagree totally. Ratios of 11 with 10c errors are not
> ratios of 11 at all. Similarly for ratios of 9. However I might lean
> towards the narrow side of an optimum 11-limit Miracle generator to
> widen those small steps a little. But no smaller than a 116.562 cent
> generator.

31-equal works for the 7-limit-plus-neutral-thirds range that's most
characteristic of miracle temperament. My low E string varies over around
10 cents depending on how hard you pick it. If you had 31 frets, you'd
always end up using some of the non-optimal intervals on them, so it'd
sound worse than if you made it equal to start with.

116.5-ish sounds okay for a Blackjack fretting. So long as it doesn't
have to work as meantone as well.

> 116.693 cents is an interesting sort of optimum for the Miracle
> generator. You might call it the phase-locking avoidance optimum. With
> this size of generator, every 11-limit ratio has an error of at least
> 0.4 cents while none has an error of more than 3.6 c. The maximin
> nearest to the minimax. In 72-EDO the minimum error drops to 0.2 cents
> (in the 6:7), but as Paul Erlich mentioned, 72-EDO is good enough in
> this regard.
>
> There is another of these kind of optima at 116.562 cents. Every
> 11-limit ratio here has an error of at least 0.3 cents while none has
> more than 5.2 cents.

Oh, this is still theoretical, then? 72-equal definitely sounds like the
optimum to me. The small steps of Blackjack become more "melodic" as you
move towards 31-equal. I've never noticed any harmonic optimum at this
point, but melody works well around there.

By the time you reach 31-equal, the chords are significantly blurred, but
still usable.

> > I was also aiming to make it easy to get to chords, by
> > starting with a lattice structure that can become an 11-limit chord
> with
> > only a few notes having to change.
>
> Yes but there are so many such chords to choose from.

I'm trying to find some good ones by ear. Guitars being guitars, this
often means some added notes creep in to the 11-limit chord. But
different things are likely to sound good with a more accurate fretting,
so perhaps you should get one of those together. Perhaps we'll need to
optimise for particular inversions.

Another aim is to keep the strings roughly evenly spread in pitch. No
pairs in my tuning should be closer than a neutral third, or further apart
than a tritone.

> > So, with whatever criteria, we have a big list of tunings. I'd
> rather
> > they were chosen on the basis of experience with playing
> Miracle-based
> > music, and so knowledge of what chords are likely to be wanted. My
> > tuning sounds okay, I think it's a good place to start.
>
> A bit of a chicken and egg thing, no? At least in the case of a
> guitar.

Yes, that's why I started with the first tuning that made sense.

> I need to decide how far down the fretboard the POS
> (point-of-symmetry) should go. i.e. what rotation of Blackjack (or
> Canasta) to put on the fretboard.

I thought of putting it around the major third from the nut. Exactly
where depends on whether you prefer 5:4 to 9:7. This should work with
different open-string tunings, including some that aren't miracle based.

> The optimum choice for this depends _totally_ on what the open string
> tuning is. If a bunch of likely-looking open string tunings all had
> the same optimum rotation, that would be good enough.

How would you define this optimum rotation? Surely it'd depend on how you
wanted the "home key" to relate to the open strings.

> There is another useful stopping point on the Miracle chain between
> Blackjack and Canasta. It has 26 notes. It is important because it
> allows JI diatonics (one major one minor) and a bagpipe scale. It is
> also one more note than is needed to allow a Pythagorean pentatonic
> (chain of 4 fifths) and the Arabic neutral diatonic (which is _not_
> the same as Mohajira, the chain of neutral thirds).

I think all these scales would sound better with 31 equal frets to the
octave. A blackjack fretting would work specifically for 11-limit
miracle-based music, and will also have frets in common with meantone. If
you have more than 21 frets, they'll get in the way, so you may as well
round up to 31, and then make them equal.

> But I'd be happy if these scales could only be played in one position
> far down the neck, making use of "parasitic" notes caused by the
> continuous frets. It would be nice if the low string could make the
> appropriate drone for the bagpipe scale, but this looks difficult.

Yes, if that's what you want, designing an open-string tuning around any
of these scales shouldn't be too difficult. Supplying an external drone
would be easier than building it into the guitar.

Graham

In-Reply-To: <memo.259702@cix.compulink.co.uk>
I wrote:

> I think all these scales would sound better with 31 equal frets to the
> octave.

To pre-empt the corrections on this, the bagpipe scale probably wouldn't.
I haven't looked at it. But the others will all have more consonances
with the meantone approximation thrown in, without important intervals
sounding much worse.

The JI diatonic will obviously become a meantone diatonic. The pentatonic
will have 5-limit meantone chords. And the Arabic neutral diatonic will
have the same 5-limit chords as it's a superset of the pentatonic.

Graham

--- In tuning@y..., graham@m... wrote:

> My low E string varies over around
> 10 cents depending on how hard you pick it.

Blech. What gauge strings are you using.

> If you had 31 frets, you'd
> always end up using some of the non-optimal intervals on them,

On what basis do you make that claim?
>
> 116.5-ish sounds okay for a Blackjack fretting. So long as it
doesn't
> have to work as meantone as well.

Canasta and meantone are totally incompatible. 81:80 is a chromatic
unison vector in the former, and a commatic unison vector in the
latter.
>
> I think all these scales would sound better with 31 equal frets to
the
> octave.

Last night I got to play with my 31-tET guitar for the first time.
Half-step 3, Whole-step 5. Half-step 3, Whole-step 5. Half-step 3,
Whole-step 5. Half-step 3, Whole-step 5. Got it! (just be sure to
have pointy callouses on your fingers.)

In-Reply-To: <9flth0+unlh@eGroups.com>
Paul Erlich wrote:

> > My low E string varies over around
> > 10 cents depending on how hard you pick it.
>
> Blech. What gauge strings are you using.

Don't know. In this case, it's the one that came with the guitar.

> > If you had 31 frets, you'd
> > always end up using some of the non-optimal intervals on them,
>
> On what basis do you make that claim?

That's what guitarists are like. They find a nice chord pattern, then go
and play it up and down the neck. With an obviously unequal fretting,
that's not a problem because you find easy chords to play around the
unequalness. But with Miracle, that equal temptation's going to be there,
even if you resist the temptation to retune the strings.

> > 116.5-ish sounds okay for a Blackjack fretting. So long as it
> doesn't
> > have to work as meantone as well.
>
> Canasta and meantone are totally incompatible. 81:80 is a chromatic
> unison vector in the former, and a commatic unison vector in the
> latter.

My meantone guitar is fretted like this, from the nut to the fifth:

r q r q r r q r r q r

Where r is a chromatic semitone and q a diesis. Or r=2 and q=1 in
31-equal.

The Blackjack fretting I was thinking of is

r q r q r q q r q r q r

In 31-equal, r and q are the same size above, and r=2q. So we can line
the two patterns up.

r q r q r r q r r q r
r q r q r q q r q r q r

Only one fret of the meantone pattern is missing from the blackjack
pattern. Two blackjack frets are missing from the meantone pattern. If
the strings are tuned no more than a fifth apart, most notes and chords
you need will be available in either system.

If you aren't fretted to 31-equal, it works less well. But a lot of
important chords will still be there before you need to rely on r=2q. You
might also be able to temper the octave to improve it.

> Last night I got to play with my 31-tET guitar for the first time.
> Half-step 3, Whole-step 5. Half-step 3, Whole-step 5. Half-step 3,
> Whole-step 5. Half-step 3, Whole-step 5. Got it! (just be sure to
> have pointy callouses on your fingers.)

And this fretting is perfectly consistent between canasta and meantone ...

Graham

--- In tuning@y..., graham@m... wrote:

> > On what basis do you make that claim?
>
> That's what guitarists are like. They find a nice chord pattern,
then go
> and play it up and down the neck.

Oh no! Look out for those naughty guitarists! They know nothing --
better "dumb down" all tuning systems for them!

> With an obviously unequal fretting,
> that's not a problem because you find easy chords to play around
the
> unequalness. But with Miracle, that equal temptation's going to be
there,
> even if you resist the temptation to retune the strings.

Ah, the equal temptation . . . must resist! Can't resist! Falling
into sin!
>
> most notes and chords
> you need will be available in either system.

Most of the ones you need . . . that depends on who's using the
systems and how _they_ want to use them.
>
> And this fretting is perfectly consistent between canasta and >
meantone ...

I can't consider 31-equal a Canasta fretting. If the 81:80 vanishes,
it's just not Canasta. But I'm sure I'll be more than happy with 31-
equal for quite some time.