Re: Designing a Blackjack guitar (was: Designing a Canasta guitar)

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

No. Not meantone as well. It's interesting to consider a keyboard tuning
that would be a kind of 31 tone well temperament; miracle in one region,
varying smoothly to meantone in another.

> Oh, this is still theoretical, then? 72-equal definitely sounds like the
> optimum to me.

Yes.

> I'm trying to find some good ones by ear.

But some will be unavailable to you _because_ of your intial choice of open
string tuning.

> Guitars being guitars, this
> often means some added notes creep in to the 11-limit chord.

Please explain. You mean because they have 6 strings and you only have 4
fingers. This of course is the design challenge.

> But different things are likely to sound good with a more accurate
fretting,

Indeed.

> so perhaps you should get one of those together. Perhaps we'll need to
> optimise for particular inversions.

Yeah like we're not very interested in chords containing 10:11, 11:12,
11:14, 11:16, 11:18, but rather 4:11, 5:11, 6:11, 7:11, 8:11, 9:11

> 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.

Do you mean no further apart than a diminished fifth (approx 7:10) 633 cents?

> I thought of putting it around the major third from the nut. Exactly
> where depends on whether you prefer 5:4 to 9:7.

Please explain.

> 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.

Yes, but I figure the scale on the top string must be considered as the
home key (whatever the rotation), because you can't choose to play these
notes on another string. All other strings only need to give you the home
key as far down the neck as the pitch of the next string (but it's better
if they go further).

By way of example, I understand your current tuning to have intervals
between strings of:

9:11 3:4 9:11 7:10 9:11
+3 -6 +3 -5 +3 in generators

Taking the high string to contain the complete home key ("r" for
reference), the strings are:

r+2 r+5 r-1 r+2 r-3 r

I think we only need to consider the string with the minimum value, r-3.
The fretting should be such that the point of symmetry on the reference
string is the same pitch class as this open string. i.e. POS is (72 -
3*7)/72 oct = 51/72 oct = 750 cents from the nut.

If the POS is note 4 (decimal) the 21 notes available on the strings would
be: (Missing home-key notes are shown with "*")

r+2 r+5 r-1 r+2 r-3 r

9 2< 6 9 4 7 nut
9> 2 6> 9> 4> 7>

0< 3< 7 0< 5 8
0 3 7> 0 5> 8>

1< 4< 8 1< 6 9
1 4 8> 1 6> 9>
*
2< 5< 9 2< 7 0<
2 5 9> 2 7> 0
*
3< 6< 0< 3< 8 1<
3 6 0 3 8> 1
*
4< 7< 1< 4< 9 2<
4 7 1 4 9> 2
* * *
5< 8< 2< 5< 0< 3<
5 8 2 5 0 3
* * *
6< 9< 3< 6< 1< 4<
6 9 3 6 1 4 POS
6> 9> 3> 6> 1> 4>
* *
7 0< 4 7 2 5
7> 0 4> 7> 2> 5>
*
8 1< 5 8 3 6
8> 1 5> 8> 3> 6>
*
9 2< 6 9 4 7 octave
9> 2 6> 9> 4> 7>

When I set it out like this I see it's tougher than I thought.

Notice that we have a "hole" between the 2nd and 3rd strings where we can't
play 4> or 5> at all.

> 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.

I think I'll leave it at 21 then (with approx 7/72 gens).

> Supplying an external drone would be easier than building it into the
guitar.

Good point.

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com