Yahoo Tuning Groups Ultimate Backup tuning Designing a Blackjack guitar

Graham's Miracle guitar thread prompted me to think more specifically about
a Blackjack (Miracle-21) guitar. I learned a few things while helping Paul
design his Shrutar.

Here are what I think are reasonable design constraints.

All frets must be continuous across the whole width of the neck.
There should only be 21 frets in the octave.
Intervals between adjacent strings should be no smaller than 7:8 and no
larger than 2:3. Maybe 7:8 is too small? Maybe 2:3 is too big?

Call the particular set of 21 pitches that we are aiming for "the reference
key" (or home key?). Because of the requirement to have all frets
continuous there will be some notes available (on some strings) that are
outside the reference key. We want to minimise the number of such notes and
we want them to occur as far from the nut as possible. So the reference key
should be playable down at least one string (called a reference string) and
also playable across all strings using the frets closest to the nut.
Ideally the highest string should be a reference string to give maximum
range for the reference key.

What this means is that the "chord" formed by all the open strings must be
as compact as possible on the chain of generators, provided this does not
have other bad consequences like making too many real chords unplayable.

Here's what an otonal hexad looks like on a chain of Miracle (7/72 oct)
generators. i.e. octaves are ignored here.

5 . . . . 7 . 1 . . . . . 3 . . . . . 9 . .11

That really only gives us two options for sets of string spacings (plus
their rotations and repetitions).

7:10 7:8 9:11 (spans five generators)
3:4 9:11 9:11 (spans six generators)

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

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

I wrote:
> Here's what an otonal hexad looks like on a chain of Miracle (7/72
oct)
> generators. i.e. octaves are ignored here.
>
> 5 . . . . 7 . 1 . . . . . 3 . . . . . 9 . .11
>
> That really only gives us two options for sets of string spacings
(plus
> their rotations and repetitions).
>
> 7:10 7:8 9:11 (spans five generators)
> 3:4 9:11 9:11 (spans six generators)

That's not true. There are many more options. The two above do not even
exhaust the possibilities for dividing an octave into 3 parts without
spanning more than 6 generators.

Although each successive interval between open strings always moves us up
in pitch, it will move either up or down the (octave-equivalent) chain of
Miracle generators.

Here's how each interval (with size between 7:8 and 2:3) moves us on the
chain of generators.

apprx
ratio cents abbr gens
---------------------
7:8 233c SM2 +2
9:11 350c n3 +3
diss 467c s4 +4
5:7 583c sd5 +5
2:3 700c P5 +6

7:10 633c d5 -5
3:4 500c P4 -6
4:5 383c M3 -7
6:7 267c sm3 -8

It seems we should exclude 4:5 and 6:7 for now, because we want to minimise
the span in generators and there are so many possibilities to explore
within a span of 6 generators.

Notice a couple of things from that list. We cannot move by only one
generator, and the only negatives are 7:10 and 3:4 which step by large
numbers of generators.

The other way of dividing an octave into 3 parts is

3:4 7:8 467c

There are also the cases where the octave is split into only 2 parts

2:3 3:4
5:7 7:10

And it is possible to avoid octaves altogether and still have 6 open
strings that don't span more than 6 generators. e.g.

7:10 9:11 9:11 7:10 9:11

If I've understood Graham Breed's current suggestion, it goes

9:11 3:4 9:11 7:10 9:11

in generators that's
+3 -6 +3 -5 +3

Taking the high string as the reference ("r") string and working from right
to left, we get the keys playable on each string as

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

So it ranges from r-3 ro r+5, a span of eight generators. I'm hoping we can
do better than that, and stay within r-3 to r+3.

Remember, the reason we want to do that is to minimise the number of notes
of the reference key that are unavailable on the non-reference strings, and
to maximise the distance from the nut before you lose such a note. A string
which is tuned to r+5 will be missing 5 notes from the reference key. It
will have an additional 5 notes which are not in the reference key.

Of course we may be able to make use of these extra notes, for example to
let us play some complete hexads. But a Blackjack chain (21 notes in a
chain of 7/72 oct generators) only needs to be extended by 2 notes to get
an otonal hexad (say r-1 to r+1) and you get another otonal hexad for every
note beyond that.

We can now enumerate all the open string spacings that remain within keys
from r-3 to r+3 when the high string is r, and have no step of less than
233c or more than 700c.

Would you believe there are 65 of them. Email me if you want them.

Some can be eliminated because the overall range of the guitar would be too
small. But that still leaves an awful lot. Can anyone suggest how could
prune the list further?

Actually, everything we've discussed so far relates equally to designing a
Canasta (Miracle-31) guitar.

Eventually we have to decide what Blackjack chords (in what inversions) it
is most important to be able to play, see what patterns they make on the
fretboard with each open-string tuning, and decide whether we have enough
fingers and can reach far enough.

If we can find a suitable open-string tuning that only ranges from r-3 to
r+3, then here is the optimum arrangement of the frets. Spacings are given
in 1/72nds of an octave. One octave is shown. The asterisk indicates the
point of symmetry of the scale.

nut 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2*2 5 2 5 2 5

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

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

🔗Paul Erlich <paul@stretch-music.com>

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

>> Do you mean no further apart than a diminished fifth (approx 7:10) 633
>> cents?
>
>Ideally no further apart than the 7:5, but that's hard to pull off

Extremely hard. And I goofed there. A 7:10 is only 617 cents (not 633).

>> > 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.
>
> It's partly to match my meantone fretting, as explained in my reply to
> Paul.

Ok. I'm not interested in meantone compatibility for a Blackjack guitar.

If your POS is that close to the nut, your current open string tuning must
have an even bigger hole (or more holes) than the one I showed. i.e. notes
missing between strings in the blackjack home key (if it were not the full
31-EDO).

> It wouldn't seem right to me for a
> guitar not to have a perfect fourth from the nut.

That's no problem.

>> I think we only need to consider the string with the minimum value, r-3.
>
>I don't think this is correct. You need to consider the strings with the
>maximum and minimum values, and make sure they can see each other.

I don't know what you mean by "see each other".

> Looks like your POS is a lot further down than mine. But it should be
> possible to place that symmetrically about the tritone. The closer it
> gets to the tritone, the more tunings should work by your criterion.
> It'll be something to do with the number of generators covered by the open
> strings, and the number you can move in either direction from the nut.

I disagree. This seems to be still ignoring gaps in the scale between strings.

Playing notes above the nut would be a neat trick! I know you didn't really
mean that, but it's the fact that you can't do this that means the optimal
POS will be as far down the neck as possible, with the constraint that for
any string which is tuned a number of generators _less_ than the top
string), all its missing notes must occur between the POS and the octave.

Any strings tuned a number of generators _more_ than the top string will
have all their missing notes immediately above the POS ("above" here means
towards the nut, not higher in pitch).

For example, if the minimum-generator string is an r-3 (as it is in your
current tuning as well as the one I propose below) then there are 3 missing
notes on this string and the POS should be 3 generators back towards the
nut from the octave fret (which is "above the nut" in an octave-equivalent
sense).

Then if the difference between max-gen string and min-gen string is only 5
generators (as in the tuning below) the missing notes on the max gen string
only come back 5 gens minus a sixth-tone, toward the nut from the octave
fret. And since this max-gen string has an interval of 7:10 (an octave less
5 gens) to the next string, then there is no gap in the home key. There is
in fact a one note overlap, i.e. the pitch of the next open string can be
duplicated on the previous string.

It might help to scroll down to the fingerboard diagram and look as where
the "*"s are.

Note that this theory of optimum POS placement also implies that the first
fret must be a small step (33c) away from the nut.

-------
Theorem
-------

If you want a 21-continuous-fret gap-free Blackjack guitar where the
intervals between strings are never smaller than a neutral third (350 c)
and never greater than a diminished fifth (617 c) then the open-string
"chord" cannot span more than 7 generators.

-----------------------
Partial Attempted Proof
-----------------------

As described above, the critical factors for being gap-free are
1. R, the range (in generators) of the open tuning, and
2. M, the largest interval (in generators) between any max-gen string and
the next string.

The first represents a problem that grows back from the octave fret and the
second a problem that grows down from the nut. When these two problems
overlap we get gaps in the scale.

With optimal POS placement, as per the above theory, the first missing note
on the max-gen string is R generators minus a sixth-tone back from the
octave fret.

An octave is 10 miracle generators plus a sixth-tone so the first missing
note is 10-R generators plus 2 sixth-tones down from the nut. e.g. if R is
5, 6 or 7 the first missing note is 5g+2s, 4g+2s or 3g+2s respectively from
the nut.

Notice that M must be negative since it is only possible to go _down_ in
generators from a max-gen string.

Given the size constraints, the available intervals between strings are:

Ratio Cents Abbr Gens
---------------------
9:11 350c n3 +3g
diss 467c s4 +4g
5:7 583c sd5 +5g

7:10 617c d5 -5g
3:4 500c P4 -6g
4:5 383c M3 -7g

So M must be -5g, -6g, or -7g.

So the note on the max-gen string that corresponds to the next open string,
will be a distance of 10+M generators plus a sixth-tone down from the nut.
i.e. it must be either 5g+s, 4g+s or 3g+s from the nut.

Notice that R must be greater than or equal to Abs(M) so we can see that
the best cases are when M = -5 and R = 5, or M = -6 and R = 6, or M = -7
and R = 7. In all of these cases you get to play the note of the next open
string.

But it's not actually _that_ note we're worried about. We only want to be
sure that we're not missing any notes _before_ that. Can we have for
example M = -5 and R = 6. The answer appears to be "Yes" in this case.

I succumbed to brain-strain here, but thought I'd post it anyway. Maybe
someone else can help. If there's anything you don't follow, please ask.

----------------------
A possible open tuning
----------------------

The following combination of open string tuning and POS looks good to me on
paper. It's the best one i can find that only spans 5 generators. Missing
notes are shown as "*".

Maybe you 31-EDO guitar guys would check it out in real life? But I think
you're gonna have to mark the non-blackjack frets somehow to make sure you
don't use them.

It has a single complete 3:4:5:7:9:11 hexad (using parasitic notes) that
might just be playable, if you're good at bending your second finger top
joint backwards. The hexad notes are shown with "@" to their left.

7:10 7:10 Intervals
9:11 5:7 9:11 between string

r-1 r+2 r-3 r+2 r-3 r Generators away from reference key

6 9 4 9 4 7 nut
6> 9> 4> 9> 4> 7>

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

8 1< 6 1< 6 9
8> 1 6> 1 6> 9>

9 2< 7 2< 7 0<
9> 2 7> 2 7> 0

0< 3< 8 3< 8 1<
0 3 8> 3 8> 1

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

The absolute tuning of the open strings should probably be:
D F] C< F] C< Eb^ or
Eb^ G< C#v G< C#v E>
depending on how much punishment you think the "B" string can take.

Legend for 72-tET notation:

A,B,C,D,E,F,G,#,b as for 12-tET
] = quarter-tone up (+50 c)
> = sixth-tone up (+33 c)
^ = twelfth-tone up (+17 c)
v = twelfth-tone down (-17 c)
< = sixth-tone down (-33 c)
[ = quarter-tone down (-50 c)

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

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

I wrote:

"The absolute tuning of the open strings should probably be:
D F] C< F] C< Eb^ or
Eb^ G< C#v G< C#v E>
depending on how much punishment you think the "B" string can take."

It could also be
D> F#v C F#v C E[

Legend for 72-tET notation:

Here's the tablature for the single parasitic hexad in this Blackjack
tuning. It also shows where the point of symmetry (POS) is.

3:4:5:7:9:11

-1 2-3 2-3 0 relative gens for open strings

|_|_|_|_+_+ 14fr
|_|_|_|_|_| POS
|_|_|_|_|_|
| | | | | |
|_|_|_|_|_|
|_|_+_+_|_|
| | | | | |
@_|_|_|_|_|
|_|_|_|_|_|
| | | | | |
|_|_|_|_|_| 21fr (8ve)
|_@_|_|_|_|

Here's a sampling of barre chords that might be playable between the nut
and the POS. In some cases I've shown five fingers (not counting the
thumb!). No need to be alarmed, it's just that I'm leaving it up to you
which string not to play.

7 limit otonality
3:4:5:7:10:14
_ _ _ _ _
+_+_+_+_+_+ odd fr
| | | | | |
@_|_|_|_|_|
|_|_|_|_|_|
| | | | | |
|_|_|_|_|_|
|_@_|_|_|_@

Major triad
5:6:8:12:16:20

+_+_+_+_+_+ even fr
|_|_|_|_|_@
| | | | | |
|_@_|_@_|_|
@_|_|_|_|_|
| | | | | |
|_|_|_|_|_|
|_|_|_|_|_|

Minor triad

+_+_+_+_+_+ even fr
|_|_|_|_|_|
| | | | | |
|_|_|_|_|_|
|_|_|_|_|_@
| | | | | |
|_@_|_@_|_|
@_|_|_|_|_|

Subminor triad
7:9:12:18:24

+_+_+_+_+_x even fr
@_|_|_|_|_|
| | | | | |
|_@_|_@_|_|
|_|_|_|_|_|
| | | | | |
|_|_|_|_|_|
|_|_|_|_|_|

9-limit tetrad
7:9:12:18:24:32

+_+_+_+_+_+ even fr
@_|_|_|_|_|
| | | | | |
|_@_|_@_|_|
|_|_|_|_|_@
| | | | | |
|_|_|_|_|_|
|_|_|_|_|_|

7:9:11 triad
9:11:14:22:28:36
_ _ _ _ _
+_+_+_+_+_+ odd fr
| | | | | |
|_|_|_|_|_@
|_|_|_|_|_|
| | | | | |
@_@_|_@_|_|
|_|_|_|_|_|
| | | | | |

1:7:9:11 tetrad
8:11:14:22:28:36
_ _ _ _ _
+_+_+_+_+_+ odd fr
| | | | | |
|_|_|_|_|_@
|_|_|_|_|_|
| | | | | |
|_@_|_@_|_|
|_|_|_|_|_|
| | | | | |

When one of you 31-EDO guitarists has some time, could you please try out
this tuning? Or at least try the fingering and tell me which of these
chords are actually playable?

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