An Eikosany guitar design

I realised, as I woke up this morning, that George Secor's MIRACLE
temperament makes a 3 of {1,3,5,7,9,11) Eikosany guitar feasible (with all
20 frets full width). One is no longer forced into the usual strict-JI
thing of only having 2:3s or 3:4s between adjacent strings, with those 2:3s
making for some big stretches.

I spent most of today searching for the optimal combination of
(a) open string tuning and
(b) scale rotation for the fretting.

I think I've found it. From low to high the open strings are
(MIRACLE-tempered versions of):

3*5*7 3*5*9 5*7*9 3*5*7 3*5*9 5*7*9
/2 *2 *2

This happens to be a supermajor triad 1/(9:7:6), repeated at the octave.

And if I tell you that the complete (MIRACLE-tempered) Eikosany only occurs
on the high string and the 3rd string, then you have enough info to
calculate the fretting.

No frets are closer together than 33 cents (same as Blackjack). There are
only 3 missing notes in the low octave (the same thing happens with the
2:3, 3:4 design). The missing notes are (octave equivalent): 1*3*9, 5*9*11,
1*5*7. If this was thought to be a serious deficiency, it could be remedied
by adding 3 more frets. This would produce two fret spacings of only 17
cents, but since they both occur within 83 cents of the nut where the
physical spacings are widest, this is feasible.

If anyone wants more details, let me know.

Has anyone ever heard of any previous attempts at an Eikosany guitar
design? Are you there Kraig Grady?

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

> I realised, as I woke up this morning, that George Secor's MIRACLE
> temperament

George Secor's?! Did I miss something?

> I spent most of today searching for the optimal combination of
> (a) open string tuning and
> (b) scale rotation for the fretting.
/.../
> If anyone wants more details, let me know.
>
> Has anyone ever heard of any previous attempts at an Eikosany guitar
> design? Are you there Kraig Grady?

As someone who only peripherally followed your blackjack guitar
thread, could you sum up the critera you use for (a) and (b)?

Are the tetrads of the eikosany playable?

What do we gain by tempering the eikosany in 72 here?

-Carl

--- In tuning@y..., carl@l... wrote:
> > I realised, as I woke up this morning, that George Secor's MIRACLE
> > temperament ...
>
> George Secor's?! Did I miss something?

You must have. Erv had Kraig point us to a certain Xenharmonikon article in
1975. See http://anaphoria.com/secor.PDF

> > I spent most of today searching for the optimal combination of
> > (a) open string tuning and
> > (b) scale rotation for the fretting.
...
> As someone who only peripherally followed your blackjack guitar
> thread, could you sum up the critera you use for (a) and (b)?

Yes. I'd love to know if anyone thinks they should be changed, and how.

They're still a bit fuzzy, but here goes:

All frets must be continuous across the full width of the neck. There
should be no more frets than are required to play the scale on a single
string. e.g. 20 per octave for the eikosany, 21 per octave for blackjack
and 30 per tritave (approx 19 per octave) for the 30 note MOS from the
Bohlen-Pierce-Stearns (BPS) temperament.

------------
Open strings
------------
1. The interval between adjacent strings should be some kind of third or
fourth, preferably from a neutral third to a perfect fourth (9:11 to 3:4,
350 to 500 cents). But at a pinch, it can go down to a subminor third or up
to a superfourth (6:7 to 8:11, 267 to 583 cents), however two such very
small or very large jumps should not occur next to each other and the
extreme open strings should not span less than 1750 cents or more than 2500
cents.

2. The intervals between open strings do not all have to be 11-limit
"consonances", but more chords are likely to be playable if most of them
are. It is preferable if the open strings roughly approximate 3:4:5:6:7:9,
3:4:5:7:9:11 or 4:5:6:7:9:11. But usually other criteria intervene.

3. The high string must contain the complete scale because these notes are
not available on any other string. I call this string and any string which
is period-equivalent to it (e.g. octave equivalent or tritave equivalent as
the case may be), a "reference string".

4. Open non-reference strings must be notes of the scale.

-------
General
-------
Any non-reference string will be missing some notes of the scale.

The aim is to find the open string tuning within the above constraints that
minimises the number of such missing notes, and to find the scale rotation
on the reference string(s) (which determines the fretting for all strings)
that puts those missing notes as far from the nut as possible.

In the case of linear temperaments, I refer to the refernce strings as "r"
and the non-reference strings as r+1, r-2 etc. where the number gives the
string's offset from the reference string in numbers of generators.

When the target scale is a contiguous chain of generators then obviously
the number of missing notes will be minimised by minimising the number of
generators that strings are offset from the reference. But usually the hard
part is making this work within the constraints given above, regarding the
intervals between adjacent strings.

The BPS-30 guitar is beautifully simple in this regard. The generator is
about 442 cents (an approx 7:9) and so the open strings are simply arranged
in a chain of generators from r-5 to r.

In the case of the MIRACLE-tempered eikosany, things are more complicated.
I found that a shift of +-6 generators (a 2:3 or 3:4) gave the minimum
number of missing notes (8), and equal second with 9 missing notes were
r+-2 and r+-8.

I then drew a state-transition diagram where the possible "states" were
represented as scattered circles labelled inside with these 7 open string
tunings r, r+6, r-6, r+2, r-2, r+8, r-8. Then whenever a transition could
be made from one string to another by an interval in the range of 267 to
583 cents, I drew an arrow between those two circles and labelled it with
the interval size. I then listed all the sequences of six open strings that
ended with a reference string. To do this, it is best to start with "r" and
follow arrows backwards (or draw them all as pointing from higher string to
lower string in the first place). Some rearranging of the diagram often
helps too.

I then chose those that minimised the number of different non-reference
string offsets. There were only two that had only two different non-ref
offsets. One had r-6 and r+8, the other had r-8 and r+6.

--------------
Scale rotation
--------------
For each open-string set under consideration I draw a (linear) fingerboard
for them using an arbitrary scale rotation (I actually just used a
spreadsheet column for each string where a row represented one step of
72-EDO or about 17 cents). I label the fret positions with the number of
generators that each note corresponds to (arbitrary zero). Then I colour
the extrascalar fret positions yellow. Any fret with no colour on it
anywhere, is a potential position for the nut, with the scale rotated
accordingly. I also colour the missing-notes' non-fret positions red.

If we can find a scale rotation (nut position) so that no string has any
missing notes occurring between the nut and the position corresponding to
the pitch of the next higher string exclusive (the critical range), then we
consider it a success, because although some notes will be missing from
some strings, no note is completely missing from the guitar
(period-specific, from the open low string to the highest fret on the high
string). For convenience of tuning, it is nice to also have frets that
corresponds to the pitch of the next higher string, (as with the Blackjack
guitar), but this is not considered essential.

So for each potential nut, we count the number of missing notes in its
critical ranges. If there are several with no such missing notes, we look
for those that have frets corresponding to the next highest open string and
if there are several of them we look for those for which the first missing
note on each string is the farthest outside of its critical range or
farthest from the nut.

In some cases, such as the {1,3,5,7,9,11} eikosany, we still have a few
notes that are completely missing, say between the open low string and the
lowest open reference string. There are 3 such in the case of the eikosany.
One can either wear that, or consider adding more frets to include them.
This may bring in the need for other optimisations which we will not
consider here, but which are satisfied by the proposed eikosany guitar.

There is a certain amount of interaction between open string tuning and
scale rotation, and it may conceivably be necessary to choose an open
string tuning that has more missing notes but allows them to be placed
outside the critical range on the non-reference strings. This possibility
has not been explored for the eikosany, but seems unlikely.

A secondary consideration is the flip-side of "missing notes", namely
"unwanted notes", i.e. notes that are outside the scale. We'll call them
extrascalar notes, since they can sometimes be useful. But in any case,
given two rotations that have the same number of missing notes in the
critical range, I would choose the one that has the fewest extrascalar
notes in the critical range.

> Are the tetrads of the eikosany playable?

I expect that most, if not all of them, are playable in some
voicing/inversion, but I haven't specifically looked at this.

Maybe you would volunteer to check it if I supply a diagram showing the
first octave of the fingerboard with all the notes marked in a*b*c*2^n form?

You can refer to Erv's Eikosany lattice on Kraig's site. If you find it,
could you post the URL?

You may even get a few extra non-eikosany JI chords introduced by the
tempering, but of course you don't have to use them.

> What do we gain by tempering the eikosany in 72 here?

Good question. First note that the design does not depend on 72-EDO in any
way. Any suitable MIRACLE generator (near 7/72 oct) can be used, e.g.
optimising some favourite weighted error. I use 72-EDO as a convenient
standard.

It is conceivable that actual tempering of the eikosany may not be
necessary, and that the MIRACLE temperament merely made the search for a
guitar solution tractable. But I rather strongly suspect that it must be
tempered in order to acheive anywhere near this many notes of the eikosany
with only 20 continuous frets per octave and no frets closer than 33 cents.
This should become obvious if I supply the abovementioned fingerboard
diagram.

Remember that the 11-limit errors in this temperament are no larger than
the typical intonation errors of even the most carefully intoned guitar
with independent bridge adjustments for each string. So for a guitar,
MIRACLE _is_ JI.

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

Here's a linear fingerboard diagram (first octave) for the eikosany guitar.

3*5*7 3*5*9 5*7*9/2 3*5*7*2 3*5*9*2 5*7*9 nut 0c

m(1*3*9)
m(1*5*7)
1*5*11 x 3*5*11 1*5*11 x 3*5*11 1fr 83c

x x 1*3*7 x x 1*3*7 2fr 117c

3*7*11 3*9*11 7*9*11 3*7*11 3*9*11 7*9*11 3fr 167c

1*3*5 1*7*11 1*5*9 1*3*5 1*7*11 1*5*9 4fr 233c

c
m(5*9*11)

1*7*9 x 3*7*9 1*7*9 x 3*7*9 5fr 317c

x 3*5*11 5*7*11 x 3*5*11 5*7*11 6fr 350c

1*3*11 x 1*9*11 1*3*11 x 1*9*11 7fr 400c

c

1*5*7 1*5*9 3*5*7 1*5*7 1*5*9 3*5*7 8fr 500c

x x 1*3*9 x x 1*3*9 9fr 550c

x 3*7*9 1*5*11 x 3*7*9 1*5*11 10fr 583c

1*7*11 1*9*11 3*7*11 1*7*11 1*9*11 3*7*11 11fr 667c

x x 1*3*5 x x 1*3*5 12fr 733c

3*5*11 x 5*9*11 3*5*11 x 5*9*11 13fr 783c

1*3*7 1*3*9 1*7*9 1*3*7 1*3*9 1*7*9 14fr 817c

x x 1*3*11 x x 1*3*11 15fr 900c

1*5*9 3*7*11 3*5*9 1*5*9 3*7*11 3*5*9 16fr 933c

x 1*3*5 1*5*7 x 1*3*5 1*5*7 17fr 1000c

1*9*11 x 3*9*11 1*9*11 x 3*9*11 18fr 1100c

x 1*3*11 1*7*11 x 1*3*11 1*7*11 19fr 1167c

3*5*7 3*5*9 5*7*9/2 3*5*7*2 3*5*9*2 5*7*9 8ve 1200c

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

Oops. I forgot to explain that in the preceding fingerboard diagram,
the x's are notes outside the eikosany (Xtrascalar). The c's are the
end of the Critical ranges, i.e. the positions that correspond to the
next open string. The m's are the positions corresponding to the 3
Missing notes.

Also, I got lazy and left out the factors of two, except at the nut
(and gave them wrongly at the octave). So just ignore them and take it
as an octave-equivalent mapping. Sorry.

-- Dave Keenan

>>>I realised, as I woke up this morning, that George Secor's MIRACLE
>>>temperament ...
>>
>> George Secor's?! Did I miss something?
>
>You must have. Erv had Kraig point us to a certain Xenharmonikon
>article in 1975. See http://anaphoria.com/secor.PDF

I read that article in '97, and own two copies of the XH issue
in which it appears. Didn't remember the number 116, though!

It may be slightly misleading to call it Secor's MIRACLE, no?
Since MIRACLE was not his name...

>>>I spent most of today searching for the optimal combination of
>>>(a) open string tuning and
>>>(b) scale rotation for the fretting.
>>...
>> As someone who only peripherally followed your blackjack guitar
>> thread, could you sum up the critera you use for (a) and (b)?
>
>Yes. I'd love to know if anyone thinks they should be changed, and
>how.
/.../
>All frets must be continuous across the full width of the neck.

Sounds good.

>There should be no more frets than are required to play the scale
>on a single string.

Sounds good.

>1. The interval between adjacent strings should be some kind of
>hird or fourth, preferably from a neutral third to a perfect fourth
>(9:11 to 3:4, 350 to 500 cents). But at a pinch, it can go down to
>a subminor third or up to a superfourth (6:7 to 8:11, 267 to 583
>cents), however two such very small or very large jumps should not
>occur next to each other and the extreme open strings should not
>span less than 1750 cents or more than 2500 cents.

Sounds reasonable (but outside of my knowledge of the guitar).

>3. The high string must contain the complete scale because these
>notes are not available on any other string. I call this string and
>any string which is period-equivalent to it (e.g. octave equivalent
>or tritave equivalent as the case may be), a "reference string".

Check.

>4. Open non-reference strings must be notes of the scale.

Check.

>Any non-reference string will be missing some notes of the scale.
>
>The aim is to find the open string tuning within the above
>constraints that minimises the number of such missing notes, and to
>find the scale rotation on the reference string(s) (which determines
>the fretting for all strings) that puts those missing notes as far
>from the nut as possible.

Makes sense to me. What do the guitar players out there think of
this? Is this the first general system for mapping alternate
tunings to the guitar, and if not, how does it compare with others?

>In the case of the MIRACLE-tempered eikosany, things are more
>complicated. I found that a shift of +-6 generators (a 2:3 or 3:4)
>gave the minimum number of missing notes (8), and equal second with
>9 missing notes were r+-2 and r+-8.
/.../
>I then chose those that minimised the number of different non-
>reference string offsets. There were only two that had only two
>different non-ref offsets. One had r-6 and r+8, the other had r-8
>and r+6.

Wild!

>In some cases, such as the {1,3,5,7,9,11} eikosany, we still have
>a few notes that are completely missing, say between the open low
>string and the lowest open reference string. There are 3 such in
>the case of the eikosany. One can either wear that, or consider
>adding more frets to include them.
>This may bring in the need for other optimisations which we will not
>consider here, but which are satisfied by the proposed eikosany
>guitar.

Oooo...

>>Are the tetrads of the eikosany playable?
>
>I expect that most, if not all of them, are playable in some
>voicing/inversion, but I haven't specifically looked at this.
>
>Maybe you would volunteer to check it if I supply a diagram showing
>the first octave of the fingerboard with all the notes marked in
>a*b*c*2^n form?

I would and do, and in any case would like to see such a diagram,
but I don't really know what makes a chord playable, as I'm not a
guitarist.

>You can refer to Erv's Eikosany lattice on Kraig's site. If you
>find it, could you post the URL?

It's in _D'Allesandro_, if that helps... I have hard copies of
all Erv's articles... plus, I have various versions of the
Wilson archive saved to my local hard drive.

>> What do we gain by tempering the eikosany in 72 here?
>
>Good question. First note that the design does not depend on
>72-EDO in any way. Any suitable MIRACLE generator (near 7/72 oct)
>can be used, e.g. optimising some favourite weighted error. I
>use 72-EDO as a convenient standard.
>
>It is conceivable that actual tempering of the eikosany may not be
>necessary, and that the MIRACLE temperament merely made the search
>for a guitar solution tractable. But I rather strongly suspect that
>it must be tempered in order to acheive anywhere near this many
>notes of the eikosany with only 20 continuous frets per octave and
>no frets closer than 33 cents. This should become obvious if I
>supply the abovementioned fingerboard diagram.

Ja.

>Remember that the 11-limit errors in this temperament are no larger
>than the typical intonation errors of even the most carefully
>intoned guitar with independent bridge adjustments for each string.
>So for a guitar, MIRACLE _is_ JI.

That may be the case for these errors (they may just be _small_
enough), but I don't think the argument is generally valid... there
may be a noticeable difference between random errors centered on JI
and consistent errors (with respect to JI) of the same size...

-Carl

--- In tuning@y..., carl@l... wrote:

> It may be slightly misleading to call it Secor's MIRACLE, no?
> Since MIRACLE was not his name...

I think MIRACLE should refer to the set of MOS scales obtainable from
the Secor generator.

--- In tuning@y..., carl@l... wrote:
> >1. The interval between adjacent strings should be some kind of
> >hird or fourth, preferably from a neutral third to a perfect fourth
> >(9:11 to 3:4, 350 to 500 cents). But at a pinch, it can go down to
> >a subminor third or up to a superfourth (6:7 to 8:11, 267 to 583
> >cents), however two such very small or very large jumps should not
> >occur next to each other and the extreme open strings should not
> >span less than 1750 cents or more than 2500 cents.
>
> Sounds reasonable (but outside of my knowledge of the guitar).

This criterion is there to ensure playability of chords.

Most chords are stacks of thirds (of some kind) or stacks of thirds
and fourths, therefore you want the open string tuning to be like that
too so that you don't ever need to depart very far from the nut or a
barre to get any chord.