A combination-product-set guitar (part 2)

Dear MMMers,

Here's part 2 of an edited version of an email conversation that Pete McRae and I had a year ago.

Part 1 appears at
/makemicromusic/topicId_13042.html#13042

In part 2 (below) the design is described and some possible problems with it are addressed.
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[Dave:]

Hi Pete,

I believe I've found a way around the philosophical impasse. I have a fretboard design for you to try. I think it's awesome. I'm dying to hear what you can do with it. Everyone will want one. ;-)

Here are some of its properties.

1) Up to the 9-limit it is strictly JI (not microtempered). So if someone objects that it's not _really_ JI because there's some tempering at the 11-limit, we can diplomatically retreat to the position that it isn't intended to be 11-limit JI, but rather 9-limit JI plus neutral thirds.

And if you don't want to call it an Eikosany guitar, you can at least call it a CPS guitar since it contains several hexanies and dekanies that are strictly just.

2) Its ratios of 11 (4:11, 5:11, 6:11, 7:11, 9:11) are all 3.6 cents wide (but that's such a small error, and a guitar string's 11th harmonic decays so rapidly, that I severely doubt anyone will be able to tell). I note that the only way to reduce this error while still having straight-across frets is by microtempering the 9-limit ratios.

3) All 30 tetrads of the 4.5.6.7.9.11 eikosany (an octave-specific 1.3.5.7.9.11 eikosany) are easily playable! By the way, the concept of an "octave-specific" CPS has Erv Wilson's blessing since he requested that my Tumbling Dekany be played at the El Paso Microtonal Festival in 2001, and I heard from Jeniffer Stapher that he greatly enjoyed it.

4) The fretboard is isomorphic. i.e. the pattern for a given type of chord (say a 7:9:11) is the same anywhere on the fingerboard (side to side as well as up and down).

5) The open strings are all tuned a neutral third (exactly half a just fifth) apart (hence the tempered ratios of 11). And therefore alternate strings are a just fifth apart.

6) It has only 19 frets! (15 to the octave) not counting any zero fret at the nut, and they are all straight and full-width, although if you wanted frets _only_ for the notes of the eikosany it would break up into fretlets. The non-eikosany positions make lots more JI chords playable, and several hexanies and dekanies.

7) The closest fret spacing is 35 cents. The only other fret spacings are 49 cents and 232 cents.

8) It has no true octaves. But it has two sub-octaves of 1172 cents.

9) One possible melodic scale consists of 7 notes to the sub-octave with a pattern of small and large steps sLsLsLs where s is 119 cents and L is 232 cents. And with all those neutral thirds you'd think we'd have to be able to get some kind of Arabic-sounding scales out of it.

10) I'm sure you'll find stuff there that I can't even imagine.

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Note re 3) above:

When I say all the tetrads are easily playable, I have only looked at the placement of four fingers for the four notes of the tetrad (all of which are a piece of cake, no stretches). I have not yet considered what to do with unused strings that occur between used strings, when one wishes to strum the chord.

I'm assuming we will always be able to either deaden them, or play them open, or stop them at a fret that will harmonise with the rest of the chord by using a barre or similar. What do you call it when you use a finger other than the index finger to stop two or three adjacent strings at the same fret? A mini-barre. :-)
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Are you interested in a fretboard that does all these things?

I'll keep you in suspense a little longer, about what the fretting actually is. :-)

I think it's one of those things that when you see it you'll say, "That's so obvious, why didn't anyone think of it before!". Although I guess the lack of octaves is the biggest thing against it.

How did you go with your luthier?

[Pete:]

He put me off for another week, and I didn't hear back from Mark Rankin, yet, either.

Well, that sounds pretty danged cool, Dave. Hopefully we'll get to hear it fairly soon.

I like the Tumbling Dekany, too.

[Dave:]

Hi Pete,

Here's the fretting. Note that all the approximate ratios of 11 are exact 9-limit ratios from the strings on either side (a neutral third away).

fret cents ratio
0 0.0 1/1
1 35.3 55/54 +3.6c (5/6 +N3)
2 84.1 22/21 +3.6c (6/7 +N3)
3 119.4 15/14
4 351.0 11/9 +3.6c (1/1 +N3)
5 386.3 5/4
6 435.1 9/7
7 470.4 55/42 +3.6c (15/14 +N3)
8 702.0 3/2
9 737.3 55/36 +3.6c (5/4 +N3)
10 786.1 11/7 +3.6c (9/7 +N3)
11 821.4 45/28
12 1052.9 11/6 +3.6c (3/2 +N3)
13 1088.3 15/8
14 1137.0 27/14
15 1172.4 165/84 +3.6c (45/28 +N3)
16 1403.9 9/4
17 1439.2 165/72 +3.6c (15/8 +N3)
18 1488.0 33/14 +3.6c (27/14 +N3)
19 1523.4 135/56

I realise that it won't be very enlightening just looking at this table.

It may help to list the spacings between frets in cents. They go

35.3
48.8
35.3
231.5
35.3
48.8
35.3
231.5
35.3
48.8
35.3
231.5
35.3
48.8
35.3
231.5
35.3
48.8
35.3

So you can see there is a repeated pattern of four frets per neutral third. Each eikosany chord is playable entirely within one of those close groups of four frets. I've shown the layout roughly below (except of course on the real fretboard the frets get closer together as you go down). I've also used it to show some otonal and utonal 11-limit hexad patterns below. All the eikosany tetrads are subsets of these patterns transposed up down or sideways.

Notice when they are transposed up and down _within_ the group of four, you lose either the 5 or 7 identity or both, as shown in the last three otonal examples below.

And of course when these patterns are transposed sideways, identities are lost off the right or left in the obvious manner.

-------7---- nut or zero fret
------------
-4---6---9-11 (otonal)
---5--------

--------/5--
11/9--/6--/4 (utonal)
------------
----/7------

-4---6---9-11
---5--------
------------
------------

------------
-------7----
------------
-4---6---9-11 sub octave 1172 c

------------
-4---6---9-11
------------
------------ farthest fret needed for eikosany 1523 c

-- Dave

[Dave:]

Hi Pete,

I've been investigating the best open string tuning for this guitar.

Because it's tuned in neutral thirds its range won't be as great as a standard guitar and I figure it's best to lose a little off the bottom and a little off the top. And we also want the fretted pitches to be notated with as few accidentals as possible. Fortunately these two things combine to give only one answer. The open strings should be (from low to high pitch)

Gb/ Bbt Db/ Ft Ab/ Ct

where
/ and \ represent 5-comma up and down
f and t represent 7-comma up and down
^ and v represent 11-M-diesis up and down (semisharp and semiflat)

Assuming we want A to be 440 Hz, the cents deviations of the open strings on a 12-ET tuner will be:

Gb+12.0c Bb-37.0c Db+13.9c F-35.1c Ab+15.9c C-33.1c

Here are the required string gauges, assuming you normally use "light" strings.

.011, .013, .018, .026, .030, .038. The first three are plain and the last three wound, as usual.

Somewhat paradoxically, you might actually get all of these gauges (or close to them) by buying one medium and one medium light set of standard strings, depending on the brand.

.011 medium E
.013 medium B or medium-light B
.018 medium G or medium-light G
.026 medium-light D
.030 medium D
.038 medium A or medium-light A

[Pete:]

Hi Dave,

Sorry I'm falling a little behind here. This looks really interesting! A couple of possibly 'dumb' questions, on the fly?

[Just as general comment, I think it would be better to go lower in the register, but that may be only because I'm using a lot of low D's and even C's on my 12-equal acoustic guitar. But I'm considering trying the Mark Rankin system on a little (!) nylon string that was my son's, because it needs fret work, anyway, and I'd like to experiment with more radical microtuning in that timbre. Your present basic open string arrangement might be really good for that. --Unfortunately, I can't afford a good acoustic to start trying it on, and until I'm pretty certain, the acoustic I have is quite valuable and useful as-is, just in case I ever need to sell it, or play jazz in a restaurant ;-), or something. There's still the option of a new bolt-on neck for a Fender-style, and I can have that made to pretty high precision, I believe, once we're decided.]

I think I'm as ready as I'll ever be to dispense with octaves, right now. Could more of that sacrifice increase the overall justness, or make more complete four-note tetrads available?

How much could microtempering of the 9-limits reduce the microtempering of the 11-limits? Or, what's the range of trade-off, if that's what's happening? (Also above, re tetrads)

--Please forgive me if I'm just not seeing the plan clearly, yet, or didn't read it carefully enough.

This is what Kraig Grady said to me about the very basic idea (before you sent these plans), just in case any of it is germane to the project. That is, feel free to ignore it, if it's not useful to you:

[Kraig:]
>More often than not pitches that land close to each other are at remote places in the tuning and if you tune them out you lose the remoteness when you move to that area. Getting rid of dissonance is not always a good thing as you know. But i need to look at it again. With the 22 tone eikosany if you get rid of notes you also destroy the melodic MOS feel and you are going to want pitches to fill those spots, which might happen already with straight across fretting.

(The 231.5-cent step is a little alarming to me as an improvising melodist and serendipitous-at-times composer, but I'll not worry you with that, until I've tried some things around it).

Hopefully, I'll steal some time to get a tuning up on my synth(s) this weekend. I'm not very handy with Scala yet, so I'll just brute it onto my DX-7II, probably. I'll try to look mainly at the possiblities from the handiness of the tetradic relationships. My left hand's pretty stretchy, so I'd also try to view it as acompanimental for a 'strummer' (or pedagogical?), as much as aspiring to its 'virtuosic' usefulness, whether as a chord-melodic OR melodic instrument. I'm no virtuoso, but I have a few chops.

Maybe if I can prove my mettle as a guitarist in this vein, everyone WILL want one! ;-)

Thanks much!

[Dave:]

>[Just as general comment, I think it would be better to go lower in the register, but that may be only because I'm using a lot of low D's and even C's on my 12-equal acoustic guitar.

OK. If you lower everything by a just fourth the notation is no worse and the lowest string will be Db/ .

But note that while a standard guitar has two octaves (2400 cents between low and high strings, this one has only 5 neutral thirds or about 1755 cents. That's 645 cents less.

> But I'm considering trying the Mark Rankin system on a little (!) nylon string that was my son's, because it needs fret work, anyway, and I'd like to experiment with more radical microtuning in that timbre.

Good idea to experiment with something inexpensive first. I suggest you remove all the frets, fill the grooves with an epoxy or polystyrene wood filler and sand the fingerboard smooth, then get some extra nylon G strings and tie them on as frets. See

http://www.mugwumps.com/FretKnot.html

The picture here is for gut strings. It will probably work for nylon if you loop through the knots many more times. The basic idea is to make a loop in one end that the other end will slide around like a pulley for tightening, then tighten it and tie the two ends together. Because nylon is so slippery you have to do about 5 turns to every knot.

>I think I'm as ready as I'll ever be to dispense with octaves, right now. Could more of that sacrifice increase the overall justness, or make more complete four-note tetrads available?

No. With the current proposal, octaves have already been sacrificed utterly. The presence of the 1172 cent sub-octaves is purely incidental.

We've got all the tetrads of the 4.5.6.7.9.11 eikosany plus lots of others, so I'm not sure what more you could want in that regard?

>How much could microtempering of the 9-limits reduce the microtempering of the 11-limits? Or, what's the range of trade-off, if that's what's happening?

Here are the errors in cents for all the intervals, at three points along the tradeoff continuum. The current proposal (untempered 9-limit) is the top row. The second row is what I'd normally recommend since it minimises the maximum absolute error over all intervals. The third row minimises the maximum beat rate in the otonalities.

Note that the two most critical intervals are 4:9 and 9:11 which are pulling in opposite directions (as you might expect since we're making half of our 2:3 act as our 9:11).

Cent errors in intervals
2:3 4:5 5:6 4:7 5:7 6:7 4:9 5:9 7:9 4:11 5:11 6:11 7:11 9:11
---------------------------------------------------------------------
0 0 0 0 0 0 0 0 0 3.6 3.6 3.6 3.6 3.6
-1.4 -1.4 0 -1.4 0 0 -2.9 -1.4 -1.4 0 1.4 1.4 1.4 2.9
-2.9 -4.0 1.1 -4.0 0 -1.1 -5.8 -1.8 -1.8 -3.7 0.3 -0.8 0.3 2.1

Incidentally, if we made everything strictly just, half the frets would break up into fretlets which would zig-zag across the fingerboard in 7.1 cent jumps (the neutral thirds kleisma 243/242).

>--Please forgive me if I'm just not seeing the plan clearly, yet, or didn't read it carefully enough.

Not a problem. I haven't really explained how it works, since I thought you implied you were in a "just give me something to try" mood.

The main ideas are:

1. We're making the chords of the 4.5.6.7.9.11 eikosany playable by spacing the strings at neutral thirds since that's the average size of the step intervals in the 4:5:6:7:9:11 chord and its utonal inversion.

2. We're turning those 7.1 cent staggered fretlets into straight frets by distributing the neutral-thirds kleisma around the place in small enough pieces that you (hopefully) don't notice it.

The problem with the 4.5.6.7.9.11 eikosany is that it is very uneven melodically, as you have noted. The problem with any octave-specific version of the 1.3.7.9.11.15 (Pascal) eikosany is that it is harder to map to the guitar since there is no even voicing of a 1:3;7:9:11:15 hexad. e.g. 4:6:7:9:11:15 has fifths on the ends and thirds in the middle. 6:8:9:11:14:15 is slightly more even since it goes fourth, second, third, third, second, but it's nowhere near as consonant with those seconds in it. There may be some reasonable solution for Pascal, but it certainly wouldn't have the pedagogical value of the simplicity and uniformity of the current proposal.

>This is what Kraig Grady said to me about the very basic idea
>
>>More often than not pitches that land close to each other are at remote places in the tuning and if you tune them out you lose the remoteness when you move to that area.

I'm not tempering out any of the small intervals between notes of the eikosany (e.g. those 35 cent intervals). Those always remain as distinct notes. What I'm doing is finding open string sets that cause many fretlets to line up or to nearly line up, and then I'm tempering out the very small misalignments of fretlets on different strings so they all line up. In our case the misalignments are 7.1 cents, and in the current proposal we're just splitting that difference. But as you can see from the table above, it is sometimes possible to spread the difference around to more places so it is spread even more thinly.

I suspect Kraig is thinking of conventional tempering where two notes which are distinct in JI might become a single note on tempering, as in the meantone diatonics. With this guitar microtempering the JI notes retain their separate identities, but may be adjusted slightly (generally no more than 3 cents) to bring fretlets into alignment.

>>Getting rid of dissonance is not always a good thing as you know.

Agreed.

>>But i need to look at it again. With the 22 tone eikosany if you get
>>rid of notes you also destroy the melodic MOS feel and you are going to
>>want pitches to fill those spots, which might happen already with straight across fretting.

Same thing. To "get rid of notes" would take much more drastic tempering than anything I am proposing.

>(The 231.5-cent step is a little alarming to me as an improvising melodist and serendipitous-at-times composer, but I'll not worry you with that, until I've tried some things around it).

Yes. They're pretty scary to me too. There's no obvious way (to me as yet) to split those gaps without destroying the pedagogical property whereby practically anything you can play will sound OK.

I'm trying to look upon them as a feature, not a bug. :-) Just think of pentatonic scales which have step intervals of a minor third. At least these are only supermajor seconds (slightly large wholetones) and you get a heptatonic scale.

Melodically I think of the outer frets of each close set of four as the notes of the scale and the inner two are merely 35 cent inflections on these as required for the harmony.

>I'm not very handy with Scala yet, so I'll just brute it onto my DX-7II, probably.

That doesn't let you tune all keys separately does it? You're stuck with octaves, right? My synth's the same. But with two synths you should be able to get most of the notes of the guitar.

By the way, there are two frets I've included (because they fit the repeating pattern) that are not actually used for the eikosany. They are the frets at 786 c and 1088 c.

>Maybe if I can prove my mettle as a guitarist in this vein, everyone WILL want one! ;-)

Yes. Sometimes limitations (such as the 232 cent steps) are good for creativity. And they are bound to give the melodies a distinctly non-12 non-diatonic sound.

[End of part 2]