A combination-product-set guitar (part 1)

Dear MMMers,

Here's an edited version of an email conversation that Pete McRae and I had a year ago. We thought it might be of interest.

Summary:

In response to Pete's desiderata, I design a guitar with all straight-across frets, that makes playable all the tetrads of Erv Wilson's (3 of 6) 4.5.6.7.9.11 eikosany (octave-specific). It achieves this by
(a) abandoning octaves (although it has some sub-octaves of 1172 cents)
(b) slightly tempering the ratios of 11 (by 3.6 cents).
Or, if you don't like to use the tempered 11's then it still plays many 9-limit subsets of the eikosany, including dekanies and hexanies, in strict JI.

It's fairly long so I'm posting it in a few parts. In part 1 (below) we discuss philosophy and settle some requirements of the design.
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[Pete:]

Hello Dave,

Paul Erlich was kind enough to tell me that you've designed a guitar
fretting for [an] Eikosany. I understand it's to be published in a
forthcoming issue of Xenharmonikon. I have an electric guitar that
I had fretted in the (3,6) 1.3.7.9.11.15 Eikosany, about 12 years
ago, copied from Kraig Grady's now defunct guitar, and encouraged by
Erv Wilson, who called me a "brave soul" for committing to it. :-)

If there's any way you'd consider letting someone have a peek at
your fretting plan right now, I'd be most humbly interested. I'm a
great devotee of Erv Wilson, and have a deep and abiding commitment
to honoring the originators of any tips or tricks I get up my
sleeve. I'm hardly an expert on the theories, being rather doltish
in the mathematical and scientific fields, but Erv's work is to me
so stunningly elegant, original, and advisedly unique, it's
impossible for me to overrate it, intuitively, I think.

If you've found a way to make it more practicable, then add that to
the very impressive list of admirable qualities I see in your work,
too.

[Dave:]

How wonderful to hear from you Pete!

You will find the Xenharmonikon paper here. http://dkeenan.com/Music/MicroGuitar.pdf
All I ask is that if anyone makes use of it, or simply enjoys reading it, they let me know, as this will give me great pleasure.

The basic idea is that by allowing a tiny bit of tempering (that is less than the typical fretting+tuning+playing errors of guitars anyway) we can get a drastically simplified fretting (fewer fretlets and mostly straight across) without having to tune any open strings as far apart as fifths or as close together as seconds, and thereby making more chords playable.

However, the actual Eikosany designs do not appear in the Xenharmonikon article, but predate it and are in the tuning list archives. The first is not _your_ Eikosany but the 1.3.5.7.9.11 (so feel free to skip it).

Note: To see the diagrams in their proper columns, hit the "Reply" button or "Forward" a copy to your own email address.
Described here.
/tuning/topicId_26614.html#26614
Fretboard diagram here.
/tuning/topicId_26614.html#26628
Errata and explanation of diagram here.
/tuning/topicId_26614.html#26629

Then Kraig told me about your guitar here
/tuning/topicId_26657.html#26657
and here.
/tuning/topicId_26657.html#26666

So I understand your guitar has (1.7.15)/3 and (3.9.11)*3 in addition to the 1.3.7.9.11.15 Eikosany. Erv calls this "Pascal" in figure 12 about halfway thru here.
http://www.anaphoria.com/dal.PDF

So then I had a go at a Pascal guitar design and got rather too carried away with my own cleverness here
/tuning/topicId_26703.html#26703
and posted a corrected version here
/tuning/topicId_26703.html#26749

We would almost certainly get a _much_ better design using the "Semithirds" microtemperament (listed in the attached paper) rather than the "Triple-kleismic" that I used in the above posts. This is because Semithirds does not try to approximate ratios of 13 and is far less complex than Triple-kleismic while having the same maximum error (2.8 cents). I didn't know about Semithirds when I did that first Pascal design.

It would be nice to "prove Kraig wrong", when he says that he suspects that Eikosanies may be "outside the language of guitars", by producing a Pascal guitar where all the tetrads are playable. Would you consider refretting your guitar if we came up with a fretting (and open string tuning) that looks significantly better than your current one?

Please let me know what you think of the design criteria described in the paper. For all I know, some of them may be completely irrelevant or wrong. We could be a good team. I do not have your musicianship and experience of playing guitars in JI tunings. It's all just theory to me.

By the way, I think Erv's a genius.

[Pete:]

And how nice to get such a prompt and generous reply!

And how long is it going to take _me_ to digest all this? :-)

Seriously, it would be a bit of a coup, I think, to expand Kraig's--and possibly thereby Erv's--thinking on the particular issue of guitars and...

Kraig has said to me that other tunings might be more felicitous for guitars, but I assumed he meant timbrally as much as technically (in terms of common guitar playing techniques). I pretty much let it drop there, having not yet researched what I might find most interesting tonally on a more or less ordinary guitar. It's the "Just give me something to play and I'll tell you what I think" sort of attitude. Relative familiarity is not an issue for me. The difficulties are just in perspective, or standpoint, I think. But I have reason to believe I may be unusual, in that respect. :-)

[Dave:]

Don't worry about learning to apply the method, I'm happy to do that. What I need from you is a better understanding of what you would like to achieve (what scales and chords playable etc) and whether you think of the 7 numbered criteria I have given in the sections entitled:
Criteria for intervals between open strings
Criteria for scale rotation and fret layout

I guess we'd add "Playability of the maximum number of tetrads".

I assume you'd like to make a guitar for one of Erv's CPSs. Do you have a particular attachment to the 1.3.7.9.11.15 Eikosany? 1.3.5.7.9.11 is probably easier to make a guitar for, but is less even melodically. Or what about something even less ambitious like a Dekany?

But like I say, don't worry too much if you don't follow the actual microtempering method for choosing fretting and open tuning. Just try to get a general feel for what I'm on about, and make sure you don't have any philosophical objection to the idea of microtempering JI.

> Relative familiarity is not an issue for me.

That's good. So you wouldn't have a problem with open tunings like
EGCEGC or EGBEGB (give or take a transposition)?

[Pete:]

Oh, no. Certainly not. I use the open strings Kraig listed in the postings you linked, except there is no G#, just D# and A# (I don't think he ever actually used anything else, either). And I've been playing around quite a bit with other 'standard' open tunings on my conventional guitars.

But in my idle musings today, it occurred to me to try to give a lot of consideration to someone who's a complete novice, or very much a hobbyist, or a Bob Dylan sort of wanna-be, you know? for whom instrumental technique may be a very low priority, or whatever, a Joni Mitchell who tunes to maj7#11 chords... :-)

One of the things Erv said to me was, "I think it would be very pleasant to listen to you play and sing." So, now I'm really thinking about how to make that more likely, and for anyone to pick up a guitar who has a willing voice, and NOT hear the same old thing.

I also have had some ambitions to be a schoolteacher, so I'd really like to come up with something to sort of "trick" American kids into hearing non-12-equal music without making any kind of a fuss about it. Anyway...

> Just try to
get a general feel for what I'm on about, and make sure you don't have any
philosophical objection to the idea of microtempering JI.

My sense from a quick scan of the paper was that you have taken all of the usual problems into consideration and dealt with them really well! I expect I'll have more to say about that specifically, later. I DO have philosophical objections to the idea of microtempering JI, but they are a.) just that: philosophical, mostly (see below), and b.) probably not relevant to what I think would be the most efficacious way to provide someone an alternative to 12-equal. Werkmeister 3 works great on my keyboards, but it's not really any kind of departure from the hegemony, to me.

Some people don't like JI, and CPS's can be quite a leap into unknown musical territory. It's been my observation that a lot of folks get bogged down in a kind of pedantry (or what Kraig hilariously called "5-limit reactionaries") about JI, or historical temperaments, or a kind of A-B-C... cause-and-effect "evolutionary" notion of expanded octaves and such. I'm still trying to decide exactly where I am with it, so this line of inquiry comes at a very good time, for me.

Part of my high praise for Erv was intended as fair warning that I consider his work somewhat sacrosanct, so I guess all I could say at this point is that I wouldn't call it an "Eikosany Guitar" if it wasn't, really. But there should be some way to do him justice as the innovator, no? Or, that a microtemperament of an Erv Wilson-derived tuning be submitted, and either endorsed or discredited by him? Maybe you've been through that, already?

Another thing I thought about was whether there's a possibility of a microtempered fretting for playability that could be untempered by some other means, like a Buzz Fieten compensated nut thing (?).

I think it would be best, for MMM, to get a guitar together, play some music, make an mp3 (or suchlike) of it and post it on the web, for the MMM-er's to peruse as a kind of exclusive.

A less ambitious Dekany might be a great start, as much as I'm fascinated by Pascal, and would love to make it more easily playable, in all its glory. :-)

[Dave:]

Hi Pete,

Well I've spent a bit of time working on the problem over the past 24 hours and the only way I can see to get all 15 otonal and 15 utonal tetrads of an 11-prime-limit eikosany to be playable on a 6 string guitar, is to use an octave-specific 4.5.6.7.9.11 eikosany and to tune the open strings in neutral thirds (alternating 9:11s and 22:27s).

Have you heard my Tumbling Dekany Excel spreadsheet. That's an example of using an octave-specific (opposite of octave-equivalent) CPS.
http://dkeenan.com/Music/StereoDekany.htm

>But in my idle musings today, it occurred to me to try to give a lot of consideration to someone who's a complete novice,

You mean you want an instrument that any one can pick up and start strumming interesting JI chords immediately, and which is obviously very different from 12-ET?

In considering the above 1.3.5.7.9.11 eikosany I assumed we're really going for maximum playable chords, in which case melodic evenness goes right out the window. The largest steps (only at the extremes of the range) are minor thirds (although they will probably end up being broken up by extending some fretlets to full width). The smallest steps are 32 cents.

We'd also be throwing octaves out the window and be transposable only by fifths (or neutral thirds). Can you handle that, or is it too radical?

The thing is, to make all 30 tetrads playable, it was easiest to make each one playable in only one inversion - its most natural and most consonant inversion (and in only one octave).

Because the 6 strings are only spaced apart by neutral thirds the range is smaller than a standard guitar (and much smaller than your current Pascal guitar) and you have to use the cutaway to play the highest tetrad on the high four strings (its maximum fret position is 1484 cents from the nut).

A beneficial effect of this is that you can play common-note chord progressions without ever having a note jump by an octave. They really are always common notes.

I'm looking at an open tuning of Dv F Av C Ev G where "v" represents a semiflat. F is 4.5.7, C is 5.6.7 and G is 5.7.9.

Unfortunately the open notes Dv Av Ev are not actually notes of the eikosany, but tuning the open strings like this has two benefits.
(a) it results in more fretlet alignments (and so more continuous frets),
(b) it makes all chord patterns transposable sideways (across the strings), as well as up and down the fretboard, thereby making the chords much easier to learn.

>I also have had some ambitions to be a schoolteacher, so I'd really like to come up with something to sort of "trick" American kids into hearing non-12-equal music without making any kind of a fuss about it.

Great! I've had similar desires myself.

So we want to make it easy to play chords that sound good. Open neutral thirds work really well for that since 6:7, 5:6, 4:5, and 7:9 intervals are within +- 85 cents (less than one 12-equal fret) from that.

>I DO have philosophical objections to the idea of microtempering JI,

Well we can look at two versions of the fret layout - the strictly JI version, and the Miracle tempered version (max 11-limit error 3.3 cents).

The real test would be to make them both up and see if anyone can tell the difference in a blind test (e.g. posted to MMM). My guess is that the errors in fret placement, open string tuning, and natural variation in finger pressure will swamp the tiny systematic errors of the microtemperament.

The trouble with this idea is that it will be much harder to make the strictly JI fingerboard since it will have a lot more fretlets. But we should draw it and see.

>Some people don't like JI, and CPS's can be quite a leap into unknown musical territory.

But that's what you want, right?

>Part of my high praise for Erv was intended as fair warning that I consider his work somewhat sacrosanct, so I guess all I could say at this point is that I wouldn't call it an "Eikosany Guitar" if it wasn't, really.

We may have a philosophical impasse on this, but here's my personal view:

Since it's obviously OK to make a guitar (without microtempering) and call it JI even though, when you actually play it there are typically intonation errors of 6 cents or more (compounded from many sources), why isn't it OK to introduce much smaller errors deliberately, to give us more continuous frets, and still call it JI?

Surely it's what it sounds like that matters, not the mathematics used to design it?

I think Kraig Grady once said that 72-ET is the smallest ET that he would consider using to approximate JI, or words to that effect.

> But there should be some way to do him justice as the innovator, no?

Absolutely. Call it an Eikosany if it sounds like one and is structured like one. We don't even need to mention the microtempering.

>Another thing I thought about was whether there's a possibility of a microtempered fretting for playability that could be untempered by some other means, like a Buzz Fieten compensated nut thing (?).

No. That's not possible because different frets on the same string will be microtempered in different directions. If there was any systematic correction available, that would be done in the optimising of the microtemperament before it was applied.

The compensated nut and bridge things are still a great idea, because they help us reduce the unintended errors, so they don't add to those of the microtemperament (of course they are equally likely to subtract from them too).

>>> I think it would be best, for MMM, to get a guitar together, play some music, make an mp3 (or suchlike) of it and post it on the web, for the MMM-er's to peruse as a kind of exclusive.

Yes. It would be insanely great to demonstrate a chord sequence cycling thru all 30 11-limit tetrads with common tones!

>A less ambitious Dekany might be a great start, as much as I'm fascinated by Pascal, and would love to make it more easily playable, in all its glory. :-)

Unfortunately it's the 11:15 intervals of Pascal that make it harder to do on a guitar.

[Pete:]

Hi Dave,

Just a jot, for now.

This is wonderful stuff. It makes me want to warn you again, though: I may be REALLY slow on the uptake of some of this, but I WILL get it. :-)

Yes, to be able to play through all the tetrads with common tone modulations would be a great thing! Then some (stellated) hexanies, I guess, and whatnot, yippee! (Erv told me, too, that he'd really like to hear the hexanies played on it.)

I anticipated a philosophical impasse of sorts after reading the tuning list stuff, somewhat cursorily. Not a big deal, anyway, as you seem to agree. Either don't mention the microtempering, or--more considerately/conservatively--call it something less obviously Erv-like, in case it turned out he DID hate it.

Just as of today, it seems, I'm truly worried that when he goes, the world will be completely lacking for a very long time someone who understands the musical universe both intuitively AND intellectually (and please pardon my grandiosity, or 'cosmic'-ness).

[End of part 1]