General > Theory

53edo,87edo,94edo

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M.Autry:
This post continues a conversation that began in the 'new member introduction' forum, between myself and Gedankenwelt, regarding various equal division of the octave. Please see that area for the previous posts if you are interested. Gedankenwelt shares a good bit of info in a very brief post, and a couple of wonderful chords.
Gedankenwelt,
I checked out the two chords that you recommended and found them quite enjoyable. I have only used 21/16 as the seventh in a V7 chord. I now here the resonance of the interval more completely. Thanks for sharing! I have been pondering the 87edo and the 94edo. I had not done so previously, as I have been thoroughly engaged in the 53edo. I see now the utility of these divisions of the octave. I am especially interested in 87edo. I think that it could be very useful for a fretted instrument. 87 frets will not fit in the octave of a guitar, but one could use the same idea that I am using to get 53 notes to the octave. One could basically place frets for 29edo, using every third degree of 87edo, then use three strings to hold all of the notes. Again, each string will only contain a third of the 87 notes, and the range would be reduced. I think a long scale eight string would work well. Ten strings might be even better since the frets get pretty crowded in the second octave.One possible tuning could be to stack minor thirds of various sizes.I have not been able to think of useful fretwork for 94edo, although I prefer the fifths and major wholes of 94edo. I will continue to contemplate 87edo, and will likely have to build an instrument. I thank you for the inspiring information, though I can already hear my wife complaining,"Another guitar?!?.  :)  That's all for now.    M.A.

Gedankenwelt:
Wow, that sounds like an awesome project! :)

I'm not very experienced with these kinds of tuning, but I think it might be worth trying to tune the strings in 8:7 steps. 87edo tempers out 1029:1024, so three stacked 8:7 make a 3:2. For 10 strings, the result would be:

2/3 - 16/21 - 7/8 - 1/1 - 8/7 - 21/16 - 3/2 - 12/7 - 63/32 - 9/4

...so it's basically a typical cello or violin tuning with four strings tuned in fifths, except that each fifth is again divided into three equal steps that represent 8:7. The temperament associated with stacking 8:7 intervals in 87edo is called rodan, and is also supported by 41edo and 46edo.


Tuning in minor thirds sounds like it might be useful, too. You can either use 6:5 minor thirds for all strings, or different kinds of minor thirds. In the first case, you'd get hanson/kleismic/countercata temperament, which is also supported by 34edo and 53edo. In this case, 6 strings make a perfect twelfth 3:1, and three strings minus six 29edo-frets make a perfect fifth 3:2. Here again the result for 10 strings:

15/26 - 9/13 - 5/6 - 1/1 - 6/5 - 13/9 - 26/15 - 25/12 - 5/2 - 3/1

I'm not sure what might be a good tuning based on different kinds of minor thirds. It's tempting to use some kind of unequal diminished seventh chord as a basis, where four strings make an octave. But then some notes become less accessible, because they're only on every fourth string, rather than on every third string.
From that perspective, it might be better to use a tuning where three successive strings always add up to a multiple of a 29edo step, which would require that either all string intervals must be a 29edo interval plus an 87edo step, or all string intervals must be a 29edo interval minus an 87edo step.

M.Autry:
I like the idea of stacked 8/7's. This would create a convenient symmetry to the fingering patterns. The main drawback to such an instrument is the reduced pitch range, in comparison to the standard guitar. However, I think that the harmonic/intonational variety more than makes up for it. One of these days I will refret a seven string Ibanez that I have with this fret array. Tuning-D 2/3,E+ 16/21,G- 7/8,A 1/1,B+ 8/7, D- 21/16,E 3/2. This is the range between the 1st and 4th strings of the standard guitar tuning.

Gedankenwelt:
Hm ... I'm a little sceptical if 7 strings are really enough with the stacked 8/7 setup. That'd mean that in chords, octaves 2/1 and major ninths 9/4 aren't very accessible (and not accessible at all when there's a fifth) except from the lowest string.
Personally, I wouldn't use a guitar with less than 8 strings, and preferably with 10, but maybe that's just me.

In case of 10 strings, it might be useful to facilitate navigation by drawing marker lines along the fretboard (i.e. orthogonal to the frets) below strings 1, 4, 7 and 10.

Man, I'd love to play on such a 13-limit wonder guitar. :)

M.Autry:
I agree with you about the limitations of a 7 string instrument tuned in stacked 8/7s. I encounter similar chordal limitations with my eight string guitars. But I have a 'beater' 7 string and refretting is pretty simple compared to building an entire instrument. It will give me an opportunity to see if the whole idea works, and many scale patterns will be available. 10 strings could give the range of  'G' on the cello to the high 'E' of a standard guitar. I am not sure how many strings is really practical. I have seen 13 string lutes. 13 strings would take the range down to the low 'C' of the cello. If the 7 string proves the utility of the fretwork and tuning scheme, then I will take it further and build something with more strings.

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