"Mean Tone" Guitar
Found this on "you tube". Check out the frets on this guys guitar. Pure Major 3rds and almost pure minor thirds very cool!
posted by Paul Seburn at Sunday, January 20, 2008
posted by Paul Seburn at Sunday, January 20, 2008
1 Comments:
Well, those zigzag frets look nifty, I guess. But straight-line frets should also considered, even on a meantone guitar. Appropriate spacing of straight-line frets can result in full-blown meantone, with its pure thirds and other delights, and with fewer key-signature limitations than on a meantone piano or organ keyboard.
In equal temperament, the frequency ratio of each semitone to its predecessor is about 1.059. In meantone, some semitones are diatonic (about 1.070) and some are chromatic (about 1.045). There are 7 diatonic and 5 chromatic semitones in each octave.
If, in equal temperament, the distance between the base fret and the first fret is about 1.4 inches, then in meantone it will be about 1.6 inches or 1.1 inches, depending on whether that first interval is to represent a diatonic or a chromatic semitone, respectively. With the two types of semitones mixed together in each octave, straight-line frets in meantone will be spaced less evenly than in equal temperament.
The following table shows one possible arrangement of straight-line frets on a meantone guitar. The first column shows the fret numbers, 00 through 12, corresponding to the twelve semitones in the first octave. The second column indicates the size of the interval, diatonic or chromatic, between each fret and its predecessor. The remaining columns list the six notes available at each fret on a six-string guitar.
00 --- E A D G B E
01 chr E# A# D# G# B# E#
02 dia F# B E A C# F#
03 dia G C F Bb D G
04 chr G# C# F# B D# G#
05 dia A D G C E A
06 chr A# D# G# C# E# A#
07 dia B E A D F# B
08 dia C F Bb Eb G C
09 chr C# F# B E G# C#
10 dia D G C F A D
11 chr D# G# C# F# A# D#
12 dia E A D G B E
The above table lists both D# (on four strings) and Eb (on the G string). In equal temperament, D# and Eb are identical. In meantone, they differ by about two-fifths of an equal-tempered semitone, with Eb being the higher. Of course, D# is necessary to make the key of E useable, while Eb is necessary for the key of Bb.
Four such enharmonic pairs are listed in the above table:
D#/Eb A#/Bb E#/F B#/C
Thus, the following ten key signatures would all be playable on a meantone guitar with straight-line frets:
Bb F C G D A E B F# C#
By contrast, on a meantone piano or organ, only the first six of these key signatures are useable, because only one-half of each enharmonic pair exists on a keyboard.
A guitar with zigzag frets would seem to present key-signature limitations similar to those on a meantone keyboard, at least if the fret arrangement is designed to consistently present only half of each enharmonic pair.
Bill Smythe
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