31 frets

Here is the neck of a guitar that I’d like to have made someday, if I should ever develop the dexterity to make it worthwhile. The blue stripes show where standard frets would be, for comparison.

The tuning is my tweaked version of meantone: compared to just intonation, each factor of 2 is sharp by 1/16 comma, each factor of 3 is flat by 1/8 comma, and each factor of 5 is sharp by 1/4 comma. (A comma is the difference between 64:81, the Pythagorean major third derived from compounding fifths, and the more harmonious 4:5.) This makes the thirds and sixths much truer than in equal temperament, and the fifths slightly truer than in traditional meantone, which puts all the error in the 3s.

This design has 31 frets in the first octave: 12 flats, 7 naturals, 12 sharps. The bent frets span the difference (~151:152) between 18 of my sharp octaves and 31 of my flat fifths.

To reduce crowding, the second octave has only three flats and three sharps. The bent frets span the difference (~50:51) between G♯ and A♭.

The charts below should look familiar to players, if you squint a little.

The dots are placed according to a slightly different scale, which divides a factor of 12 into 111 equal steps; this is a local optimum among cyclic scales by the same criterion I used to choose the non-cyclic intervals for the frets.

2 thoughts on “31 frets”

  1. This temperament, though optimized for 5-limit, gives many further acceptable intervals “for free”; for example
    F♭:G♯♯ ≈ 7:9
    B:F ≈ 7:10
    F:B ≈ 5:7
    C♭:E♯ ≈ 8:11
    C♯:E♭ ≈ 7:8
    G♭:E♯ ≈ 6:11
    B♯:F♭ ≈ 10:13
    E♯:C♭ ≈ 11:16
    E♭:C♯ ≈ 4:7
    G♯♯:A♭♭ ≈ 7:13
    A♯:D♭ ≈ 9:11
    B♭:F♯ ≈ 7:11
    E♭:G♯ ≈ 13:17
    B♭♭:F♯♯ ≈ 7:12
    A♯:G♭ ≈ 8:13
    C♯:B♭ ≈ 7:12

  2. E A D G B E
    F♭ B♭♭ E♭♭ A♭♭ C♭ F♭
    E♯ A♯ D♯ G♯ B♯ E♯
    F B♭ E♭ A♭ C F
    G♭♭ A♯♯ D♯♯ G♯♯ D♭♭ G♭♭
    F♯ B E A C♯ F♯
    G♭ C♭ F♭ B♭♭ D♭ G♭
    F♯♯ B♯ E♯ A♯ C♯♯ F♯♯
    G C F B♭ D G
    A♭♭ D♭♭ G♭♭ A♯♯ E♭♭ A♭♭
    G♯ C♯ F♯ B D♯ G♯
    A♭ D♭ G♭ C♭ E♭ A♭
    G♯♯ C♯♯ F♯♯ B♯ D♯♯ G♯♯
    A D G C E A

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