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Author Topic: Neutral thirds tuning for quartertones on conventionally fretted guitar  (Read 2706 times)

Mat

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This tuning allows you to play quartertone scales on a conventionally fretted guitar or bass guitar. The modification is cheap and reversible, only requiring a custom set of single strings. This is 'stealth microtonality', no-one would know by looking at your guitar it is microtonal.

Introduction to 24EDO

The quartertone scale is also known as: 24 Equal Divisions of the Octave (24EDO), 24 Equal Temperament (24ET), 24 Tone Equal Temperament (24TET) or just 24 equal. Use of the word 'temperament' is sloppy, the quartertone scale is not necessarily a temperament, so I use 24EDO. 24EDO is a division of the octave into 24 equal steps of pitch, each step being a quartertone = half a semitone = 50 cents. Our standard system of 12 equal temperament is contained within 24EDO. 24EDO is the standard tones plus 12 quartertones, each one added halfway between the standard tones.

When naming the new intervals, super = quartertone sharp, sub = quartertone flat, neutral = halfway between major and minor.

A popular ASCII notation is to use up and down arrows: ^ = quartertone sharp, v = quartertone flat.

Semitones - Interval name - Abbreviation - Notation example

12 ..... Octave .......................................................... 8 ....................... C
11.5 .. Supermajor seventh .................................. ^7 ..................... B^
11 ..... Major seventh .............................................. 7 ...................... B
10.5 .. Neutral seventh .......................................... n7 ..................... Bv
10 ..... Minor seventh .............................................. b7 ..................... Bb
9.5 .... Supermajor sixth / Subminor seventh ..... ^6 / vb7 ........... A^ / Bbv
9 ....... Major sixth ..................................................... 6 ...................... A
8.5 .... Neutral sixth ................................................. n6 ..................... Av
8 ....... Minor sixth .................................................... b6 ..................... Ab
7.5 .... Superfifth / Subminor sixth ........................ ^5 / vb6 ............ G^ / Abv
7 ....... Fifth ................................................................ 5 ........................ G
6.5 .... Subfifth ......................................................... v5 ....................... Gv
6 ....... Augmented fourth / Diminished fifth ....... #4 / b5 ................ F# / Gb
5.5 .... Superfourth ................................................. ^4 ....................... F^
5 ....... Fourth ........................................................... 4 ......................... F
4.5 .... Supermajor third / Subfourth .................... ^3 / v4 ............... E^ / Fv
4 ....... Major third .................................................... 3 ........................ E
3.5 .... Neutral third ................................................. n3 ..................... Ev
3 ....... Minor third .................................................... b3 ..................... Eb
2.5 .... Supermajor second / Subminor third ..... ^2 / vb3 ............ D^ / Ebv
2 ....... Major second ............................................... 2 ....................... D
1.5 .... Neutral second ........................................... n2 ..................... Dv
1 ....... Minor second .............................................. b2 ..................... Db
0.5 .... Subminor second ...................................... vb2 .................... Dbv
0 ....... Unison .......................................................... 1 ....................... C

Neutral thirds tuning

The intervals between the open strings are all neutral thirds of 3.5 semitones = half a fifth. Therefore chords and scales can be transposed to any position on the fretboard without changing shape, modulation to any key is possible. Although the pitch range of a guitar is reduced, there are now 24 usable tones per octave, so the total number of notes actually increases.

Open string tuning
Semitones - Interval

High
12+5.5 ..... Supereleveth
12+2 ........ Ninth
10.5 ......... Neutral seventh
7 .............. Fifth
3.5 ........... Neutral third
0 .............. Unison
Low

The tuning follows the conventional chord structure of root - third - fifth - seventh - ninth - eleventh. The strings alternate between standard tones and quartertones. Playing the open strings or chords straight across one fret creates neutral versions of the conventional chords: 3 adjacent strings create a neutral triad, where the major or minor third is replaced by a neutral third halfway between at 3.5 semitones. 4 adjacent strings create a neutral seventh, where additionally the major or minor seventh has been replaced by a neutral seventh halfway between at 10.5 semitones. 5 adjacent strings add a ninth to the these chords and 6 adjacent strings add a supereleventh to this neutral ninth chord.

String gauges and restringing

I have used the D'Addario tension charts to create a sequence of gauges suitable for this tuning and with a slight, steady fall in tension from lowest to highest string. This tuning reduces the range of the instrument so choose a sequence of gauges that covers the required range. Acoustic guitars have a fixed intonation bridge so you need to use similar gauges to an acoustic set, the closest match is 48 38 30 24w 17p 13p. Using strings thinner than the nut slots is not problematic as long as there is downforce at the nut and the slot floors retain their curvature.

Guitar
74 59 48 38 30 24w 17p 13p 10p 8p

Bass
125 95 75 60 45 35 25w 17p 13p 10p 8p

For example 6 string electric guitar equivalent to a 10-46 set:

10p D^
13p B (standard B)
17p G^
24w E
30 C^
38 A (standard A)

4 string bass equivalent to 40-100 set (FACE tuning)

45 Ev
60 C
75 Av
95 F

Since with these gauge sequences the lowest string is the tightest, restring the lowest string first and tune it up to the highest tension you are comfortable with, this will set the pitch of your tuning. Because the gauges have changed you will need to move the saddles to reset the intonation.

Tuning the quartertone strings

Most guitar tuners display pitch from -50 cents to +50 cents. -50 cents = quartertone flat , +50 cents = quartertone sharp. When correctly tuned to the quartertone midway between 2 semitones the display may jump back and forth from the lower semitone +50 cents to the higher semitone -50 cents, as these are the same pitch.

Scale example 1 n2 n3 ^4 5 n6 n7 8

This scale introduces all 4 neutral intervals and the superfourth / 11th harmonic.
Scales can be defined as the pattern of a 1 octave run played across the strings. Red circles are the tonics.







To cover the entire fretboard with the scale, first draw the tonics and use them to position copies of the pattern above.







A 2 octave run through the scale on a 9 string guitar, showing how to play across the strings.







2 octave run through the scale playing up the strings. This involves skipping between 2 strings.







A diagonal 3 octave run through the scale on a 6 string guitar.

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Mat

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Re: Neutral thirds tuning for quartertones on conventionally fretted guitar
« Reply #1 on: February 21, 2012, 04:52:49 PM »

Chords

Red circles are the root tones.

Suspended superfourth chord, sus^4. A fundamental chord combining the
superfourth / 11th harmonic and the fifth / 3rd harmonic.







Neutral triad, n. Precisely midway between a major and a minor triad.







Suspended neutral second chord, susn2







Suspended superfourth neutral seventh chord, sus^4 n7







Suspended superfourth supermajor seventh chord, sus^4 ^7







Suspended superfourth subminor ninth chord, sus^4 vb9







Suspended superfourth neutral ninth chord, sus^4 n9







Neutral seventh chord, n7 . A neutral triad plus a neutral seventh.







Neutral triad supermajor seventh chord, n ^7







Neutral triad subminor ninth chord, n vb9







Neutral triad neutral ninth chord, n n9







Suspended neutral second neutral sixth chord, susn2 n6







Neutral ninth chord, n9. A neutral seventh chord plus a ninth. A ninth can be added to all previous chords.







Neutral ninth supereleventh chord, n9 ^11. A neutral ninth plus a supereleventh. A supereleventh can be added to all previous chords.

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Mat

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Re: Neutral thirds tuning for quartertones on conventionally fretted guitar
« Reply #2 on: February 21, 2012, 04:55:53 PM »

Creating and visualising scales

Here is the scale pattern master diagram with all 24 tones.





A scale can be defined by the pattern it creates within the black outline, the pattern is a 1 octave run played across the strings. For example ...





To cover the entire fretboard you 'tile' the fretboard with the shape outlined in black. I have added 2 tonics to the master diagram to help position copies of the pattern.

15 tones of 24EDO closely approximate a Just Intonation tonal system constructed from the 3rd and 11th harmonics (no need to understand what that means). The 15 tones contain 7 standard tones plus 8 quartertones.





If you choose your scale from the 15 tones shown the resulting scale will have a certain tonal consistency.

To narrow the choice further, here are the 11 most consonant tones of that Just Intonation system.





This is a good way to start designing scales and to acclimatise your ears to 24EDO. Choose your scale from the 11 tones shown.
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Mat

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Re: Neutral thirds tuning for quartertones on conventionally fretted guitar
« Reply #3 on: February 21, 2012, 04:57:59 PM »

Just Intonation

I'll now quickly describe the ability of the 24EDO neutral thirds tuning to be slightly retuned to play Just Intonation (JI) scales. It's not essential to understand this additional theory as it is independent from the theory above.

This is the Just Intonation major scale on a 7 string guitar. Intervals between the strings are alternating JI major and minor thirds (to within 2 cents).





Moving the tonic to strings 2, 4 or 6 results in JI minor scales. Here the tonic has moved to the major third of the scale above.





So with this open string tuning it is possible to modulate between 12 JI major and 12 JI minor keys, within certain limitations.

Retuning the intervals between the strings to alternating 267 cents and 433 cents, septimal JI scales can be played (that include the subminor third and subminor seventh / 7th harmonic, 'blue notes' of jazz and blues). The strings never need to be retuned by more than a semitone either way to play different JI tonal systems. It's possible to tune the JI intervals using harmonics, so a special microtonal tuner is still not needed.

Tuning by harmonics

The root, fifth, ninth etc. strings are tuned normally using a guitar tuner or by tuning to the 7th fret. The strings inbetween, above each of those by JI thirds, are tuned using harmonics:

JI major third, 386c. Tune the 4th harmonic of the higher string to the 5th harmonic of the lower (frequencies in ratio 5:4).
JI minor third 316c. Tune the 5th harmonic of the higher string to the 6th harmonic of the lower (frequencies in ratio 6:5).

Septimal JI subminor third, 267c. Tune the 6th harmonic of the higher string to the 7th harmonic of the lower (frequencies in ratio 7:6).
Septimal JI supermajor third, 435c. Tune the 7th harmonic of the higher string to the 9th harmonic of the lower (frequencies in ratio 9:7).
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Mat

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Re: Neutral thirds tuning for quartertones on conventionally fretted guitar
« Reply #4 on: February 21, 2012, 05:04:55 PM »

The first 3 posts above are attached to this post as a pdf file if you want to print out the diagrams ;)
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