Just Intonation on a Normal Guitar by Restringing and Retuning

[ixlramp]

samsegarra,
With this system, because it uses normal frets, the entire scale pattern can be freely shifted up and down the strings without altering the system.
So yes, you can modulate to the same JI scale on any root note (something conventional JI guitars cannot do). However, keep in mind that the root note will be moving by a 12TET interval, not by a JI interval. Not really a problem but if you are being a JI purist you might complain =)
Of course, a root note movement by a 12TET major second, fourth or fifth is very close to being a JI root note movement, because those 12TET intervals are very close to being JI intervals. So to a degree, JI modulation is possible.

 

[ixlramp]

I'm trying to figure out if I could play in JI in multiple keys if I use the suggested gauges and tuning, or only in F?

The system does not force a root note of F, the root note of a JI scale can be placed anywhere on the so-called 'tonic' strings.
Please read the section 'Modulation to other scales' in post 14, that seems to answer your questions.

 

[bostjan]

I suppose that the only system where arbitrarily modulated JI is possible is fretless.

Last year, I was playing around with something. I wanted JI major and minor (Indian Classical) fretting, then I started wanting to be able to modulate up a fifth, which shifts a few notes, but the frets were going to be impractically close together, and I thought "what if I compromise between the two fret positions and place the fret exactly halfway in between - no one will notice" and then realized that I just foolishly "reinvented" ET.

 

[ixlramp]

Since this thread is currently being bumped, i will use this opportunity to post that i realised that the optimum 'base tuning' (the initial 12TET tuning of the guitar) for this system is all-minor-thirds. Playing melodically through scales across the strings works very well, and chords have ergonomic shapes, everything is beautifully laid out on the fretboard. Of course, this slightly reduces the range of the guitar compared to the other base tunings.
I intend to post in this thread details of the minor thirds version of this system.

The second best base tuning is all-major-thirds, detailed in post #18 of this thread, which slightly increases the range of the guitar at the expense of slightly less playable scales and chords.

 

[Pyramid Gallery]

Wow, I'm unwilling to change my tuning, but I would love to hear what this sounds like on a keyboard, tuned to the JI methodology. Does it sound Middle Eastern-like?

 

[bostjan]

Wow, I'm unwilling to change my tuning, but I would love to hear what this sounds like on a keyboard, tuned to the JI methodology. Does it sound Middle Eastern-like?

Not really. JI isn't exactly a tuning, it's more loosely an approach to tuning. The system outlined in th OP is based off of just major and minor thirds and 12edo 2nds and 5ths, so it sounds like a sweeter version of "normal" music- the isn't anything Middle Eastern about it. Middle Eastern, or "maqam" music uses quarter steps added into standard tuning, so that some scales have neutral intervals instead of major or minor. That's usually achieved by taking 12edo standard guitar tuning, and adding tones exactly halfway in between, either by detuning a string 50 cents or by adding frets placed in between the existing frets.

 

[ixlramp]

I'm unwilling to change my tuning

This system works with any tuning, but larger interval tunings like all-fourths make melodic playing awkward, requiring large shifts.
In all-fourths, triad chords are playable but other chords are much less playable.

For all-fourths this system would be:

* Retune standard tuning to all-fourths by raising the top B E strings by 1 semitone (no setup required for a fixed bridge).
* Slightly retune every alternate string by +14 cents (to play the JI Minor scale), or -14 cents (to play the JI Major scale).
* Then you play the 3rds, 6ths, 7ths of the scale on the 'slightly retuned' strings, and play the 2nds, 4ths, 5ths of the scale on the 'normal' strings.
* Do not complain that it is awkward to play melodically, i did warn you

This system could be considered as being a development of an entire playable system from the fairly common trick of playing a triad chord and retuning the string the 3rd of the chord is on slightly until the 3rd has 'perfect harmony' (therefore becoming a Just Intonation 3rd).
If you do this trick, which can be done in standard tuning, then you will partially experience what this system does and how it sounds.

 

[waitati]

Just Intonation
-------------------

What 'Just Intonation' (JI) is, is best explained in detail elsewhere, but briefly:

It is 'perfectly tuned harmony'.

The modern dominant worldwide tonal system is called '12 Tone Equal Temperament' (12TET). It has 12 equally-spaced pitches within an octave. However, these pitches are actually only approximations to perfectly tuned intervals.
The octave is exact.
The 'fourth' and 'fifth' are very close approximations, with an error of only 2 cents.
The 'second' has a non-problematic error of 4 cents.
'Third's, 'sixth's and 'seventh's have significant errors of 12-18 cents.
('cent' = 1/100th of a semitone.)

If you have had the experience of thirds sounding out of tune, but when you retune them by ear they are then out of tune with 12TET, this is why.

The JI major third is 3.86 semitones, 14 cents flat of the 12TET major third.
The JI minor third is 3.16 semitones, 16 cents sharp of the 12TET minor third.

Perfectly tuned intervals are not equally spaced within an octave, however it was generally decided that the advantages of equally spaced pitches (and regularly spaced frets on a guitar) outweigh the disadvantages of imperfect harmony.

///////

The system
---------------

My system of restringing and retuning allows various JI scales to be played on a normal guitar with normal 12TET frets. The alternative is to use a customised guitar with JI fretting, for example:

View attachment 68444

^ Owned by Matthew Grasso

My system is a thirds tuning created by 2 interlaced 12TET fifths tunings.
For example on a 7 string guitar: Strings 1, 3, 5, 7 are one 12TET fifths tuning, strings 2, 4, 6 are the other.
These 2 12TET fifths tunings have a non-12TET pitch offset between them, which can be altered to switch between various JI scales.

JI major scale
------------------

For the first JI scale detailed in this thread, the JI major scale, this pitch offset is 3.86 semitones, the JI major third.
The result is a thirds tuning composed of 2 alternating thirds of sizes 3.86 semitones (JI major third) and 3.14 semitones (very close to the JI minor third).
Moving across by 2 strings always results in a 12TET fifth (7.00 semitones).

My system closely approximates the JI major scale to within 4 cents.

View attachment 68445

^ JI major scale.
Numbers on the left are the intervals between the open strings in cents.

The 2 interlaced fifths tunings are considered to be 2 'groups' of strings.
One group contains 12TET pitches, the other group contains pitches that are 14 cents flat of 12TET.
On the '12TET group' of strings the following scale degrees of the JI major scale are played:
Tonic, second, fourth, fifth, octave.
On the 'flattened group' of strings the following scale degrees of the JI major scale are played:
Third, sixth, seventh.

These scale degrees are shown on the diagram above. The red circles with no number are the tonic or octave, the other scale degrees are shown as numbers 2-7.

Playing 3 strings straight across a fret plays the JI major triad chord, 4 strings straight across a fret plays the JI major seventh chord.

///////

String gauges
------------------

Obviously a custom set of gauges is needed. I have used D'Addario tension data to create suitable sequences of gauges to choose a subsequence from according to the pitch range you desire.

PL0085 or PL009
PL0105 or PL011
PL013
PL016
NW022
NW028
NW036
NW046
NW056
NW070

For bass guitar:

PSG022w
PSB028w
PSB035
PSB045
PSB055
PSB070
PSB085
PSB105
PSB130

Here's a suggestion for a 6 string guitar tuned to FACEGB. The tension is similar to the common 9-42 and 10-46 sets. It shares 3 strings with standard tuning.

B PL013 standard B
G PL016 standard G
E NW022
C NW028
A NW036 standard A
F NW046 standard E + 1 semitone

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Tuning
---------

As a starting point, tune the strings to a 12TET tuning of alternating major and minor 12TET thirds, for example FACEGB.

The next step is to slightly detune all the major third intervals, by 14 cents, to make them JI major third intervals.
For the FACEGB example, this means detuning the A E B strings (which themselves are a 12TET fifths tuning, and are the 'flattened group' of strings described above).
Once 1 string has been detuned, the other strings in the 'flattened group' can be easily tuned in 12TET fifths to that string.

So a method of tuning 2 strings to a JI major third interval of 3.86 semitones is required. Most guitar tuners cannot do this.

Tuning a JI major third using harmonics
--------------------------------------------------

First, practice playing open string harmonics to acheive clear harmonics with good sustain.
The specific harmonics to practice are the 4th harmonic, located precisely at 'fret 5', and the 5th harmonic, located at roughly 'fret 3.9' (not at 'fret 4' as many sources suggest).

* Start with 2 adjacent strings tuned to a 12TET major third.
* Play the 5th harmonic of the lower string and the 4th harmonic of the higher string, such that the 2 harmonics sustain together.
* They will be similar pitches but slightly out of tune.
* Repeatedly play the harmonics, very slowly and carefully detune the higher string until the 2 harmonics are identical in pitch. The higher string only needs detuning by 14 cents to do this.
* The 2 open strings will then be tuned to a JI major third.

///////

JI minor scale
------------------

Once your guitar is tuned it can also play the JI minor scale, by simply placing the tonic note on an adjacent string, moving the whole pattern across by 1 string:

View attachment 68446

^ JI minor.
'b' means 'flattened' or 'minor'.

Effectively, the 2 groups of strings now have the JI minor third as the pitch offest between them.

Or, retune to swap the positions of the JI major and JI minor thirds.

Playing 3 strings straight across a fret plays the JI minor triad chord, 4 strings straight across a fret plays the JI minor seventh chord.

The string gauges recommended are chosen to work for either of major-minor or minor-major alternating JI thirds.

Hi Matthew,

I computed the frequencies and identified notes for your suggested 6 string tuning FACEBG. Can you check. It is easy to do other tunings if you would like those.

Thanks
Nick
waitati@gmail.com

 

[ixlramp]

I computed the frequencies and identified notes for your suggested 6 string tuning FACEBG. Can you check. It is easy to do other tunings if you would like those.

Hi. I sent you a private message. Sign in and click the 'envelope' icon ('Conversations') in the top right of the screen to view it.

 

[ixlramp]

I computed the frequencies and identified notes for your suggested 6 string tuning FACEBG.

What you suggested about there being two different systems was not correct. Both forum threads describe the same system.

The system can be applied to any Equal Temperament guitar tuning (but some tunings are much more practical than others).
This older thread describes the system applied to an 'alternating major and minor thirds' tuning (of which FACEGB is one example).
The newer thread specifically describes the system applied to an 'all minor thirds' tuning.

So discussion of the FACEGB tuning is not suitable for the newer thread, we should discuss that here.
I will reply further here later.

 

[ixlramp]

@waitati
The Just intervals my system intends to approximate are shown in the tables in posts #2 and #3 in the newer thread.
These Just intervals are the same for all 'base tunings' (alternating major and minor thirds tuning, all major thirds tuning, all minor thirds tuning, or any other base tuning).

Looking at your analysis, the Just intervals you used for your analysis are partially different to the Just intervals my system intends to approximate, this is not your fault because you based your analysis on this older thread which does not list them, so you had to guess what they were.
Also, your analysis seems unusual and incompatible to how my system functions.
So your analysis does not apply to how my system functions. But if you personally find it useful, that is cool