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DF-Mark

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Reply with quote  #1 
For many years I've been less than happy with intonation issues stemming from the realities of using 12TET with our fretted, string instrument design ... often it seems a completely forgone conclusion that the only fretboard of any use will be the standard we have come to know and love where the octave is divided into 12 'equal' (logarithmically speaking) fret divisions. Obviously there are a lot of other interesting microtonal possibilities but my concern is mainly with making a more harmonious noise by moving towards JI (Just Intonation).

As a player the obvious discrepancies in 3rds and 6th intervals with 12TET can be a real nuisance and it seems clear that the lower tension of tappers only exaggerate this. The limitation of being stuck in certain keys with JI is a real concern, but in my experience I lean heavily towards playing in certain keys anyway ... so perhaps this wouldn't be such a big deal after all.

As a builder its very easy to just go with the flow and stick to 12TET, yet its still kind of tempting to make an instrument (or several!) with an alternative fret layout to explore other possibilities. Unfortunately in my experience the extra (unpaid) investment of time and energy in such a project makes it destined to never happen in the usual routine of life and work. I've tentatively explored ideas of making instruments with interchangeable fretboards, interchangeable fret positions, bent frets, partial frets etc. ... but nothing has convinced me enough (in terms of keeping things simple and useful to the general public) to actually go ahead and make it ... yet!

Recently the inspiration has been revived with some comments and input from @ixlramp ... and I'm particularly interested in 2 possible ideas:

1. Using standard fret positions and re-tuning individual strings to create more harmonious fretted notes at certain positions (to effectively achieve the results of JI within a restricted range of notes). This seems kind of obvious in retrospect (as no alteration to the instrument is required), but frankly it had never occurred to me.

2. Constructing alternative fretboards with irregular fret spacing calculated to maximise Just Intonation for the appropriate key(s)/tuning(s) one is interested in ... but sticking with straight frets across the whole width of the fretboard and being careful/clever about the tunings used

Soooo ... apart from me being just getting that out of my system, I guess I'm wondering if this of interest to anyone else out there ... or if anyone has tried anything like this out and whether they think its worth doing. I look forward to any comments or suggestions. Thanks! 

For those interested theres a great link to the work of Jim Snow (making a Just Intonation guitar) that @ixlramp shared with me - http://jsnow.bootlegether.net/cbg/justintonation.html

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Big George Waters

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Reply with quote  #2 

I'm reading this before I turn in for the night, I hope I remember it tomorrow when I get up, as there's some real interesting ideas here.

I seem to think that Turkish and Indian instruments - some of the stringed instruments have movable frets - perhaps for this very reason ??

Trey Gun once remarked about the Godin A-11 fretless classical guitar which had 11 strings, with all but one [lowest note] being paired.

Trey mentioned this below in an article called "Gear I can't Live Without"

The Godin Glissentar. I first came across this around 2000 and I had to try it. It is modeled on the oud, which I love the sound of. And I loved the string configuration: Eleven strings in six groups. The top five strings are doubled in unison and the low string is on it's own. I had never played a fretless instrument before and felt a bit dubious about taking it on now. I had heard so many different musicians complaining about fretless bass players that played out of tune. But... screw it, I had to try this guy and ordered one. It wasn't so hard to play in tune at all. OK, I'm sure I am not perfect on it and I can totally hear when I am really out, but it is wonderfully fun to play. I tune it in fifths like a cello so the low string is a C. Oh wait, no it isn't. I tune it down a whole step from that, so the low string is a Bb. That is pretty wierd for me, but it works. The thing I like most about it is that it makes me think and hear so much more melodically than any other instrument. Plus it is real ear opener to to hear what a true major 3rd or minor 7th sounds like. You can't do that on a fretted instrument

 

Within a couple of weeks, I had one myself !!

But totally - what is being mentioned here is very interesting to me, as I am always game for a new way of doing things.


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Big George W

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ixlramp

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Reply with quote  #3 
For idea 2:

I'm quite interested in trying to figure out full-width fret positions for a JI tapper. It will depend on what intervals you want to be able to play, and the primary concern is avoiding frets so close together that they become difficult to tap between (tapping close frets may be more difficult than just fretting them). The open string tuning will be decided by these factors as well as making chords playable (although having 2 hands to tap a difficult shaped chord will help a lot).

Such a fretting will be limited to playing your chosen scales in 1 key, which is the limitation which encouraged the use of modern 12 Tone Equal Temperament. However, different modes of your chosen scales will be playable.
An alternative is cutting full width slots but inserting partial-length frets, should be quite easy.
A very useful tool for designing is 'FretFind2D' fret calculator which can accept JI scales and model partial frets http://www.ekips.org/tools/guitar/fretfind2d/
It's a little tricky to use so i need to get re-aquianted then i will post some tips.

Perhaps a simple starting point would be the intervals of JI major and minor scales:

Name. Frequency

Unison 1/1
Major 2nd 9/8
Minor 3rd 6/5
Major 3rd 5/4
4th 4/3
5th 3/2
Minor 6th 8/5
Major 6th 5/3
Minor 7th 9/5
Major 7th 15/8

Maybe also Minor 2nd 16/15.

More information, see https://en.wikipedia.org/wiki/Five-limit_tuning
especailly the 'Twelve tone scale' section, my suggestion above is the 'asymmetric' system minus the tritione.

For idea 1:

This has been done quite often but i have designed a particular system that i haven't seen used anywhere else.
My system is detailed at the sevenstring forum here https://sevenstring.org/threads/just-intonation-on-a-normal-guitar-by-restringing-and-retuning.335492/
You have to join the forum to see the diagrams full size, however i intend to transfer this thread to a .pdf later to share more widely.
My system has some advantages and disadvantages, but is worth trying as it only requires a custom string set and a spare guitar.
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ixlramp

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Reply with quote  #4 
Here's how to use FretFind2D, i worked through using the suggested JI tonal system above.

Go to http://www.ekips.org/tools/guitar/fretfind2d/
Unfortunately you will need to enable Adobe Flash Player but you can configure it to be enabled on this site only.

Here's the data to enter for a conventional 6 string guitar, 25.5" scale, string spacing 7mm to 10,5mm, overhang 3.5mm for a 42mm nut.

The scale format is 1st line scale name, 2nd line number of tones, following lines are the JI tone frequencies expressed as fractions, excluding the tonic '1/1' and including the octave '2/1'.

'Tuning' is the scale degree of the open string where '0' is the tonic. For this example we set every other string to the 6th degree '3/2' for a CGCGCG type tuning which minimises the number of partial frets.

To make sure the diagram is updated the final step should be entering a value in 'tuning' and pressing enter. 
ff2d.png 

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ixlramp

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Reply with quote  #5 
I exported the image as SVG, opened it in an image editor, gave it a white background:

JI_5lim_11tone.png 

The program only shows the partial frets needed for the scale. If this were to be fretted using full width frets, a full width fret would be used where there are partial frets lined up.
This particular fretting would create extemely close frets in 4 places if full width frets were used, so in these places i would suggest using partial frets in full width slots, all other frets can be full width.

However this is just an example, i'm not suggesting this fretting is suitable for you. It looks like there is room for a 12th tone, the JI tritone 45/32, to complete the JI chromatic scale.

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ixlramp

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Reply with quote  #6 
Formatted scale:

JI 5-limit 12 tone
12
16/15
9/8
6/5
5/4
4/3
45/32
3/2
8/5
5/3
9/5
15/8
2/1

Number of frets:

24

Tuning:

7
0
7
0
7
0

JI_5lim_12tone.png 

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ixlramp

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Reply with quote  #7 
This 12 tone JI system seems suitable for what you want, it is the JI version of 12TET, or actually, 12TET evolved from it.

1/1 Tonic
16/15 minor second
9/8 major second
6/5 minor third
5/4 major third
4/3 fourth
45/32 augmented fourth / diminished fifth
3/2 fifth
8/5 minor sixth
5/3 major sixth
9/5 minor seventh
15/8 major seventh

(2/1 octave)

///////////////////////////////////


JI intervals are always stated as a fractional frequency, this is the frequency of the higher note if the lower note has frequency 1.
The simpler the fraction the more consonant the interval, for example the octave is 2/1, the fifth is 3/2.

If the fraction is changed to a ratio, this ratio is the ratio of the frequencies.
3/2 -> 3:2, a fifth interval is 2 pitches with frequencies in ratio 3:2, such as 300Hz and 200Hz.

A JI major triad is actually 3 notes whose frequencies are in ratio 4:5:6, this is where the major triad came from and why it is special.

///////////////////////////////////

With this tonal system, major triads and major seventh chords will be playable with the root note on:

Tonic
Fourth
Fifth
Minor second
Minor sixth
Minor third

Minor triads will be playable with the root note on:

Tonic
Fourth
Fifth

Major sixth
Major third
Major seventh

Minor seventh chords will be playable with the root note on:

Tonic
Fourth

Major sixth
Major third

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DF-Mark

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Reply with quote  #8 
Thanks very much for your input @Big George Waters and @ixlramp. Its all great stuff!  ... the fret calculator is seriously fantastic now that I understand how to use it!
I realise that you've done your homework @ixlramp and the resulting fret positions (from the JI 5-limit 12 tone scale) are pretty elegant! You've really helped open it up for me!! I'm not super convinced by either the partial fret or the full fret solution to resolve the major 2nd/ major 6th discrepancy though, and wonder whether splitting the difference would sound too bad in those positions (or at least the ones further up the fretboard).
On the other hand I really think its time I sat down put some decent work into properly to working through other tuning possibilities and things I'd like to explore. At this stage I'm thinking I might actually be ok with sacrificing versatility for an even simpler fretboard ... at least to try some things out. It may prove to be a waste of time but it'll help me get an even better handle on the whole idea and hopefully I'll be back soon with some other ideas and queries. 
What does anyone else think?

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Big George Waters

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Reply with quote  #9 
Is going fretless an option ??
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digitech, T.C. Electronics, ART, Lexicon, GK, Markbass, EA, Bag End, Guild/Hartke, SWR, EV, Radial, Furman

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ixlramp

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Reply with quote  #10 
High strings are very difficult to tap on a fretless as they have so little energy or sustain.
Very difficult to play all the notes of a chord in tune on a fretless, one note is hard enough.

///////////////////////////////////

This interests me a lot too so i am partly working through this for myself =)
I have converted a couple of guitars to quartertone fretting but haven't done a JI fretboard yet.
Quote:
Originally Posted by DF-Mark
I'm not super convinced by either the partial fret or the full fret solution to resolve the major 2nd/ major 6th discrepancy though, and wonder whether splitting the difference would sound too bad in those positions (or at least the ones further up the fretboard).

I encourage you to stick with exact fret positions, after all if you accept small errors you might as well just use 12TET. Partial frets in a full width slot should be easy to do.

Well worth trying various open tunings. It's possible that with a certain open tuning that the partial frets would be spaced far enough away from other frets that the partial frets could be implemented using a full width fret.

With my example above, the CGCGC tuning means triads would have to be played in root-fifth-third form with the third up an octave, likewise for the seventh chords.

Quote:
Originally Posted by DF-Mark
At this stage I'm thinking I might actually be ok with sacrificing versatility for an even simpler fretboard

Well worth trying removing certain intervals from the 12 tone JI system.

////////////////////////////////

Considering playable scales, assuming the 12 tone JI system is used:
Just like 12TET, there will be a vast number of scales playable by choosing combinations of intervals.

I find that the 'limitations' of JI are overstated, just because you can't freely modulate a scale to a different key unchanged.
Here's a different way to look at it: When you move the tonic note you end up with a new set of JI intervals relative to it. And so an additional vast number of scales becomes playable from the combinations of those intervals.
So the variety of scales greatly increases, at the cost of not being able to modulate a scale to another key unchanged. But then, when you move the tonic note it will be interesting to modulate to a different scale.
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DF-Mark

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Reply with quote  #11 
OK ... I'm with you on all that ixlramp [thumb] 

Is going fretless an option ??

We've made fretless models for clients and they're a lot of fun ... but certainly not as versatile in terms of tapping in my opinion, and as ixlramp indicates, tapping complete chord shapes on a fretless instrument is an extremely challenging art! 

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Big George Waters

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Reply with quote  #12 
Oh it definitely is but on the other hand you can do things chordwise on a fretless instrument which are impossible on a fretted instrument.
For many years I played a custom made Mojo Vibemaster that was based on a 1962 Fender Jazz bass but with an unlined no dots ebony fretless fingerboard.
It also had 3 pickups and telecaster controls anyhow, playing chords I could if I was slightly off intentionally I could get a beating sound which would almost be like a natural chorus.
It was quite exciting !!
That is a reasonable answer, thanks !!

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digitech, T.C. Electronics, ART, Lexicon, GK, Markbass, EA, Bag End, Guild/Hartke, SWR, EV, Radial, Furman

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ixlramp

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Reply with quote  #13 
For the sake of documenting some investigation ...

Avoiding partial frets in the system of post 6 https://www.tappistry.org/post/show_single_post?pid=1308716656&postcount=6&forum=406209
requires removing one tone from each of these pairs:

Maj 2nd 9/8, maj 6th 5/3
Aug 4th 45/32, min 2nd 16/15
Min 7th 9/5, 4th 4/3

These turn out to be the 3 impure 5ths (5ths that are not a 3/2 interval) of the asymmetric 12 tone scale:
D A
F# Db
Bb F

These root-5th pairs are the 3 pairs that are not horizontally next to each other:

asym.png 
So do not appear lined up on the root-fifth string pairs of the open tuning.

Removing a tone and using a full width fret adds back a new tone a fifth above or below the tone that is preserved. As follows:

Removed tone. New tone.

Maj 2nd 9/8. Maj 2nd 10/9
Maj 6th 5/3. Maj 6th 27/16

Aug 4th 45/32. Aug 4th 64/45
Min 2nd 16/15. Min 2nd 135/128

Min 7th 9/5. Min 7th 16/9
4th 4/3. 4th 27/20

Good news: The new tones are fairly common JI tones. Bad news: The new tones can't be present on 3 the strings as they would add new partial frets very close to other frets.

So if i choose to remove Maj 6th, Aug 4th, 4th from the scale:

JI 5-limit 9 tone
9
16/15
9/8
6/5
5/4
3/2
8/5
9/5
15/8
2/1

18 frets (2 octave fretboard)

fretboard.png

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