Fret Placement for Just Intonation/Temperament

[AV8R_GA]

I am a retired engineer, and will, as a confession to the group and my dear loving wife, admit that sometimes I obsess about things that really don't matter in the big picture of life. But here we are...

As a side project, I've been trying to work out the fret positions for just intonation (more correctly called just tempered scale) fret spacing and note creation, which is based on mathematical harmonics rather than the current universally used even tempered scale. Frets 1 thru 5, fret 7, fret 11, and fret 12 come out dead on as fractions of the octave for a given string, as predicted by harmonic string theory (nodes, anti-nodes, and all that stuff from first year physics).

Frets 6, 8, 9, and 10, on the other hand, are driving me nuts. These represent the tritone, minor 6th, major 6th, and minor 7th for a given root note. Can anyone help me find the formulas for them?

[Robin, Wales]

I have build a few instruments in just intonation using Ptolemy's ratios to produce a fret placement calculator. But they have all been purely diatonic. Additionally, the instruments have been essentially single string melody instruments restricted to one key rather than chord playing multi-key capable instruments. I have simply played the melody against pure root and 5th drones. Note pairs blend beautifully as long as the strings providing those pairs as tuned to the same note, but you cannot fret a string tuned to a different note and expect it to blend. Also, the fret placement only really works for the mode for which it was calculated. So you cannot play a Dorian scale starting on the fret after the root fret if your frets are set for the Ionian scale. You can "just" get away with Mixolidian and Aeolian at a push.

The nature of the 6 string guitar (chord playing and key changing) is going to make JI fretting very complex and also key limiting. You'll constantly be solving one problem and creating another - which is why guitars have settled for equal temperament.

On the other hand, for mountain dulcimers (and other fretted zither type folk instruments) where a melody is played against drones, just intonation makes sense. It is very easy to set frets by ear in just intonation using the drones as an audio guide; in fact it is the fret placement system you will naturally arrive at (I know because I have done it). Whereas, it is almost impossible to set frets by ear in equal temperament.

If you do want the formula for chromatic fret placements in just intonation then have a look at Indian music scales. Again, the instruments of India using this system of JI frets are played against root and 5th drones and do not change key, except through re-tuning.

So you could build an instrument with chromatic just intonation frets - but it is not going to be a guitar - or at least not one that will work from standard tuning.

[JonWint]

Be careful. You might end up with something like this:

[IMG][/IMG]

This is Kenny Hill's version. Also search "well tempered" or "wiggly frets".

[Rudy4]

Originally Posted by AV8R_GA

I am a retired engineer, and will, as a confession to the group and my dear loving wife, admit that sometimes I obsess about things that really don't matter in the big picture of life. But here we are...

As a side project, I've been trying to work out the fret positions for just intonation (more correctly called just tempered scale) fret spacing and note creation, which is based on mathematical harmonics rather than the current universally used even tempered scale. Frets 1 thru 5, fret 7, fret 11, and fret 12 come out dead on as fractions of the octave for a given string, as predicted by harmonic string theory (nodes, anti-nodes, and all that stuff from first year physics).

Frets 6, 8, 9, and 10, on the other hand, are driving me nuts. These represent the tritone, minor 6th, major 6th, and minor 7th for a given root note. Can anyone help me find the formulas for them?

I've studied fretted instrument temperament for several years. Just temperament is key specific, so don't put a lot of effort into this. Here's the internet archive of an older web page I wrote relating to intonation and temperament:

http://web.archive.org/web/201603282...tionTips3.html

[Robin, Wales]

I forgot that I had made this video 5 years ago looking at the difference between equal temperament, just intonation and quarter comma meantone on mountain dulcimers.

[AV8R_GA]

Originally Posted by Robin, Wales

The nature of the 6 string guitar (chord playing and key changing) is going to make JI fretting very complex and also key limiting. You'll constantly be solving one problem and creating another - which is why guitars have settled for equal temperament.

On the other hand, for mountain dulcimers (and other fretted zither type folk instruments) where a melody is played against drones, just intonation makes sense. It is very easy to set frets by ear in just intonation using the drones as an audio guide; in fact it is the fret placement system you will naturally arrive at (I know because I have done it). Whereas, it is almost impossible to set frets by ear in equal temperament.

My apologies. I should have begun my post with the caveat that this attempt at just temperament will be for a 3-string cigar box guitar that will use a simple G-D-G open tuning (low G, Middle D, high G). I will take a look at the Indian music scale.

[Robin, Wales]

Originally Posted by AV8R_GA

My apologies. I should have begun my post with the caveat that this attempt at just temperament will be for a 3-string cigar box guitar that will use a simple G-D-G open tuning (low G, Middle D, high G). I will take a look at the Indian music scale.

You will have to think about where you are going to start your scale from, and what scale that is going to be - i.e. Ionian from the nut or perhaps from the 5th fret as the root note? If you intend to form chords by fretting both the two G and one D strings at different points on the neck then you could run into trouble as the intervals on the D string will differ comparatively with the intervals on the G strings. Some chord shapes will be OK and others "out".

I would suggest, once you have decided on where you are going to start your scale, that you play around with string gauges and tunings. You may find that G,d,d would work well, if you are starting your scale at the 5th fret or perhaps G,g,g or D,d,d if your scale starts at the nut. If you fret two strings tuned to the same note but on different frets you will still get perfect intervals. The problems always arise when you fret two strings of different tunings.

[RJVB]