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: ^ 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. ^ 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 /////// 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: ^ 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.