# Just Intonation and True Temperament Tuning

#### Introduction

I ended my previous article on tuning issues and the Circle of Fifths with two conclusions:

Conclusion 1: All tuning is an approximation

Conclusion 2: Tuning changes according to the key

In this article, I would like to explore these conclusions more, from another perspective.  This time I want to look at a system of “Just Intonation”, or tuning by ratios.  I will then investigate what “true temperament” means, and whether true temperament frets reall solve the tuning problems for guitars.

#### Just Intonation

Historically, there have been many attempts at determining the best system for tuning the 12 notes of the Western musical scale (other musical scales have different numbers of notes and I may look at that in another article).  We saw with the Circle of Fifths that the interval from root note to fifth is a 50% increase in frequency, or a 3:2 ratio.  Hence a fifth above A440 (A4, standard concert pitch) is E5 with a frequency of 660hz.

The reason for this is that because of the simple mathematical ratio, the two waveforms combine in a way that is pleasing to the ear.  Logically, other simple ratios will also produce intervals that sound good, and indeed we see that.  For example, a ratio of 4:3 is a perfect fourth, 5:4 is a major third and 5:3 is a major sixth.

One common tuning system based on this principle is 5-limit tuning, which uses ratios based on multiples of the prime numbers up to five (2, 3 & 5).  This gives a specific ratio for each note as a multiple of the root note.  As well as the ratios mentioned above, for example, a semitone is 16:15 and a diminished fifth (the dissonant six-semitone interval) is the most complex ratio of all, 45:32.

The table below shows the scale from A4 to A5 again, calculated using these ratios in three different keys.  The first key, Root A, takes 440 as the root frequency and calculates the values of the other notes using the ratios shown.  The second key, Root A#, takes A# on that scale as the root note and calculates the other notes using the ratios applied to that note; similarly for the third column, which takes B and applies the ratios.  In the final column, I have copied in the Standard tuning values for reference.

The red numbers show notes where the frequency differs from the Root A values.  Note that all of the notes in Standard tuning are different, apart from A and the octave!

This exercise proves the two conclusions that we started with.  Tuning is different for each key, and therefore if you want to be able to play in different keys on the same instrument, all tuning must be an approximation.

#### True Temperament

Guitarists may have seen the “True Temperament” frets that claim to solve this problem.  These are frets that are shaped differently in order to make small adjustments to the intervals, and the claim is that this solves tuning problems.  Some well-known guitarists have used these systems (including Steve Vai, so you don’t need more references than that!); they look a bit strange but apparently it doesn’t particularly affect how the instrument plays.

Standard fret distances are calculated using the 12th root of two, which follows the standard tuning (whereby every semitone is the same multiple of the previous semitone, a little under 1.06).  According to the True Temperament website, standard frets only account for the length of the strings, but a string’s pitch is a function of its length, mass and tension.  The true temperament frets are shaped to take into account the additional factors of mass and tension (each of which is different for each string) in order to achieve more accurate tuning.  However, the goal here is simply to achieve a better standard tuning; it does not, and cannot (as they acknowledge – their FAQ page is worth a read), provide just intonation in every key.

I have not tried a true temperament fretboard myself so I can’t comment on them directly.  I doubt whether my technique is good or consistent enough to really benefit from them; I would certainly be interested to try one but not sufficiently to justify buying a new guitar only for that!

#### More conclusions

This article hasn’t really generated any extra conclusions but it has supported my two previous conclusions.  Perfect tuning is impossible, and the true temperament fret system can only increase the accuracy in relation to a scale which itself is an approximation.  This is a worthy goal, but its effect is only to leave you with a smaller problem than you have with a standard fretboard.

I hope this article and the previous one have been interesting and clear.  Please subscribe and comment on this article; I have only scratched the surface of this topic and there is much more that one could say, but I think that going into more detail will only complicate the issue without altering the conclusions.