How many notes in an octave?

Short answer? As many as you like. But some choices are better than others. The common “western” system is known as 12-TET (12-tone equal temperament), which is a modification of earlier geometrical tunings (such as the Pythagorean and quarter-comma meantone temperaments) where the frequency of each note is defined in terms of a simple ratio with respect to a reference note (the frequencies of C and G are in the ratio 2:3, for example). There are problems with such systems – frequency ratios between arbitrary notes are not simple in general, and any geometrical tuning system will have a wolf interval, whose dissonant sound must be avoided in musical composition.

By contrast, “equal temperament” systems define the frequency ratio between any two adjacent notes to be a constant, but this constant cannot be a rational number and note frequencies which were once simple ratios of each other are now very slightly detuned. Analogue instruments have a natural variability in their tuning, so this slight “error” usually goes unnoticed.

The division of the octave into twelve equal intervals is a natural progression of the 12-interval Pythagorean system, but is not the only one. The “ideal” tuning system is one where all intervals are based on simple frequency ratios, but this is impossible to realise in practise without either providing an infinite number of notes or severely limiting the musical range of one’s compositions. Many of these ideal intervals are very badly approximated by 12-TET, notably the minor third (the interval between C and E♭). 31-TET is a better match overall, however it more than doubles the number of notes required on an instrument, and many of these will not be used in the vast majority of compositions because they do not approximate a useful interval.

Compromise tunings which do not require so many redundant notes include the alien-sounding 22-TET and the more mellow 19-TET. The latter gives improved major (C-E) and minor (C-E♭) thirds compared to 12-TET, at the expense of a slightly less accurate fifth (C-G). It also has few enough notes that 19-TET instruments are quite practical to construct and play. I had to try.

19TET refretted Encore strat copy

I refretted this guitar two years ago and forgot to blog about it at the time. I was fully convinced that I had, up until the point when I tried to find the appropriate link to help spice up my entry into this competition. I’m addressing my earlier failing now.

I have three guitars, though technically one of them belongs to my brother. The cheapest and crappest of my two was duly sacrificed in the name of science. I bought a fretsaw and obtained a length of fretwire from a local instrument repair shop. The existing frets on the guitar were easy enough to prise out of their grooves, and new grooves in the appropriate places* were easy to cut. To cut down my workload, I only refretted one octave worth of the neck – this isn’t as big a sacrifice as it sounds, as the 19-TET fret spacing close to the body would be far too narrow to play comfortably. The biggest difficulty came in cutting and shaping the new frets – I ended up having to file off a few offending corners and in doing so scratched the edges of the fretboard. Overall I’m pleased with the result though – it plays easily and looks just like a “real” guitar until you get too close.

How do you adjust your playing style to a completely different fretboard?  The pearl inlays from its 12-TET days are still present, and provide a useful visual clue for where the fingers should go. If that doesn’t work, I usually just fiddle with the chords until it sounds right. As a rule of thumb, anything requiring a one-fret interval should now use two, and a two-fret interval becomes three.

One advantage that 19-TET has over (say) 22-TET is that standard music terminology can be (ab)used. The white-note names are unchanged, black flats and sharps are now distinct notes (C♯ is not D♭), and B♯=C♭ and E♯=F♭ are now notes in their own right:

C C♯ D♭ D D♯ E♭ E (E♯=F♭) F F♯
  G♭ G G♯ A♭ A A♯ B♭ B (B♯=C♭) C

If you are familiar with the piano, imagine that each black note has been replaced by two black notes, and the gaps between B-C and E-F are now filled by one black note each.

How does it sound, I hear you ask? Unfortunately I can’t make you a recording at the moment, as the bridge recently came apart (someone remind me to go to the repair shop this weekend please?). However, some fantastic examples of 19-TET music can be found on Jeff Harrington‘s site – my favourite is this one. He’s a much better musician than me anyway.

* D(n) = D(0)/2^(n/19) where the distances D are measured from the bridge and D(0) is the distance to the nut.

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