Adaptuner note names

What’s the problem with notations for just intonation, anyway?

Ask yourself what the difference between, say, \(G\sharp\) and \(A\flat\) is. For most modern musicians (especially keyboard players) the answer is probably that it’s a matter of orthography that has something to do with the harmonic context.

This “harmonic context” is determined by how we imagine chords and melodies as stacks of intervals. For instance, a major chord is a minor third stacked on top of a major third. In tempered tunings, any difference from that spelling merely conveys intent: You won’t press a different key on the piano, but you might press it differently.

In just intonation, on the other hand, we assume a set of “basis” intervals – octaves, pure fifths, pure major thirds, natural sevenths, and so on – and actually construct all other intervals as stacks of basis intervals. Different constructions of the “same” interval (i.e. the same keys on the piano) will yield different pitches. In a sense, you could say that just intonation takes the “harmonic context” idea seriously.

This means that a notation for just intonation must distinguish not only \(G\sharp\) and \(A\flat\), but even which specific \(G\sharp\) or \(A\flat\) we mean. It must be able to accommodate all of the (infinitely many) classes of intervals that can be obtained as stacks of basis intervals. The challenge is to find a system that does so in a non-confusing, musically useful way.

Why Ben Johnston’s note names?

The adaptuner uses the note names invented by Ben Johnston. Compared to some other notations for just intonation, this system has the drawback that it is slightly less regular – it’s not what a mathematician would invent. However, it has a number of very good properties that in my opinion make up for any initial irritation:

• It is completely unambiguous: There’s exactly one way to write every note. In particular, there are no enharmonic equivalents.
• It is relatively light on additional accidentals for music that emphasises “normal” harmonies. By and large, most pieces of music won’t look too strange with their pitches explicated by Ben Johnston’s notation, and where they do look strange, this is mostly warranted by actual harmonic complexities.
• Great music has been written in it.

Explaining Ben Johnston’s note names

The notes of the C major scale

The \(C\) major, \(F\) major, and \(G\) major triads are defined to be pure major chords (i.e. they have frequency ratio \(4:5:6\)). This defines all notes of the \(C\) major scale, and this tuning is known by many names.

We could now calculate the frequency ratios of all notes, but I don’t think they’re a useful thing to think about when making music. So, here are a few facts that I find useful:

• \(F-A-C-E-G-B-D\) is an alternating sequence of pure major thirds and pure minor thirds.
• \(F-C-G-D\) and \(A-E-B\) are stacks of pure fifths.
• \(D-F\) is not a pure minor third.
• \(D-A\) is not a pure fifth.
• The intervals \(C-D\), \(F-G\), and \(A-B\) are bigger than an equally tempered whole tone. They have frequency ratio \(\frac{9}{8}\), an interval sometimes called the major tone.
• The intervals \(D-E\) and \(G-A\) are smaller than an equally tempered whole tone. They have frequency ratio \(\frac{10}{9}\), an interval sometimes called a minor tone.
• The semitones \(E-F\) and \(B-C\) are of the same size, which is exactly the difference between a pure fourth and a pure major third. This is interval is sometimes called a diatonic semitone and is bigger than an equally tempered semitone.
• The diminished fifth \(B-F\) is slightly bigger than the augmented fourth \(F-B\).

Sharps and flats

Sharps and flats denote the difference between a pure major third and a pure minor third (i.e. the frequency ratio \(\frac{25}{24}\)). Again, a few hopefully useful facts:

• \(F-A\flat-C-E\flat-G-B\flat-D\) and \(F\sharp-A-C\sharp-E-G\sharp-B-D\sharp\) are alternating sequences of pure minor and major thirds.
• Notes with a sharp are lower than their “enharmonic equivalents” with a flat. The difference can be quite startling: the interval \(G\sharp-A\flat\) is about 40 cents wide, \(C\sharp-D\flat\) is about 60 cents!
• The semitones you get by adding an accidental to major tones are very big. These are intervals like \(C\sharp-D\) or \(A-B\flat\).
• The semitones you get by by adding an accidental to minor tones are diatonic semitones, and are therefore bigger than an equally tempered semitone. These are intervals like \(D\sharp-E\) or \(G-A\flat\).
• Augmented unisons like \(C-C\sharp\) are smaller than an equally tempered semitone. This interval is sometimes called a chromatic semitone.

Plus and minus

The nerd’s way to explain the accidentals \(+\) and \(-\) is to say that they denote the syntonic comma (i.e. the frequency ratio \(\frac{81}{80}\)). For the working musician, this means that we can extend the alternating sequence \(F-A-C-E-G-B-D\) of pure major and minor thirds by using plus and minus signs like so:

\[ \cdots G^{-}-B\flat^{-}-D^{-}- F-A-C-E-G-B-D- F\sharp^{+}-A^{+}-C\sharp^{+} \cdots \]

This allows us to construct many more pitches, which I’ll illustrate with a few examples:

• The fifth of \(D\) major is \(A^+\). The third of \(F\) major is \(A\), which is also the fifth of \(D^-\) major.
• \(C-E^+\) is a pythagorean major third.
• \(C-D\flat^-\) is a diatonic semitone.
• \(D-E^+-F\sharp^+-G-A^+-B-C\sharp^+\) are the notes of the \(D\) major scale, if it is tuned like the “accidental-free” \(C\) major scale.
• A frequently-discussed tuning of all twelve notes of the chromatic scale adds to the \(C\) major scale the notes \(D\flat^-\), \(E\flat\), \(F\sharp^+\) or \(G\flat^-\), \(A\flat\), and \(B\flat^-\). For example, Paul Hindemith constructs this scale at the outset of his Unterweisung im Tonsatz.

Notations for higher harmonics

At the moment, the GUI of the adaptuner only exposes the fragment of just intonation known as five-limit just intonation: All of the intervals you can get by stacking octaves, pure fifths, and pure major thirds. They can all be notated with the part of Ben Johnston’s system I described so far. Once I start exposing the higher harmonics, I’ll add an explanation of their notation as well.