In the context of characterizing a completely arbitrary instrument or piece of world music, when trying to describe the number of notes or tones per octave we tend quickly to get grounded in notions such as temperament or intonation.

I need to find a universal term for the number of notes (or tones) per octave which is completely independent of the above: a term equally applicable to

  • equal temperament as just, Pythagorean, well, meantone, syntonic or tempered (and any others you may care to name)
  • any number of tones per octave.

Clearly, I can have (say) nn notes or tones in an octave more or less irrespective of which temperament or intonation is in use, making nnTET, nnEDO, nnJI subsets of this higher-level definition.

I've come across cobbled-together or not-quite fitting terms such as:

  • xenharmonic index (--> 'other-than-12TET')
  • n tone per octave index (no abbreviation)
  • tones per temperament or intonation (too specific, no abbreviation)

There must be a term that is all-encompassing, widely accepted, and has an abbreviation, right?

  • I've not heard of one, but something like "octal population" seems naively appealing.
    – Nick
    Jul 30 '15 at 14:58

Why do you require abbreviation? If there's a perfectly good term for this that doesn't use an abbreviation, will it be acceptable? "Notes per octave" or "pitches per octave" seem pretty widely used, universally understood, and tuning-agnostic.

As an extension of this, scales themselves can be described as n-tonic, where n is a Greek number (as in, "pentatonic", "hepatonic", "dodecatonic", etc...). This can also be translated to English (as in "a 12-tone scale").

Mathematically, if you think of a scale as a set of pitch classes, you could refer to the cardinality (or size) of that set, although it is distinctly possible that only mathematicians would know what you're referring to. This doesn't have an abbreviation per se, but there is a shorthand mathematical notation. If you have a set of pitch classes (a scale) S, you can denote it's cardinality as |S|. I'm not familiar enough with musical set-theory to know if this is actually a correct use of terminology.

As a further complication, since you're looking for a universal term, you shouldn't necessarily assume that all scales cover an octave. I believe there may be scales for which this is not the case.

Update: My suggested term "cardinality" is used by tonalsoft, an online "encyclopedia of microtonal music theory", and also forms the title and central topic of the following paper: Favored Cardinality Of Scales.

  • Why an abbreviation? Any tool -virtual or physical- that tries to escape the bounds of western musical notation by presenting a user with free choice of temperament/intonation and number of notes or tones per octave will be confronted by this terminology issue. As an analogy, were the abbreviation 'MIDI' not available for the vast majority of existing tools, user interfaces would be considerably more cluttered. Aug 3 '15 at 10:14
  • I get that standards love their TLA's (or FLA in this case), but that doesn't mean everything has to be an abbreviation to be a standard. I mean, if you don't have enough room to write out the word "Cardinality" on your interface without making it feel cluttered, you may want to reconsider your design. Even the mod wheel on my keyboard is still labeled "modulation" despite the common shortened form "mod" and the near ubiquity of such such controls. I'd expect that a cardinality control would have to become commonplace before any abbreviation becomes useful. Aug 3 '15 at 13:41
  • You are very likely right, but with cardinality somehow begging explanation and own curiosity not quite stilled, I think I'll leave the answer unaccepted for just a little while. Aug 3 '15 at 21:08

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