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It seems to me that in current musical practice, we are often in a 12-TET situation where effectively, we have an (octave repeating) set of 12 notes that make up the chromatic scale, each of which can then have various names - for example, I might point to a particular note on the keyboard and call it 'F♯' in one key; I might call it 'G♭' in another. Calling it one of those names implies you're not playing in any of a certain set of keys in which that wouldn't be the name for the note.

However, if I just want to refer to the note in its own right, without implying any sense of key, what do I call it? Obviously pointing to the key on the instrument works, if it's physically in front of you, but that's not much use if it isn't; MIDI note numbers are logically similar to what I'm talking about, except it's rather odd to use them if you're not using MIDI.

(If you're wondering why I'm asking, it's because I wanted to write an answer to Is the Western system of notes and intervals essentially two-dimensional?, but in order to write a response clearly, I needed to be able to refer to the idea of 'a note that can be played' without implying a specific tonal context.)

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  • BTW, the Sinatra fan I referred to in chat was BB King, not Satchmo. (I don't think it makes much difference relevant to that subject)
    – Stinkfoot
    Apr 20, 2018 at 5:05
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    He's Live & Well here at my place, although the club bearing his name is closing here.
    – Stinkfoot
    Apr 20, 2018 at 9:13

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The standard is to use integers from 0–11 (actually, 0–e or 0–B as I’ll explain in a second) to represent each of the 12 possible pitch classes.

“Pitch” refers to a particular note: Gb4 or F#2. “Pitch Class” refers to the family of pitches that are enharmonically and/or octave equivalent to each other. For example F# in any octave is the same family as Gb in any octave. In the unlikely event of an Ex, that too is in the same family. To refer to the entire family generally, we have arbitrarily assigned numbers to each one, starting with the pitch class that includes C as 0.

So, any B#s, Cs or Dbbs are members of pitch class 0. Any C#s or Dbs are part of pitch class 1. Any Cxs, Ds or Ebbs are pitch class 2, etc. This means that A#s and Bbs are PC 10 and Bs and Cbs are PC 11, but that can be confusing since it’s often hard to see the difference between 1 0 (meaning, perhaps, a Db followed by a C) and 10 (meaning, perhaps, A#). Instead most analysts refer to 10 and 11 with the symbols t and e, or A and B.

EDIT TO ADD: As Tim points out in the comments, this system does NOT specify octave, so if you're looking for a way to specify a particular octave without specifying a particular enharmonic spelling, it's more difficult. For the most part, this isn't an issue because of the way interval is discussed in set theory. I'll provide a quick précis of the four ways that interval is discussed, but will leave details for a different question:

1) If I care about the full size of an interval AND its direction, I will refer to the "ordered pitch interval" or opi using number of semitones and + for up and - for down. The distance from middle C up to Db a minor ninth above is +13. The distance from middle C down to B below is -13.

2) If I care about the full size but I DON'T care about direction (which is another way of saying that I don't care which note came first), then I will refer refer to the "unordered pitch interval" or upi using only number of semitones. Both of the examples I referred to above would be upi 13.

3) If I don't care about octave distances, but I do care about the order of notes, then I use "ordered pitch class intervals" or opci. This is probably the hardest one to envision at first. I will count the number of half steps it would take me to get from the first note up to the next note even if the interval went down in the actual music. Going from any C to any Db (that is to say from 0 to 1) would always be opci 1. Going from any Db to any C (i.e. 1 to 0) would always be 11. Going from D to G (2 to 7) would be 5 and G to D (7 to 2) would be 7. The main benefit of using numbers instead of note names (and, importantly, of starting our numbering system with 0 instead of 1) is that intervals can be easily figured out by simple subtraction. Subtract the first number from the second number (mod 12) and you have the opci. My four examples above: 1-0=1; 0-1=-1 which is 11 mod 12; 7-2=5; 2-7=-5 which is 7 mod 12. The mod 12 part may be unfamiliar to you, but it's something you use every time you say that something happening three hours after 11 AM will happen at 2 PM. 11+3 is actually 14 of course, but we treat time like a circle, and roll back to 2.

4) Finally, and most commonly, we can refer to the interval in the most general way possible: the interval class (sometimes called the "unordered pitch class interval", but IC is pithier and has an obvious similarity to "pitch class." In this situation we just want to know the how to get from one note to the other in semitones in the closest way possible. The closest way to get from a Db (1) to a C (0) if we don't care which comes first is 1. All half steps, major sevenths, minor ninths, etc. will belong to this same family of intervals: interval class 1. Whole steps, minor sevenths, major 9ths, etc. are part of IC 2; minor thirds, major sixths etc. are IC 3; and so on. There are only 6 possible intervals classes (unless you want to count unisons and octaves).

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  • So, it's more of a duodecimal (12 different 'digits') than decimal system? And would middle C be known as 04?
    – Tim
    Apr 19, 2018 at 13:37
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    @Tim I kind of obliquely answered that in my #3 point above: we start with 0 in this case because it makes the math far easier. If we called a unison a 0th instead of a 1st, everything would actually make more sense: as it is, an octave means going up seven steps rather than the eight that "oct-" prefixes imply. As a result is double octave is actually a 15th (or quintessima) even though you would think that double octave should equal 16. By calling our arbitrary starting point 0, we eliminate that kind of confusion. (although, admittedly, probably just cause new confusion!)
    – Pat Muchmore
    Apr 19, 2018 at 15:05
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    @ToddWilcox It is usually just called "set theory" unfortunately, despite the potential confusion with math. Actually, it's ultimately the same basic abstract world: this form of music theory is simply an application of mathematical set theoretical ideas to the pitches of 12-TET music. I like your idea of called it "musical set theory" in order to distinguish it most clearly. It was sort of independently explored by composers like Elliott Carter, Allen Forte and George Perle. The current (more-or-less) standardized form is well-explicated in Straus' Intro to Post-Tonal Theory.
    – Pat Muchmore
    Apr 19, 2018 at 16:59
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    Actually, the only really value-free way of specifying pitches is by frequency. Any number (0-11) or letter (A-G#) system is kowtowing to some prejudicial notion about preferential systems of tones.
    – Scott Wallace
    Apr 19, 2018 at 18:15
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    @topomorto It would be odd now. There was a time when several analysts were using more of a "movable do" or "movable 0" system, but C as 0 is 99.9% standard in everything I've seen. Since there's no longer any automatic implication of greater significance for some notes in music without tonal centers, the choice is arbitrary and is a lot easier when we all just agree.
    – Pat Muchmore
    Apr 19, 2018 at 21:11
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There is no convention that I have used, except naming them with the way they are used most, in any Western music. Thus, F# wins over Gb, Bb over A# etc. But - it leaves a dead heat with G#/Ab !!

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    ...and the reason it's a dead heat with G#/Ab is the same reason that D is the problem note in just intonation: look at a keyboard. D and G#/Ab are the two symmetrical keys. Think about it.
    – Scott Wallace
    Apr 19, 2018 at 18:11
  • @ScottWallace - I'm missing the point of D being problematic. Yes, it's a centre point of symmetry, like Ab/G#, but that in itself isn't a reason to be a problem in J.I. Especially if the instrument is tuned to play in D !
    – Tim
    Apr 20, 2018 at 6:55
  • I agree with this answer. F♯, E♭ or G do not suggest any particular tonality because they occur in so many commonly used keys. Unfortunately, G♯ does suggest “probably A-major, E-major or their relatives” whereas A♭ suggests C-minor or F-minor, but that doesn't matter if there are enough other non-tonal accidentals around.
    – leftaroundabout
    Apr 20, 2018 at 9:32
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    Regarding Scott's point – the issue is this: it's quite unambiguous how to go about constructing most of the Ptolemaic C scale by first picking the F and G as the (Pythagorean) neighbours in the circle and fifths, and then adding E, A and B as the just major third to each. None of these notes can be constructed in a contradicting way that relates as closely to C. But for the D, you have the option of constructing it either as a minor third below F or as a perfect fifth above G. Both have a “construction distance” of 2 from C, but they result in different pitches (a syntonic comma apart).
    – leftaroundabout
    Apr 20, 2018 at 9:39
  • @leftaroundabout - G# could equally suggest F#m or C#m just as Ab could equally suggest Eb or Ab major. I've missed your point, sorry. Regarding the Ptolemaic scales, I don't understand why it's any more difficult - if that's the word - to produce one in key D than key C. Surely the start point will dictate the pitches of all subsequent notes. I understand that using 'C' notes in a 'D' key won't work.
    – Tim
    Apr 20, 2018 at 9:49
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I know of no standard, but perhaps it's most common to use naturals and sharps: C, C#, D, D#, etc. I don't see any problem in this context in terms of implying any sense of key.

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At risk of people looking at me blankly, I'd be tempted to say something like "261.63hz".

https://pages.mtu.edu/~suits/notefreqs.html

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  • That's value-free in the context of tonal music. But it's favoring the second over other possible units of time, and favoring Hertz over Cycles Per Second.
    – Scott Wallace
    Apr 20, 2018 at 15:54
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You can't. It's because music is two-dimensional: the note and the underlying frequency.

If you think that music is universally transposable, that's true in theory but never in practice. If this were not true, why would Bach write "The Well-Tempered Klavier"?

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  • I'm not quite sure I understand the why would Bach write "The Well-Tempered Klavier bit?
    – Нет войне
    Apr 19, 2018 at 22:20
  • I'm saying that we use Bach as a reference for music theory all the time. If transposition was immaterial, then the Well-Tempered Klavier would be the same played in one key as it is in 12 keys... thus we know that Bach acknowledged that it makes a difference when you transpose something, somehow.
    – elliot svensson
    Apr 19, 2018 at 22:37
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    I thought the current music history thinking is that the well-tempered clavier was not written for equal temperament, in which case it isn't surprising that transposition would make a difference - en.wikipedia.org/wiki/…. "It is sometimes assumed that by "well-tempered" Bach intended equal temperament, the standard modern keyboard tuning which became popular after Bach's death, but modern scholars suggest instead a form of well temperament." Perhaps I should ask a different question about that!
    – Нет войне
    Apr 19, 2018 at 22:41
  • @topomorto - AFAIK, the well tempered system was one of the attempts at improving intonation, to make keyboards playable in all keys - before equal temperament came along. (Other systems had avoid keys - I believe Ab was considered the worst - because they were not well intoned.) Bach was a proponent of the well tempered system and embraced it because it made it possible for him to compose keyboard works in all keys without worry about avoid keys. That resulted in "The Well Tempered Clavier".
    – Stinkfoot
    Apr 20, 2018 at 9:22
  • elliot svensson- you are quite correct about Bach's music: whatever "wohltemperirt" meant to him (it's not known for sure), it wasn't equal temperament, so transposition meant a different set of intervals. But we are talking about equal temperament here, as far as I know.
    – Scott Wallace
    Apr 20, 2018 at 16:01
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Maybe it's the mathematician in me: Establish a consistent, unambiguous notation in the context of the problem. Use any system you like that clearly communicates your meaning. Don't re-use it in the future unless it is helpful again.

You could go with "Alice, Bob, Charlie, ...", but that is a lot of semi-structured labels to keep in mind. I'd probably use the integers as labels, place 0 at middle C, and label subsequent notes by subsequent integers, positive increments corresponding to rising pitch and negative increments corresponding to falling pitch. This also captures the order properties of the pitches and is at least as easy to count with as Pat Muchmore's answer.

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