# Definition of the note C

That I'm using the note C is arbitrary, I could be using any other. How would you define a C? If you define it as the tone with frequency 261.6 Hz (or 2^n * 261.6) then you run into problems when you change tuning - the note C in pythagorean tuning does not have the same frequency as in equal temperament, yet regardless they are both called C - so frequency can't be the answer.

I'm very new to music theory, but from what I can gather, most tunings use letters A-G along with the sharps and flats, so that there are 12 different symbols which are assigned to 12 notes of different frequencies. The letter C just represents the 4th note (ordered in ascending pitch) of these 12 notes. Is this correct?

I'm just a bit confused because if I talk, for example, about the note G# on a piano keyboard, in a certain context it cannot be called G# and must be called A flat, or so I hear.

The very simplest answer is that A is 440Hz (* 2^n, as you say) and that C is a minor third higher than that (523.251 Hz).

However, the mapping of absolute pitches to note names is only a convention, and in reality the absolute pitch of C only needs to be agreed between the people performing a piece of music.

• When I pick up my guitar, for me C is the pitch I get when I play the third fret of the A string. Even if the guitar is only "in tune" relative to itself.
• If I tune my guitar by ear, without external reference, then as long as the E string feels about right, I'll probably tune everything else relative to that. So C is 4 semitones lower than whatever arbitrary pitch the E string is at.
• If I tune my guitar with a tuning fork or an electronic tuner, then C is about 523Hz -- unless I change my tuner to a difference reference pitch.
• If I decide a song would be easier to sing if I put a capo on the 2nd fret, I have a choice. When I play the third fret relative to the capo, I can call it D -- so all the chord shapes I know have changed their names -- or I can continue to call it C, even though its pitch has risen by a tone.
• If I'm in a band, and we all decide it would sound better if we all dropped our tunings by a tone, we have a choice. We can define C according to the fingerings we're used to on our instruments -- which most people are going to find easier -- or we can say that C is still 523Hz, in which case if someone says "play C", I must fret the D position.
• In an orchestra, C is whatever comes out of the oboe when the oboist fingers a C note. Everyone else tunes to the oboe.

... and so on.

To complicate matters more, you could end up in a situation where different band members have different definitions of C. For example, Adam is band-leader, Bill is a keyboardist at standard tuning, Charlie is a novice guitarist with a capo on the 2nd fret. "Right," says Adam, "Bill, you're playing D,G,G,D." Then turning to Charlie: "You're playing C,F,F,C".

Exactly this happens in orchestras, since some (mostly brass and wind) instruments are "transposing instruments" -- What is a transposing instrument?

As for G# vs Ab, it's simplest to think of these as the same pitch. That pitch is given different names for convenience when working in a particular key.

A western major or minor key consists of 7 notes out of the 12 available. For example C major uses A,B,C,D,E,F,G and omits C#,D#,F#,G#,A#.

D major is: D E F# G A B C#

Why isn't D major: D E Gb G A B Db ?

Well, it's more difficult to think about, described in that way. It has two Gs, one natural and one flattened, and to Ds in the same way. It's much simpler if we organise things so that a scale has all the letters in it, and we can remember that a particular note letter is always flattened or sharpened. Hence, D major goes DEFGABC, with F and C sharpened.

To be super-pedantic, players of some instruments will play Gb and F# as slightly different pitches, and if you get into the maths of tuning in more depth, you'll find out why.

This is a very good question, and you're on exactly the right track to be looking away from frequency. I think the place to start is with taking apart some concepts that, for convenience, most of us group together most of the time. This means defining some terms, but I'll try to keep it to the most essential.

When you ask to define "a C", you are asking to define a pitch-class, which isn't the same as a pitch. "Middle C" (aka `C4`) is an example of a pitch, and one octave above it (`C5`) is a different pitch. When we talk about pitch class, we are asserting pitches that form perfect octaves are fundamentally equivalent (which is a really well-supported, useful concept, even though in some contexts the differences between different members of the pitch-class `C`).

The concept of pitch-class is particularly important to the atonal, 12-tone music of the 20th century, but it is not limited to that context. Importantly, the concept of pitch-class does not depend on another concept associated with that kind of music (and musical analysis): enharmonic equivalence. Enharmonic equivalence is the principle of the piano keyboard: that `C#` = `Db`. If you are going to consider the music of someone like Webern, you would be entirely lost without using the concept of enharmonic equivalence; however, as you noted in the last sentence of your reply, enharmonic equivalence definitely does not always apply. For our broader purposes, therefore, we want to consider `C#` to be a different pitch-class than `Db` (whereas in 12-tone music, both would be part of `pc 0`). This means we have a lot more than 12 pitch-classes: in fact, 35 (7*5, i.e. `Cbb`, `Cb`, `C`, `C#`, `C##`, `Dbb`, etc.).

To define what a pitch-class is, then, we need to look at what makes `G#` (to use your example) different from `Ab`. Let's imagine a piece in the key of C major:

• `G#` in that key, would be most likely as part of a secondary dominant chord (E-major) leading to the submediant chord (A-minor). Imagine a (bad, awkward) melody starting in C major that is going along and comes to `... B G# C`. These notes might be harmonized by `iii` `V/vi` `vi` (i.e. `{E G B}` `{E G# B}` `{A C E}`), which could begin a modulation into the relative minor key, A minor.

• `Ab`, on the other hand, might occur as an added seventh in a fully-diminished B chord (`viiº7`). Consider the same melody, but spelled as `... B Ab C`. Fully-diminished chords normally occur in minor keys, so here, the harmony would likely be `iii viiº7 i` (i.e. `{B D F Ab}` `{C Eb G}`) and imply a modulation into C minor, the parallel key.

Even if played on a piano, where in isolation they would be indistinguishable from one another, `G#` and `Ab` convey very different musical meanings about the further direction of the piece.

By calling a pitch-class `C`, then, what we are doing is representing through a notation certain kinds of relationships between members of pitch-class `C` and members of pitch-classes `G#`, `Ab`, and so on. Any `C` and any `G#` will have the same kind of relationships to one another (of course, there are other relationships that will be specific to context.

It is impossible to make a comprehensive list of all these relationships, especially because many of them derive from the socially-constructed understanding of how `C`s have been used in music throughout the centuries. Trying to understand these functional relationships is one of the major tasks of the field of music theory. One example: the relationship between `C` and `G` is very like the relationship between `G` and `D`, and we call that kind of relationship a "perfect 5th".

These relationships are very strongly related to frequency ratios and the physics of sound/overtones, but as you observed, they are not identical, for two reasons:

• First, the trivial: tunings and temperaments define the ratios between frequencies, but don't specify an absolute reference frequency. For much of history, this was not at all standardized, beyond perhaps whatever the local organ happened to be tuned to. Even today, while `A = 440 Hz` is very prevalent, `A = 415 Hz` is common in performances of early music, and some orchestras are now tuning sharper (e.g. to `A = 443 Hz`).

• Secondly, the ratios themselves are adjusted from the "pure" small-whole-number forms to fit the needs of the tuning system. Even the octave, which because it's so critical is kept in its perfect `2:1` ratio in virtually every system, can in principle be adjusted. In Equal Temperament, every ratio except the octave is adjusted away from its ideal value — yet we consider the relationship between `C` and `G` (or whatever) to be more alike than different, and the Well-Tempered Clavier to still be the same piece of music when played in Equal Temperament.

So in brief, a `C` (or any other pitch-class) is an abstract category which signifies that its members possess certain types of relationships to each of the other pitch-classes.

How would you define a C?

You define it using frequency, like you said. But usually, people don't calculate the frequency of the note C, but the frequency of the note A. The 'standard' tuning pitch that is used nowadays for most Western music, is 440 Hz, is named a′ or A4.

most tunings use letters A-G

This is correct, but I think you are a bit confused. The letters A-G are used to represent notes; the tunings in any instruments are notes, so they do use the letters A-G.

In some other countries/languages/cultures, instead of the letters A-G they use do-re-mi-fa-sol-la-si, each of which corresponds to a letter.

The letter C just represents the 4th note (ordered in ascending pitch) of these 12 notes. Is this correct?

Yes, it is.If you see the letters A-G, with A as the first letter, the C is the 4th note, on the chromatic scale. A (1st), A# (2nd), B(3rd) and then C (4th), but it is the 3rd if you use the Aminor/Cmajor scale, since there is not A# in it.

I'm just a bit confused because if I talk, for example, about the note G# on a piano keyboard, in a certain context it cannot be called G# and must be called A flat, or so I hear.

Yes, this is correct. Sometimes you write it G# and sometimes Ab, and that depends on the content. For more insight on this topic, look at this thread:

• It's important to point out that a 12-note scale is only one type of scale. Other composers use 24 or 48-note scales, dividing the octave into many microtones. It is also important to remember that tuning systems have changed dramatically over the past few hundred years, and even vary by country today. Really, the frequency for a note encompasses a range of frequencies, and not any one particle single number. This question is similar to "why is the sky called Blue?" – jjmusicnotes Sep 3 '14 at 13:09

Music is completely different from math, and therefore there is no right (in the sense of correct) choice. Even if you may have set up up your instrument tuner to the desired reference frequency (however you determined it) and it flashes the green light, as soon as your buddy on the piano has a different C, you lost since more likely than not, none of you is able to re-tune a piano. Note, that there is a trend to increase the base frequency hertz by hertz: for greater brilliance or without plausible reason for this nonsense, depending on whom you ask. So 442 Hz is quite common for an orchestra, but 444 Hz is also not unheard of.

How would you define a C?

It's a note (a set of notes, one for each octave) in the scale. If you specify the scale AND the tuning, then you'll have a frequency for it. As you rightly noted, there's no single frequency. But...

The beauty of equal temperament is that the frequency for C is fixed, regardless of the root note and scale.

• I understand what you mean about a fixed frequency for C using equal temperament, but strictly speaking it's fixed relative to whatever your overall concert pitch is. Implicitly this would be the current standard of A440, but this need not (and has not) always been the case. Sorry to nitpick... – Bob Broadley Sep 2 '14 at 20:38
• Might be better to describe it as a fixed frequency relationship with other pitches when using equal temperament. – Bob Broadley Sep 2 '14 at 20:41

I assume you are working with the chromatic scale. The chromatic scale has an evenly distributed 12 semitones per octave. This means that going up 12 semitones equals doubling the frequency. Going up 1 semitone it therefore equal to multiplying the frequency with

2^(1/12) ≈ 1.06

If you have a base frequency of 440Hz for A4, then C3, which is 9 semitones lower, will have a frequency of

440*2^(-9/12) ≈ 261.6Hz

This way you can calculate any frequency based on your base frequency.

The way you define C is that its a note between previous and next note. (Unless on 1 tone tunings where with only C note). The C means nothing more than this as everything else is arbritary.