Understanding Musical Notes and their Notation
Reading music is really all about memorization. Your painfully learn how to look at a printed note (on paper) and immediately finger the instrument in the correct way. On a piano, each note lands on exactly one of the 88 keys, and this is fantastic. On a guitar, a given note could land on up to five separate strings, and you can always finger a lower fret and bend said strings, and so forth. So, it might not be unfair to say that you could find given note in perhaps fifteen distinct locations on a guitar neck---perhaps more! So, learning to navigate the guitar neck can be a real problem, and it is very important to keep the note names clear as one learns this complex layout.
The painful process of associating a symbol (graphical representation of a note) with a note (a tangible fret that you finger) is made much easier if you have a unique name to use to help you reason about note placement and memorization. This article describes the naming of notes, and how to keep track of them.
Some of my friends who are real musicians have complained about the way that I have burned away so much theory and tradition to get such a minimalist view of notes. Their complaints invariably revolve around, "you've removed a perspective that I use to compose." They start to get upset at the point where one goes to a chromatic (semitone number) representation of all notes, and I'm sure that they're right. That said, one can think in any style that they want depending on their circumstance, and thinking about semitone number 5 (me) is not incompatible with "the second note in the Phrygian mode" (them). Really, you can see it both ways at the same time. Also, I see little point in worrying about issues like "modes" until one is a great "typist" of existing musical manuscripts---most compositional perspectives will do nothing for you in this important regard. Therefore, without further ado or apology, let's look into naming musical notes.
Pretend that you had a piston, and it is open to the air. Now, let's say that you start it moving back-and-forth, with a smooth sinusoidal motion, at a given frequency, f, i.e., a particular number of cycles per second. If f=440Hz, i.e., 440 times per second, you'll hear "concert A" (also known as A4), coming out of your loudspeaker!
If you experimented with the knob for f, you'd discover that you'd only be able to hear sound between about f=20Hz and f=20kHz (20 thousand Hz). Your dog will be able to hear well above you, and your pet bat even further.
Octaves, and the Notes within
The range of hearing is divided up into octaves. According to convention, each octave holds exactly twelve notes, and there are about eleven octaves.
Counting the Octaves
An octave is a defined as the interval between a frequency f, and twice f. The 0th octave is taken to start at a note called C0, where f(C0) = 16.3516Hz. C0 is the lowest note discernible to the trained ear. According to the numbering by The Acoustical Society of America,
|Octave||Starts with||at||ends just before||Comment|
|0||C0||16.3516Hz||C1||pipe organs and electronic instruments only!|
|2||C2||65.4064Hz||C3||C2 is "low C"|
|4||C4||261.6256Hz||C5||C4 is "middle C"|
|6||C6||1,046.5024Hz||C7||C6 is "high C"|
|8||C8||4,186.0096Hz||C9||C8 is the last note on a piano or piccolo|
|9||C9||8,372.0192Hz||C10||Never used (as fundamental).|
|10||C10||16,744.0384Hz||C11||Never used (as fundamental).|
So, we have learned that when somebody speaks of "C", we should shout, "which one?!" (Unless what they're saying holds for every C, and if that's the case, they should make it clear.)
What are the notes in a given octave?
Now that we have learned that there are actually eleven C's, let's pick one specific octave and stay within it. How many notes are in this octave? It turns out that each octave has exactly twelve notes. We can number the notes by means of their value as semitones, where the notes go up in frequency as we go:
|0||C or B#|
|1||C# or Db|
|3||D# or Eb|
|6||F# or Gb|
|8||G# or Ab|
|10||A# or Bb|
|11||B or Cb|
Look at the uniformity of the numbers on the left, and the crazy sharps (#) and flats (b) on the right! What is going on here? Well, in the early days of Western music, there were only seven distinct notes: C, D, E, F, G, A, and B. (Each exists in each octave, of course.) Over time, it was realized that five more notes should be added, eventually enabling the transposition of music (i.e., raising or lowering the pitch of an entire piece to meet the limited abilities of particular singers, etc.). The new notes fit between C/D, D/E, F/G, G/A, A/B. So, the problem became, "how does one notate the new note between C and D, for example," but in such a way as not to destroy existing sheet music? (Yes, the legacy problem rears its head....) One notation option is C# ("C sharp, the new note above C), and another is Db ("D flat, the new note below D"). (Sadly, I don't have the proper character to make a flat, so I need to use the letter "b".) Adding further obscurity to this notation, while there is no new note between E/F, E# is taken to be F, for example. There are canonical naming conventions that dictate if you should view a note as (say) A# or Bb in a given key, and you can learn all about them in music theory books---if you care, that is!
Putting it all together
Reading left-to-right, top-to-bottom, we have enumerated every single possible note that you can hear (with human ears) in Western music (sorted by increasing frequency):
(where E#=F, Fb=E, D#=Eb, and so forth).
Why is there no F10, for example? Notes above D10 can only be heard by your pets! And even notes above C8 don't sound very good---that's the highest key on a piano or piccolo! So, we probably should ignore the bottom three rows of the above table, and perhaps even most of the first row! In other words, the 8th, 9th, and 10th octaves are probably worthless (except as harmonics of lower frequencies, of course).
What are the ranges of some instruments?
Here are the ranges of some popular instruments
|seven string guitar||B1||E6|
And here is the bass (lower) and trebble (upper) staff, which can be thought of as one giant staff eleven-line staff, except C4 is not shown:
So, when you talk to somebody who calls his guitar's E2 string "low E," just slap him, and shout, "E2, buddy!" Most musicians, because of the intense demands of their partying lifestyle, have little time to study their octaves and learn that every E has a unique name. But one warning: Guitar music is notated one octave up, i.e., when you're supposed to play C4, it is traditional to write C5.
The people who become conductors and writers know perfectly well that each note has a unique name, but they use some crazy notation systems. For example, C6 can be notated as c''' (Helmholtz), C64 (piano key number), C^3 (organ notation).
Why does C4 on a piano sound different than C4 on a guitar?
A particular note is defined by its fundamental frequency; for example, E2 is 82.4096Hz. But when you pluck the E2 string on a guitar, you get a lot more notes back, in addition to E2. You'll get the E2 plus some number of semitones relative to the note that you plucked:
|Additional relative semitones||N|
|+0 (fundamental)||1 (fundamental)|
|+12 (one octave above fundamental)||2|
|+24 (two octaves above fundamental)||4|
|+36 (three octaves above fundamental)||8|
So, you'll get E2, E2 plus 12 semitones (E3), E2 plus 19 semitones (B3), E2 plus 24 semitones (E4), E2 plus 27.86 semitones (something between G4 and G4#), E2 plus 31 semitones (B4), E2 plus 36 semitones (E5), and so forth. (And this holds for any note you pluck, not just E2.) The additional contributions (harmonics) are weaker than the fundamental, but their presence can be heard; so, when you pluck a string, it is really a chord, i.e., multiple notes come out. The strength of the harmonics is dictated by how/where you pluck it, and this is what makes a piano sound different than a guitar---it's just that they'll have different amounts of each harmonic.
Note that the harmonics start an octave above the fundamental that you pluck---a harmonic can never be lower than the fundamental, or even in the same octave.
So, turning back to C9, you might hear it as an octave if you plucked C5, perhaps faintly, but nobody would ever try to make a C9 string and pluck that.
For those interested in "why," f(n + m)/f(n) = exp((m/12) ln(2)). If you divide the length L of the string by N, where N is an integer, N = exp((m/12) ln(2)), or m = (12/ln(2)) ln(N). So, if you have a harmonic that cuts the string into N parts, it is m semitones above the fundamental.
My personal, totally non-standard notation is the best, in my opinion. If I want to talk about a note, like E, I'll just say 4, since it is the 4th semitone in any octave. If I mean to talk about E2, I'll say 2_4 (actually, just 24 in base 12). And if I'm doing computations (as you'll see later), I view 2_4 as its absolute semitone number, 2*12 + 4 = 28. (Note that o_s = 12o + s, o_(12 + s) = (o+1)_s, and so forth.) For those of you who remember base twelve arithmetic from elementary school (it was probably taught around the time of "clock arithmetic"), yes, you're right, the ultimate thing is to just remember the numbers in base twelve, and this is actually what I do. (If you don't know what the prior statement means, ignore it for now.)
I tune my guitar uniformly in fourths, so the notes on it are (low to high) E2, A2, D3, G3, C4, F4. But this isn't the way that I think about it; to me, they're 2_4, 2_9, 3_2, 3_7, 4_0, 4_5. Your guitar is probably tuned E2, A2, D3, G3, B3, E4, or 2_4, 2_9, 3_2, 3_7, 3_11, 4_4.
If I'm thinking of an F3, I might be lazy and only say 5, but I should never lose track that it's 3_5 I'm talking about. Some people might think this is overly complicated, but I find it easier to think of 6 than F# or Gb. In other words, there is now only one name for a particular note. And it is easy for me to answer questions like, "what is a perfect fourth above C?" It's F! I know because a "perfect fourth" means five semitones (more on this later), and 0 (= C) + 5 (= perfect fourth) = 5 (= F). The following sections on intervals make this kind of work easy.
A critical factor in reading sheet music is to memorize of the exact note. Don't just think "C" when you look at the staff---if you're looking at middle C, think "C4," "4_0," or at the very least "middle C," but not just "C." The same should go when you look at a "C" on your guitar neck---you should know the octave, both on the written sheet music and the instrument neck.
Making sense of common scales
|C anhemitonic pentatonic||0||2||5||7||9|
|C anhemitonic pentatonic (variant)||0||2||4||7||9|
|C hemitonic pentatonic (rare)||0||4||5||7||11|
For example, do you remember, "a major scale has intervals WWHWWWH?" W means +2 semitones, H means +1 semitones, so the notes in E major would be 4, 6, 8, 9, 11, 13 (= 1), 3.
You can also note that the pentatonic scales are just a subset of the major scale---just view them as a major scale without the 11, and a 4 or 5.
Figuring out key signatures
As a quick reason to use semitones, "what key has N flats?" The answer is, key = 5 * N mod 12. So, for example, if N=3 (three flats), the key is 5*3 mod 12 = 15 mod 12 = 3 = Eb. Cool, huh?
"What key has N sharps?" The answer is, key = -5 * N mod 12. So, for example, if N=2 (two sharps), key = -5*2 mod 12 = -10 mod 12 = 2 = D.
"What is D# minor as a major key?" Well, minor = major + 9 (mod 12). If minor = 3, major = 6. (Note 6 + 9 = 15, 15 mod 12 = 3.)
And, if you tune your guitar in 4ths, there is a very easy way to move your finger to do this computation (more in the section on tuning in 4ths).
(For those of you who do not remember what "mod" means, "a mod b" means that you can add/subtract b's from a until you get the smallest positive number possible. So, for example, 24 mod 12 = 0, 30 mod 12 = 6, etc.)
What are common intervals?
Sadly, the names of musical intervals are full of rich historical legacy. Basically, if you view two notes as an absolute semitone, and subtract them, this difference is the interval.
|Semitones difference||Traditional Name|
|+1||flat second, minor second, diminished second|
|+6||augmented fourth, diminished fifth, tritone, flat fifth|
|+8||flat sixth, minor sixth, augmented fifth|
|+10||flat seventh, dominant seventh, augmented sixth|
How can I compose chords?
(This is coming shortly.)
What is the frequency of a given note?
Let f(n) be the frequency of a given semitone (numbered as above). A given note will vibrate at a given number of cycles per second, a Hertz (Hz). The standard is A4 = 4_9 = 440Hz, or f(9+4*12) = f(57) = 440Hz.
There is only one other rule, an octave is twice the frequency, i.e., f(n+12) = 2f(n).
Most of you will try to guess a solution of the form f(n)=a exp(b n), and with the back of an envelope you will be able to determine that f(n) = C0 * exp((n/12)*ln(2)), and C0 = 440Hz/exp(57/12 ln(2)) = 16.3516Hz.
So, for a practical example, given that a phone passes frequencies below about 3kHz, the highest semitone it can hear is around 90, or F7#.
Where do you put the frets on a guitar neck?
If you have a string of length L, put the first fret at (94.38%)L, and the next fret at (94.38%)^2L, and so forth, until you get close enough to the bridge to stop.
Why? The frequency of a string goes as the reciprocal of its length. So, pretend that length L gets frequency f(n). You want a shorter length, k * L (where k < 1), to get frequency f(n+1). Therefore, you can say, where m is some (arbitrary) constant, f(n) = m/L. f(n+1) = m/(k L). Therefore, k f(n) = f(n+1). You can easily solve for k = exp(-(ln(2))/12) = 0.9438743 = 94.39%.
|Semitone||My Notation||Conventional Notation||Frequency (Hz)|
The above table has lots of worthless notes; you'd never consider anything above C8 in practice unless you were writing, "The Canine Symphony." The notes are just here for completeness.
If you prefer to read this web page in Dutch (it's so exciting to say that!), check out http://www.gitaar.net/les/notenlezen3/, with thanks to Rob van Putten.
I wish to thank David Espinosa for helping me with many of these issues over time. I also wish to thank wildcowboy for pointing out a critical typing mistake!
Kleanthes Koniaris, email.