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.

Frequencies

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,

OctaveStarts withatends just beforeComment
0C016.3516HzC1pipe organs and electronic instruments only!
1C132.7032HzC2 
2C265.4064HzC3C2 is "low C"
3C3130.8128HzC4 
4C4261.6256HzC5C4 is "middle C"
5C5523.2512HzC6 
6C61,046.5024HzC7C6 is "high C"
7C72,093.0048HzC8 
8C84,186.0096HzC9C8 is the last note on a piano or piccolo
9C98,372.0192HzC10Never used (as fundamental).
10C1016,744.0384HzC11Never 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:

SemitoneName
0C or B#
1C# or Db
2D
3D# or Eb
4E
5F
6F# or Gb
7G
8G# or Ab
9A
10A# or Bb
11B 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):

C0C0#D0D0#E0F0F0#G0G0#A0A0#B0
C1C1#D1D1#E1F1F1#G1G1#A1A1#B1
C2C2#D2D2#E2F2F2#G2G2#A2A2#B2
C3C3#D3D3#E3F3F3#G3G3#A3A3#B3
C4C4#D4D4#E4F4F4#G4G4#A4A4#B4
C5C5#D5D5#E5F5F5#G5G5#A5A5#B5
C6C6#D6D6#E6F6F6#G6G6#A6A6#B6
C7C7#D7D7#E7F7F7#G7G7#A7A7#B7
C8C8#D8D8#E8F8F8#G8G8#A8A8#B8
C9C9#D9D9#E9F9F9#G9G9#A9A9#B9
C10C10#D10         

(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

InstrumentStartsStops
guitarE2E6
seven string guitarB1E6
celloC2A6
bass guitarB0?D4?
pianoA0C8
piccoloC5C8
violinG3E7

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: The notes on the staff

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 semitonesN
+0 (fundamental)1 (fundamental)
+12 (one octave above fundamental)2
+193
+24 (two octaves above fundamental)4
+27.865
+316
+33.697
+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.

Using Semitones

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

Name0?1?2?3?4?5?6?7?8?9?10?11?
C major0 2 45 7 9 11
C minor0 23 5 78 10 
C anhemitonic pentatonic0 2  5 7 9  
C anhemitonic pentatonic (variant)0 2 4  7 9  
C hemitonic pentatonic (rare)0   45 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 differenceTraditional Name
+0unison
+1flat second, minor second, diminished second
+2(major) second
+3minor third
+4(major) third
+5(perfect) fourth
+6augmented fourth, diminished fifth, tritone, flat fifth
+7(perfect) fifth
+8flat sixth, minor sixth, augmented fifth
+9(major) sixth
+10flat seventh, dominant seventh, augmented sixth
+11(major) seventh
+12octave

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%.

Every note....

SemitoneMy NotationConventional NotationFrequency (Hz)
00_0C016.3516
10_1C#017.3239
20_2D018.3541
30_3D#019.4454
40_4E020.6017
50_5F021.8268
60_6F#023.1247
70_7G024.4997
80_8G#025.9565
90_9A027.5000
100_10A#029.1352
110_11B030.8677
121_0C132.7032
131_1C#134.6478
141_2D136.7081
151_3D#138.8909
161_4E141.2035
171_5F143.6535
181_6F#146.2493
191_7G148.9994
201_8G#151.9131
211_9A155.0000
221_10A#158.2705
231_11B161.7354
242_0C265.4064
252_1C#269.2957
262_2D273.4162
272_3D#277.7818
282_4E282.4069
292_5F287.3071
302_6F#292.4986
312_7G297.9989
322_8G#2103.8262
332_9A2110.0000
342_10A#2116.5410
352_11B2123.4709
363_0C3130.8128
373_1C#3138.5914
383_2D3146.8324
393_3D#3155.5635
403_4E3164.8138
413_5F3174.6142
423_6F#3184.9973
433_7G3195.9978
443_8G#3207.6524
453_9A3220.0001
463_10A#3233.0819
473_11B3246.9417
484_0C4261.6256
494_1C#4277.1826
504_2D4293.6649
514_3D#4311.1271
524_4E4329.6277
534_5F4349.2283
544_6F#4369.9945
554_7G4391.9955
564_8G#4415.3048
574_9A4440.0000
584_10A#4466.1640
594_11B4493.8835
605_0C5523.2512
615_1C#5554.3654
625_2D5587.3297
635_3D#5622.2542
645_4E5659.2552
655_5F5698.4565
665_6F#5739.9891
675_7G5783.9911
685_8G#5830.6097
695_9A5880.0003
705_10A#5932.3280
715_11B5987.7669
726_0C61046.5024
736_1C#61108.7306
746_2D61174.6595
756_3D#61244.5083
766_4E61318.5105
776_5F61396.9136
786_6F#61479.9783
796_7G61567.9822
806_8G#61661.2191
816_9A61760.0001
826_10A#61864.6559
836_11B61975.5338
847_0C72093.0059
857_1C#72217.4624
867_2D72349.3190
877_3D#72489.0166
887_4E72637.0210
897_5F72793.8260
907_6F#72959.9565
917_7G73135.9644
927_8G#73322.4382
937_9A73520.0002
947_10A#73729.3118
957_11B73951.0676
968_0C84186.0100
978_1C#84434.9224
988_2D84698.6360
998_3D#84978.0356
1008_4E85274.0444
1018_5F85587.6543
1028_6F#85919.9130
1038_7G86271.9287
1048_8G#86644.8800
1058_9A87040.0040
1068_10A#87458.6235
1078_11B87902.1353
1089_0C98372.0200
1099_1C#98869.8450
1109_2D99397.2770
1119_3D#99956.0660
1129_4E910548.0840
1139_5F911175.3040
1149_6F#911839.8210
1159_7G912543.8630
1169_8G#913289.7600
1179_9A914080.0080
1189_10A#914917.2470
1199_11B915804.2705
12010_0C1016744.0400
12110_1C#1017739.7000
12210_2D1018794.5550
12310_3D#1019912.1330
12410_4E1021096.1680
12510_5F1022350.6070
12610_6F#1023679.6520
12710_7G1025087.7150
12810_8G#1026579.5060
12910_9A1028160.0020
13010_10A#1029834.4790
13110_11B1031608.5570

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.