A grandfather and his granddaughter sing the same song together. His voice sits an octave below hers, the two lines never touch, and yet nobody in the room thinks they are singing different songs. Play the tune on a piano an octave higher and it is still the tune. Play it a little faster and it survives. Play it with every note shifted up by a fixed amount, in a new key, and a four-year-old will still say "that's the one."
What survived in each case was not the sounds. It was the relationships: this note is higher than the last by about this much, and lasts about twice as long as that one. Music fundamentals is the study of those relationships and of how to write them down, so that a stranger a century away can recover the tune. It is a very small course with a very large reach. Everything else in the major assumes it.
No prior knowledge is assumed here. If a word is new, it is explained on the spot. Sound plays only after you tap something, the volume is kept low, and every sound interactive also shows its numbers, so it works with the sound off.
A sound is a pressure wave that repeats. How many times it repeats per second is its frequency, measured in hertz (Hz). We hear high frequency as high pitch. The note an orchestra tunes to, the A above middle C, is set by international agreement at 440 Hz: the air pressure at your ear goes through 440 full cycles every second (ISO 16 fixed this standard, after a 1939 conference in London; orchestras often tune a little higher, and period-instrument groups playing baroque music commonly tune to 415 Hz).
Here is the surprising part. Your ear does not add; it compares. Double the frequency and you hear the same note, higher. 220 Hz, 440 Hz and 880 Hz are all called A. The distance between one A and the next is an octave. Music divides each octave into 12 equal steps called semitones, and since "equal" to the ear means "equal ratio," each step multiplies the frequency by the same number.
Worked: middle C is n = 60, so f = 440 × 2−9/12 = 261.63 Hz. E above it (n = 64) is 329.63 Hz, G (n = 67) is 392.00 Hz, and the A an octave above 440 is 880.00 Hz. The keyboard below is two octaves long. Tap a key and watch the numbers. The sequence of names repeats after 7 white keys and 12 keys in all, because 12 semitones is exactly one doubling.
Equal temperament, A4 = 440 Hz. Letter names (C D E) and fixed-do solfège (do re mi) are both shown, because much of Latin America and Europe uses solfège as the name of the note itself.
Why 12 and not 10 or 15? There is no single answer. One reason is arithmetic. The most consonant interval after the octave is the fifth, a frequency ratio of 3:2. Stack twelve fifths and you land almost exactly on seven octaves: (3/2)12 = 129.746, while 27 = 128. The gap, about 23.46 cents (a cent is one hundredth of a semitone), is the Pythagorean comma. Twelve equal steps spread that small error around so that every key sounds acceptable. Other cultures divided the octave into five, seven or other numbers of notes, and some tune their scales differently from piece to piece. Acoustics, Tuning & Timbre (35/68) and Music Traditions of the World (35/69) go there.
Notation is a graph. The vertical axis is pitch: higher on the page is higher in sound. The horizontal axis is time, left to right. Five horizontal lines, the staff, give you reference rows; notes sit on a line or in the space between two lines. Each step up from a line to the next space, or space to line, moves to the next letter name: A B C D E F G, and then A again, because the pattern repeats at the octave.
Which line is which note? A clef at the left tells you. The treble clef (the curl) circles the second line from the bottom and says "this line is G." The bass clef places its two dots around the fourth line and says "this line is F." Piano music uses both, one for each hand. Middle C sits between them, one short ledger line below the treble staff or above the bass staff. That is the same key either way.
Musicians remember the lines and spaces with mnemonics: the treble lines are E G B D F ("Every Good Boy Does Fine"), the spaces F A C E, the bass lines G B D F A and the spaces A C E G. Small signs change a note by one semitone: a sharp (♯) raises it, a flat (♭) lowers it and a natural (♮) cancels either. That is why a black key has two names: the key between F and G is F♯ or G♭, depending on the key of the music (more below).
Up to eight notes are kept. Names are shown as letter, fixed-do and number.
Worked: in the treble clef, a note on the fourth line is D5. Add a sharp and it is D♯5 (or E♭5; the ear can't tell them apart on a piano, though the notation can). Its frequency is 440 × 2(75−69)/12 = 622.25 Hz. The octave number counts from the C. The C an octave above middle C is C5.
The distance between two notes is an interval, and it is the unit the ear actually tracks. A song stays recognizable when you move it up because every interval is kept. Intervals have a number and a quality. The number counts letter names, both ends included: C up to E is a third (C, D, E); C up to G is a fifth. The quality says how many semitones that is: a third can be major (4 semitones) or minor (3), and the fourth, fifth and octave come in a perfect form, which is the same in both major and minor scales. Spelling matters: C to F♯ is an augmented fourth; C to G♭ is a diminished fifth. They sound the same on a piano, and they mean different things in the music.
On a piano, each interval's ratio is 2(semitones/12). A perfect fifth is 7 semitones, a ratio of 1.4983 (a pure 3:2 is 1.5, about 2 cents wider). A major third is 4 semitones, a ratio of 1.2599; the pure 5:4 third that choirs often sing is 1.25, which is 13.7 cents narrower than the piano's. The tool below shows both, so you can see that "in tune" depends on the tuning system. Acoustics, Tuning & Timbre (35/68) explains where the pure ratios come from.
Just ratios are the simple-fraction (5-limit) values; where theorists disagree on one ratio (the tritone, the minor seventh) the table says so. Equal temperament, A4 = 440 Hz.
A scale is a handful of notes picked from the 12 and arranged in order inside an octave. The one most people mean by "do re mi" is the major scale. What defines it is not which keys it uses but the pattern of steps. A whole step is two semitones (W), a half step is one (H). The major scale is always W W H W W W H, from any starting note. Start on C and that pattern uses only white keys. Start on G and the pattern forces one black key, F♯, to keep the same shape.
The note you start on is the tonic, and the pattern gives the scale its character. The natural minor scale has the pattern W H W W H W W: it sounds darker because its third is a semitone lower. The harmonic minor raises the seventh note to make a stronger pull back to the tonic. A pentatonic ("five-note") scale keeps only five notes of the seven, with no half steps, so it is hard to play a clash; it turns up in folk traditions on every continent, from Scotland to China to West Africa.
A key signature is a shortcut: the sharps or flats a piece uses all the time are written once at the start of each line instead of next to every note. Count them and you know the key. G major has one sharp (F♯); D major has two; F major has one flat (B♭). Each step round the circle of fifths adds one. Try the builder below. It spells every scale by the rule that each letter name is used once, which is why you will see E♯ and B♯ in keys with many sharps.
W = whole step (2 semitones), H = half step (1), W+H = 3 semitones. The gold key is the tonic. Pentatonics are spelled as selections from the matching seven-note scale. The key signature drawn is that of the major or natural-minor key the scale belongs to.
Notice that G major and the A minor next to it are different scales, but A natural minor uses the same seven notes as C major, just starting on A. The two are relatives and share a key signature. That is the whole reason there are only 15 signatures for 30 keys.
Rhythm has two parts: how long each note lasts, and how those lengths are grouped into repeating bars (also called measures). Durations are fractions, relative to each other and never in seconds. A whole note lasts as long as two half notes, or four quarter notes, or eight eighth notes. Each is twice the one below it. A dot after a note adds half again: a dotted quarter is a quarter plus an eighth. A rest is a silence with the same lengths.
Seconds enter through tempo, usually in beats per minute (bpm). At 100 bpm one beat lasts 60/100 = 0.6 s. Tempo is often a word: adagio for slow, allegro for fast. The words are Italian because that is the language the first printed scores used. The time signature sits at the start of the piece. The top number counts the beats in each bar; the bottom number says which note value gets one beat (4 = quarter note, 8 = eighth). So 4/4 is four quarter-note beats per bar. 6/8 is six eighth notes, which most people hear as two beats of three.
Notation also joins the flags of short notes with beams, grouped by beat so the eye can see where the beats fall. Try filling a bar. The tool refuses a note that doesn't fit, which is how you learn what "a full bar" means.
Each sixteenth note is one unit. In 6/8 the beat is a dotted quarter, so the bpm counts those. The line is a single-line percussion staff: height carries no pitch here.
Worked: put one half note, one quarter and two eighths in 4/4. That is 2 + 1 + ½ + ½ = 4 beats, so the bar is full. The same two eighths in 6/8 would be a third of a bar.
Everything above is Western staff notation: one way of writing music, which grew in medieval European monasteries. The monk Guido of Arezzo, writing around 1025–26, is credited with a clear form of the staff. He also taught singers with syllables from a Latin hymn, ut re mi fa sol la. "Ut" became "do" in Italy in the 1600s, and a seventh syllable was added later (si, or ti in English-speaking countries, a change credited to Sarah Glover in the nineteenth century).
Solfège came in two flavors. In fixed-do, used across Romance-language and Slavic countries (Spain, France, Italy, Brazil, Russia, Israel), do is always C: it is a note's name, as "C" is in English. In movable-do, common in the US, UK and Germanic countries, do is the tonic of whatever key you are in. Both teach the same thing from opposite sides. Other cultures wrote melody differently. Numbered notation (1 2 3 …) was proposed in 17th- and 18th-century France, and spread to Japan and then China, where it is called jianpu; most Chinese songbooks use it. Indian music teaches with sargam, the syllables Sa Re Ga Ma Pa Dha Ni, where Sa is wherever the singer places the tonic. No notation writes everything: staff notation says little about the bends, ornaments and timing that define a performance in many traditions, and cultures that pass music on by ear often teach and record it by other means.
The tune is the first phrase of the traditional French melody known in English as "Twinkle, Twinkle" (first published in France in the 1760s; public domain). Sargam names assume Sa is the starting note, which is how a beginner meets it; a real singer sets Sa to suit the voice. Jianpu normally adds dots above and below the numbers to mark octaves.
Try the three keys. In fixed-do the names change when the key changes (G becomes sol), while movable-do, numbers and sargam stay the same because the tune's relationships are unchanged. It is the grandfather and his granddaughter again: the tune is the pattern, and the names are the labels.
Music notation is two graphs, pitch and time. Pitch is frequency and is heard as ratio, so 12 equal steps make an octave; a scale is a pattern of those steps; a bar is a sum of fractions. The names and the symbols are conventions, and learning them gives you the alphabet every other music course uses.
Try this: sing or hum any song you know, then hum it again a little higher or lower. Notice what stayed the same. Then tap its steady beat with one finger and count how many beats come before the pattern starts over: that is its bar.
Sources. Equal-temperament frequencies and cents are mathematics, computed and checked in node (A4 = 440 Hz; MIDI numbers; 21/12). The A440 standard: ISO R 16 (1955), ISO 16 (1975), and the 1939 London conference; baroque pitch of 415 Hz (encyclopaedia summaries of A440). Just ratios from the standard 5-limit table. The Pythagorean comma, 1200·log2((3/2)12/27) = 23.46 cents, was computed. Guido of Arezzo (Micrologus, c. 1025–26; some details of his legend are disputed); solfège, fixed and movable do and Sarah Glover (encyclopaedia summaries); numbered notation and jianpu (encyclopaedia summary; Souhaitty, Rousseau 1742, Galin-Paris-Chevé). The tune is the traditional "Ah! vous dirai-je, Maman" (public domain). Course placement is described in general terms. All interactive numbers were computed in node.