Music Theory
INTERVALS
Unison
Character
The unison isn't an interval in the strict sense — it's the simultaneous sounding of two identical pitches. It's considered the most perfect of all consonances: the only one that produces literally no beating, no friction, no tension. Both notes vibrate at exactly the same frequency in a 1:1 ratio and merge into a single, undivided sound.
In music theory the unison is the starting point for measuring every other interval — all distances are calculated from it. In composition it's used deliberately when several voices or instruments play the same pitch: to reinforce a melodic line (unison playing) or to blend timbres through doubling (e.g. violin + flute on the same note).
Although the unison creates no harmonic motion, it matters compositionally: it provides the resting point from which any melodic or harmonic journey begins — and to which it ideally returns.
Fingering
The string vibrates at its full, unshortened length. You don't fret a note — the open string sounds freely. The string-length ratio is 1/1 (no shortening), so the frequency stays unchanged (factor × 1.000).
On guitar this means striking an open string. On the high E (329.63 Hz) the pitch stays exactly 329.63 Hz — no change. This is also the lowest pitch that string can produce; every higher note is created by shortening the vibrating length.
Harmonic
Strictly speaking, no harmonic is possible because the open string IS the first harmonic (the fundamental). All higher harmonics — octave, fifth, major third, minor seventh and so on — are already present, inaudibly, as overtones within that fundamental and shape its timbre.
If you listen carefully (or use a spectrum analyzer), you'll discover that a plucked string doesn't just produce its fundamental: a whole series of overtones rings along with it. That overtone structure is exactly what distinguishes a violin from a trumpet or a human voice. The unison, then, is the complete sound from which all other intervals can in a sense be extracted.
Minor 2nd
Character
The minor 2nd is the smallest step of the Western diatonic scale — a half-step (semitone). With its 16:15 ratio it produces strong friction between the two notes: they sit so close together that the ear hears them as tense, unstable, and urgent.
Its most important role in tonal music is as the leading-tone: the 7th degree of a major scale (e.g. B in C major) lies a semitone below the tonic (C) and audibly „pulls” toward it. This pull is the basis of every cadence: the dominant chord (G major in the key of C) contains the leading-tone B, which inevitably resolves to C — generating the resolution into the tonic.
Melodically, the minor 2nd often feels painful, plaintive, or threatening. Famous examples: the „shark motif” from Jaws (two alternating semitones create immediate menace), or the opening notes of Beethoven's Moonlight Sonata. In jazz and blues, minor 2nds are an important source of tension.
Fingering
At 15/16 of the string length, the next semitone sounds — the string is shortened by only about 6.25%. On guitar this corresponds to exactly one fret. Fretting the high E (329.63 Hz) at the 1st fret produces F (349.23 Hz). The frequency rises by a factor of 16/15 ≈ 1.067.
The fret is positioned so that the vibrating length between the fret and the bridge matches this 15/16 ratio. This construction allows precise semitones in equal temperament without needing to retune.
Harmonic
As an isolated harmonic, the minor 2nd is essentially unplayable. The 16:15 ratio corresponds roughly to the gap between the 15th and 16th harmonics — these overtones lie so high and so close together that they can hardly be cleanly isolated on a normal guitar string.
In just intonation the minor 2nd is derived from the relationship of fifth (3:2) and major third (5:4), not directly from the harmonic series. This is one reason it's considered an „artificial” interval — it arises only through the construction of the scale, not through the natural overtone structure.
Major 2nd
Character
The major 2nd — the „whole tone” — is the backbone of every melody. With a 9:8 ratio it sounds mildly dissonant: distinctly more tense than a third or fifth, but nowhere near as friction-filled as the minor 2nd. It's mobile, flowing, forward-driving — the ideal step for unfolding a line.
Almost every melody moves predominantly in 2nds. The major scale consists of five whole tones and two half tones (pattern: W-W-H-W-W-W-H), and the minor scale is similar. Two consecutive whole tones, incidentally, almost equal a major third: 9/8 × 9/8 = 81/64, very close to the just major third (5/4 = 80/64) — the difference is called the „syntonic comma”.
Well-known melodies that begin with a major 2nd: „Frère Jacques” (C–D), „Happy Birthday” (the upbeat G-G–A), „Mary Had a Little Lamb” (E-D-C-D-E…). The 2nd is so omnipresent that we barely notice it consciously as listeners — it's the basic motion of music itself.
Fingering
At 8/9 of the string length, the whole tone sounds — the string is shortened by about 11.1%. On guitar this corresponds to a span of two frets. Fretting the high E (329.63 Hz) at the 2nd fret produces F♯ (369.99 Hz). The frequency rises by a factor of 9/8 = 1.125.
Whole-tone steps are the standard motion in scale playing: in a major scale you move exactly two frets at five out of seven points. This regular pattern is what makes scale playing on stringed instruments so predictable and learnable.
Harmonic
The 9th harmonic appears when you very lightly touch the string at exactly 1/9 of its length (or at 2/9, 3/9, etc.) and pluck simultaneously. This harmonic stands in exactly a 9:8 relationship to the 8th harmonic — describing the pure whole tone as an overtone.
On a high E string the 9th harmonic sits roughly where a theoretical 28th fret would be — well above the fretboard, near the soundhole (acoustic guitar) or pickup (electric guitar). It sounds three octaves plus a major 2nd above the fundamental — a very high, glassy, almost flute-like tone.
Minor 3rd
Character
The minor 3rd is the defining third of the minor sound. With its 6:5 ratio it sounds warm, soft, and slightly melancholic — exactly that subdued sweetness that distinguishes minor from major. It's the foundation of every minor triad (e.g. A minor: A-C-E, where A→C is a minor third).
In functional harmony the minor 3rd is the most important means of characterizing a key. When a piece switches from major to minor (or vice versa), it does so by altering this single interval — the third note above the root. C major (C-E-G) becomes C minor (C-E♭-G) by lowering the E to E♭. This tiny shift of just one semitone overturns the entire emotional impact.
The minor 3rd is the inversion of the major 6th (5:3) — added together, they form exactly an octave. Famous melodies starting with an ascending minor 3rd: „Greensleeves” (E–G), „Smoke on the Water” (the iconic riff begins with a minor 3rd: G–B♭), „Brahms' Lullaby” (E♭–G in the original key).
Fingering
At 5/6 of the string length, the minor 3rd sounds — the string is shortened by about 16.7%. On guitar this corresponds to a span of three frets. Fretting the high E (329.63 Hz) at the 3rd fret produces G (392.00 Hz). The frequency rises by a factor of 6/5 = 1.2.
Three frets is a frequently used spacing for minor-tinted power chords and blues licks. In pentatonic playing the minor 3rd is one of the most characteristic intervals.
Harmonic
The 6th harmonic arises when you touch the string at 1/6, 2/6, 3/6, 4/6, or 5/6 of its length. Relative to the 5th harmonic (the major third) it stands at exactly 6:5 — sounding a minor 3rd higher than that one.
The 6th harmonic lies two octaves plus a perfect fifth above the fundamental. On the high E string it sounds as a very high B (about 1976 Hz). This harmonic is still reasonably clearly producible on guitar but already sounds thin and bell-like — typical for the higher harmonics.
Major 3rd
Character
The major 3rd is the radiant third of the major sound. With its 5:4 ratio it sounds bright, clear, and joyful. It forms the harmonic heart of every major triad: C major is C-E-G, where the interval C→E is a major 3rd. This third conveys the typical „bright” mood of major.
The pure major 3rd (5:4) sounds brighter and purer than the tempered third used in equal temperament or the older Pythagorean third (81:64). Historical performances using just intonation therefore sound especially „luminous” in chordal passages — many listeners initially find this unusual but then deeply pleasing.
In music history, the major 3rd was long contested: in the Middle Ages it was considered impure and avoided; only from the Renaissance onward was it accepted as a full consonance. Today it's the most important „pleasant” interval of Western music. Famous melodies with an ascending major 3rd: „Oh, when the saints” (C–E), „Kumbaya” (C–E at the start).
Fingering
At 4/5 of the string length, the major 3rd sounds — the string is shortened by 20%. On guitar this corresponds to four frets apart. Fretting the high E (329.63 Hz) at the 4th fret produces G♯ (415.30 Hz). The frequency rises by a factor of 5/4 = 1.25.
Four frets is a typical spacing for major chord voicings on two adjacent strings. In standard guitar tuning the G string sits exactly a major 3rd below the B string — the only „third gap” in the otherwise fourths-tuned guitar, and the reason some chords have to be fingered slightly unusually on guitar.
Harmonic
The 5th harmonic is one of the most characteristic and easily produced harmonics on any string instrument! It appears when you lightly touch the string at 1/5, 2/5, 3/5, or 4/5 of its length.
Relative to the 4th harmonic it stands at exactly 5:4 — describing the pure major 3rd as an overtone. The 5th harmonic lies two octaves plus a major 3rd above the fundamental. On the high E string it sounds as a very high G♯ (about 1660 Hz). This harmonic is widely used in classical guitar and especially on violin or cello — it sounds pure, bell-like, and particularly radiant.
Perfect 4th
Character
The perfect 4th is one of the oldest and most venerable consonances. With its 4:3 ratio it sounds open, suspended, and powerful. In medieval organum (from roughly the 9th century onward), the 4th — together with the 5th and the octave — formed the foundation of the very first Western polyphony.
Its harmonic role is famously ambivalent: in classical counterpoint, the 4th between bass and upper voice is treated as a dissonance requiring resolution — even though it sounds perfectly consonant between other voices. This peculiarity makes it one of the theoretically most interesting intervals.
In modern music the 4th has flourished again: sus4 chords (e.g. Csus4 = C-F-G) replace the third with a fourth and create an open, suspended sound that can later resolve to a major or minor third. Famous examples: the opening motif of „Here Comes the Bride” (C–F = ascending 4th), „Amazing Grace” (G–C at the start), or „Auld Lang Syne” (G–C). The Star Wars main theme also opens with an ascending 4th.
Fingering
At 3/4 of the string length, the 4th sounds — the string is shortened by 25%. On guitar this corresponds to five frets apart. Fretting the high E (329.63 Hz) at the 5th fret produces A (440.00 Hz, the standard tuning pitch!). The frequency rises by a factor of 4/3 ≈ 1.333.
The 5th fret has special significance on guitar: with the exception of the G string, every string at the 5th fret produces the same pitch as the next higher open string (E at 5th fret = open A, A at 5th fret = open D, etc.). This is traditionally used for tuning the guitar by ear.
Harmonic
The 4th harmonic arises when you touch the string at 1/4, 2/4 (= 1/2!), or 3/4 of its length. It sits two octaves above the fundamental — the string vibrates in four equal segments simultaneously.
The 4:3 ratio itself doesn't describe the 4th harmonic relative to the fundamental (that would be 4:1, two octaves), but the distance from the 3rd to the 4th harmonic — from „one octave + fifth above the fundamental” to „two octaves above the fundamental”. A 4th sits exactly between them. On the high E string the 4th harmonic sounds as a high E (about 1319 Hz) — very clearly producible and one of the most beautiful harmonics on guitar.
Tritone
Character
The tritone — three whole tones stacked — is the notorious „Diabolus in Musica” („the devil in music”). Medieval theorists considered it impure because of its tense, unstable sound, and avoided it in sacred music. With the complex 45:32 ratio, it sits exactly at the midpoint of the octave and divides it symmetrically into two equal halves — a geometric property that sets it apart from every other interval.
Its very tension makes it indispensable, however: in the dominant 7th chord (e.g. G7 = G-B-D-F in C major), the major 3rd (B) and the minor 7th (F) form a tritone. This built-in point of tension is the driving force of all Western tonal music — it imperatively demands resolution to the tonic (B pulls up to C, F pulls down to E).
In the 20th century the tritone was fully rehabilitated by classical composers (Wagner, Debussy) and especially by jazz. Famous tritone uses: the opening of Wagner's Tristan chord, „Maria” from West Side Story (Bernstein deliberately uses an ascending tritone to make the longing audible), and the opening notes of „The Simpsons” theme.
Fingering
At 32/45 (≈ 0.711) of the string length, the tritone sounds. On guitar this corresponds to six frets apart. Fretting the high E (329.63 Hz) at the 6th fret produces B♭ (466.16 Hz). The frequency rises by a factor of 45/32 ≈ 1.406.
The 45/32 ratio results from three stacked whole tones: 9/8 × 9/8 × 9/8 = 729/512 — approximately 45/32 (with a small Pythagorean discrepancy). The tritone is unique in that its inversion is again a tritone: 6 + 6 = 12 semitones (an octave). That makes it the only interval that is symmetrical with itself.
Harmonic
The tritone cannot be produced as a simple harmonic — the 45:32 ratio is not part of the natural overtone series. It's one of the most dissonant and mathematically complex ratios in just intonation.
The nearest available natural pitch is the 11th harmonic (11:8 to the fundamental), which forms a so-called „neutral tritone” — slightly flatter than the tempered tritone, with a characteristically alien sound that you sometimes hear in alpine yodeling or in the blues. The „real” equal-tempered tritone arises only through the mathematical construction of the scale and is in some sense an „artificial” interval.
Perfect 5th
Character
The perfect 5th, after the octave, is the purest and strongest consonance there is. With its plain 3:2 ratio it sounds powerful, open, almost „empty” — precisely because there's no third to give it a major or minor coloring. It's the foundation of the circle of fifths, which structures the entire Western tonal system.
The 5th is one of the most important intervals in music history: Pythagoras derived his scale from it. In Gregorian organum (ca. 9th–13th century), two voices sang in parallel fifths — Europe's first polyphonic singing. In rock and metal, power chords are played using only root and 5th (e.g. „A5” = A + E): the deliberate omission of the third allows heavy distortion without the sound becoming muddy.
Famous ascending fifth openings: „Twinkle, Twinkle, Little Star” (C–C–G–G), the 20th Century Fox fanfare, and the Star Wars main theme prominently uses fifths. Strauss' opening to „Also sprach Zarathustra” (the 2001: A Space Odyssey theme) is built on an ascending fifth as well.
Fingering
At 2/3 of the string length, the 5th sounds — the string is shortened by about 33.3%. On guitar this corresponds to exactly seven frets apart. Fretting the high E (329.63 Hz) at the 7th fret produces B (493.88 Hz). The frequency rises by a factor of 3/2 = 1.5.
The 7th fret is a magical spot on the guitar: not only the 5th sits here, but also the second-easiest harmonic to produce (see below). The 7th fret also serves traditionally as a second tuning point (alternative to the 5th fret).
Harmonic
The 3rd harmonic is, after the octave, the best-known harmonic on any string instrument. It appears when you lightly touch the string at 1/3 or 2/3 of its length — directly over the 7th (or 19th) fret on guitar.
The 3rd harmonic sits one octave plus a perfect 5th above the fundamental. Relative to the 2nd harmonic (octave) it stands at exactly 3:2 — describing the pure 5th as an overtone. On the high E string the 3rd harmonic sounds as a very high B (about 988 Hz). This harmonic is particularly resonant on guitar and very widely used — in classical guitar literature, in Pink Floyd's „Run Like Hell”, and in the famous opening of Yes' „Roundabout”.
Minor 6th
Character
The minor 6th is the inversion of the major 3rd: together they form exactly an octave (4 + 8 = 12 semitones, or 5/4 × 8/5 = 2/1). With its 8:5 ratio it sounds yearning, melancholic, and soulful — there's something plaintive about it, but also gentle and consoling.
In the Romantic era the minor 6th became a favorite interval of many composers. Brahms, Schumann, and Tchaikovsky used it to express painful longing or unrequited love. In film music too it's often used to underscore emotional climaxes or tender love scenes.
Famous melodies with an ascending minor 6th: „The Entertainer” (Scott Joplin, the leap in the main motif), „Black Orpheus” (A Felicidade, the famous bossa nova piece), and the opening of „Where Do I Begin” (Love Story theme) uses the 6th very emotionally. In many jazz standards like „Body and Soul” you also find the minor 6th as an expressive leap.
Fingering
At 5/8 of the string length, the minor 6th sounds — the string is shortened by 37.5%. On guitar this corresponds to eight frets apart. Fretting the high E (329.63 Hz) at the 8th fret produces C (523.25 Hz). The frequency rises by a factor of 8/5 = 1.6.
Eight frets is a wide spacing rarely fingered within one hand position; it's usually realized through a position shift or by moving to another string. On two adjacent strings the minor 6th can often be fingered with just one fret offset.
Harmonic
The 8:5 ratio describes the distance from the 5th to the 8th harmonic. The 8th harmonic (touch at 1/8 of the string length) lies three octaves above the fundamental. The 5th harmonic (see „major 3rd”) lies two octaves plus a major 3rd above. The interval between them is exactly a minor 6th — that is, 8:5.
As a standalone harmonic, the minor 6th is hard to isolate. It emerges rather from the comparison of higher harmonics with one another. On the high E string the 8th harmonic sounds as a very high E (about 2637 Hz) — far too high for most musical applications, but technically producible.
Major 6th
Character
The major 6th is the inversion of the minor 3rd and one of the warmest and most cheerful intervals. With its 5:3 ratio it sounds bright, open, joyful, and inviting. It's a typical „folk-song interval” — many folk melodies from around the world begin with an ascending major 6th.
In functional harmony the major 6th is central to the VI chord of a major key (e.g. the relative-minor parallel Am in C major, whose root A lies a major 6th above C). In jazz it also plays a key role in 6th chords (e.g. C6 = C-E-G-A) and in Hawaiian music.
Famous examples of ascending major 6ths at melodic openings: „My Bonnie Lies Over the Ocean” (G–E at the start), the main theme of Dvořák's New World Symphony, and the NBC network logo („G-E-C”, two 6ths in succession). Even the famous Beatles song „Hey Jude” prominently uses the 6th in its melodic shaping.
Fingering
At 3/5 of the string length, the major 6th sounds — the string is shortened by 40%. On guitar this corresponds to nine frets apart. Fretting the high E (329.63 Hz) at the 9th fret produces C♯ (554.37 Hz). The frequency rises by a factor of 5/3 ≈ 1.667.
Like the minor 6th, the major 6th is rarely played as a stretch across nine frets but usually realized through string changes or position shifts. In sixth-chord voicings the major 6th is often the topmost note (e.g. in a Maj7 or 6 chord).
Harmonic
The 5:3 ratio describes the distance from the 3rd to the 5th harmonic. The 3rd harmonic is the perfect 5th (an octave + fifth above the fundamental, see „perfect 5th”). The 5th harmonic is the pure major 3rd (two octaves + major 3rd above the fundamental, see „major 3rd”). The interval between them is exactly a major 6th.
This ratio is especially beautiful to hear when you successively pluck the 3rd harmonic (over the 7th fret) and the 5th harmonic (over the 4th fret) on a guitar — the leap between them sounds immediately like a pure, glass-clear consonant major 6th.
Minor 7th
Character
The minor 7th is one of the most exciting intervals in Western music. With its 9:5 ratio (or, in the harmonic series, approximately 7:4) it sounds soft-dissonant, earthy, and full of yearning for resolution. It's the defining interval of the dominant 7th chord (e.g. G7 = G-B-D-F in C major, where G→F is a minor 7th).
In blues and jazz the minor 7th has a particularly important place: it's the typical „blue note” and gives the blues its unmistakable character. Unlike the tempered minor 7th, the seventh sung or played in blues often corresponds more closely to the natural 7th harmonic (7:4 ratio), which sits slightly lower than the equal-tempered B♭ — and produces that characteristic „bluesy” sound.
Famous examples: practically every blues piece begins or ends with a 7th chord. In pop the minor 7th is also ubiquitous — it's the foundation of every „bluesy” coloration. A famous ascending leap of a minor 7th appears at the beginning of „Somewhere” from West Side Story (Bernstein) and in the main theme of „Star Trek”.
Fingering
At 5/9 of the string length, the minor 7th sounds — the string is shortened by about 44.4%. On guitar this corresponds to ten frets apart. Fretting the high E (329.63 Hz) at the 10th fret produces D (587.33 Hz). The frequency rises by a factor of 9/5 = 1.8.
The minor 7th is prominently fingered in many guitar chord voicings — for instance in the classic dominant 7th chord (e.g. C7 with B♭ as the minor 7th above the root C). In jazz it's a central sound in practically every II-V-I scheme.
Harmonic
Closely related to the 7th harmonic, which arises by touching the string at 1/7 (or 2/7, 3/7, etc.) of its length. At a 7:4 ratio to the fundamental this harmonic sits slightly lower than the tempered minor 7th — about 31 cents (just under a third of a semitone) flatter.
This slightly-flat 7th harmonic is the „blue note” — a sound that gives the blues its unmistakable, yearning character. If you play the natural harmonic on the high E string in the area of the 7th harmonic (between the 8th and 9th frets), you hear exactly this slightly-flatter-than-expected pitch — many listeners find it „interestingly foreign” because it deviates from equal temperament.
Major 7th
Character
The major 7th is the „leading-tone seventh” — it sits just one semitone below the octave and pulls upward with great tensile force. With its 15:8 ratio it sounds strongly dissonant, almost painful, and demands resolution to the octave.
In traditional harmony the major 7th was avoided for centuries as an unresolved dissonance. Only jazz turned it into a characteristic pleasant sound: the maj7 chord (e.g. Cmaj7 = C-E-G-B) became the trademark of bossa nova, smooth jazz, and romantic ballads. Here the seventh's tension is no longer resolved but enjoyed as a tone color — a mild, soft, shimmering sound.
Famous examples of maj7 colors: „The Girl from Ipanema” (Tom Jobim, throughout with maj7 chords), „Just the Way You Are” (Billy Joel), the opening of „Take Five” (Brubeck), and countless film music themes. As a melodic leap the major 7th is rare — when it occurs, it often feels yearning or dramatic (e.g. in the opening of Bernstein's „Somewhere” from West Side Story, which combines a sixth leap and a major 7th).
Fingering
At 8/15 of the string length, the major 7th sounds — the string is shortened by about 46.7%. On guitar this corresponds to eleven frets apart. Fretting the high E (329.63 Hz) at the 11th fret produces D♯ (622.25 Hz). The frequency rises by a factor of 15/8 = 1.875.
The 11th fret sits very close to the 12th fret (the octave) — the major 7th is „almost” the octave, just a semitone lower. Precisely this proximity creates the strong tension and the urge to resolve. In maj7 chord voicings the major 7th is often found as the topmost note, giving the chord its characteristic shimmering glow.
Harmonic
The 15:8 ratio sits between the 8th and 16th harmonics. Theoretically the 15th harmonic could be produced at 1/15 of the string length, but practically it's hard to isolate — the position lies very close to the bridge, and higher harmonics are energetically very weak.
The major 7th is one of the most complex ratios in the natural overtone series and counts as a „derived” consonance: it arises only by stacking lower harmonics. In just intonation the major 7th is traditionally formed from the ratio 15:8 (5/4 × 3/2 = 15/8 — that is, major 3rd plus perfect 5th). On guitar as a harmonic it's barely practicable.
Octave
Character
The octave is the most perfect consonance after the unison and is considered the most fundamental of all intervals. With its plain 2:1 ratio (the very first overtone), two notes an octave apart sound so closely related that the human ear perceives them as the same note at different pitches. That's exactly why notes an octave apart share the same letter name (C, c, c′, c″, etc.).
The octave is universal: in practically every musical culture in the world — from Western classical to Indian raga, from Gregorian chant to Balinese gamelan — the octave is perceived as „the same” pitch and used to structure the tonal space. This isn't a cultural convention but a consequence of physics: a string vibrating at twice the frequency simply produces the „same” note one step higher.
Famous examples of ascending octaves at melodic openings: „Somewhere Over the Rainbow” (the famous „Some-where” — an octave leap!), „Singin' in the Rain”, „Bali Hai” from South Pacific. The main motif of „Star Trek: The Next Generation” also contains a prominent octave leap.
Fingering
At exactly 1/2 of the string length, the octave sounds — the string is exactly halved. On guitar this corresponds to exactly twelve frets apart. Fretting the high E (329.63 Hz) at the 12th fret produces the next-higher E (659.26 Hz). The frequency exactly doubles (factor × 2.0).
The 12th fret is marked on every guitar with a double inlay because it's the „midpoint” of the string. Everything that happens from the 12th fret onward sounds an octave higher than what happens between the open string and the 11th fret — fret spacings from the 12th fret on become identical, just at half the length.
Harmonic
The 2nd harmonic — the best-known and easiest-to-produce harmonic on any string instrument. Touch the string exactly in the middle (over the 12th fret) very lightly and pluck: the same pitch sounds as when fretting at the 12th fret — one octave higher than the open string.
What's happening physically: the string no longer vibrates as a whole but in two equal halves simultaneously — like two small strings producing the same higher pitch together. This is the simplest form of an overtone vibration. Every note played already contains this octave as its strongest overtone in its timbre — even if you don't consciously hear it, it's always there, shaping what we call „tone color”.