Exploring Microtonal Guitar in 22 EDO: Intervals and Demonstration

Equal temperament divides the octave into 12 equal semitone intervals using the twelfth root of 2 (~1.059463). This number, when multiplied 12 times, equals exactly 2. Cents measure intervals as hundredths of a semitone using the 1200th root of 2. The formula f(n) = 440 × 2^((n-49)/12) calculates any note's frequency from A4 = 440 Hz, where n is the key number on an 88-key piano.

Modern pianos and keyboards use the equal temperament scale, which divides the entire octave into 12 equal sections of 100 cents each. Sa starts at 0 cents, Komal Rishab at 100 cents, Shudha Gandhar at 200 cents, Komal Dhaivat at 300 cents, Shudha Nishad at 400 cents, and so on up to 1200 cents. This system was developed as a convenient method for tuning and is widely used even today to tune pianos.

Musical intervals can be measured in cents, a logarithmic unit where one octave (2:1 frequency ratio) equals 1200 cents; to convert a frequency ratio to cents, multiply 1200 by the base-2 logarithm of the ratio, or equivalently multiply 3986 by the base-10 logarithm of the ratio, and to convert cents back to frequency, multiply the base frequency by 2 raised to the power of cents divided by 1200.

The octave contains 12 semitones, each separated by 100 cents in equal temperament. The cent is a logarithmic unit for measuring musical intervals. When dividing the octave into 10 parts, each semitone becomes 120 cents. When dividing into 24 parts, each semitone becomes 50 cents. These systems create intervals larger than the standard semitone (macrotonal) or smaller (microtonal). Indian music uses unequal temperament systems where intervals vary between 100, 26, and 131 cents. Azerbaijani music also features unique microtonal systems with distinctive notation. This demonstrates how mathematical division of the octave creates diverse musical possibilities.

In musical acoustics, cents are the smallest standard unit for measuring musical intervals, where one cent equals one-hundredth of a tempered semitone. The equal temperament system divides an octave into 12 equal semitones, and each semitone is further divided into 100 cents, making an octave equal to 1200 cents. The formula to calculate cents for any interval is: cents = 1200 × log₁₀(interval ratio). For example, a semitone equals 100 cents, a whole tone equals 200 cents, and an octave equals 1200 cents. This logarithmic measurement system was invented in 1885 by British mathematician Alexander Ellis, who conducted extensive measurements of musical instruments worldwide to compare and standardize scales across different musical traditions.

The guitar fretboard can be treated as a coordinate system using X (frets between notes) and Y (strings between notes). In all-fourths tuning (EADGCF), semitone distance equals 5 times strings plus 1 times frets. Count by fives when changing strings and by ones when changing frets. Negative numbers represent downward movement. Standard tuning (EADGBE) has an odd interval between G and B strings: a major third (4 semitones), called the 'warp.' When moving fingerings across this line, they shift by one fret. To maintain the counting method, count along the diagonal for the next five semitones, then count up two more frets to reach 12 for an octave.

Intonation on fretted instruments like guitars is fundamentally limited by physics: perfect intonation is impossible because guitars are designed around equal temperament, which creates systematic tuning deviations (fifths are ~2 cents flat, major thirds are ~14 cents sharp, minor thirds are ~16 cents flat), compounded by string stretching, inharmonicity, and physical movement over time; luthiers compensate for these factors through slanted saddles and compensated nuts/zeros, but can only achieve 'close to perfect' intonation rather than true perfection.

This section covers standard guitar tuning and fretboard navigation. Standard guitar tuning uses only whole notes: 6th string (mi), 5th string (si), 4th string (sol), 3rd string (re), 2nd string (la), 1st string (mi), with the difference between 6th and 1st strings being two octaves. To find any note on the fretboard, start from a known note and use the octave system. For example, if you know where do is on the 6th string, you can find the same note on the 5th string by moving down 5 frets (the interval of a fourth). The octave system allows you to find the same note on a different string by moving across strings and adding 2 frets. For the 6th string: open string is mi, then at frets 1, 3, 5 you have fa, sol, la. At frets 7, 8, 10 you have si, do, re. For the 5th string: open string is si, then at frets 1, 3, 5 you have do, re, mi. At frets 7, 8, 10 you have fa, sol, la.

The guitar is tuned in fourths (E-A-D-G-B-E), but the G to B string creates a 'kink' because the distance is a third (G-A-B) instead of a fourth. To compensate, add one fret when moving between G and B strings. The fretboard repeats every 12 frets, so memorize only the first 12. The order of flats follows 'Battle ends and down goes Charles's father' (B-E-A-D-C-G-F). When moving from F to B, you encounter a sharp fourth requiring additional compensation. When crossing both G-B and F-B strings simultaneously, add two frets total (double compensation). The guitar is tuned to C major, but the open strings correspond to E minor (the relative minor). The seven modes have specific formulas: Phrygian (root, flat 2, flat 3, 4, 5, flat 6, flat 7), Aeolian (root, 2, flat 3, 4, 5, flat 6, flat 7), Dorian (root, 2, flat 3, 4, 5, 6, flat 7), Ionian (root, 2, 3, 4, 5, 6, 7), Mixolydian (root, 2, 3, 4, 5, 6, flat 7), and Lydian (root, 2, 3, augmented 4, 5, 6, 7). Practice scales in three-note-per-string patterns to internalize the fretboard layout.

This lesson introduces the fundamental layout of the guitar fretboard, explaining that the sixth string (thickest, lowest pitch) is at the top of diagrams while the first string (thinnest, highest pitch) is at the bottom, with pitch increasing both horizontally toward the body and vertically toward the first string. The fretboard anatomy includes the nut (top vertical line), fret wires (vertical dividers), frets (spaces between wires), and fret markers (dots at 3rd, 5th, 7th, 9th, and 12th frets). Open circles represent open strings, while filled circles indicate fretted notes. The lesson also covers musical intervals: a half step (semitone) is the smallest distance between adjacent frets, while a whole step (whole tone) spans two frets with one fret in between, following the musical alphabet pattern (A to A#/Bb is a half step, A to B is a whole step, B to C is a half step, etc.).

Equal temperament provides consistent intervals across all keys by dividing the octave into 12 equal parts using the twelfth root of 2. Just intonation uses simple frequency ratios (2, 3, 5) for purer-sounding intervals but creates inconsistencies when transposing. 5-limit tuning calculates intervals like major thirds as 5/4 ratios. Pythagorean tuning uses perfect fifths with 3:2 ratios, creating the circle of fifths but introducing the Pythagorean comma—the small frequency difference between reaching the same note through 12 perfect fifths versus 7 octaves. Equal temperament was adopted over just intonations because it allows music to be played in any key with consistent intervals and solves problems like the Pythagorean comma, making it ideal for keyboard instruments and facilitating musical collaboration.

Just intonation is an ancient tuning system that creates musical intervals using simple whole-number ratios (such as 3/2, 4/3, and 5/4) multiplied by a base frequency, producing pure, consonant sounds that differ significantly from the equally spaced intervals of 12-tone equal temperament; unlike equal temperament which became dominant in Western music starting in the 15th century for easier transposition, just intonation offers expanded harmonic possibilities and can be implemented through trained performers, adapted instruments, or computer software, allowing musicians to explore new musical languages while maintaining intuitive, non-technical approaches to composition.

Modern pianos use tempered tuning with twelve equally spaced notes, enabling key changes but introducing slight dissonance in chords. Just intonation uses exact integer ratios (3:2 for perfect fifths) to create pure, dissonance-free chords. While just intonation sounds more natural and pleasant, it requires constantly retuning notes based on the key, making it impractical for fixed-pitch instruments like pianos.

Just intonation is a tuning system where the distance of every note in a scale relative to the root is expressed in relatively clean whole-number frequency ratios. The advantage of just intonation is that harmony produced in this system has kinder, more consonant sound waves because the frequencies align in these cleaner ratios. For example, dominant seventh chords appear verbatim in the harmonic series with a ratio of 4:5:6:7 between frequencies, producing a crisp and pure sound compared to the wobbly quality of 12-TET dominant seventh chords. Barber shop quartets intuitively sing in just intonation for this very reason.