Exploring Complex Chords in 22-Tone Equal Temperament

The octave is divided into 12 notes because of the mathematical relationship between frequencies: when one note has twice the frequency of another (2:1 ratio), they sound pure together. Using simple frequency ratios between 1 and 2 (such as 3/2, 5/4, 4/3, 5/3, 6/5, and 8/5), we can derive 6 consonant notes that sound harmonious with the base note. When these 6 notes plus the 2 octave notes are combined, they create a 12-note scale called just intonation, where consonant notes have exact frequency ratios and sound smooth, while dissonant notes have awkward ratios and sound harsh.

Just intonation uses simple whole-number frequency ratios for pure intervals: octave (2:1), perfect fifth (3:2), perfect fourth (4:3), major third (5:4), minor third (6:5), major sixth (8:5), and minor sixth (5:3). The Lima software allows users to create custom tuning systems by inputting these ratios. Equal temperament approximates these intervals but with deviations: the major third is about 14 cents flat, and the minor third is about 14 cents sharp. By adjusting notes by small amounts (cents), musicians can achieve purer-sounding intervals, demonstrating that equal temperament is an approximation.

Just intonation is a tuning method where fret positions are determined by listening to where harmonics naturally occur on a string, resulting in more stable and musically resonant chords compared to equal temperament, which divides the octave into equal segments as an approximation.

In just intonation, musical intervals are defined by their frequency ratios derived from the overtone series, where the most harmonious intervals have the simplest ratios: the perfect fifth is 3:2 (1.5), the perfect fourth is 4:3 (1.333), the major third is 5:4 (1.25), and the minor third is 6:5 (1.2); as intervals become more complex with larger numbers in their ratios, they become progressively less harmonious and more dissonant, which is why equal temperament was developed to divide the octave into twelve equal semitones using the twelfth root of 2 (~1.05946) for consistent tuning across all keys.

Xenharmonic music explores tuning systems beyond the standard 12EDO (12 Equal Division of Octave), where the octave is divided into equal parts using ratios from the harmonic series; key concepts include just intonation (using pure harmonic ratios), equal temperament (dividing intervals into equal parts like 19EDO or 100cET), and cents as a measurement unit where one semitone equals 100 cents.

Microtonal and xenharmonic theory explores how all musical harmony originates from the harmonic series, where every sound wave consists of harmonics at integer multiples of the fundamental frequency; when notes are played together, they construct the harmonic series of a note at half the frequency, creating consonance or dissonance based on how closely their frequency ratios match ideal harmonic intervals like 2:1 (octave), 3:2 (perfect fifth), and 5:4 (major third); temperaments like 12-tone equal temperament approximate these ratios by dividing the octave into equal steps, introducing small errors called commas that affect how chords sound, beat, and behave under distortion, making understanding prime number approximations essential for composing in alternative tunings.

The cent is a logarithmic unit used to measure musical intervals, where twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each; an octave spans 1200 cents with a frequency ratio of 2:1, and the ratio between frequencies one cent apart is precisely 2^(1/1200) ≈ 1.0005777895, allowing precise comparison of interval sizes across different tuning systems.

Xenharmonic or microtonal music explores tunings, scales, and chords beyond the standard 12-tone equal tempered scale used by most Western musicians. This approach opens new musical possibilities ranging from strange and alien sounds to fresh interpretations of familiar chords and scales. Two primary approaches exist: just intonation using intervals derived from the harmonic series, and equal temperaments dividing the octave into equal steps (such as 22-tone equal temperament). The harmonic series reveals that when an instrument plays a note, we actually hear a chord of multiple frequencies (partials or harmonics) occurring simultaneously. The fundamental frequency determines perceived pitch, while partials create timbre. Harmonic sounds follow integer multiples of the fundamental, while inharmonic sounds have chaotic structures. Just intonation derives intervals from frequency ratios like 5:4 (major third, 14 cents flatter than equal temperament) and 7:4 (minor seventh, 31 cents lower). These ratios represent simple mathematical relationships, with simpler ratios generally producing more consonant intervals. This foundation enables creation of microtonal tunings that sound consonant and natural, reflecting inherent acoustic relationships.

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.

All chords, including seventh chords and extended chords, are built upon triadic structures. Complex chords with four or more notes can be understood as combinations of triads. For example, an F# minor 11 chord in the key of E contains seven sets of triads: E major, F# major, G# minor, A major, B major, C# minor, and E flat/D# diminished. This triadic foundation has been central to every great style of music including rock, funk, R&B, soul, heavy metal, and reggae. Understanding triads provides a fundamental framework for understanding all chord structures in music.

All chords in music are built using tertian intervals (major and minor thirds), and there are four fundamental triad types: major triads (major third + minor third), minor triads (minor third + major third), diminished triads (two minor thirds), and augmented triads (two major thirds); these basic triads form the foundation for all more complex chords like seventh chords and extended chords.

Seventh chords are four-note chords that add depth to basic triads. A major seventh chord is built by adding the seventh note of the scale to a major triad. For example, C major seventh is C-E-G-Bb. Minor seventh chords are built by adding the seventh note to a minor triad (C-Eb-G-Bb). Extensions are the ninth (2nd), eleventh (4th), and thirteenth (6th) of the scale. For a C major chord, these are D (9th), F (11th), and A (13th). Playing these extensions transforms a simple chord into a rich, professional-sounding chord. The basic extensions are the 9th, 11th, and 13th. Advanced extensions include flatting the ninth (b9) and sharpening the ninth (#9) for dominant seventh chords. You can only change the ninth in dominant seventh chords, not in major or minor seventh chords. You can also flatten or sharpen the fifth in chords. Chords with extensions are named by adding the extension numbers to the chord name. For example, a C major chord with the 9th is called Cmaj9.

This section covers the foundational chord types and their labeling conventions. Triads include major (note name only), minor (m, min, or -), diminished (dim or °), and augmented (aug or +). Power chords (common in rock) omit the third and are labeled with '5' or left unmarked. Suspended chords replace the third with a perfect fourth (sus4) or major second (sus2). Sixth chords come in major (note+6) and minor (m6, min6, -6) varieties. Seventh chords include dominant (7), minor (m7, min7, -7), major (Maj7, △7), half-diminished (m7♭5, ø7), and fully diminished (dim7, °7). These basic chord types form the building blocks for more complex extended chords.

Sophisticated chords are complex chords with more than three notes, and their sophistication is context-dependent. The foundation of all chords lies in triads, which have four basic qualities: major (tonic, major third, perfect fifth), minor (tonic, minor third, perfect fifth), augmented (tonic, major third, augmented fifth), and diminished (tonic, minor third, diminished fifth). Seventh chords are formed by adding a seventh note to triads, with the main qualities being major seventh, minor seventh, and dominant seventh. Understanding these basic building blocks is essential before moving to more complex chord structures.