Just Intonation vs Equal Temperament: A Comparative Analysis

Equal temperament divides the octave into 12 equal semitones using the 12th root of 2 (~1.06) as the ratio between adjacent notes. This enables perfect transposition between any keys—a revolutionary advantage for Western music. The fifth approximates 3:2 with only 0.1% error, nearly indistinguishable to expert ears. However, all intervals except the octave are slightly compromised, particularly the third (~0.8% error). This system sacrifices the unique character of each key present in historical temperaments, fundamentally changing the musical landscape since the mid-19th century.

The advantage of equal temperament is that intervals are consistent across all keys, enabling free transposition. The downside is that except for the octave, no interval is mathematically perfect. You can hear the difference between a major chord in equal temperament and the same chord with pure ratios. This represents a fundamental trade-off between mathematical perfection and practical usability.

Just intonation produces sweeter, more harmonious chords based on pure mathematical ratios. Equal temperament divides the octave into 12 equal semitones, allowing modulation to any key but resulting in slightly less pure intervals. This system was developed as a compromise solution that sacrifices some harmonic purity for the flexibility to play in any musical key without retuning instruments.

Equal temperament was developed to solve the practical limitations of previous systems by allowing all 12 notes to be used in any composition. In this system, the 12 notes are equally spaced, meaning each adjacent key on a piano has the same interval distance. This eliminates the wolf interval but sacrifices perfect consonance—every chord sounds slightly out of tune. The trade-off enables harmonic freedom, allowing composers like Beethoven, Stravinsky, and Britten to experiment with complex harmonies. This represents the fundamental tension in music between mathematical perfection and creative flexibility.

Pure tuning (just intonation) creates perfectly consonant intervals but cannot accommodate modulation between keys. This limitation led to equal temperament, which sacrifices perfect consonance in individual intervals to enable free modulation across all keys. While equal temperament creates intervals that are not perfectly consonant (with some differing by up to 14 cents from pure tuning), it allows for free modulation between keys. Johann Sebastian Bach's Well-Tempered Clavier explicitly demonstrates these possibilities, with its title directly referencing this tuning system. The adoption of equal temperament fundamentally transformed musical development by providing composers with unprecedented freedom in modulation, enabling the emergence of diverse musical styles including bossa nova.

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.

In 31-tone equal temperament (31EDO), dividing an octave into 31 equally spaced notes expands traditional Western harmony by enabling new chord types such as neutral chords (between major and minor), subminor chords (flattened minor thirds), and supermajor chords (raised major thirds), which offer more precise tuning and greater harmonic flexibility than the 12-note equal temperament system commonly used in Western music.

Microtonal music theory explores tuning systems beyond the standard 12-note equal temperament, such as 31 EDO (Equal Division of the Octave), which contains 31 equally spaced notes per octave. Within this system, subsets like Orwell 9 (a nonatonic scale with nine notes) and Orwell 4 (a tetratonic scale with four notes) are constructed by stacking alternating step patterns of four and three steps, creating intervals like neutral seconds (~155 cents) and minor seconds (~116 cents) that don't exist in 12-note tuning. These scales enable unique harmonic possibilities including sub-minor thirds, super-major thirds, and neutral intervals, expanding the traditional understanding of musical harmony and chord structures.

In 31 equal divisions of the octave (31edo), there are six basic triads: Sus4 (root, perfect fourth, perfect fifth at EDO steps 0, 13, 18), Supermajor (root, supermajor third, fifth at 0, 11, 18), Major (root, major third, fifth at 0, 10, 18), Neutral (root, neutral third, fifth at 0, 9, 18), Minor (root, minor third, fifth at 0, 8, 18), and Subminor (root, subminor third, fifth at 0, 7, 18). Unlike 12edo, 31edo provides two variations of major and minor chords plus a neutral chord, offering richer harmonic possibilities through its superior approximation of just intonation ratios like 5:4 and 7:6.

Microtonal music is defined as music written in tuning systems other than the standard 12-note system, with more than 12 notes per octave creating notes between semitones. The 19-EDO system divides an octave into 19 equal divisions, adding 7 notes and making previously enharmonically equivalent notes (like C and Db) distinct. The 24-EDO system doubles notes per octave, introducing quarter tones and requiring new accidentals (half-sharp, half-flat) for notation. In 24-EDO, C half-sharp is enharmonically equivalent to Db half, and D half-flat is equivalent to C# half. These systems demonstrate how increasing notes per octave creates new harmonic relationships and notation requirements.

This extended segment details the technical process of adding microtonal capabilities to digital audio software. The creator explains that standard samplers assume the 12-tone equal temperament scale, which cannot accurately reproduce microtonal instruments like the chipi. To solve this, the creator implements microtuning support in their software (Decent Sampler), enabling users to load custom scale files in formats like Scala. The implementation allows users to apply alternative tuning systems by simply dragging scale files onto the sampler interface. The final result is a chipi sample library offering both the standard chromatic scale and an authentic Albanian scale option, demonstrating how software development can preserve and make accessible traditional musical knowledge for contemporary composition.

The Destiny X synthesizer's SD multisample mode supports microtonal tunings using Scala files, which are placed on the microSD card in designated folders (SEL for scales, KVM for keyboard maps). Users can select from default scales like equal temperament or the R & Johnson 710 seven-tone scale, and must match the keyboard map to the scale to avoid mismatch warnings. Custom Scala files can be generated online and exported to enable microtonal tuning on the sample player.

This tutorial demonstrates how to use the Vienna Synchron Player software to implement microtonal tuning, including features like cent deviation adjustment on the piano roll, support for various tuning systems (22 EDO, just intonation), and keyboard mapping for non-standard scales. The presenter shows practical techniques such as applying the same cent deviation across all octaves of a note, creating custom Scala files for scales like 22 EDO, and combining multiple software instances to create interesting microtonal effects.

The Lima software exports tuning systems as Scala files (.scl format). The Base Station 2 has MIDI Tuning System (MTS) support, allowing users to upload custom tuning tables. The Peak synthesizer has 16 built-in tuning tables that can be saved and recalled. Users can create custom tuning systems in Lima by adding notes at arbitrary frequencies, then export the Scala file and upload it to the Base Station 2. This enables musicians to implement custom microtonal tunings on hardware synthesizers, demonstrating the practical application of microtonal theory.

Just intonation is a tuning theory using simple ratios to determine pitch frequencies, producing consonant intervals. Unlike 12 tone equal temperament (standard since Bach), it offers unique intervals like 7/6 (septimal minor third), 14/9 (quarter-tone sharper fifth), and 7/4 (minor seventh). This provides originality since 230 years of equal temperament have exhausted many possibilities. Equal temperament was adopted for key flexibility on fixed-pitch instruments like pianos, but is less relevant for synthesizers. To ensure consonance, calculate all intervals between notes using a matrix method. Adding 4/3 (perfect fourth) creates a consonant pentatonic scale impossible in 12 tone equal temperament.