“Mapping” has double meaning. A road atlas measures and records all the routes through a given territory. But we also call “mapping” the creative act of planning out a journey, using map information to choose between many possible routes. Composers use an array of processes to map out a musical journey. Designing a piece entails making a storytelling rhetoric, a pacing plan, and an architecture of interrelated components.
Each Map Lab in Mapping the Music Universe presents step-by-step recipes to compose simple pieces based on models of different musical genres. Each lab also includes an original sample piece following the Map Lab guidelines, illustrating one possible creative path and outcome.
Try your own experiment with any of these lab projects:
Imagine a piece of music exploring texture in time, made of single sounds and sonorities occurring one at a time in sustained resonance. Then imagine the points of sound are separated by rests, silence. As the texture drifts in and out of a resonant cloud, the sound events remain unconnected. Suddenly, their pace explodes into a torrent of notes. That describes the following powerful piece by my UNT colleague, Joseph Klein.
Joseph Klein – Pathways IV: Rhymes & Spirals (2024)
Sound color
Our next music map shows a simple color-coding graphic system for classifying most musical timbres, informally the tone quality of sounds. The map intuitively chooses colors of the rainbow. While the color spectrum orders the frequencies of light (another manifestation of periodicity), our sound-color classifying map does not imply any ordered quantification of timbral complexity.
instrumental color rainbow
Though we think first of an orchestra for a rainbow of color, chamber music can incorporate a variety of instrumental colors, each produced in vivid isolation by one instrument, standing out or changeably mixed with other colors.
Augusta Read Thomas wrote Dance Mobile in 2021, scored for 13 instruments: Woodwind quartet (Flute, Oboe, Clarinet, Bassoon); Trombone; String quintet: (2 Violins, Viola, Cello, Contrabass); Piano; 2 Percussion (vibraphone/metal, marimba/wood, drums).
The piece starts with a single pitch, blending several colors that swell in intensity. Then ensues a kaleidoscopic dance of at least seven distinct color combinations, of two basic types:
Sustained sounds – strings; high woodwinds; lone brass of the trombone
Sparks – pizzicato strings; ringing metal sounds; drum strokes; staccato piano
Augusta Read Thomas – Dance Mobile (2021)
Though the piece is dedicated “in memoriam Oliver Knussen,” the memory is a joyous dance of color.
Symmetry
In the exposition of Webern’s Symphony, Op. 21, we saw that each contrapuntal line duplicates the exact rhythm of the lead line, with each entrance one bar later — a classic canon. But each contrapuntal line presents a different succession of instrumental colors:
The German term for this is so elegant, we’ll use it here:
KLANGFARBENMELODIE — melodic or contrapuntal line expressed by a string of changing tone colors
Webern placed each pitch in every line in a particular fixed octave, except Eb that appears in two different octaves. This makes a striking, symmetrical 13-pitch constellation with a palindromic array, the same array going down as going up.
Webern 13-pitch constellation
Not only was he obsessed with symmetry in this piece, but this constellation’s symmetry also proves that he was thinking specifically about the chord voicing in what I have identified in successive interval array form.
We can use this constellation as a Y-axis for a graph mapping the timbres as they appear in the various parts in canonic lines in pitch space for the first 9 bars. This farben color map looks like one of the later geometric paintings of Piet Mondrian.
Op. 21 color map
Pointillism
Though we often share musical terms and concepts with visual art, we sometimes mean different things by the same term. In painting, a technique developed in the Impressionist style period of the late 19th century that became known as pointillism. The most famous example is Georges Seurat’s “A Sunday Afternoon on the Island of La Grande Jatte” at the Chicago Art Institute. Instead of sweeping brush strokes and palette-blended colors, it used small separate spots of subtly varied colors to make a texture that, when viewed from a distance, seems to merge into a color cloud, giving the impression of animated light.
Musical pointillism, unlike painting, separates sounds in time and pitch space, not to blend them into a texture so much as to highlight the different qualities of each unique sound event. Webern was a pioneer of musical pointillism in works such as Op. 21. Let’s graph the first 10 bars of this fabric using our timbre color-coding (BLUE = wind, ORANGE = percussion, VIOLET = plucked string) on a broadly distinguished 6-octave pitch range. We get something as colorful as a Mondrian painting!
Andromeda sound color map
As a musical fabric, isolation — using the vast available range of pitch and the empty time of rests and silence — is a fitting analog for the vast, mostly empty space of a galaxy. Let’s use it for a demonstration etude.
Andromeda is the nearest large galaxy, 2.5 million light-years from our own Milky Way galaxy. Our sound color demonstration study uses every sound quality on our sound color spectrum except red. Here is a score of the first 10 bars.
Notice that the green woodwind notes are doubled with a synthesized vocal-type sound. Yellow brass notes are punctuated by orange metallic percussion attacks. Likewise, blue string notes are articulated by the plucked string sounds of harp.
Here is the whole colorfully pointillistic 3-minute study:
Sound Mass
At a time when electronic music was emerging in the 1950s, new instrumental resources were also developing a new style that was all about animating massive layers of sound.
German experimentalist Karlheinz Stockhausen composed two early, influential sound mass works, Gruppen (1957) for three orchestras, and Carré (1960) for four orchestras and four choirs. The scores were huge, dense, 12-tone, and monolithic in form.
A 2002 piece by John Adams, On the Transmigration of Souls, harkens back to a mid-century masterpiece of the Avant Garde. In 1961, Polish composer Krzysztof Penderecki wrote a piece for a massive score of 52 string instruments. Conceived as an abstract, freeform, dense massing of animated and intense musical fabrics, it represents a pioneer in the genre of sound mass music, winning the UNESCO Prize that year. Only after it was heard in performance, he said, “I was struck by the emotional charge of the work … I searched for associations and decided to dedicate it to the Hiroshima victims” — thus the title, Tren Ofiarom Hiroszimy (translated Threnody for the Victims of Hiroshima).
As a young composer in the ‘70s, I reflected this approach in some pieces titled Animated Landscapes. (The title was inspired by John Cage’s famous Imaginary Landscapes no. 4 for 12 radios.) Beyond referring to the painting genre of landscapes, the title sets the imagination for solid, continuous textures like viewing the shapes of a mountain range, but set into rhythmic motion. (This approach became prevalent in ensemble music, especially of Midwestern composers such as Donald Erb.)
Considerably predating the music mentioned above, Schoenberg’s Fünf Orchesterstücke, Op. 16 (1909), was originally scored for a large orchestra of 37 parts. It is not thought of as sound mass music, as its five movements each have Expressionist or Impressionist titles: “Vorgefühle” (“Premonitions”); “Vergangenes” (“The Past”); “Farben” (“Summer Morning by a Lake”); ”Peripetie” (“Peripeteia”); “Das obligate Rezitativ”(“The Obligato Recitative”). The third movement, Farben, is of special interest not only for its exquisite mixed-palette painting of orchestral timbres, but also for its thick though delicate fabric of sustained sounds. At the start, nothing moves, the subtle shimmer of instrumental colors fading in and out of a continuous fabric of delicate, faint sounds. (A sound mass can be delicate, not necessarily “massive.”)
Here is a score of the first page, showing sounding concert pitches for all instruments.
Schoenberg Farben scoring
Each measure presents one constellation, recolored with different instruments in the second half of the measure. For the first three bars, the constellation does not change, and then only subtly in the next five bars, maintaining the constant C pedal point in the low strings.
Farben constellations
The bass clarinet’s F3 in bar 7 is considered an ornamental non-harmonic pitch. While you can see many recurring smaller constellations imbedded within these changing large constellations, such as 5 5, 3 5 and its inversion 5 3 (which are triads), and some transformations of smaller constituent constellations: 8 3 redistributed to 9 2, 4 7 shrinking to 4 5 (another triad), and 3 4 (also a triad) shrinking to 2 4.
Though there are many triads embedded in the constellations, the overall quality of the sonorities is complex, as the triads are framed within critical dissonances:
framing dissonances
Foreground / background
Most landscape paintings, distant textures of forest, mountains, sky, waves on the sea, or clouds, have some sharp focal point. Often on the horizon (in itself a focusing anchor of the visual display), it may be a barn, a setting sun, a boat, a farmer and dog. If we consider proportion and symmetry in a visual composition, the focal point is best not dead center. A more interesting balance, according to expert photographers, follows the Rule of Thirds, placed one-third from the left or right, one third from the top, or both. Two-thirds is a ratio of 0.667. The Greeks famously defined the Golden Ratio, an ideal ratio dividing a whole length or height into two parts such that the ratio of the smaller part to the larger is the same as the ratio of the larger part to the whole. The ratio is 1.618:1, the solution to the equation: x2 – x – 1 = 0; a 62% and 38% division.
In a simple traditional musical texture, an accompanying harmonic texture is designed as a background for the focal element of a melody. Sound masses may lack such focus, like the forest or sea waves. When there is to be perceived a standout element of the texture, Schoenberg called this focal element of the musical fabric the Hauptstimme. Though that might translate “highest voice,” the melody or other focal events are not necessary to be higher in the pitch range of the fabric than other elements. But there must be some isolation or distinction setting them off from background in at least one of the parameters mentioned above. The Hauptstimme focal line or textural element can be:
in a pitch range isolated from background
a color isolated as a single timbre, not a mixed diffusion of background colors
slower or faster than background
more rhythmically elastic, varied than background
not synchronized with background
loudest line (the most obvious)
Schoenberg devised a special symbol for the focal Hauptstimme line of a fabric, a boldface stylized capital H, which you see marking the bass clarinet entrance in bar 7 of the Farben example. Here is how that principal Hauptstimme line continues, a Klangfarbenmelodie of changing color, from bass clarinet to clarinet with trombone to three solo contrabasses.
Hauptstimme handoffs
Notice the aggressive rhythmic motive, each time stepping down 2 semitones; and the 7 7 7 quintal-chord constellations in the contrabasses. (The rhythmically aligned clarinet and trombone are separated by 14 semitones, 7 + 7.)
Beyond color isolation, Learning to Compose makes a distinction for a timbre mixed with itself or other colors spread over some pitch register (“diffuse”) or reinforcing itself in a narrow, confined pitch space (“concentrated”). While Farben’sbackground is diffuse, its Hauptstimme color is isolated in the low pitch register of the bass clarinet and then also concentrated with the three solo contrabasses.
In the first movement of Anthracite Fields (2015) by Julia Wolfe, the bass clarinet emerges as a focal sound by its loudness and singularity of pitch in a cloud mass of softer sound. Then aggressively loud clusters suddenly interrupt the steady-state background, yielding eventually to repetitive sung chords and floating vocal duets. The sound fabric maintains a three-dimensional depth of contrasting intensities.
Julia Wolfe – Anthracite Fields I: Foundations (2015)
Galaxy groups
Our sample etude composition for sound mass is a thick score of 10 wind parts and harp, with a fabric the opposite of pointillism: everything sustains and overlaps. There are basically no pauses or holes in the continuous 2-minute sound fabric. Its title, Laniakea, is the name of the supercluster of galaxies that includes the Milky Way.
Laniakea score excerpt
Having shown the score with all its notational details, to better illustrate the main point of the example, sound mass, here is a graphic rendering of that actual second system of notes. We can reveal its pointillism by increasing the contrast in a negative image of light on dark. That makes the attack beginning of each sound show up but not the staff lines or sustained resonances . . . a fanciful art image of Laniakea, a vast empty part of the universe dotted with millions of galaxies.
Some points of starlight are actually double stars or star clusters, as revealed through a sufficiently powerful telescope. Borrowing that term, one particular type of musical pitch constellation arising in the 20th century involves sounding adjacent scale steps together as a simultaneity.
CLUSTER — a constellation presented harmonically consisting of adjacent scale steps separated by small scalar intervals of one or two semitones.
If the intervals are one-semitone half-steps from the chromatic scale, the resulting harmony is intense, dark, dissonant. If the separating intervals are mostly whole steps, the quality can tend to be like bright glowing light.
It turns out that the diatonic scale is rich with cluster possibilities.
diatonic clusters
The names are borrowed from the Greek names of modes. One cluster array that can be readily found throughout the octatonic scale but not possible within a diatonic scale is the 1 2 1 array, since the diatonic scale has no one-semitone intervals that close to each other. One other cluster array not found in the diatonic scale is the intense, gritty 1 1 1 chromatic-scale array described above.
Harmonic constellations built from clusters can be still and radiant or animated by rhythmically active lines close together in pitch space. Here is a composed example that does both.
Photons excerpt
Symmetrical arrays
Dorian and Lydian clusters are symmetrical. Their scale-pattern arrays — 2 1 2 and 2 2 2, respectively — are the same when inverted or reversed. Many other interesting pitch constellations have this property.
Here is a sampling of other, taller symmetrical constellation arrays, reading the same from top to bottom as bottom to top:
many symmetrical arrays
Note that although each is a 4-pitch constellation, two of them (4 4 4 and 8 4 8) contain an octave and thus only three unique pitch classes.
The example below explores the first four arrays listed above. The stacked interval array of 4d is: 7 4 7. Note too that this constellation contains two 11-semitone intervals (+7+4 and +4+7) — in example 4d C up to B and G up to F#; and one very large interval of 18 semitones, C up to F# in the higher octave.
four symmetrical arrays
Three of the four could be analyzed in triadic harmony: 1c as an Ab Major-Major 7th chord in first inversion; 2c as an A minor-minor 7th in first inversion, or a C Major triad with jazz added 6; and 4c as a C Major 11th chord with 3rd and 9th missing! And 3c could be seen as a rearranged voicing of a segment of the “Circle of Fifths”! Obviously, I don’t recommend such contortions of traditional harmonic analysis to explain these beautiful, symmetrical constellations.
Each constellation’s successive-interval array, its stack, is symmetrical under mirror-inversion or palindromic — that is, its interval stack reads the same lowest-to-highest or highest-to-lowest: 7 1 7 and 7 4 7 are two of my favorite constellation stacks.
These interval stacks were chosen here for two of my particular interests. Each features the “Perfect-5th” 7-semitone interval at top or bottom, with a smaller interval in the middle. The Perfect 5th has a stable, rooted quality, but with two “roots” in the harmony, the overall stability of the sonority is compromised — complex yet balanced. It is like a “double star” in astronomy, to further pursue my constellation metaphor.
Acoustic quality
We started with clusters, sonorities that would be traditionally considered highly dissonant. To assess a constellation’s quality or sound character, we will transition from the concept of consonance and dissonance to an assessment of acoustical complexity.
When two pitches sound together, they make a harmonic interval, but also their distinct overtone series are interacting. In the purer sounding intervals, this interaction is mainly a closely compatible one, with some overtones matching. An example, take a Perfect 5th, C up to G. Their overtones are:
overtone match for a Perfect Fifth
The matching or interfering overtones make more of a difference with the lowest partials, as the higher overtones are fainter and fainter higher up in the series. So the c’ 4th partial of C interferes with the 5th partial b’ of the G overtones; but that interference is fainter than the lower g-to-g match. This gets scientifically and mathematically complex to calculate, as we will tackle in a while below.
For now, the ancient classification for counterpoint is accurate enough to adapt: Perfect Consonance, Imperfect Consonance, Dissonance . . . though I will distinguish between mild dissonance and strong dissonance.
PERFECT CONSONANCE — intervals P8, P5 and P4 (12, 7 and 5 semitones) — “pure”
IMPERFECT CONSONANCE — intervals of major and minor 3rds and 6ths (3, 4, 8 or 9 semitones) — “triadic”
MILD DISSONANCE — intervals 2 semitones different in size from a unison, octave, or double octave (2, 10, 14, 22, 26 semitones) and the tritone (6 semitones)
We can use these distinctions to come up with an assessment of the general harmonic complexity of a constellation’s intervals.
PURE — containing no intervals except perfect consonances
SIMPLE — containing no mild or strong dissonances
MODERATE COMPLEXITY — containing at least one mild dissonance but no strong dissonance
STRONGLY COMPLEX — containing at least one strong dissonance
Examples of increasing complexity:
It is important to note that strongly complex does not mean “unpleasantly dissonant.” The Major-Major 7th chord in this category (C E G B) is quite a beautiful harmony. And the last two complex examples, quartal and quintal chords, are the sturdy mainstay of 20th-century American composers such as Copland.
In a recorded excerpt of Tyshawn Sorey’s Pulitzer Prize-winning Adagio (2023) for saxophone and orchestra, beautiful sonorities are quietly complex, tensely dissonant, dark and mysterious in their lyric unfolding. Like dark clouds, some morph to reveal brighter sounds, even simple triads. While there is no sense of any “chord progression,” there is a feeling of impending change in the air.
In notes on a recent composition, Frost Serenade, I described “changing tonal temperature.” Here is a deep dive into what that meant.
The metaphor of tonal color and temperature has to do with what we normally call consonance and dissonance in a chord or other harmonic entity. Centuries-old tradition classified musical pitch-intervals as pure, perfect consonances (“Perfect Fifth” and “Perfect Octave” for example); major or minor (exp. “Major Third” or “minor Sixth”); or problematic (“Augmented Fourth” and “diminished Fifth”). Some major and minor intervals (thirds and sixths) were considered imperfect consonances; the others (seconds and sevenths) were considered dissonant. Every music student learns these categories while studying 16th-century model counterpoint.
Using the color spectrum in temperature order:
harmonic color spectrum
Let’s convert the consonance/dissonance concept, going back to a pitch-interval’s acoustic complexity. Reviewing what was explained above: every musical tone has a fundamental pitch, plus faint overtones that give the sound its color. They are of fading intensity and felt (as color) more than actually heard as distinct pitches. Discovered by Pythagoras as partial vibrations in whole-number fractions, the overtones are always in a fixed interval ladder, rising from the fundamental: Up an octave, then a Perfect Fifth, then a Perfect Fourth, a Major third, minorthird, then to the eccentric seventh partial, which is out of tune by our scale-trained pitch perception (and shown a pale gray below), and on to the eighth partial, which is three octaves above the fundamental. (An octave is a multiply-by-2 operator, so partials 2, 4, 8, and 16 of the C overtone series are also the pitch-class C. Likewise, partials 3, 6, and 12 are all octave related.)
Two different fundamental pitches sounding together each bring into the acoustical mix their distinct overtones. The overtones from one either match (simple) or clash with (complex) overtones of the other. This is what makes the sonic complexity or perceived purity of the interval between two fundamental pitches. Using this relationship, we theorize that the higher we need to go to start finding matching overtones between the two pitches, the more complex is the interval. Following this logic, here is an overtone-match analysis of all harmonic intervals smaller than an octave. (We’ll show these horizontally to fit better what would otherwise be very tall slender graphics!) Each interval is shown from a fundamental pitch C up to a higher pitch.
PERFECT CONSONANCES
overtone match for perfect consonances
The rather pure Perfect Fifth interval between fundamental pitches, C up to G, matches overtones at G’s partial 2, a low level in the series, matching the C’s partial 3. The interval makes four such matches in this lowest-two-octaves span. The pitch match up of the G’s 2nd partial with the C’s 3rd partial (both are the same pitch, G) will be duplicated in all higher octaves, making this an acoustically simple interval. The two pitches’ overtones mostly match and don’t interfere with each other much.
IMPERFECT CONSONANCES
overtone match for imperfect consonances
The triadic consonant Major 3rd interval between fundamental pitches, C up to E, matches overtones at a somewhat higher level in the series, partial 4, and makes two matches in this lowest-two-octaves comparison.
DISSONANCES
overtone match for dissonances
The dissonant minor 7th interval between fundamental pitches, C up to Bb, matches overtones makes only one match in this lowest-two-octaves comparison, at partial 5. That means its harmonic quality is more complex, with most of the lower overtones interfering, not matching. Not a strong dissonance, but more complex than the others.
By contrast, with the more complex Major Seventh interval (ex. C up to B), you have to go all the way up three octaves to the B’s 8th partial (matching the C’s 15th partial!) to find an overtone that matches and doesn’t conflict/interfere. The Major 7th interval can be considered much more complex at a rating of 8 than a Perfect 5th at rating 2.
The most complex interval analyzed, the minor 2nd, clashes all the way up until the 15th partial.
A colorful summary depiction of this Pythagorean analysis of harmonic intervals looks rather like a modern-day sound mixing board.
overtone matching for 13 intervals
Summarizing the analysis with a complexity rating number for each interval:
interval complexity ratings
Now we can add up the ratings of each interval in a chord and take an average complexity quotient. And we can think of complex as darker than simple, or we can invoke the color spectrum. In digital photo imaging, we use a temperature metaphor, seeing red as warmest (infrared heat) down through orange, yellow, green, down to blue, the coolest. The “hottest,” most complex harmonic interval is the minor 2nd. The “coolest,” purest (other than the octave) is the Perfect 5th.
The intervals in the following example are shown in semitones. Each chord has four pitch classes and six intervals between them. The Blue chord has an average complexity rating of 3.8. Green chord is slightly more complex, at 4.3. Yellow, which includes the more complex 11-semitone Major 7ths, rates 5.5. And Orange, with the only minor 2nd 1-semitone hot dissonance, is warmest at 6.2. Try to hear the differences. (No attempt here to demonstrate the red-hot complexity of 10 or higher for a cluster chord!)
examples of four temperatures
The following demonstration phrase uses hose four chord types to build a progression of tonal temperature colors. Again, as you listen, try to feel the temperature warm up then cool back down.
you must understandthe language in which it’s written,
the language of Mathematics.”
— Stephen Hawking
Galileo revolutionized astronomy, in part by using a new tool: the telescope.
Schoenberg revolutionized harmony by evolving an existing concept, the chromatic scale, into a new tool: the 12-tone scale, and devised a new compositional tool of the 12-tone row.
Allen Forte took Schoenberg’s ideas to another level of abstraction: defining Pitch Class and applying basic math to the 12-tone universe.
Chrisman focused on the interval essence of pitch patterns: defining the “successive interval array.”
I am merely another explorer using their maps but choosing my own creative path. In doing so, I will define some of my own terms, while adapting and clarifying some established terms that fit what I’m thinking and expressing.
From Tuning to Tonality
We think of traditional common-practice Tonality of the 17th through 19th centuries being synonymous with the major and minor scales. But there’s more to traditional common-practice Tonality than just the scale. Here are the four basic factors that determine any tonal design:
SOURCE SCALE
HARMONIC TYPE
TONAL CENTER
ancient mode
perfect intervals
fixed by mode
Major / minor
triad
modulatory shifting
extended chromatic
extended triad
polytonal centers
exotic / synthetic
non-triad
established contextually
12-tone
diverse
none
tonal design factors
As you can see, there is much to explore: scales, modes, intervals, consonance . . .
Tuning
Taking the overtone series and partial vibrations as a natural acoustical model, Pythagoras identified pitch intervals as simple integer ratios of lengths of a vibrating string. The same ratios describe frequency ratios.
fundamental pitch C and overtones
For example, what we call a Perfect Fifth, the interval of the Third Partial to the Second Partial of a natural overtone series, is a 3:2 ratio. Such natural tuning is always employed by orchestras, bands, and a cappella choirs.
Octave = 2:1
Perfect 5th = 3:2
Perfect 4th = 4:3
Major 3rd = 5:4
Minor 3rd = 6:5
Major 6th = 5:3
Minor 6th = 8:5
Major 2nd = 9:8
This approach requires, however, that intonation be constantly adjusted as the key changes or tonal context shifts. For a keyboard that can’t make those adjustments, the fixed tuning devised in the 18th century, called Equal Temperament, compromises the Perfect Fifth, shrinking it from a 1.5 ratio to 1.498307 so that it and all other intervals are very slightly but equally mis-tuned in every possible key or tonal context. The ratio for a semitone is derived mathematically from the 12th root of 2: 1.059643094. That ratio, multiplied by itself 12 times, results in 2.000, the ratio of the octave.
comparing tuning systems
While “chromatic” historically meant extending a key with accidentals — temporary extra sharps or flats — now we refer to the 12-half-step scale as the chromatic scale. Two pitch names for the same piano key — C-sharp or D-flat — are said to be enharmonic and considered equivalent, almost interchangeable.
Equal Temperament became the basis for the 20th-century system of 12 equal semitones per octave, the basis not only for all keyboard instruments but also for harmonic theory in the post-tonal world of 12-tone music. We should not forget, however, that choirs, orchestras and bands still use the purer natural tuning, even with music that has no key signature.
Other tuning systems
Long before equal temperament, the Chinese culture developed several systems. A fascinating history is described in Gene Jinsiong Cho’s monograph, LU-LU: A study of Its Historical, Acoustical and Symbolic Signification (Caves Books, Ltd., Taipei, 1989). Cho (a music theory professor colleague at the University of North Texas) explains the LU system from the Chin Dynasty, which extended beyond 12 increments in an octave as far as to the arcane realm of Jing fang’s sixty LU series.
In the West and into the 20th century, two American composers experimented with microtonal tunings splitting the octave into finer increments than our 12 semitones.
Working with American Lou Harrison, California composer Harry Partch (1901-1974) devised his own tuning system with 43 increments, described in Genesis of a Music (1947). The system necessitated invention of specialized percussion and string instruments to precisely intone the sounds, which felt exotic both in tuning and sound quality.
Harry Partch – Castor & Pollux (1952)
University of Illinois professor Ben Johnston (1926-2019) wrote music for standard orchestral string instruments using the ancient just intonations of Pythagorus. This involved specifying pitches microtonally slightly higher or lower than the equal-tempered standard pitch classes – a notational challenge of pitch-adjustment symbols.
Ben Johnston – String Quartet No. 7 (1984)
In the 21st century, Japanese composer norokusi has produced a broad catalog of microtonal music, apparently using a 17-increment division of the octave.
norokusi – Piano Sonata n.718 (2018) 17EDO/TET
Such complex systems as described above never became mainstream. The vast bulk of 20th-century and now 21st-century music is based on the equal-tempered 12-increment system found on a well-tuned piano, with subtle adjustments by orchestral strings, wind bands and a cappella choirs to momentarily purify some sonorities.
MAPPING MUSIC 