Centricity Without Hierarchy World Pitch and Interval Chart, Part L

 Centricity Without Hierarchy

Pitch Cells, Intervals, and Cents Values Across World Musical Cultures

Peter Thoegersen — 2026

All traditions below organize pitch around a nuclear center without triadic hierarchy, dominant-tonic function, or voice-leading obligation. Cent values (¢) are measured against the equal-tempered semitone (100¢) and whole tone (200¢).

Region	Culture / Tradition	Ambitus	Semitones	Key Cents Values	Pitch Cell	Characteristic Interval	Tags	Description
Africa	San (Bushmen)	Minor 3rd or less	1–3 st	~50–300¢ (variable)	2–3 pitches	Neutral 2nds, microtonal inflections	Microtonal, neutral intervals	Kalahari San vocal music operates within a minor third or less. Pitch placement shifts microtonally — values typically 50–300¢ above the fundamental, variable between singers. The center is felt gravitationally without a fixed frequency. Yodel-like register breaks represent timbral contrast, not harmonic stratification.
Africa	BaAka / Mbuti Pygmies	Seconds to minor 3rd	1–3 st	~100–300¢	2–3 interlocking cells	2nds and 3rds (non-functional)	Neutral intervals, close-interval	Dense polyphony built from interlocking cells within a 2nd or 3rd (~100–300¢). Vertical intervals are not governed by the consonance-dissonance hierarchy. Centricity is multiplicative — each voice carries its own nuclear center.
Africa	Ethiopian Ge'ez chant	3rd to 4th, stepwise	2–5 st	~90–500¢ (modal)	3–5 pitches	Steps and modal 3rds	Neutral intervals	Three modes (Ge'ez, Ezil, Araray) with stepwise motion. Modal intervals ~90–500¢, no leading-tone mechanism. One of the three branches of Christian chant is organized on modal-centric rather than triadic-hierarchical principles.
Europe	Slavic birch tree songs	Major 2nd or unison	0–2 st	0–200¢	1–2 pitches	Major 2nd oscillation (~200¢)	Close-interval	Troitskie and semitskie pesni oscillate between two pitches a major second (~200¢) apart. All musical weight is carried by rhythm, text, and timbre. Prichitaniya (laments) may reduce to a single pitch with approach-tone inflections of 20–50¢.
Europe	Bulgarian Rhodope	2nd to major 3rd	1–4 st	~180–220¢ (non-tempered)	2–3 pitches per voice	Major 2nd as stable consonance	Microtonal, close-interval, neutral	Two voices sustain a major second (~180–220¢, non-tempered) as a deliberately stable sonority — not a suspension. Neutral intervals throughout. The major second as primary consonance is an alternative axiom within the system's own terms.
Europe	Ganga (Herzegovina/Bosnia)	Minor 2nd / near-unison	0–1 st	0–100¢ (beating zone)	1–2 pitches	Minor 2nd / unison ± ~20–80¢	Close-interval, microtonal	Two or three voices sit within a minor second (0–100¢), sustaining acoustic beating as the primary sonic material. Ambitus approaches zero. The entire music lives in the beating zone between near-unisons — the most extreme narrow-range tradition in Europe.
Chant	Gregorian chant	3rd to 4th per phrase	2–5 st	~90–500¢ (modal, just-adjacent)	4–8 pitches (modal octave)	Steps ~200¢; 3rds ~300–315¢; no leading tone	Neutral intervals	Eight modes are organized around finalis and tenor (reciting tone). Individual phrases move within a 3rd or 4th (~300–500¢). Modal degrees carry no harmonic function. Incorporated directly in Peter Thoegersen's Symphony No. 4, where chant pitch logic coexists with polytempic polymicrotonal layers as another instance of centricity without hierarchy.
Chant	Byzantine chant	3rd to 4th with microtones	2–5 st	Neutral 2nds ~150¢; microtonal 3rds ~270¢	4–7 pitches	Neutral 2nds (~150¢), microtonal 3rds (~270¢)	Microtonal, neutral intervals	Eight echoes with microtonal inflections. Neutral 2nds (~150¢) and microtonal 3rds (~270¢) place Byzantine intonation between Western semitone (100¢) and whole tone (200¢). Structurally adjacent to non-Western neutral-interval traditions across Africa and Asia.
Oceania	Aboriginal Australian	Under a perfect 4th	1–4 st	Neutral 3rds ~260–280¢; neutral 2nds ~150–180¢	2–4 pitches	Neutral 3rds (~260–280¢), neutral 2nds (~150–180¢)	Neutral intervals, drone present	Near-universal narrow range. Intervals are frequently neutral — neither major nor minor in equal-tempered terms. Neutral 3rds ~260–280¢ and neutral 2nds ~150–180¢. The didgeridoo drone creates a non-triadic environment in which voice operates with its own independent logic.
Americas	Pueblo peoples (Hopi/Zuni/Tewa)	Compressed, drone-like	2–5 st	~240¢ avg. step (non-functional pentatonic)	3–5 pitches	Non-functional pentatonic (~240¢ avg. step)	Neutral intervals, drone present	Ceremonial music with compressed range. Non-functional pentatonic with average steps ~240¢ — wider than equal-tempered whole tones, narrower than minor thirds, organized around a nucleus without hierarchical function.
Americas	Navajo / Apache	2–4 pitch set	0–3 st	Semitone oscillation ~100¢	2–4 pitches	Semitone oscillation (~100¢), non-tonal	Close-interval, neutral intervals	Chantway songs operate on 2–4 pitch sets with semitone oscillation (~100¢). The minor 2nd functions as primary melodic material without harmonic urgency — the center is the oscillation itself, not either pitch in isolation.
Americas	Inuit katajjaq	Step or two	0–2 st	0–200¢	1–2 pitches	Step (~100–200¢) / unison alternation	Close-interval	Game-like interlocking vocal practice. Pitches only a step apart (100–200¢). Primary interest is breath, timbre, and competitive-cooperative rhythm. Music is organized below the threshold at which Western pitch hierarchy becomes possible.
Americas	California Indigenous (Pomo/Yurok/Wintu)	Rarely exceeds a 5th	2–7 st	Neutral intervals, shifting ~20–50¢ between repetitions	3–5 pitches	Neutral intervals, shifting 20–50¢	Neutral intervals, microtonal	Tonal vagueness: pitches shift 20–50¢ between repetitions, intervals neutral, ambitus rarely exceeds a fifth (~700¢). Not imprecision but a different precision: pitch organized around a gravitational nucleus whose exact frequency matters less than its position within the narrow cell.
Asia & India	Gamelan Slendro (Java/Bali)	~5 notes, stretched pseudo-octave	4–12 st	~240¢ per step (≈5-EDO); octave +10–20¢	5 pitches	~240¢ per step (between maj. 2nd & min. 3rd)	Microtonal, inharmonic partials, drone	Slendro divides the pseudo-octave into 5 steps of ~240¢ each. The octave is stretched 10–20¢ beyond 1200¢. Crucially, each gamelan has its own unique tuning — to copy another ensemble's pitches insults the spirit of that instrument. The inharmonic partials of bronze metallophones mean the overtone series does not produce triadic implications: non-triadic pitch organization is built into the physics of the instruments, not merely cultural convention.
Asia & India	Gamelan Pelog (Java/Bali)	7 tones, 5 used per pathet	4–12 st	Small steps ~133¢; large steps ~267¢ (≈9-EDO)	5 of 7 pitches per pathet	~133¢ (small) and ~267¢ (large)	Microtonal, inharmonic partials, drone	Pelog's 7 unequal pitches approximate 9-EDO: small steps ~133¢ and large steps ~267¢. Each pathet (mode) uses 5 of the 7, selected around a nuclear center without functional harmonic hierarchy. Inharmonic bronze partials mean equal-temperament intonation is physically irrelevant to the instrument's acoustic structure.
Asia & India	Indian raga/shruti system	Octave, 22 unequal shruti positions	0–12 st	Pramana ~22¢; nyuna ~70¢; purna ~90¢	5–7 swaras per raga	3 shruti sizes: ~22¢, ~70¢, ~90¢; raga-specific	Microtonal, drone present	The 22-shruti system (Bharata's Natyashastra, ~200 BCE) divides the octave into 22 unequal microtonal positions. Three shruti sizes: pramana (~22¢), nyuna (~70¢), purna (~90¢). The vadi (nuclear swara) functions as the gravitational center of the raga — a centric nucleus, not a harmonic root. The tanpura drone sustains Sa and Pa continuously; all melodic material is relational to this fixed center, with no voice-leading obligations between scale degrees.
Asia & India	Japanese shakuhachi/gagaku	Pentatonic, non-equal	2–5 st	Meri inflections ~25¢ flat from 12-TET	5 pitches (pentatonic)	Non-equal pentatonic; meri ~25¢ flat	Microtonal, neutral intervals	Non-equal pentatonic scales (ryo and ritsu modes). The shakuhachi's meri technique produces notes ~25¢ flatter than nearest 12-TET equivalent — structurally integral, not ornamental. Ma (間, negative space) between pitches is compositionally significant. No harmonic dimension in traditional hogaku performance.
Asia & India	Mongolian / Tuvan khoomei	Overtone melody above fixed drone	0–24+	Partials 6–13: +702¢, +968¢, +1200¢, +1386¢, +1583¢, +1902¢, +2098¢	Overtone series partials 6–13	Harmonic series: includes 7th partial ~968¢ (32¢ below equal-tempered m7)	Drone, overtone melody, microtonal	The singer sustains a fixed fundamental drone and simultaneously amplifies specific overtone partials (6th–13th) to create melody. The 7th partial (~968¢) is ~32¢ flatter than an equal-tempered minor 7th. The 11th partial (~1751¢) has no equivalent in Western equal temperament. These just-intonation ratios are not triadic, not equal-tempered, and include intervals entirely outside Western pitch theory.
Asia & India	Balinese kecak / vocal interlocking	Narrow cells, interlocking	1–3 st	~100–300¢ (gamelan slendro-derived)	2–3 interlocking vocal cells	2nds and 3rds from slendro (~240¢)	Close-interval, inharmonic-derived	Kecak vocal interlocking derives its pitch cells from gamelan slendro intervals (~240¢ steps). Individual voices operate in narrow 2nd-to-3rd ranges, interlocking in hocket patterns. The collective rhythmic pulse is the primary organizing center — pitch is nuclear to each cell, rhythm is nuclear to the ensemble.

Detailed Notes by Culture

Africa

San (Bushmen)

Ambitus	Cents values	Semitones	Pitch cell	Nuclear center
Minor 3rd or less	~50–300¢ (variable)	1–3 st	2–3 pitches	Single nucleus tone

Kalahari San vocal music operates within a minor third or less. Pitch placement shifts microtonally — values typically 50–300¢ above the fundamental, variable between singers. The center is felt gravitationally without a fixed frequency. Yodel-like register breaks represent timbral contrast, not harmonic stratification.

BaAka / Mbuti Pygmies

Ambitus	Cents values	Semitones	Pitch cell	Nuclear center
Seconds to minor 3rd	~100–300¢	1–3 st	2–3 interlocking cells	Each voice independent

Dense polyphony built from interlocking cells within a 2nd or 3rd (~100–300¢). Vertical intervals are not governed by the consonance-dissonance hierarchy. Centricity is multiplicative — each voice carries its own nuclear center.

Ethiopian Ge'ez chant

Ambitus	Cents values	Semitones	Pitch cell	Nuclear center
3rd to 4th, stepwise	~90–500¢ (modal)	2–5 st	3–5 pitches	Modal finalis

Three modes (Ge'ez, Ezil, Araray) with stepwise motion. Modal intervals ~90–500¢, no leading-tone mechanism. One of the three branches of Christian chant is organized on modal-centric rather than triadic-hierarchical principles.

Europe

Slavic birch tree songs

Ambitus	Cents values	Semitones	Pitch cell	Nuclear center
Major 2nd or unison	0–200¢	0–2 st	1–2 pitches	Single nucleus/oscillation

Troitskie and semitskie pesni oscillate between two pitches a major second (~200¢) apart. All musical weight is carried by rhythm, text, and timbre. Prichitaniya (laments) may reduce to a single pitch with approach-tone inflections of 20–50¢.

Bulgarian Rhodope

Ambitus	Cents values	Semitones	Pitch cell	Nuclear center
2nd to major 3rd	~180–220¢ (non-tempered)	1–4 st	2–3 pitches per voice	Shared sustained center

Two voices sustain a major second (~180–220¢, non-tempered) as a deliberately stable sonority — not a suspension. Neutral intervals throughout. The major second as primary consonance is an alternative axiom within the system's own terms.

Ganga (Herzegovina/Bosnia)

Ambitus	Cents values	Semitones	Pitch cell	Nuclear center
Minor 2nd / near-unison	0–100¢ (beating zone)	0–1 st	1–2 pitches	Beating between near-unisons

Two or three voices sit within a minor second (0–100¢), sustaining acoustic beating as the primary sonic material. Ambitus approaches zero. The entire music lives in the beating zone between near-unisons — the most extreme narrow-range tradition in Europe.

Chant

Gregorian chant

Ambitus	Cents values	Semitones	Pitch cell	Nuclear center
3rd to 4th per phrase	~90–500¢ (modal, just-adjacent)	2–5 st	4–8 pitches (modal octave)	Finalis (modal)

Eight modes are organized around finalis and tenor (reciting tone). Individual phrases move within a 3rd or 4th (~300–500¢). Modal degrees carry no harmonic function. Incorporated directly in Peter Thoegersen's Symphony No. 4, where chant pitch logic coexists with polytempic polymicrotonal layers as another instance of centricity without hierarchy.

Byzantine chant

Ambitus	Cents values	Semitones	Pitch cell	Nuclear center
3rd to 4th with microtones	Neutral 2nds ~150¢; microtonal 3rds ~270¢	2–5 st	4–7 pitches	Echos (modal center)

Eight echoes with microtonal inflections. Neutral 2nds (~150¢) and microtonal 3rds (~270¢) place Byzantine intonation between Western semitone (100¢) and whole tone (200¢). Structurally adjacent to non-Western neutral-interval traditions across Africa and Asia.

Oceania

Aboriginal Australian

Ambitus	Cents values	Semitones	Pitch cell	Nuclear center
Under a perfect 4th	Neutral 3rds ~260–280¢; neutral 2nds ~150–180¢	1–4 st	2–4 pitches	Resting tone (tumbling strain)

Near-universal narrow range. Intervals are frequently neutral — neither major nor minor in equal-tempered terms. Neutral 3rds ~260–280¢ and neutral 2nds ~150–180¢. The didgeridoo drone creates a non-triadic environment in which voice operates with its own independent logic.

Americas

Pueblo peoples (Hopi/Zuni/Tewa)

Ambitus	Cents values	Semitones	Pitch cell	Nuclear center
Compressed, drone-like	~240¢ avg. step (non-functional pentatonic)	2–5 st	3–5 pitches	Nuclear center, drone-adjacent

Ceremonial music with compressed range. Non-functional pentatonic with average steps ~240¢ — wider than equal-tempered whole tones, narrower than minor thirds, organized around a nucleus without hierarchical function.

Navajo / Apache

Ambitus	Cents values	Semitones	Pitch cell	Nuclear center
2–4 pitch set	Semitone oscillation ~100¢	0–3 st	2–4 pitches	Oscillation center

Chantway songs operate on 2–4 pitch sets with semitone oscillation (~100¢). The minor 2nd functions as primary melodic material without harmonic urgency — the center is the oscillation itself, not either pitch in isolation.

Inuit katajjaq

Ambitus	Cents values	Semitones	Pitch cell	Nuclear center
Step or two	0–200¢	0–2 st	1–2 pitches	Interlocked breath rhythm

Game-like interlocking vocal practice. Pitches only a step apart (100–200¢). Primary interest is breath, timbre, and competitive-cooperative rhythm. Music is organized below the threshold at which Western pitch hierarchy becomes possible.

California Indigenous (Pomo/Yurok/Wintu)

Ambitus	Cents values	Semitones	Pitch cell	Nuclear center
Rarely exceeds a 5th	Neutral intervals, shifting ~20–50¢ between repetitions	2–7 st	3–5 pitches	Gravitational nucleus

Tonal vagueness: pitches shift 20–50¢ between repetitions, intervals neutral, ambitus rarely exceeds a fifth (~700¢). Not imprecision but a different precision: pitch organized around a gravitational nucleus whose exact frequency matters less than its position within the narrow cell.

Asia & India

Gamelan Slendro (Java/Bali)

Ambitus	Cents values	Semitones	Pitch cell	Nuclear center
~5 notes, stretched pseudo-octave	~240¢ per step (≈5-EDO); octave +10–20¢	4–12 st	5 pitches	Gong / ensemble-specific

Slendro divides the pseudo-octave into 5 steps of ~240¢ each. The octave is stretched 10–20¢ beyond 1200¢. Crucially, each gamelan has its own unique tuning — to copy another ensemble's pitches insults the spirit of that instrument. The inharmonic partials of bronze metallophones mean the overtone series does not produce triadic implications: non-triadic pitch organization is built into the physics of the instruments, not merely cultural convention.

Gamelan Pelog (Java/Bali)

Ambitus	Cents values	Semitones	Pitch cell	Nuclear center
7 tones, 5 used per pathet	Small steps ~133¢; large steps ~267¢ (≈9-EDO)	4–12 st	5 of 7 pitches per pathet	Gong / pathet (mode)

Pelog's 7 unequal pitches approximate 9-EDO: small steps ~133¢ and large steps ~267¢. Each pathet (mode) uses 5 of the 7, selected around a nuclear center without functional harmonic hierarchy. Inharmonic bronze partials mean equal-temperament intonation is physically irrelevant to the instrument's acoustic structure.

Indian raga/shruti system

Ambitus	Cents values	Semitones	Pitch cell	Nuclear center
Octave, 22 unequal shruti positions	Pramana ~22¢; nyuna ~70¢; purna ~90¢	0–12 st	5–7 swaras per raga	Sa (tonic); vadi (nuclear swara)

The 22-shruti system (Bharata's Natyashastra, ~200 BCE) divides the octave into 22 unequal microtonal positions. Three shruti sizes: pramana (~22¢), nyuna (~70¢), purna (~90¢). The vadi (nuclear swara) functions as the gravitational center of the raga — a centric nucleus, not a harmonic root. The tanpura drone sustains Sa and Pa continuously; all melodic material is relational to this fixed center, with no voice-leading obligations between scale degrees.

Japanese shakuhachi/gagaku

Ambitus	Cents values	Semitones	Pitch cell	Nuclear center
Pentatonic, non-equal	Meri inflections ~25¢ flat from 12-TET	2–5 st	5 pitches (pentatonic)	Modal finalis (choshi)

Non-equal pentatonic scales (ryo and ritsu modes). The shakuhachi's meri technique produces notes ~25¢ flatter than nearest 12-TET equivalent — structurally integral, not ornamental. Ma (間, negative space) between pitches is compositionally significant. No harmonic dimension in traditional hogaku performance.

Mongolian / Tuvan khoomei

Ambitus	Cents values	Semitones	Pitch cell	Nuclear center
Overtone melody above fixed drone	Partials 6–13: +702¢, +968¢, +1200¢, +1386¢, +1583¢, +1902¢, +2098¢	0–24+	Overtone series partials 6–13	Fixed fundamental drone

The singer sustains a fixed fundamental drone and simultaneously amplifies specific overtone partials (6th–13th) to create melody. The 7th partial (~968¢) is ~32¢ flatter than an equal-tempered minor 7th. The 11th partial (~1751¢) has no equivalent in Western equal temperament. These just-intonation ratios are not triadic, not equal-tempered, and include intervals entirely outside Western pitch theory.

Balinese kecak / vocal interlocking

Ambitus	Cents values	Semitones	Pitch cell	Nuclear center
Narrow cells, interlocking	~100–300¢ (gamelan slendro-derived)	1–3 st	2–3 interlocking vocal cells	Collective rhythmic nucleus

Kecak vocal interlocking derives its pitch cells from gamelan slendro intervals (~240¢ steps). Individual voices operate in narrow 2nd-to-3rd ranges, interlocking in hocket patterns. The collective rhythmic pulse is the primary organizing center — pitch is nuclear to each cell, rhythm is nuclear to the ensemble.

Note on independence: No pitch materials, scales, instruments, timbres, tuning systems, or aesthetic surfaces from any of the traditions documented here have been borrowed, adapted, or employed in the composer's work. The convergences described are structural rhymes arising from independent derivation — evidence that nucleus-based centricity reflects something deep in the possibilities of organized sound, not a cultural borrowing. The sole exception is Gregorian chant as incorporated in Symphony No. 4, where the source is fully acknowledged.