Begin with small-integer ratios in Croton, then divide the octave into twelve equal multiplications along independent routes in Qinyang and Leiden. Move through twenty-four keys in Köthen, sinusoids in Paris, transmission limits in New York, digital synthesis in Murray Hill, and compression in Erlangen—hearing the choices between exact ratios and usable music.
QUESTION FOR THE ROUTE
If all beautiful interval ratios cannot fit one keyboard at once, what do mathematics, instruments, ears, and cultures preserve—and what do they compromise?
WHAT THIS RIVER DOES NOT CLAIM
The river is not geography. Distance downstream stands for time passing, and the light turns from dawn to dusk as the centuries go by. The objects by each stele are symbols of the kind of event and of how each century band wrote and calculated; they do not reconstruct any real artefact. The land around each stop sketches the natural geography of the scene’s real place, and an iconic building appears only if it already stood in that year. Each scene keeps its real place and evidence basis; open it on the map to read where it happened. This route does not reduce music to universal integer ratios or the Western twelve-tone scale. It separates the hammer legend from string experiments, just intonation from equal temperament, well temperament from modern twelve-tone equal temperament, Fourier’s heat research from later acoustics, Nyquist’s telegraph paper from the completed sampling theorem, and computer-assisted composition from digital synthesis. The map line is an edited comparison of independent calculation, translation, instruments, and standards—not one scale’s linear conquest of the world.
WHAT YOU SEE ON THIS RIVER
- Diagram in the sky
- A string vibrating in 1:2, 2:3 and 3:4
- Emblem at the source
- A single-string monochord
- The real place around each stop
- Around each stele the land takes on the natural geography of that scene’s real place — sea or lake, plain, hills or mountains, the colour of the ground and its common trees — and, where one defines the place, its landform: a volcano, snow peaks, granite domes, a mesa, dunes, a fjord, islands, a rock hill, a gorge or loess terraces. The water near the stop takes the colour of the real river or sea, and the haze the place’s climate. A small globe on the stele marks where it is, with the route from the previous place. Where a city has an iconic building that already stood in the scene’s year, its schematic silhouette rises behind the stop and is named on the card. The land follows today’s terrain and climate as a sketch and the silhouettes are not measured reconstructions. Between stops the river itself stays symbolic.
- A figure board at every stop
- Each board draws the mathematics of that scene. When the boat arrives, the construction is drawn in and the key result rises in red. The drawings are schematic reconstructions, not historical manuscripts.
- Century bands along the banks
- to 499 · Sandstone stele · braziers · earthen villages and beacons · rafts · flocks of birds
- 500–1449 · Stone stele · paper lanterns · single-arch bridge · watermills and villages · lateen boats
- 1450–1749 · Marble stele · iron lanterns · three-arch bridge · windmills and clock-tower towns · sailing ships
- 1750–1899 · Cast-iron plaque · gas lamps · iron truss bridge · factory chimneys, railway and steam train · steamboats
- 1900–1969 · Concrete marker · electric streetlights · concrete bridge · pylons, apartment blocks and radio masts · barges · aircraft
- 1970 onward · Glass marker · LED lights · cable-stayed bridge · glass towers, wind turbines and data centres · ferries · satellites
Where the century band changes, the boat passes under a bridge of the new band. Villages, mills, factories, pylons and towers stand for the technology of each century, not for any real place or architectural style.
01·c. 500 BCE·Crotone(basis: Main activity)
Hearing Octaves and Fifths in String-Length Ratios — the Pythagorean Tradition
For otherwise identical strings, halving length doubles frequency and produces an octave; a 2:3 length ratio produces the 3:2 frequency ratio of a perfect fifth. These small integers made a powerful bridge between number and sound. The story that Pythagoras discovered them from blacksmiths’ hammers appears centuries later, and hammer weight does not determine pitch in the simple proportions of the legend.
- Pause and ask
- Why do string-length ratios 1:2 and 2:3 sound like an octave and a perfect fifth?
- How thinking changed
- Turn vague pitch difference into a repeatable relationship between length and frequency ratios.
- What we cannot claim
- No writing by Pythagoras survives, and the blacksmith-hammer story is a later legend; hammer weight does not map to pitch by those simple ratios.
- This place
- The Pythagorean community at Croton is remembered as a working setting where number, music, education, and ritual met. (Ionian coast · olives · low hills · 39.1°N 17.1°E)
- Figure board
- Three identical strings stopped at full, half and two-thirds length: 1 : 2 sounds an octave, 2 : 3 a perfect fifth.
- On the river
- A place of ongoing work, marked only by the route’s emblem · to 499 (Sandstone stele · braziers · earthen villages and beacons · rafts · flocks of birds)
02·c. 150 CE·Alexandria(basis: Composition)
Testing Calculated Ratios Again with Ears and Instruments — Ptolemy
Ptolemy’s Harmonics argued with numerical ratios while insisting that reason be checked and adjusted through perception and instruments. It moved beyond the simple claim that correct numbers alone make music. It is not projected forward as the birth of all modern psychoacoustics or a direct design for today’s scales.
- Pause and ask
- When a calculated ratio and heard sound disagree, what should be revised?
- How thinking changed
- Move from the authority of ratio alone to a method in which reason and perception test one another.
- What we cannot claim
- The Harmonics is not labeled the whole of modern psychoacoustics or a direct blueprint for today’s scales.
- This place
- Alexandria’s mathematical, astronomical, and instrumental traditions connected tables and argument with devices such as the monochord. (Mediterranean coast · date palms · flat sand · 31.2°N 29.9°E · landmark: Lighthouse of Alexandria (Pharos) (280 BCE))
- Figure board
- A bridge set by the computed ratio m : n is heard again and shifted: reason and perception test each other in a loop.
- On the river
- A desk holding a written record · to 499 (Sandstone stele · braziers · earthen villages and beacons · rafts · flocks of birds)
03·c. 950 CE·Baghdad(basis: Composition)
Reworking Greek Ratios through the Oud and Living Melody — al-Farabi
Al-Farabi’s Great Book of Music did not merely store translated Greek theory; it analyzed intervals, rhythm, hearing, and instruments such as the oud together. Baghdad’s translation, scholarly, and court networks let different languages and performing traditions meet. Not every idea in the book began in one city or moved along one straight route to Europe.
- Pause and ask
- How did translated Greek ratios change when they met oud frets and living melody?
- How thinking changed
- Move beyond preserving theory to recalculating and classifying it through instruments, rhythm, and hearing.
- What we cannot claim
- The Great Book’s contents are not made al-Farabi’s lone invention or one straight transmission line toward Europe.
- This place
- Baghdad’s networks of translators, scholars, and court musicians supplied conditions for languages and performance practices to meet. (Tigris banks · date palms · flat plain · 33.3°N 44.4°E)
- Figure board
- Frets re-set by ratio on an oud neck, with a melody moving across them and a cycle of rhythmic beats beside it.
- On the river
- A desk holding a written record · 500–1449 (Stone stele · paper lanterns · single-arch bridge · watermills and villages · lateen boats)
04·1558 CE·Venice(basis: Publication)
Organizing Polyphonic Harmony with Purer Ratios — Zarlino
Zarlino’s Le istitutioni harmoniche accepted ratios involving five, connecting a just major third of 5:4 with the theory of polyphony. Venetian printing and church-music networks spread the system widely. Yet intervals pure in one key drift in another, so just intonation did not solve every instrument and modulation at once.
- Pause and ask
- Why does making a pure major third in one key create trouble when moving to another?
- How thinking changed
- See that purity of individual intervals and consistency across keys are different objectives.
- What we cannot claim
- Just intonation is not perfect tuning; it is pure for selected harmonies and exposes discrepancies under modulation.
- This place
- Venice’s music printing, churches, and polyphonic ecosystem linked harmonic theory to composition and a broad readership. (Venetian lagoon · islands · broadleaf trees · 45.4°N 12.3°E · landmark: St Mark’s Basilica (1094), St Mark’s Campanile (1514))
- Figure board
- On a pitch line the 5 : 4 third and 3 : 2 fifth land cleanly, but a pure fifth built from another note misses the fixed pitch.
- On the river
- A stack of books · 1450–1749 (Marble stele · iron lanterns · three-arch bridge · windmills and clock-tower towns · sailing ships)
05·1584 CE·Qinyang(basis: Main activity)
Dividing an Octave into Twelve Equal Multiplications — Zhu Zaiyu
Zhu Zaiyu calculated extremely precise values equivalent to the twelfth root of two and embodied them in pitch pipes, dividing the octave’s 2:1 ratio into twelve equal semitones. Long computation at a Ming princely establishment linked calendrics, ritual, and tuning. His work predates Stevin’s, but direct transmission between the two calculations has not been established.
- Pause and ask
- What number closes a 2:1 octave in exactly twelve equal multiplications?
- How thinking changed
- Move from choosing small integer ratios to repeatedly computing an irrational ratio.
- What we cannot claim
- Its precedence over Stevin matters, but evidence of direct transmission from China to Europe is a separate question.
- This place
- The Ming princely establishment in Huaiqing gave Zhu Zaiyu resources and duties to study calendrics, ritual, tuning, and pitch pipes together. (Riverside · loess terraces · Taihang foothills · 35.1°N 113.0°E)
- Figure board
- Thirteen pitch pipes shrinking by one constant ratio close the 2 : 1 octave in twelve equal multiplications.
- On the river
- A place of ongoing work, marked only by the route’s emblem · 1450–1749 (Marble stele · iron lanterns · three-arch bridge · windmills and clock-tower towns · sailing ships)
06·c. 1605 CE·Leiden(basis: Composition)
Calculating the Same Twelve Equal Steps by Another Route — Stevin
Simon Stevin presented a calculation for dividing the octave into twelve equal ratios in the unfinished manuscript Van de Spiegheling der singconst. It is an important independent European scene, but the manuscript was unpublished in his lifetime and surfaced only in 1884. Leiden marks his educational and practical network, not a proven writing room or immediate influence.
- Pause and ask
- How much can an accurate but unpublished calculation change tuning practice?
- How thinking changed
- Separate mathematical arrival from historical influence, which depends on publication, instruments, and readers.
- What we cannot claim
- Leiden is a network anchor, not a proven writing room; the unpublished manuscript is not made a direct cause of seventeenth-century practice.
- This place
- Leiden’s engineering, surveying, and educational environment linked Stevin’s decimal and practical mathematics with music. (Old Rhine · broadleaf trees · flat polders · 52.2°N 4.5°E)
- Figure board
- A manuscript string divided into twelve equal ratios, and the long gap until the manuscript surfaced in 1884.
- On the river
- A desk holding a written record · 1450–1749 (Marble stele · iron lanterns · three-arch bridge · windmills and clock-tower towns · sailing ships)
07·1636 CE·Paris(basis: Publication)
Measuring String Length, Tension, Density, and Frequency — Mersenne
Mersenne’s Harmonie universelle organized experimental and numerical relations among a vibrating string’s length, tension, linear density, and pitch. Instrument makers, performers, scholars, and his correspondence network turned ratios into measurable physics. Modern formulas use later notation, and Mersenne did not complete all of acoustics alone.
- Pause and ask
- Can length, tension, and thickness be joined in one account of a string’s pitch?
- How thinking changed
- Turn harmonic ratios into measurable frequency and controllable physical quantities.
- What we cannot claim
- The compact modern ‘Mersenne laws’ use later notation, and one book did not complete acoustics.
- This place
- Paris connected Mersenne’s monastic base and European correspondence with instrument makers and scholars. (Seine banks · broadleaf trees · flat basin · 48.9°N 2.4°E · landmark: Notre-Dame de Paris (1250))
- Figure board
- Four weighted strings vary length, tension and thickness one at a time; traces at right compare their vibration rates.
- On the river
- A stack of books · 1450–1749 (Marble stele · iron lanterns · three-arch bridge · windmills and clock-tower towns · sailing ships)
08·1722 CE·Köthen(basis: Composition)
Circling All Twenty-Four Major and Minor Keys on One Instrument — Bach
Bach assembled Book I of The Well-Tempered Clavier in Köthen, moving through twenty-four major and minor keys and demonstrating what a usable circulating temperament could make musical. Wohltemperiert refers to a broad family that made every key serviceable. The evidence does not establish modern exact twelve-tone equal temperament or make the work its invention.
- Pause and ask
- Does making every key usable mean that every semitone is exactly equal?
- How thinking changed
- Separate the historical category ‘well-tempered’ from modern twelve-tone equal temperament.
- What we cannot claim
- The Well-Tempered Clavier does not establish modern equal temperament or its invention, and Bach’s exact tuning remains debated.
- This place
- The Köthen court supplied players, keyboards, and patronage for Bach’s concentrated instrumental work across keys. (Open farmland · broadleaf trees · flat land · 51.8°N 12.0°E)
- Figure board
- A two-ring circle of twelve pitches is travelled through all twenty-four keys, though the steps between them need not all be equal.
- On the river
- A desk holding a written record · 1450–1749 (Marble stele · iron lanterns · three-arch bridge · windmills and clock-tower towns · sailing ships)
09·1739 CE·Saint Petersburg(basis: Publication)
Measuring Consonant Pleasure through the Complexity of Prime Factors — Euler
Euler’s Tentamen novae theoriae musicae factored interval and chord ratios to rank their perceived agreeableness mathematically. It was a bold number-theoretic experiment—famously too mathematical for musicians and too musical for mathematicians. No single index determines taste across timbre, culture, and context.
- Pause and ask
- If a chord ratio has simple prime factors, can musical pleasure be fixed by one number?
- How thinking changed
- Model taste with computable structure while separating a model from a universal law.
- What we cannot claim
- Prime-factor complexity alone is not treated as determining every listener’s taste across timbre, learning, culture, and context.
- This place
- The Saint Petersburg Academy gave Euler a publishing base for crossing number theory, analysis, mechanics, and music. (Neva delta and gulf · birch and pine · flat · 59.9°N 30.3°E · landmark: Peter and Paul Cathedral (1733))
- Figure board
- Prime factors of the ratios 1 : 1 to 5 : 6 stacked as cells rank agreeableness — more complex, less agreeable.
- On the river
- A stack of books · 1450–1749 (Marble stele · iron lanterns · three-arch bridge · windmills and clock-tower towns · sailing ships)
10·1822 CE·Paris(basis: Publication)
Analyzing Complex Shapes as Sums of Sines and Cosines — Fourier
Fourier developed trigonometric expansions while solving the diffusion of heat. The language later became central to analyzing sound by frequency components. His 1822 book was not about music or digital signals, and the conditions and meaning of convergence required further mathematical work.
- Pause and ask
- Can a complex waveform be seen again as amplitudes and phases of simple sinusoids?
- How thinking changed
- Move from a shape over time to a representation by its frequency components.
- What we cannot claim
- The 1822 book is about heat, and it does not imply unconditional pointwise recovery of every signal as a sinusoidal sum.
- This place
- Parisian academy, print, and debate networks reviewed, challenged, and circulated series methods for the heat equation. (Seine banks · broadleaf trees · flat basin · 48.9°N 2.4°E · landmark: Notre-Dame de Paris (1250), Dôme des Invalides (1706))
- Figure board
- A complex periodic shape is split into three sines and cosines, then rewritten at right as the strength of each component.
- On the river
- A stack of books · 1750–1899 (Cast-iron plaque · gas lamps · iron truss bridge · factory chimneys, railway and steam train · steamboats)
11·1863 CE·Heidelberg(basis: Publication)
Separating Partials, Beats, and the Ear’s Response with Resonators — Helmholtz
Helmholtz used resonators to isolate partials in complex tones and connected beats and auditory physiology with consonance and dissonance. He built an experimental bridge between mathematical waves and hearing. Physiological roughness matters, but timbre, learning, culture, and context keep musical preference from reducing to one natural law.
- Pause and ask
- Why can the same frequency ratio feel different when timbre and beating change?
- How thinking changed
- Experimentally connect abstract wave components with resonance, physiology, and perception in the ear.
- What we cannot claim
- Beats and physiological roughness are important parts of consonance, not a complete explanation of aesthetic judgment in every musical culture.
- This place
- Physiology and physics at Heidelberg, together with instrument and resonator making, made hearing experimentally tractable. (Neckar banks · broadleaf trees · wooded hills · 49.4°N 8.7°E · landmark: Heidelberg Castle (1214), Old Bridge (Karl Theodor Bridge) (1788))
- Figure board
- A resonator singles one partial out of a complex tone, and two close tones add into beats whose loudness swells and fades.
- On the river
- A stack of books · 1750–1899 (Cast-iron plaque · gas lamps · iron truss bridge · factory chimneys, railway and steam train · steamboats)
12·1928 CE·New York(basis: Publication)
Asking How Fast a Telegraph Channel Can Carry Signals — Nyquist
At Bell System, Harry Nyquist analyzed the relationship between pulse rate and bandwidth in a noiseless telegraph channel. The work became an important shoulder for later sampling theory. It did not by itself invent digital music or the complete theorem now named for Nyquist and Shannon.
- Pause and ask
- How fast can signals be sent through limited bandwidth without running into one another?
- How thinking changed
- Turn a continuous waveform into conditions on symbol rate, bandwidth, and later sampling.
- What we cannot claim
- The 1928 paper studies telegraph transmission rate; it is not the completed sampling theorem or a lone invention of digital music.
- This place
- Bell System research and telegraph networks in New York turned abstract wave theory into measured design at continental scale. (Harbor and Hudson · broadleaf trees · low land · 40.7°N 74.0°W · landmark: Manhattan skyline (1913), Brooklyn Bridge (1883))
- Figure board
- Pulses fitted to band B and spaced T apart: at each pulse’s peak every other pulse passes through zero, so none run together.
- On the river
- A stack of books · 1900–1969 (Concrete marker · electric streetlights · concrete bridge · pylons, apartment blocks and radio masts · barges · aircraft)
13·1957 CE·Urbana(basis: Main activity)
Using Probability Rules and a Computer to Make a String-Quartet Score — the ILLIAC Suite
Hiller and Isaacson used the University of Illinois ILLIAC computer to test rules and probabilistic procedures for score material in the ILLIAC Suite, then had human players perform it. Computation entered compositional choice, but this was neither the first computer-synthesized sound nor an autonomous AI composer.
- Pause and ask
- If probability and a computer choose some compositional rules, who is the creator?
- How thinking changed
- Use a computer not as a sound player but as an assistant for testing rules of compositional choice.
- What we cannot claim
- The ILLIAC Suite is computer-assisted composition, not the first digital sound synthesis or an autonomous AI composer.
- This place
- The University of Illinois linked ILLIAC, musicians, a chemist, programmers, and string players in one computational-performance project. (Prairie · broadleaf trees · flat farmland · 40.1°N 88.2°W)
- Figure board
- A die proposes candidate notes, rules strike some out, and the survivors join into a melody on the staff.
- On the river
- A place of ongoing work, marked only by the route’s emblem · 1900–1969 (Concrete marker · electric streetlights · concrete bridge · pylons, apartment blocks and radio masts · barges · aircraft)
14·1957 CE·Murray Hill(basis: Main activity)
Calculating a Waveform So a Computer Produces Sound — Max Mathews
At Bell Labs, Max Mathews and colleagues used MUSIC I to calculate digital waveforms; Newman Guttman’s short Silver Scale became an early result. This is a foundational scene in digital sound synthesis, not the first electronic or machine-made sound of every kind. Slow computation, conversion hardware, and industrial-laboratory resources all mattered.
- Pause and ask
- How can a numerically calculated sequence become actual vibration in the air?
- How thinking changed
- Move from selecting notes to digitally calculating and converting the waveform sample by sample.
- What we cannot claim
- This is a key origin of digital sound synthesis, not the first electronic music or machine-made sound of every kind.
- This place
- Bell Labs’ mainframes, converters, telephony acoustics, and long-term industrial support made slow early synthesis possible. (Woods · broadleaf trees · low ridges · 40.7°N 74.4°W)
- Figure board
- A column of computed numbers becomes samples one by one, then passes a converter into a stepped, real waveform.
- On the river
- A place of ongoing work, marked only by the route’s emblem · 1900–1969 (Concrete marker · electric streetlights · concrete bridge · pylons, apartment blocks and radio masts · barges · aircraft)
15·1995 CE·Erlangen(basis: Standards agreement)
Shrinking Music Files with Hearing Limits and Frequency Transforms — MP3
Fraunhofer researchers and the international MPEG collaboration standardized MPEG-1 Audio Layer III by combining filter banks, transforms, psychoacoustic models, quantization, coding, and listening tests; the .mp3 extension was selected in 1995. It does not merely remove ‘inaudible frequencies,’ and lossy settings can create audible artifacts. The result belongs to a long university, industry, and standards network rather than one inventor.
- Pause and ask
- How much musical identity can be preserved while reducing information the ear notices less?
- How thinking changed
- Instead of storing a waveform directly, jointly use frequency transforms, hearing models, quantization, and coding.
- What we cannot claim
- MP3 does not merely delete inaudible frequencies; settings can create audible artifacts, and it is not one person’s invention.
- This place
- Fraunhofer research in Erlangen and the international MPEG standards network turned experimental codecs into an interoperable format. (Regnitz riverside · pines · flat valley · 49.6°N 11.0°E)
- Figure board
- A waveform split into frequency bands; bands under the hearing model’s masking curve are quantized coarsely, not deleted, then coded as bits.
- On the river
- A standard bar and a balance · 1970 onward (Glass marker · LED lights · cable-stayed bridge · glass towers, wind turbines and data centres · ferries · satellites)