From Newsgroup: rec.puzzles
David Entwistle <
qnivq.ragjvfgyr@ogvagrearg.pbz> wrote or quoted:
I'm curious about the frequency (and wavelength) relationship between the >notes of the musical scale,
Here's a generated list of infos, I also added something about birds.
The Fundamentals of Pitch and Ratios
Musical notes are vibrations measured in Hertz (Hz).
The relationship between two notes is called an interrCE
val, which is expressed as a mathematical frequency
ratio.
- Octaves double frequency: Two notes an octave apart
always have a 2:1 frequency ratio.
- Unisons are identical: A 1:1 ratio means two notes
share the exact same frequency.
- Integers create consonance: Small, simple integer
ratios sound pleasant and stable to the human ear.
- Complexity creates dissonance: Large, complex matherCE
matical ratios sound tense, unstable, and harsh.
- Pitch is logarithmic: Human perception of pitch
scales logarithmically, not linearly, relative to
frequency.
The Mathematics of Major Intervals (Just Intonation)
In pure tuning systems based on natural harmonics,
intervals are defined by clean, uncompromised fracrCE
tions.
- Perfect Fifth (3:2): The most stable interval after
the octave, vibrating three times for every two of
the base note.
- Perfect Fourth (4:3): The inversion of a fifth,
vibrating four times for every three of the base
note.
- Major Third (5:4): The bright, foundational interval
of a major triad, vibrating five times for every
four.
- Minor Third (6:5): The dark, foundational interval
of a minor triad, vibrating six times for every
five.
- Major Sixth (5:3): A highly consonant interval,
vibrating five times for every three of the fundarCE
mental.
- Major Seventh (15:8): A highly tense, complex interrCE
val that strongly desires to resolve upward to the
octave.
Tuning Systems and Scale Construction
Over centuries, musicians developed different systems
to organize these frequency relationships into prerCE
dictable scales.
- Pythagorean tuning builds fifths: This ancient sysrCE
tem calculates every note by stacking pure 3:2 perrCE
fect fifths.
- The Pythagorean comma exists: Stacking 12 perfect
fifths does not perfectly match 7 octaves, creating
a small mathematical overshoot.
- Just Intonation uses fractions: This system tunes
all notes to clean, wholerCEnumber ratios based on a
single fundamental note.
- Just Intonation limits modulation: A scale tuned
perfectly to one key sounds completely out of tune
in another key.
- Equal Temperament compromises physics: Modern
12rCETone Equal Temperament (12rCETET) mathematically
divides an octave into 12 identical steps.
- 12rCETET uses exponents: The frequency of each consecrCE
utive semitone is multiplied by the twelfth root of
two.
- Only octaves stay pure: In modern 12rCETET, every sinrCE
gle interval except the octave is slightly out of
tune with natural physics.
- Modern fifths are narrow: EqualrCEtempered fifths are
roughly 2 cents flat compared to a pure 3:2 ratio.
- Modern thirds are sharp: EqualrCEtempered major thirds
are nearly 14 cents sharp compared to a pure 5:4
ratio.
Harmonics and the Overtone Series
Frequency relationships within scales are derived
directly from the physics of vibrating strings and air
columns.
- Fundamentals dictate the pitch: The lowest, loudest
frequency of a sound determines its perceived musirCE
cal note.
- Overtones are integer multiples: A vibrating string
simultaneously produces frequencies at 2x, 3x, 4x,
and 5x the fundamental.
- Scales mirror the series: The major scale naturally
emerges from the upper partials of the overtone
series.
- Timbre relies on overtones: The relative volume of
these different frequency relationships gives
instruments their unique sound character.
Psychoacoustics and Perception
How human brains interpret the physical interaction of
these frequencies defines musical emotion and tension.
- Critical bands cause roughness: When two frequencies
are too close together, the ear cannot separate
them, creating physical friction.
- Beating measures close pitches: Two frequencies
slightly out of tune create a physical pulsing senrCE
sation called beating.
- Beat speed equals difference: The number of beats
heard per second is exactly equal to the difference
in Hz between the two notes.
- Combination tones appear naturally: When two loud
notes are played, the brain naturally perceives a
third "difference tone" (f2-f1).
The Hidden Scales of Birdsong
To human ears, birds may seem to chirp "between the
cracks" of our musical notes, but deep acoustic analyrCE
sis reveals strict underlying patterns:
- The Harmonic Series: Species like the Hermit Thrush
explicitly select notes that follow a harmonic
series governed by smallrCEinteger ratios - the exact
same mathematical distribution that builds human
musical scales.
- Pentatonic and Consonant Intervals: Many birds
default to consonant intervals (pleasant combinarCE
tions) and pentatonic structures rather than random
frequencies.
- Absolute Pitch vs. Spectral Shape: Humans easily
recognize a melody if it is transposed to a differrCE
ent octave (relative pitch). Songbirds recognize
melodies by "spectral shape" (timbre and texture)
rather than moving a melody up or down a scale.
Shared Rhythms at High Speeds
Avian rhythm is incredibly precise, though it can easrCE
ily overwhelm human perception:
- Too Fast to Track: Birds can sing up to four times
faster than human music. When a 2rCEsecond wren call
is slowed down digitally, it unfolds into a beautirCE
fully timed, rhythmically distinct composition.
- Isochronous Beats: Many songbirds utilize
isochronous rhythms, where the spacing between notes
is perfectly equidistant - just like a steady
metronome or a drum beat.
- Categorical Rhythms: Research published in Current
Biology shows that songbirds like the Thrush
Nightingale cluster their notes into rhythmic caterCE
gories shared by human musicians, using structured
pacing to pass songs down through generations.
The Biomechanics of the Duet
Birds possess a vocal structure completely different
from our own. Instead of a larynx, they use a syrinx
located at the base of their trachea. Because this
organ splits into two bronchi, a songbird can produce
two different notes simultaneously, essentially singing
their own harmony or producing complex chords that
sound like a rapid blur to us.
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