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| From | ram@zedat.fu-berlin.de (Stefan Ram) |
|---|---|
| Newsgroups | rec.puzzles |
| Subject | Re: Musical notes... |
| Date | 2026-05-27 16:25 +0000 |
| Organization | Stefan Ram |
| Message-ID | <infos-20260527172331@ram.dialup.fu-berlin.de> (permalink) |
| References | <10v69g9$2k4n9$2@dont-email.me> |
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 inter‐ 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 mathe‐ 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 frac‐ 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 funda‐ mental. - Major Seventh (15:8): A highly tense, complex inter‐ 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 pre‐ dictable scales. - Pythagorean tuning builds fifths: This ancient sys‐ tem calculates every note by stacking pure 3:2 per‐ 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, whole‐number 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 12‐Tone Equal Temperament (12‐TET) mathematically divides an octave into 12 identical steps. - 12‐TET uses exponents: The frequency of each consec‐ utive semitone is multiplied by the twelfth root of two. - Only octaves stay pure: In modern 12‐TET, every sin‐ gle interval except the octave is slightly out of tune with natural physics. - Modern fifths are narrow: Equal‐tempered fifths are roughly 2 cents flat compared to a pure 3:2 ratio. - Modern thirds are sharp: Equal‐tempered 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 musi‐ 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 sen‐ 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 analy‐ sis reveals strict underlying patterns: - The Harmonic Series: Species like the Hermit Thrush explicitly select notes that follow a harmonic series governed by small‐integer ratios - the exact same mathematical distribution that builds human musical scales. - Pentatonic and Consonant Intervals: Many birds default to consonant intervals (pleasant combina‐ tions) and pentatonic structures rather than random frequencies. - Absolute Pitch vs. Spectral Shape: Humans easily recognize a melody if it is transposed to a differ‐ 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 eas‐ ily overwhelm human perception: - Too Fast to Track: Birds can sing up to four times faster than human music. When a 2‐second wren call is slowed down digitally, it unfolds into a beauti‐ 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 cate‐ 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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Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-27 08:20 +0000
Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-27 12:58 +0000
Re: Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-27 18:06 +0000
Re: Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-28 10:15 +0000
Re: Musical notes... ram@zedat.fu-berlin.de (Stefan Ram) - 2026-05-27 16:25 +0000
Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-27 17:10 +0000
Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-28 12:30 +0000
Re: Musical notes... HenHanna@NewsGrouper <user4055@newsgrouper.org.invalid> - 2026-05-28 18:21 +0000
Re: Musical notes... msb@vex.net (Mark Brader) - 2026-05-28 18:50 +0000
Re: Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-29 17:19 +0000
Re: Musical notes... Charlie Roberts <croberts@gmail.com> - 2026-05-29 15:48 -0400
Re: Musical notes... ram@zedat.fu-berlin.de (Stefan Ram) - 2026-05-29 20:59 +0000
Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-30 12:07 +0000
Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-30 12:25 +0000
Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-30 12:33 +0000
Re: Musical notes... ram@zedat.fu-berlin.de (Stefan Ram) - 2026-05-30 12:56 +0000
Re: Musical notes... Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-05-29 22:38 +0100
Re: Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-30 06:38 +0000
Re: Musical notes... Phil Carmody <pc+usenet@asdf.org> - 2026-06-05 16:41 +0300
Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-06-05 14:54 +0000
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