Studies in
Music by Wim van den Dungen |
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Ars Inveniendi The Farben Project |
OMNIA STANT HARMONIA. Anarmonia cadunt omnia. Nec erigitur,
reficitur, restituitur quidquam, nisi ad Harmoniam relatum atque
redactum. |
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Intro & Definitions § 1 Musical Sound § 2 The Tone § 3 Naturalizing Master Pitch § 4 Color Spectrum and Pitch § 5 The Harmonic Series § 6 Defining Tonal Scales § 7 The Tuning System § 8 Psycho-Acoustic Definitions In "Farben Project", the
German word "Farbe" ("color") refers to
"Klangfarbenmelodie", a musical technique involving splitting
a melody between several instruments, rather than assigning it to
just one instrument (or set of instruments). The aim was Music, derived from the Greek " In the definition of
what is music, both intrinsic & intentional approaches will be
combined. From the intrinsic strategy we retain the presence of There are two types of
sound : musical sound & noise. Let this crucial initial distinction
be rooted in the physical (material, efficient) difference between
irregular & regular sound waves.
In the common aural
chaos surrounding us, regular sound-ripples are very noticable, they
stand out, are uncommon. Always working together, these waves
reinforce each other, and generate strong resonance in any sonic
environment. When, as in a musical performance, surrounded by total
silence, these regular sound waves immediately reveal their
exceptional status. But in the sonic environment, these well-formed
sounds constitute but In music theory, the
word "note" refers to a sign, say G, calling for pitch (frequency), relative loudness (amplitude),
duration (rhythm) & timbre (quality or shape of the
frequency spectrum of the sound wave). In acoustics, G implies
Harmonic Series of C (with offset in cents, 1 octave = 1200 cents) When the frequencies of the partials are
Is there an
"The eye perceives in light which has been split up by a prism a
natural series of vibration frequencies. The light of the sun always
produces the same immutable series of colors, familiar to us in the
rainbow. Now, just as light consists of graduated colors of the
spectrum, so a tone consists of many partial tones. Spectrum of the
world of sound is the harmonic or overtone series. A tone produced
by a voice or instrument carries with it a greater or lesser number
of barely audible overtones. Their order is not arbitrary : it is
determined by a strict law, and is as immutable as the color series
of the rainbow."
Hindemith
(1942, Book 1, p.16). Newton's
Color Circle ( In Newton's circle, the
spectral colors from red to violet are divided by the notes of the
musical scale, starting with D (Dorian). The seven spectral bands are
associated with the seven tones (dividing the octave). The circle completes a full octave,
from D to D. Newton's circle places red, at one end of the spectrum, next
to violet, at the other (non-spectral purple colors are observed when red
and violet light are mixed). Newton's name for "indigo" is what today is
called "blue", whereas his name for "blue" is what today is called "cyan".
Why D is used remains unclear. This, together with the absence of
numerical correspondences between colors & tones (frequency), makes
Newton's scheme somewhat arbitrary. color spectrum of visible light Compare the
wavelengths of visible light (defining color), measured in
nanometers, with the
frequencies (defining tone pitch), measured in Hertz. The wavelength bands
given below give rise to a "centre of color gravity" (Newton)
or "color center wavelength", the
chromatic scale (starting with G) with color correspondences
"The structure of the true octave is derived
from the overtone series, from which nothing more complete than the
comprehensive building material furnished by the chromatic
twelve-tone scale can be developed." -
Hindemith (1942, Book 1, p.52). In music, the harmonic
series is an acoustical phenomenon defined by a sequence of pitches
related to a lower pitch. This is a sequence of successive intervals
: octave (2:1), fifth (3:2), fourth (4:3), major third (5:4), minor
third (6:5), two intervals between a minor third and a whole tone
(7:6 & 8:7), a large whole tone (9:8), a small whole tone (10:9), a
minor second (16:15) and progressively smaller intervals
the Harmonic Series of a string Suppose this is a string vibrating at 110 Hz. When
it divides itself up in two halves, these vibrate at 220 Hz. Divided
into three thirds, each vibrates at 330 Hz. Divided into four
quaters, each vibrate at 440 Hz. etc. All these vibrations and lots
more happen
There are two types of
harmonics : odd-numbered (1th, 3th, 5th, 7th & 9th etc.) and
even-numbered (2th, 4th, 6th, 8th, 10th etc.). The presence & power
of these in the frequency spectrum defines the "timbre" of
instruments. In clarinetes, the typical harmonics are the
odd-numbered, whereas in the flutes, even-numbered predominate.
Harmonic unison,
harmonic octave, harmonic second, major & minor harmonic third,
harmonic fourth, harmonic fifth, major & minor harmonic sixth are Blind
Harpist with Harp tuned to a Heptatonic Scale ? Noise is a sound with an irregular, amorph frequency spectrum.
A musical tone (note) is a sound defined by the Harmonic Series and its integer
frequency patterns. Besides defining timbre or the "instrument
color" of a tone when sounded, the Harmonic Series refers to six
important frequency
ratio's : 1/1, 2/1, 3/2, 4/3, 5/4, 6/5 (or the harmonic intervals of
unison, octave, fifth, fourth, major third & minor third to the
When two strings are
tuned
The relative absence of
harmonic push-and-pull dynamism (voice leading or harmonic
progression) in the pentatonic system, gave
rise to further subdivisions of the octave, introducing two new
intervals (from C as first tone) : the fourth (F) & seventh (B), together
dividing the octave in seven steps (for C) : C, D, E, F, G, A & B ;
the heptatonic scale.
The Greek modes were
used in the Middle Ages and associated with spirito-acoustic
properties. They continued to be used in Late Hellenism and the
Middle Ages. In the Renaissance (starting in painting as early as
the 13th century), individuality, coloration and
stronger expressiveness begin to bloom, giving rise (in Modern
Times), to the idea of specific "key colors", each family of tones
expressing an overall timbre linked with musical ideas. The
difference between a major & minor third, defined by a semitone, played a
crucial role in this. Shades of loud beats ("major") and shades of
softer beats ("minor") could be generated. The scales themselves had
characteristic content & dynamic push-and-pull behaviors. While frequency in Hertz
(the nominal lower frequency limit of human hearing being 20 Hz with
a capacity to differentiate between two tones 1 to 2 Hz apart)
gives the precise "location" of the tones, to further define
scales, especially in terms of their
When the intervals are expressed as cents, the differences between
the intervals of different tuning systems can be easily compared. In Equal
Temperament, the standard tuning system, the intervals are always
multiples of one hundred. This equalization of the microintervals is
the cause of the "grayness" of this tuning system, eliminating the
specific "key colors" of the tonal scales (eliminating the
difference between the chromatic & the diatonic semitone, defining
the semitone as 50c, tempering all octave-dividing intervals). Its main advantage is
tuning facility (especially on keyboards) and standardization.
There is no
Tuning Systems A tuning system consists of
three components : a) PYTHAGOREAN Tuning SCALAR DIMENSION : 12
However, no matter how many
fifths (3/2 intervals) one takes, either above or below a given
note, one 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 = 531.441/41096 = 129,746 2/1 x 2/1 x 2/1 x 2/1 x 2/1 x 2/1 x 2/1 = 128,0 To tackle this discrepancy, derive a complete chromatic scale taking
a series of 11 perfect fifths. Indeed, no stack of 3:2 intervals fits exactly into any stack of 2:1 intervals (octaves). Take
C, move up 12 fifth and then 7 octaves down. This brings one to B#,
almost exactely where one started. The crucial discrepancy is the
small interval (or comma) existing in Pythagorean tuning between two
enharmonically equivalent notes such as C and B# (or Ab and G#). It
is equal to the frequency ratio 3
In the case the notes G#
and Eb need to sound together, the position of the wolf fifth
can be changed. For example, a C-based Pythagorean tuning would
produce a stack of fifths running from Db to F#, making F# - Db the
wolf interval. As there is always one wolf fifth in Pythagorean
tuning, playing all keys in tune is impossible. b) JUST Intonation (or Just Tuning) SCALAR DIMENSION : 12
Various systems of Just Intonation were designed. Usually, a limit is imposed on how complex the ratios used in Just Intonation are. Just Intonation 7-limit implies there is no prime number larger than 7 featuring in the ratios.
Wolf intervals also occur and hence the same problems as with Pythagorean Tuning persist. c) MEANTONE Temperament SCALAR DIMENSION : 12
and more
In Meantone Temperament,
the whole tone is precisely half of the harmonic major third
(386,314c/2 = 193,157c). It rejects the wide Pythagorean third and
seeks to establish the harmonic major third by tempering certain
fifths. Eventually, Meantone Temperament sought to "close" the
circle and eliminate the wolf interval. The wolf remained.
In the 16th century, it
was suggested to spread the comma over five or six fifths, causing
less differentiation between the four "bad" thirds. The eight "good"
thirds (393c) were no longer harmonic thirds (386c), but sounded
good while the fifths were better. These are the one-fifths or
one-sixth-comma Meantone Temperaments.
d) IRREGULAR Temperaments SCALAR DIMENSION : 12
Irregular temperaments
Thomas Young's
Well Temperament (N°2) Young N°2 starts the set of tempered fifths with C as "scalar center", making the key of C# (Db) beat the most and the key of G the less. If the scale would start with F, F# (Gb) would be the worst and C the best, etc. The fifth (for example : C - G) is tempered with 1/6th of a Pythagorean comma (or 23,460c/6 = 3.910c), or a perfect fifth (3:2) of 701,956c - 3.91c = 689.046c.
Young N°2 defines the
minor third harmonically (32:27). d) EQUAL Temperament SCALAR DIMENSION : 12
The advantage of this
temperament is easy tuning and standardisation, but the cost is loss
of harmony, or as some rightly claim, the "ruin" of harmony as such
... § 8 Psycho-Acoustic Definitions The impact of music on
the human being, besides auditive input, is mainly through
These correspondences may be used to compose music intended to have psycho-acoustic effects. The question then is : How to translate the criteria into an intented effect and this into a piece of music ? |
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©
Wim van den Dungen,
Antwerp - 2018 -
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