| Pýthagorás ze Samu , c. 575-509 BC, the ancient Greek mathematician, founder of the philosophical school. According to his teachings, harmony rules to music, to the movements of the heavenly bodies, to microworld and to the human soul itself. |
Many phenomena in nature occur periodically - tides regularly start, solar activity fluctuates, the position of the moon and planets repeats, electrons oscillate around the nuclei of atoms ... Periodic motion has two basic characteristics - frequency f and amplitude A. The frequency states number of oscillations per second and is measured in Hertz [Hz = 1 / s], amplitude determines the largest possible swing and is measured in meters.
The motion of the planets fall into the category of mechanical movements and is governed by Newton's laws. This category of movements includes also the sound.
The pair of parameters (f, A) completes for the sound with the third parameter - duration t, which indicates the time of sounding in seconds.
Aristotelés ze Stageiry| Aristotelés ze Stageiry [], 384-322 př.n.l., Greek philosopher, scholar, polyhistor, one of the largest thinkers of antiquity. Dealt with the fundamental principles of all being. He knew the nature of sound. Music considered as an imitation of movement. Tones are perceived sensually, but their relationship intellectually. To establish mental harmony is necessary to choose the proper rhythms and modes. |
The movement composite of several periodic motions can be periodic only then, if the frequencies of the motions are in a ratio of integers.
Pythagoras noticed that sounds whose frequencies are in the ratio of small integers creates consonant (euphonic) harmony.
He considered this so important that dared to generalize it on the movements in the whole universe. According to Pythagoras, the motion of planets creates harmony - "Harmony of the spheres," but the sound of this cosmic harmony cannot be heard, because we had been (from birth) accustomed to it.
The term sound means any stimulus that we can capture by hearing.
We distinction own sound (objective existence) from its subsequent perception (subjective existence). Perception of sound is caused by changes in air pressure. But the actual change in air pressure in not enough, it must take place periodically with some frequency - wave pressure changes must follow in quick succession.
Leonardo da Vinci| Leonardo da Vinci [], -, 15.stol. examined the reflection of sound, formulated the principle of independence of the spread sound waves from various sources. |
The conductor of the sound is material environment. In his absence (eg. on the Moon, where there is no atmosphere) sound does not spread (contrary to the movement of the electro-magnetic waves that spreads also in vacuum). Mechanical movement is carried from one point of environment to another by sound waves. It is assumed that all particles, which surround the sounding body, oscillat. Because compression and dilution of air change in the direction of movement, we are talking about the longitudinal waves. (In the solids where particles are not freely movable, transverse wave - i.e. oscillation perpendicular to the direction of wave motion - may exist.). Wavelength is the distance that get particles again in the same situation relative to the axis about which they oscillate (e.g. the highest guitar string with a frequency of about 330 Hz has a wavelength in air about 1 m).
Huygens Christian| Huygens Christian [hojchens], 1629-1695, Dutch physicist, mathematician and astronomer, author of the wave theory of light. Derived 31-piece music system. |
The waves do not affect each other. Each wave progresses by space so as if it was alone. The sum of the waves computes by vectors and can therefore be less than two addends (bute zero-sum, is difficult to explain be flux of particles ...).
The basic properties of waves (albeit different nature) are common - the Snell's law, the Fermat's principle and the Huyghens' principle - originally discovered for light - can be applied also to sound (but the same does not apply to laws of light polarization, etc.).
| Taylor Brook [], 1695-1731, English mathematician, dealt with analysis, optics, astronomy, ballistics. Compiled formula for the development of the functions in square series. |
Sounding body (wire, plate, ..) usually has a basic natural frequency of oscillation.
Also, each person has their "own" tone. L.Janáček found still the same tones in speaking his neighbors. A large part folk songs end with the begining tone.
Body vibrate not only as a whole but also its parts. The emerging tones are called partial tones (overtones). If frequencies of the tones are integer multiples of the fundamental frequency f (2f, 3f, ...) we speak about (higher) harmonic constituents, aliquots. Spectrum of sounding tones depends on the shape, dimensions and materials of sounding body. In the case of plates and especially bells, where the partial tones are not multiples of the fundamental frequency, we're talking about non-harmonic components.
Guido z Arreza| Guido z Arreza [], asi 990-1050, Italian musical theorist, a Benedictine monk. Perfected neumatic notation by introducing staves, used organ mixture. |
Composed sound means the fundamental tone and partial tones (arising from one given body).
Artificial colors, e.g. in organ mixture achieves by amplification of selected harmonics.
The tone is composed sound, retaining for a certain period of time constant value of the fundamental frequency. In practical music tones are used with maximum frequency of about 5 kHz (piccolo, violin flageolets). But also higher tones sounds yet (as part of the tone) due to the existence overtones.
The spectrum of harmonics is considered to be the color (timbre) of the tone.
Ohm Georg| Ohm Georg [Óm], 1787-1854, German physicist known especially for his experiments with electricity. Author of electric, magnetic and acoustic law. |
Color is (next to the frequency, amplitude and duration) the fourth parameter of the sounding tone.
According to Ohm's theory - sound quality of tones depends just on amplitudes of aliquots, not on their mutual phases.
Hearing scans (physiologically) all sounds - according to Fourier analysis - to tones of simple (sinusoidal) waveform.
N-sound is the sounding of several tones. For two tones is used term interval, for more tones term chord.
We distinguish paralel sounds one from the other (according to their pitch and color).
Tartini Giuseppe| Tartini Giuseppe , 1692-1770, Italian violin virtuoso, composer, teacher and music theorist. He considered natural septime as a consonant. Noticed of subjective differential tones (1754), e.g. sounding d 1 -f # 1 also sounds D. |
With the current sounding of the two tones with frequencies f 1 and f 2 on one sounding body arise real differential tones with frequencies:
Gain, substitution of basic tone spectrum (fn–fn–1=f0).
- E.g. c g - amplification of tone c (formally).
Deep organ registers - differential tones of two short whistles,, g0c0 àc–1 .
Martenot M.| Martenot M. [], -, French composer and educator. He constructed (1928), an electronic musical instrument using interference between the two frequencies. The tool allow to create separate vibrato - called Marten's waves. |
In addition to the differential tones, other - so called combinatorial tones - arise on the sounding body. Sum combination tone is very weak. It arises as a differential tone of aliquote tone and differential tone (first aliquot - first difference tone):
Summary tone (unlike differential tone) is usually dissonant with regard to played tones.
Parallel sound of the two slightly different tones brings differential tones outside the boundary of hearing and cause fluctuations in the volume of the resulting tone (fake trilll on organ fH fC = 8) or emit unpleasant sound (in the range of c.30-60Hz).
Young T.| Young T. [], 1773-1829, discovered the interference of sound waves, differential tones are faster beats (>40 Hz). |
These are so called beats. Helmholtz - against T.Young - considers beats and differential tones as different (principle of linear superposition). According to his theory beats affect euphony of chords (consonance).
Beats are used for tuning of instruments - tuning is performed until the beats completely disappear. Beats do not arise in trill or tremolo and are not perceptible between instruments of different colors or tones with different intensity.
Sound can cause healing (bz heats, destroys microorganisms) or pain or dead (infrasound can disrupt biological rhythms - alpha rhythm of brain, heart rate, ..)
Hearing evolved only in higher creatures, simple animals do not have it, or it is very weak. Smaller animals hear usually better higher sounds, larger animals deeper sounds. But there are exceptions (dolphins communicate by ultrasonic waves ...).
The human hearing system is considered to be a pressure audio receiver. It consists of a series of sections each of which has its special function. The outer ear captures signals and directs them into the middle ear. As the self auditory organ is considered snail i.e a hollow formation filled with liquid and longitudinally divided by the basal (basic) membrane and the closed oval window. On basal membrane begin (at the point of so called organ of Corti) fine fibers of the auditory nerve, which transmits signals to the brain. Organ of Corti with a number of differently thick and long fibrils which stiffen basal membrane.
The pressure of the sound wave advancing to the inner ear increases. The middle ear reduces the amplitude of the oscillations, but increases acoustic density.
The set of possible frequencies and intensities are bordered by the limits of audibility (frequency 16 Hz to 20 kHz, pressure 2 ∙ 10-5 Pa to 10 Pa). The limits of sensitivity determine which sounds man perceives most sensitively (1 kHz-4 kHz). Limits of pleasantness determine when the sound still does not cause pain (pain threshold of 20 Pa, fatigue limit of hearing c.5 seconds) and which is tolerated by the particular impurity (tolerance of unpleasantness, of dissonance of chord, of vibrato in voice ...)
To distinguish different sensations, they must be differentiated by a minimum value, limit of distinctiveness. Distinctiveness is assessed either psychologically under ideal conditions (in laboratory), or aesthetically listening artistic production. Limits of distinctiveness relates to the transmission of information, which in humans is c. 12-25 bodes i.e bits/second. Eye can discern c.10 to 20 images per second, and c.5 to 20 sounds per second (depending on frequency).
If the sound persists for a short time (it does not complete even one whole period) it is not possible in principle to determine the exact frequency value. (A similar problem occurs in astronomy - how to determine exactly e.g orbital period of the body, which was seen only briefly).
Fourier Jean-Baptiste de| Fourier Jean-Baptiste de 1768-1830, French mathematician who showed how to decompose each complex periodic motion to a number of harmonics. He was engaged by the mathematical physics. |
Instruments having sparse spectrum of particulate tones sounds soft, shallow, hollowly (organ whistles, height tones of the piano). Presence of aliquots stronger than the basic tone (oboe, bassoon, deep strings) makes timbre to be more pronounced than the timbre with weaker aliquots (flute, harp, horn).
The more harmonic components sound contains, the fuller it is (e.g. strings as a whole). Higher harmonic components make sound growing stronger (trumpet, trombone).
Continuous spectrum (drums).
Even - softly, clearly. Odd - hollow, nasal, hard, shrill nasal, moodily (clarinet, covered organ pipes, picked strings in the middle?)
The sound with a decreasing intensity of aliquots sounds fully (principal organ registers).
Timbre is associated also with other phenomena (onset, dynamics, formants, ...). Onset means the order of coming of harmonic tones (from below - clarinet, from above - piano, strings, ..). Stronger dynamics makes weaker harmonic tones audible. Formants prefer selected bands of frequencies. (deeper singing voices have louder aliquots with higher numbers, ...).
Békésy Georg von| Békésy Georg von [], 1899-1972, American physician of Hungarian origin. Nobel Prize winner for study of physiological processes of auditory system. |
According to the Helmholtz resonance theory each tone raises irritation of certain areas on the inner auditory system. It is assumed that sound traps basial membrane. Helmholtz thought that the fibrils in the blanc (with lengths of? 0.04 to 0.5 millimeters) are similar to the resonating strings. He assumed that the fibrils need not to be dimensioned as strings because oscillate in a thick liquid.
Ewald J.R.| Ewald J.R. , -, acoustic, author of the theory of sound images - competitive theory to the Helmholtz resonance theory. Each tone thrills the whole basial membrane wherein the different heights and various sound colors appear as "images" on the membrane. |
G.Bekésy came with another theory. Sound vibrations generate vortex pairs in the liquid, loser or deeper in the cochlea - according to tone pitch. At the appropriate location pressure creates and irritates the correct nerve.
Hearing reacts instantly to very short pulses and record their correct height. Do sharply tuned resonant fiber suffice to fully vibrate?
Sensitive nerves are deployed along the cochlea so lavishly that given sound activates just large number of signals at once. Yet man is able to detect even slight differences in frequencies.
Basial blanche (assuming its complete flexibility) could, oscillate simultaneously at several frequencies and receive different tones, including harmonic components.
| Sorge Georg Andreas [], 1703-1758, German organist, who first noticed the subjective differential tones (1715). He was against the construction of (4-tone)chords from thirds, D 7 is given by the natural chord. |
Certain resonances occurs during the transmission of the sound to the organs of human hearing. This creates additional overtones called subjective aliquots. Also subjective difference tones arise in the organs of hearing during the simultaneous sounding of the two tones.
Tones covers (mask) each other if they frequencies are closer. (The weaker tone is covered by the stronger one) - the maximum is at a certain distance (then beats). E.g. listening to (200Hz, 80dB) we do not hear (800Hz, 70 dB).
Primary, real-sounding tones, remind external forces producing strain in our ears. Secondary tones, i.e our perceptions, are tones that we hear.
The set of secondary tones may not be the same as the set of primary tones (hearing, harmonic bindings, masking, ...) Somewhere in the middle of the journey tones interact, some tones place obstacles in the path of another tones. Then we hear only the result (only tones that have undergone barriers).
Elementary acoustic stimulus (f, A, t), where f is frequency, A amplitude and t time, reflects in our senses like sensation (v, h, d), where v is pitch (height), h loudness (volume) and d duration (length).
Weber, E.H.| Weber, E.H. -, physicist and physiologist |
Physically distinct sounds can be psychologically very similar.
Weber found that a noticeable increase in the perception must gain stimulus proportional to the previous stimulus.
Fechner Theodor Gustav| Fechner Theodor Gustav [], 1801-1887, psychophysics, founder of the experimental aesthetic school.. |
Fechner clarified this - the intensity of our perception grows proportionally to the logarithm of the energy stimulus.
Logarithmic relationship between the intensity of physical stimulus, and intensity of sensation is valid for hearing and also for vision..
In music:
To derive the loudness h from intensity I was by international agreement established equation:
where L = q ln I/I0 ( = q/ln10 log10 I/I0 ) is sound level in dB , q coefficient of sensitivity of the human ear for the given frekvency ( f= 1kHz => q=ln10) ,
I0 threshold intensity. If r = ln I/I0, then H = e0.069qr–2.76.
At the same time loudness slightly increases with the frequency of the pressure change.
We perceive the frequency ratio rij (objective interval) as the difference of pitches hij (subjective interval):
r = f2/ f1 (f2 > f1)
∆V = V2– V1 (V2 > V1) log f2/f1 = log f2-log f1
Révész Géza| Révész Géza, -, music theorist, author so called two-component theory to explain the phenomenon of the octave. |
A special case of interaction is the resonance ratio of 2:1, ie. octave identity. It goes back to the very basics of music systems - tones with frequencies in the ratio 2: 1 tones are perceived as one class and named with the same letters.
Tone system, which thus arises is called formal system. In the formal system are all tones in terms of frequency invariant to 2n. Interaction simplifies, though with some distortion.
Octave identity and tone identity at all is the strongest and most distinctive harmonic relationships. And as the strongest binding in the music it takes a different meaning than other bindings.
Handschin Jacques| Handschin Jacques [], -, first strong criticism of the two-component theory (1946). |
In the color of light - similarly as in case of the sound - distinguishes height (quality, tone) according to the frequency and the intensity (brightness) according to the intensity of lighting. In addition, so called saturation (purity) can be meassured in the light,\ depending whether it is mixed with another color.
An analogy of saturation of a certain color by white color is brightening of tones towards higher octaves in music. According G.Révész the pitch is a complex of two independent factors:
tone kvality (znělosti, tónovosti) C,D,… kvantity světlosti C,c,c1,…
In analogy with the light are then e.g all D tones of yellow, while d is dark and d 2 pale yellow. With this idea work alse E.M.Hornostel, M. Kolinsky, C.Stumpf and others.
Fade out of piano - higher tones 1 s, basses 50 s. More massive string sound longer (sound is fuller) - modern pianos have thicker and more tense strings. Long strings are single -to not give a strong sound, the shorter strings are duplicated, triplicated. Instruments: plucked - less damping to sound longer; strings - greater damping - to make changes possible. The tone sounds after its completion still 0.1 s in 1/10 of primordial force.
Damped oscillations (plucked instruments, piano-hammer blows) intensity gradually decreases (resistance of environment, poor flexibility of string).
Undamped oscillations (string and wind instruments) almost equal intensity throughout duration of the sound (constant power supply).
Stumpf Carl| Stumpf Carl , 1848-1936, German musicologist and psychologist, he evaluated merging of tones statistically and proved that formants form a whole bands of frequencies. |
Aristotle believed that the deep tones are spreading faster than high, however, it has been experimentally denied. But deep tones tend to have greater tendency to be root. The cause of this could be overtone serie, but there are also other possible causes. Josef Micka has written in the book Hudební dynamika (1943):
>> ... deep tones tend to strong dynamics, high tones to weak dynamics. One should seek the cause in the acoustic ratios. Dynamics depend on the height of the amplitude (proportional to the square of the amplitude). Deep tones have a smaller number of oscillations per second, needing therefore substantially less energy than high tones ... When the same amount of energy acts on the one hand during the creation of a high tone and on the other hand for a deep tone, then the excess quantity, that results by savings of energy from the difference between the number of oscillations may express only towards increasing of the amplitude: deeper tone is stronger. .. We can therefore say, that the deeper tone is, the more capable it is to dynamic gradations. <<
This argument seems convincing. But strings that are used for the lower tones are also thicker and heavier. Therefore, we use for their sound a little more energy (thus gaining greater inertia i.e. momentum).
Momentum is physically defined as the product of mass and velocity. If one attributes the lower and louder tones greater "weight", then their movement will get greater momentum. Let us make such an attempt.
Let mj represents "mass of tone":
where Ei is energy and fi frekvency.
If we define the interaction of tones (impulse) by relationship:
i=m1∙m2/s, where mi= E/fi,
pak i = (E/f1∙E/f2)/s = E²/f1f2s,
which could explain better audibility of the same pitch differences at higher altitudes than at lower ones. And when we denote the rhythmic motility "v" (speed of music changes) Then we can define the momentum of tonal changes, like in physics:
We endure rather slightly bad tuning than unison,
but the higher tone is, the smaller must be the bandwidth
(with a small number of instruments - in chamber music - it is better
to avoid unison entirely, particularly in the bass position).
| Eukl(e)ides z Alexandrie , 365-300 př.n.l, Greek mathematician, physicist, and music theorist. He tried to derive all geometric theorems by deductive procedure from several basic axioms and definitions. His book 'Fundamentals', became one of the most read books of all time. |
When the chord works nicely (is pleasant, melodious, calm, soft ..) we talk about "consonance". Otherwise (cacophony, tension, disparity, sharpness, difficult perception, ..) we says the chord is "dissonance".
|
d |
Interval |
N1/n2 |
Primes |
∑ |
∏ |
|
0 |
Unison, octave |
1/1 |
- |
2 |
1 |
|
0 |
Octave |
2/1 |
2 |
3 |
2 |
|
7 |
Fifth |
3/2 |
2,3 |
5 |
6 |
|
5 |
Quart |
4/3 |
2²,3 |
7 |
12 |
|
4 |
Major third |
5/4 |
2²,5 |
9 |
20 |
|
8 |
Minor sixth |
8/5 |
2³,5 |
13 |
40 |
|
9 |
Major sixth |
5/3 |
3,5 |
8 |
15 |
|
3 |
Minor third |
6/5 |
2,3,5 |
11 |
30 |
Two consonant tones are - according to Euclid - able to join together in one unit, of which is considered - they belong to each other. This is not possible with dissonant tones.
Known music systems usually include ratios 2:1 (octave) and 3:2 (fifth).
Ratios 1/1, 2/1, 3/2, 4/3 (derived from primes 2 and 3)
were used since ancient times and still prevailed at the time of early polyphony
i.e in 9 to 11 century. Pythagoras and Aristotle considered third (6/5, 5/4) as dissonant.
Thirds and sixth started to appear in music up in the 12th century and, theoretically, they have been recognized in the 18th century (J.P.Rameau).
We observe (continuous?) shift of the boundary between consonance and dissonance.
Question consonance and dissonance has not yet been definitively solved. Also other factors may be the cause of dissonance, e.g.: differential tones (Hindemith), subjective aliquotes (Husman), division of frequencies after transmission to the central nervous system (Hornbostel).
All mathematical theories that try to evaluate sonance of chords, seek to express simple numerical ratios.
In astronomy, the simplicity resonance is assessed according to sum (∑) of quotients (from simple ratio it is the sum of the numerator and denominator of the fraction), e.g. 5/4 has value ∑ equal to 9. Similarly it is possible to multiply the individual coefficients (in abbreviated form), product ∏ is so called coefficient consonance, in our example 5.4 = 20, see table.
|
Interval |
Fraction |
2i |
3j |
5k |
ψ |
G |
|
Prime |
1/1 |
< |
1 |
|||
|
Octave |
2/1 |
21 |
2 |
|||
|
Quint |
3/2 |
2–1 |
31 |
3 |
||
|
Quart |
4/3 |
2² |
3–1 |
3 |
||
|
Major third |
5/4 |
2–2 |
51 |
5 |
||
|
Minor sext |
8/5 |
2³ |
5–1 |
5 |
||
|
Major sext |
5/3 |
3–1 |
51 |
15 |
||
|
Minor third |
6/5 |
21 |
31 |
5–1 |
15 |
| Euler, Leonard [oiler], 1707-1783, Swiss mathematician and physicist, one of the greatest mathematicians of all time. He entered into all areas of mathematics, often by a quite original way. He also tried to establish music theory on mathematical relationships. |
Music seems to differ from the mechanics and astronomy, since ratios 1: 2n are favored in some way. Minor sixth 8/5 has consonance factor: 5∙8=40, while major sixth 5/3: 5∙3=15. So major sixth (formally the same as the minor third) would be then more consonant then minor sixth (formally major third). But this - it seems - does not reflect reality. The ratios currently regarded as consonant satisfy relationship: 2c ∙ p/q, where p,q ε {1,3,5}, c ε {–2,–1,0,1,2,3}. Complete unification of all octave positions can be achieved by removing coefficients 2n from Π, we talk about formal characteristic ψ:
L. Euler proposed for ratio rεQ (rational numbers) with decomposition r = ∏ pj: to product of primes pj, with exponents ajεZ (whole numbers) degree of consonance ("Gradus suavitatis"):
Euler assumes octave identity only for the nearest tones, according to his relationship one octave (21) does not affect the result of the calculations (1∙(2–1)=1) but the interval 4/1 is more dissonant than 2/1.
Now we will try to generalize coefficient consonance and formal characteristic for multi-tone chords. For chord with frequency ratios: f1/f2/f3,... = n1/n2/n3,..., where n1,n2,...ε N (small integers) and let us define the value ∏ as least common multiple of numbers ni:
|
Chord |
n1/n2/n3 |
∏=nsn(n1,n2,n3) |
ψ |
|
Major [c,e,g] |
2/3/5 |
30 |
15 |
|
Minor [c,eb,g] |
6/10/15 |
30 |
15 |
|
Augmented [c,e,g#] |
1/5/25 |
25 |
25 |
Augmented triad has better coefficient of consonance, but not formal characteristic.
Mathematical theories were the subject of a number of critics, but not always justifiably. The objection that disonance increases significantly by a slight change of tones fails, because it does not consider tolerance of human hearing, on which is (successfully) based tempered tuning.
The objection - that empty septime is more disonant than the septime filled with thirds - ignore (psychological) spreading of attention (and thus also dizonance) to all existing bindings. (Septime contains 1 binding, tetrad 6 bindings, behind which the dizonance of the septime hides).
D'Alembert, Jean-Baptiste| D'Alembert, Jean-Baptiste Le Rond [dalambér], 1717-1783, French mathematician, physicist, scientist and enlightenment philosopher. By his treatise on the dynamics he get an European fame. |
For chord cegc and cegh comes the same degree of dissonance
(cegc = 4:5:6:8 = 120 units cegh = 8:10:12:15 = 120 units).
The objection does not work with octave identity and with all the ratios among the chord tones..
Chords cegc and cegh have the same resonant characteristik
r(120) = r(2³∙3∙5) = 15,
because
n(c,e,g,c) = nsn(4,5,6,8) = 120 a n(c,e,g,h) = nsn(8,10,12,15) = 120.
Interval h-c does not resonate (it has impulse characteristic) and causes (perceived) disonance of chord cegh. Consonance of chords can not be derived only from the resonant conditions.
| Helmholtz Hermann , 1821-1894, German physicist, physiologist and mathematician, expanded and mathematically formulated the law of conservation of energy. He placed the physiological basics of music theory. He derived results of musical sensations from the particle tones and explained chords interference by beats. By his discoveries he significantly influenced the further development of science. Measured the speed of propagation of impulses in nerve tissue, developed a theory of color vision. He studied the hydrodynamic analogy of electricity and magnetism. |
Helmholtz theory builds on the assumption that sonance is determined by beats occurring between aliquots (he concider 1 to 6 aliquote).
The presumption is confirmed by the well-known fact, that the combination of some instruments sound soften - by using of different colors of tones beats between aliquots do not arise.
Helmholtz considers critical band for beats with frequencies 7-276 Hz, and these are most uncomfortable in the area c. 33 Hz. Such beats arise e.g between
(To eliminate beats vote in their respective octaves greater intervals.)
Deriving: c1 h1 232.5 Hz and pleasantness? No, but the inconvenience caused by aliquote c2 ( c1 b1 204.5Hz x b1 c2 57.0 Hz dtto).
In close proximity (see Euler) of the consonant intervals are located bands with a great degree of tension (halftone, tritonus, neutral third?)
Objections to the Helmholtz theory: from the beats between aliquotes results minor sixth and quart as dissonant. The same dissonant interval (e.g cd in two octaves) has an octave above twice the number of beats. Helmholtz theory does not work with octave identity..
Musical intervals comes from ratios of frequencies. It is therefore somewhat strange to base the determination of dissonance on the differences of frequencies!
From the viewpoint of music a semitone remains semitone, although we push the two tones up or down.
Disonance value could be rather a function of the shape mi∙mj/g(fi/fj), where g() is some function-e.g.logaritmic, and would be defined similarly as impulse.
We distinguish - according K.Risinger - consonant and dissonant intervals also with tones without aliquots.
Each tone is part of a series, and the source of his own series of harmonics. According to the distribution of intensity of harmonics, density, ... Chord + differential tones (= meta-chord).
Degree of consonance and dissonance is affected by sound color. Even single tone can be felt as dissonant. (Musical intervals that appear in the bird singing, is often difficult to recognize. This is due to the richness and variability of tone colors.
The opposite of consonance and dissonance acquires the meaning of purity and impurity and we talk about acoustic consonance and dissonance. The aim is to avoid acoustic dissonances (impurity), that is studied by the science of instrumentation. Already H.Riemann distinguished acoustic and musical consonances.
Hindemith Paul| Hindemith Paul , 1895-1963, German composer and theorist (since 1939 in the US) important representative of 20th century music. He defined series of relatedness of tones. When i-th aliquote of tone (e.g.c) is identical to the j-th aliquote of the another tone (e.g.x), then c and x are of relatedness (i, j). The basic tone of the chord is the tone, for which the number of identical aliquotes with all the other tones is supreme. |
Consonance is related to arrangement of the chord, it is given by a certain distribution of energy. This layout affects the resonances between the tones generated due to simple numerical ratios. In connection to disorderliness whose rate is entropy, dissonance is associated to greater information content.
Dissonances carry more information (greater information content).
Dissonance may depend also on the arrangement of tones. Consistency (consonance) of multiple frequencies is not aimed directly by ratios of small whole numbers. Simple ratios affect transmission of energy (continuity). Rate of consistency is then a rate of a certain arrangement of energy.