Studies in Music

Aesthetics - Ars Inveniendi - the Farben Project

by Wim van den Dungen

Introduction

Neurophilosophy of Sensation

Aesthetics of Music

Ars Inveniendi

The Farben Project

Compositions

Biblio

OMNIA STANT HARMONIA. Anarmonia cadunt omnia. Nec erigitur, reficitur, restituitur quidquam, nisi ad Harmoniam relatum atque redactum.

All exists through Harmony. Disharmony destroys all. Nothing can be built, nothing created, nothing restored that is not related to and based on Harmony.

Tout existe à travers l'Harmonie. La disharmonie détruit tout. Rien ne peut être construit, ni créé, ni restauré qui n'est pas lié à et basé sur l'Harmonie.

Georg Philipp Telemann (1681 - 1767)

The Farben Project

retuning to Nature & scalar coloration
music & psycho-acoustics

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


Introduction & 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 to add color (timbre) and therefore alternative texture to the melodic line. This technique derives from Arnold Schoenberg and his "timbre-structures" and was also used by Anton Webern.

With the "Farben Project", my first aim is to integrate color into music and this by three methods : (a) aligning standard pitch with natural process, (b) relating the chromatic scale (and on the basis of this also the harmonic series of each tone) to the visible spectrum of light and (c) allowing scalar coloration, or the return of "key characteristics" to the 24 keys of the tonal system of harmony (by reintroducing pure intervals). Secondly, on the basis of this naturalization and reintroducion of scalar coloration, define a psycho-acoustic model, in other words, establish guidelines allowing music to entrain the minds of its listeners (AVS) in the direction of a life more in harmony with the beauty of the natural order.


Music, derived from the Greek "mousike" or "art of the Muses", is an artistic phenomenon whose medium is well-formed sound in combination with its absence (silence). This definition of music begs the question : "What is art ?", a topic involving a complex network of concepts studied in Aesthetics (2013).

Three ways to define "music" can be identified : (1) intrinsic, (2) (inter)subjective and (3) intentional.

• Intrinsic, objectivist definitions try to pinpoint the objective aural properties of musical sound as opposed to mere noise, for example specific frequencies. But if we want to include "out-of-tune" sound-sources (like machines), different Western & non-Western musical (microtonal) scales, "untuned" percussion and works lacking basic "musical" features (such as pitch or rhythm) -like Yoko Ono's Toilet Piece/Unknown- then the intrinsic definition becomes too extended, for indeed both non-musical noise and musical sounds have frequency. However, it cannot be denied "music" has certain "musical" properties, of which frequency (pitch) and temporal pulse (rhythm) are the most basic, and this together with dynamics (loudness, intensity), timbre (tone-color), verticality (harmony), horizontality (melody, counterpoint) and format.

• (Inter)subjective definitions identify music, regardless of intrinsic properties, as whatever sounds like or is perceived by a listener or a group of listeners as music. Can one say Beethoven's Fifth Symphony stops being music for listeners ignorant of Western culture ? Can I reasonably deny Rembrandt's The Night Watch to be a painting ? Can I miraculously transform the noise of a drilling machine into music by merely hearing them in a certain way ? And what about music emitted by a radio nobody is attending ? Does this stops being music (while it remains music for the neighbour next door) ? Clearly all of this is counterintuitive. Too much subjectivism (like exclusive objectivism) leads to problems.

• Intentional definitions focus on music-making intentions of those responsible for the noises or sounds. Not the listener is important, but the intent of the human sound-source. Intentional action has been defined as "the execution and realization of a plan, where the agent effectively follows and is guided by the plan in performing actions which, in manifesting sufficient levels of skill and control, bring about the intented (i.e. planned) outcome." (Livingston, 2005, p.14). This definition helps to include works of silence, like John Cage's work 4'33". So-called music produced by machines (CD players, iPods, etc.) is rooted in the intentions of the musicians behind the sounds. Even computer-generated scores are designed by someone, even if the outcome is unpredictable. The same goes for aleatoric music (or passages) and improvisations. So these are covered by the definition. Sounds generated by animals however, although they sound like music, lack the capacity for complex intentions and so do not improvise or invent new melodies. These sounds fall outside the definition. They are music-like without actually being music.


In the definition of what is music, both intrinsic & intentional approaches will be combined. From the intrinsic strategy we retain the presence of objective musical features. This begs the question : What are objective musical features ? From the intentional approach, the importance of a human source for acoustic events is adopted. An extended & strict form of the definition is proposed. In sensu lato, all types of sound (combined with silence) are allowed, whereas in sensu stricto, only well-formed sounds are taken into account.

• Extended definition of music :

Music is any event intentionally produced to be heard, either to possess intrinsic musical features or to be listened to for such features.

By saying "any event intended to be heard", the definition includes both sound & silence. If we want to include machine-noise etc. intended to be part of a musical work (like Honegger's Pacific 231), then the musical event is not limited to well-formed sound. The disjunction in the definition allows one to call works like Toilet Piece/Unknown "musical" for indeed the toilet is flushed with the intent to discover musical features, even if such features are objectively lacking (but projected upon the acoustic phenomenon by intent). According to this definition, some sonic works of art are not musical in any way (cf. John Cage's Williams Mix for eight simultaneously played independent quarter-inch magnetic tapes). Hence, not all sound art is music !

• Strict definition of music :

Music is a well-formed sound (combined with silence), intentionally produced to be heard.

The restriction here is the condition of well-formedness, excluding all sonic works of art lacking all or some intrinsic musical features such as pitch, rhythm, dynamics, timbre, melody, harmony & format. Unpitched instruments form an exception, and may be included, but only in combination with pitched instruments. A sonic artwork exclusively generated by unpitched instruments is excluded. So is Toilet Piece/Unknown, despite Yoko Ono's intent, as well as Cage's 4'33". This definition includes Western & non-Western sonic works making use of microtonality (Indian Râgas, works of Harry Partch). If the concept of well-formedness is stretched to include pitched or unpitched noise (like that produced by machines, trains, traffic etc.), then music is at hand if and only if this noise is combined with less random (read : more traditional) sources of sound (like musical instruments). In this case, the overall balance between noise & sound needs to be checked in order to know whether music is at hand or not.


§ 1 Musical Sound

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.

Everything around us, our body included, vibrates in a certain pattern and transfers these vibrations to the surrounding medium. When these vibrations disturb the medium in a periodic way, a portion is repeated over and over again, i.e. with repetitions at equal time intervals. One cycle of this periodic motion is called a wave, characterized by a high point (crest) and a low point (trough).

Sound is a longitudinal wave, i.e. an oscillation of pressure transmitted through a medium (like air or water) causing the particles in the medium to vibrate parallel to the direction in which the wave is travelling, and composed of frequencies within the range of hearing. The speed of sound is proportional to the square root of the ratio of the stiffness of the medium to its density. In dry air at 20° C (68 °F), the speed of sound is 343.2 metres per second (1.126 ft/s), or about a kilometre in three seconds (a mile in five seconds). The moment musical is a single temporal unit not exceeding three seconds, the time span of immediate consciousness. This is the specious present, the time duration wherein one's perceptions are considered to be in the present. In this moment, earlier may be distinguished from what is later, but both past & future are directly given, simultaneously present to consciousness. So the "present", events perceived in the first time, is not merely a non-existent gap between before and after.

When an object vibrates, waves of increased and decreased pressure in the air are caused. It is not the case of a particular molecule of air moving in the direction of the wave at this speed. Rather, the local disturbance to the pressure propagates at this speed. This is similar to water. No particular piece of water moves along, but the disturbance of the surface propagates. However, in the latter case the local movements are up and down, and so at right angles to the direction of propagation (transverse waves). Sound moves in the same direction as the propagation (longitudinal waves).

A sound wave travels through the air, hits the eardrum and allows one to hear the sound. Human hearing is normally limited to frequencies between about 20 Hz and 20.000 Hz. Compare this with a turtle (20 - 1000 Hz), a dog (50 - 45.000 Hz), a cat (30 - 50.000 Hz) and a dolphin (1000 - 130.000 Hz). The human ear can distinguish a difference of 1 - 2 Hz.

Sound waves (and waves in general) are characterized by frequency, amplitude, wavelength & period. The frequency (f) is the number of waves passing a given point per second and measured in Hertz (Hz), with 1 Hz = 1 vibration/second.  The amplitude (A) of a wave is the maximum displacement from the equilibrium position (the original position of the sounding body). The wavelength (λ) is the distance between any point on the wave and the corresponding point on the next one (or the distance the wave travels in one cycle). The period (T) is the time it takes for one whole cycle to pass a given point. Period & frequency are reciprocals, with T = 1/f.

A sonic environment is composed of many different sound waves emitted by a variety of sound sources. Interacting, they usually produce irregular sound-ripples, causing a cacophony. This mixture of sounds is discordant & harsh. It is generated by vibrations producing irregular sound-ripples. These complex pressure ripples have no or little relationship to each other and therefore clash. Instead of co-operating, as in a team, they as it were "beat each other up", increasing the irregularity of their interactions. Irregular sound waves are never well-formed. Hence they are called "noise".

sound waves in a sonic environment regular ripples musical sound uncommon cultural
irregular ripples noise common natural, cultural

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 a small subset of all possible sounds. And to generate these musical sound waves special tools are necessary. Although music imitates (cf. the Harmonic Series) fundamental natural phenomena (like φ), these sound waves are always the symptom of a degree of culture accommodating the production of cultural forms, in casu, musical works of art, the first of which were very likely composed ca.30.000 BCE (date of the oldest instruments - Davies, 2014, p.70).

So music can be understood as a finite number of musical sounds generating regular sound-ripples interacting & working together to produce an organized and hence well-formed sound. The objective musical features or attributes are therefore certain phenomena contributing to a musical sensation as opposed to other aural sensations, such as sonic works of art using noise & various degrees of noise.

The history of music around the globe evidences seven musical features denoting music in objective terms : pitch, rhythm, dynamics, timbre, horizontality (melody, counterpoint), verticality (harmony) and format (musical architecture, overall layout or plan of a piece). Together, these seven intrinsic features define music as an objective, sounding form (cf. Hanslick's "tönend bewegte Formen"). This as opposed to noise, or unmusical sound.

Of these seven objective features, two are fundamental : pitch & rhythm. Pitch is fundamental, for it is the qualitative attribute of auditory sensation denoting highness or lowness primarily conditioned on the frequency of sound waves. We do not hear this rate of vibration of musical sound waves, but only a "musical tone". Pitch determines spatial organization. Next comes temporal motility, defining the durations of pitches, known in music as rhythm (tempo, time). With both, the foundational spatio-temporal sphere of music is set. Loudness speaks of the intensity of certain sound waves relative to others, whereas timbre is a function of the instruments generating the musical sound. The horizontal, vertical and integral features are meta-properties based on the distribution of pitch, rhythm, dynamics & timbre. Format is called "integral" because it involves the organized layout of all the musical events constituting a work of music.

Well-formed sound (and its absence, silence) are the medium of music. The musical phenomenon is an object of human valuation fulfilling specific determined aims. Its conscious perception (acoustic & kinesthetic) -as well as music's unconscious impact on the whole body through resonance- leads towards the elaboration of conjectured meanings, in casu : what is a well-formed sound ? Music is an intentional phenomenon. It is the object of human valuation because it belongs to the interests of an individual, a group or humanity as a whole to define what is music and what it is not.

Let us focus on pitch, the most objective of the intrinsic properties of music.


§ 2 The Tone

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 a pitched musical sound, i.e. a specific steady periodic oscillation called "tone". As a note, G always calls for a functional discussion about its relationship with other notes. The note G is the tonic in the key of G or g, the mediant in E or e and the dominant in C. In each case, for the same G, another functional relationship with the other notes of the scale is given. The concept "note" is therefore contextual. G as a tone is just a matter of frequency, and this independent of context.

Pitch, rhythm, loudness & timbre are the fundamental dimensions of the single sounding tone. They answer the four questions denoting the primary features of its sound wave, related to its physical existence :

(1) spatial : how high or low is the frequency ?
(2) temporal : how long is the frequency operational ?
(3) intensity : how strong, how much energy carries the frequency ?
(4) timbre : which instrument generates the frequency ?

The common element is frequency, the number of waves passing a given point per second (in Hz). When tones are combined, the meta-dimensions come in : harmony, counterpoint & format. Tones may sound simultaneously, generating a vertical relationship (harmony), or one after the other, inviting a horizontal dialogue (melody and by extension counterpoint). Finally, format defines layout and plan. This is the last dimension : architecture. All together they define an actual musical form existing in the moment musical.

In musical practice, secondary frequency features are also present, such as attack transients (high amplitude, short-duration sound at the beginning of the wave), vibrato (a musical effect consisting of a regular, pulsating change of pitch), envelope modulation (the so-called ADSR-factor, or the "Attack Decay Sustain Release" specifics of an instrument) and reverb (impact of performance space on the sound). They are used with great care & moderation.

Sounding a single tone is not yet music, but merely the most elementary building-stone of music, a well-formed sound generated with vibrations collaborating to produce a regularly repeating sound-ripple, i.e. vibrations strongly related to each other and joining together in an organized way. The most forward way to generate a single tone is making a single string (monochord) vibrate or a column of air do the same.

All musical tones, besides electronically generated "pure" tones, are complex, i.e. formed by adding an infinite number of component waves, each having a different frequency. This infinite set of frequencies is divided into two : the fundamental frequency and the others. The fundamental is the dominant frequency, the one best heard and somewhat overpowering the others. These are always integer multiples of the dominant frequency.

A musical tone consists of a fundamental frequency component plus other frequency components (in "pure" tones, only the fundamental frequency is present). These frequency components (fundamental + other frequency components) are called "partials" (or "parts" of the total, infinite spectrum of frequencies constituting a musical tone). The xth partial of a tone is the xth frequency component counted from the bottom. So the first partial, the one with the smallest frequency, is the fundamental frequency or fundamental (f) of the tone. All frequency components besides the fundamental frequency (first partial) are "overtones" ; they never include the fundamental frequency.

As the frequencies of the overtones of a tone are integer multiples of its fundamental frequency (2f, 3f, 4f, ...), order & balance is evidenced and so all partials (all frequency components) are called "harmonics" ; the fundamental frequency being the first harmonic. The component with frequency xf is called the xth harmonic or (x - 1)th overtone. With x = 1, we obtain the first harmonic (= first partial), i.e. the fundamental. There is no overtone (x = 1 - 1 = 0). The first overtone is the second harmonic or the second partial. With x = 3, we obtain the 3th harmonic or second overtone. Simply put : overtones = harmonic - 1 and so harmonics = overtones + 1. The series of harmonics, from the first harmonic up to the xth harmonic is called the Harmonic Series. In theory, an infinite number of harmonics exist, but in practice the first sixteen harmonics are considered, of which the first six are crucial to tone-formation and vertical (simultaneous) interactions between tones (harmony).

When the totality of the frequency spectrum of a tone is at hand the term "harmonics" will be used (as in acoustics), but when the dominant influence of the fundamental frequency is considered, "overtones" are indicated (as in sound engineering).

From the physical point of view, musical sounds are "musical" because these pitches have regular ripple-patterns. This regularity is obvious, for all frequency components besides the fundamental component are integer multiples of the latter. Of all the frequency components of a tone, the fundamental frequency, first harmonic or first partial is the dominant frequency. From the informational & sentient perspective, "music" also operates the other six dimensions besides pitch, namely duration, tone quality (sonance), verticality, horizontality and integrality. Let it be clear sound waves with irregular ripple-patterns or noise lack this harmonic feature. Noise is inharmonic. All the frequencies of a single musical tone or pitch are in "harmony", i.e. work together in a unity promoting difference. The fundamental frequency defines the foundation of a tone, but also contains overtones (the fourth & the fifth) generating the basic structure of the triadic relationships at work in verticality (tonal harmony). The Harmonic Series of C contains E & G, forming the tonic triad of C Major.

Harmonic Series of C (with offset in cents, 1 octave = 1200 cents)

When the frequencies of the partials are not integer multiples of the fundamental frequency, an inharmonic is at hand. When a violin plays a tone, the vibrating string is made up of its individual harmonics coming together as "one". But when the string is plucked or struck, it exhibits inharmonicity. At that moment, the sound waves do not blend together, and small irregular ripple-patterns occur (often of very short duration). In the strict definition of music, inharmonic tones are not the rule. Music is primarily based on tones with a well-balanced Harmonic Series (a set of blending harmonics). The "common use" of noise in Western music may punctuate rhythm or be called in to put in sharp contrast, loud accentuation or climax. But such inharmonicity, in terms of the strict definition of music, is always used sparingly. In the extended definition, inharmonicity may be a prime component of music, or be part of elaborated & recurrent aleatoric devices ...

Two tones with the same fundamental frequency are in unison and totally blend together, producing a fusion with no interference between them. In general terms, when two sound waves interfere, they sometimes combine, making a louder sound, and sometimes partially or fully cancel each other out, producing a softer sound. This pulsating pattern of louds & softs is called beats. For example, given sound wave X has a pitch of 50 Hz and sound wave X' a pitch of 55 Hz, then 5 beats per second are heard. Now when the pitch of one of two tones of the same fundamental frequency is slightly increased, then the number of beats heard per second also increases, creating discord. Maximum discord or dissonance is generated when the beat frequency (the difference between the two fundamental frequencies) is about 25 or 30 Hz. Beyond that, this decreases, to finally disappear. 

When one tone has a fundamental frequency double that of another tone, they form an octave and no beats are heard either. Unison & octave are the two universally recognized fundamental consonant intervals between two tones. Defining other tone-relationships than unison & octave is answering the question of how to divide the octave. This is another crucial initial distinction.

"Accordingly, in all known tonal systems, the basic basic scale-patterns, with few exceptions, fill in the space between two tones an octave apart." - Hindemith (1942, Book 1, p.15).


§ 3 Naturalizing Master Pitch

Is there an absolute way to know whether two tones played on a single instrument or two tones played by several instruments simultaneously are "in tune" relative to one another ? To answer this, we need to know the fundamental frequencies of the tones in question. This depends on the frequencies attributed to them and this is necessarily defined by a standard. In terms of notation, this means we need to know which frequencies to attribute to which note.

Put technically, and by consensus, we need to know the fundamental frequency of A above middle C, the so-called "master tuning" or "standard pitch". This is the frequency all instruments are set to (sound this A). This frequency is usually "locked" in a tuning fork, a relatively stable tuning device invented by John Shore in 1711, and used to tune instruments before Hertz came along (tuning forks aim to let the fundamental frequency ring through at the expense of the other partials). The fact these forks survive, allow us to reconstruct these historical master tunings.  In the orchestra, the oboe is used to "sound the A" because this instrument has fixed, factory-tuned tones.

At a conference held in May 1939, the British Standards Institute endorsed A4 above middle C (C4) to correspond to 440 Hz. One has only to recall history to realize this was a conventional decision. In Germany, prior to 1600, organ pitch is thought to have varied from A at 567 Hz for the first simple pipe organs of the Middle Ages, to A at 377 Hz for the early modern German organ around 1511. Handel (1751) favoured A = 422,5 Hz, Mozart (1780) A = 421 Hz (Mendel, 1968). Scientific pitch, also known as "philosophical pitch" or "Sauveur pitch", with A = 430,54 Hz, was first proposed in 1713 by French physicist Joseph Sauveur. Pleyel's piano's (1836) were at A = 446, Giuseppe Verdi considered A = 432 Hz to be better. In 1859, the French Government made the "diapason normal" or A = 435 Hz law, whereas in 1896, Britain adopted "philharmonic pitch" or  A = 439 Hz. In 1925, the American music industry adopted A = 440 Hz. In November 1955 and in January 1975, the International Organization for Standardization reaffirmed A = 440 Hz. In 1989, over a dozen of opera singers -including Placido Domingo & Luciano Pavarotti- added their names to a petition before the Italian government, asking them to lower the standard pitch from A = 440 Hz to A = 432 Hz. In their view, A = 440 Hz is one of the main reasons for the crisis in singing, giving rise to "hybrid voices" unable to perform the repertoire assigned to them ... Apparently, the issue of standard pitch is a fundamental discussion in music.

The objective solution to this problem proposed here is derived from the structure & function of the human ear, in particular the φ shape of the outer ear and the cochlea (inner ear), as well as the function of the latter (cf. φ dampening). The Golden Ratio φ = (1+√5/2) ≈ 1,618033988749894848204586834 ... is part of so many natural processes it has been called the "Divine proportion". Indeed, the ratio is expressed in the arrangement of branches along the stems of plants, in the veins in leaves, in the skeletons of animals, in the proportions of chemical compounds, in the geometry of crystals, in the magnetic resonance of spins in cobalt niobate crystals, in the human genome DNA, etc.  In Zeising's Law (1854), the Golden Ratio permeates all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical and finds it full realization in the human form. φ is also closely related to the Fibonacci numbers (Fn), Fibonacci series or Fibonacci sequence. These are the numbers in the following integer sequence Fn : 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... defined by Fn = F(n-1) + F (n-2), whereas Fn = φn-(-φ)-n/√5.

In the inner ear, the purpose of the φ-shaped cochlea is to separate out sound into various frequency components before passing it unto the nerves. This makes the functioning of this body of great interest in terms of the harmonic content of a single note in particular, as well as music in general. Can it therefore be mere coincidence E3 = 161,817 Hz (or φ) when A4 = 432 Hz ? Clearly not. No other Master Tuning generates φ.

Because A = 432 Hz, or "Verdi Pitch", aligns with φ, this Master Tuning may be considered an objectivation of standard pitch on the basis of acoustic law in general and the mechanism of human frequency processing in particular. These are natural phenomena, not cultural. Master Tuning A = 432 Hz is therefore the Natural Master Tuning. When the objective & subjective parameters of acoustics are not followed (moving away from A = 432 Hz), we interfere with these natural lines of communication between sound-source & sound-receptor, thus placing additional stress on the sense of hearing, and this for reasons beyond those related to the optimalization of Master Tuning in terms of musical sound. Tones are always harmonics and our ear differentiates frequencies on the basis of φ. As we humans are capable of hearing a difference of 1 to 2 Hz, one cannot claim this matter futile.

Given the Harmonic Series (on the side of the object, the sound-source) and the hearing apparatus (on the side of the subject, the sound-receiver, the hearer) are defined by φ, it stands to reason music should best adopt Verdi Pitch and not some arbitrary standard facilitating an industry. Given most, if not all, music today is tuned to A = 440 Hz, we may digitally retune (using quality software like Melodyne Editor). The fact top singers prefer A = 432 Hz should not be considered whimisical.


§ 4 Color Spectrum and Pitch

"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).

As the Greeks (following Pythagoras & Plato) considered all phenomena to be ruled by numbers, the correspondence between (regular) musical sound (defined by fractions) and the regular movement of the planets seemed self-evident. Today, this notion of "music of the spheres" (Musica Universalis), even cherished by Kepler, may be used as a heuristic tool to help us discover significant (in a material, information, sentient sense) unexpected correlations between natural phenomena, in casu, between visual elements, such as colors, and aural phenomena, such as tones.

Can pitch be related to color ?

To extend the Pythagorean idea of the "Harmony of the Spheres", encompassing planets, tones and colors, Plato linked the intervals of a major second and a perfect fifth to yellow and the perfect fourth to red. Aristotle also suggested a parallel between the harmony of colors and the harmony of musical intervals. To directly attribute color to notes, various systems have come into existence, either based on an objective measure (Newton in his Opticks, 1704) or on the subjective experience of color. The latter tradition is based on Goethe (Zur Farbenlehre, 1810).

"Goethe's theory of the origin of the spectrum isn't a theory of its origin that has proved unsatisfactory ; it is really not a theory at all. Nothing can be predicted by means of it. It is, rather, a vague schematic outline, of the sort we find in James's psychology. There is no experimentum crucis for Goethe's theory of color." - Wittgenstein, L. : Bemerkungen über die Farben, 1950.

Newton's Color Circle (Opticks - 1704)

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.

The question before us : can the notes of the chromatic scale, each representing a specific tone and therefore frequency, be correlated with the spectral colors ?

An intriguing numerical correspondence between the pitch or frequency of the notes of the chromatic scale & the colors of the visible light spectrum can indeed be observed. This is the basis of the objective measure proposed here.

The visible light spectrum is a section of the electromagnetic radiation spectrum visible to the human eye. It ranges from ca.380 nm (3,8 x 10-7m) to ca.750 nm (7,5 x 10-7m). The wavelength of light or λ, measured in nanometers (nm) or Ångstroms (1 nm = 10 Å) is what determines the perceived color. The frequency of light (f) defines the number of waves occurring in a given amount of time. Hence, waves become shorter as the frequency becomes higher, or : λ = c/f, with the speed of light c =  299.705 km/s.

As the color bands blend, their precise lengths vary a little. Hence, their boundaries are approximate, for the color spectrum is continuous. Nevertheless, frontiers are present as anyone can see, although a small measure of arbitrariness cannot be avoided.

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 mean value of the color band. This wavelength is numerically compared with the frequencies of the notes at International Pitch (A4 = 440 Hz) & Verdi Pitch (A4 = 432 Hz). Their measure of numerical correspondence is then calculated.

TABLE 1 : Defining the Colors of the Notes
Wavelengths of Visible Spectrum
(in nm)
Centres of Color Gravity Tone Names
Middle C = C4
Verdi
Tone Pitches
(in Hz)
A4 = 432Hz
QBL Yoga
Ch'i
International
Tone Pitches
(in Hz)
A4 = 440Hz
VII VIOLET 400 to 380 390 G4 384,868 / 5,132 (*)
Δ% = 98,7 (**)
1 Ur 7 391,995 / 1,995
Δ% = 99,5
Dark Blue 420 to 400 410 G#4
Ab4
407,754 / 2,246
Δ% = 99,4
2 Ne 6 415,305 / 5,305
Δ% = 98,7
VI BLUE 450 to 420 435 A4 432 / 3
Δ% = 99,3
440 / 5
Δ% = 98,9
Light Blue 475 to 450 462,5 A#4
Bb4
457,688 / 4,812
Δ% = 98,96
3 Pl 5 466,164 / 3,664
Δ% = 99,2
V CYAN 495 to 475 485 B4 484,904 / 0,096
Δ% = 99,98
493,883 / 8,883
Δ% = 98,2
IV GREEN 520 to 495 507,5 C5 513,737 / 6,237
Δ% = 39098,8
4 Ju
6 Su
4 523,251 / 15,751
Δ% = 9
6,99
Yellow Green 565 to 520 542,5 C#5
Db5
544,286 / 1,786
Δ% = 99,7
5 Ma
8 Me
3 554,365 / 11,865
Δ% = 9
7,86
III YELLOW 575 to 565 570 D5 576,651 / 6,651
Δ% = 99,5
587,33 / 17,33
Δ% = 97,05
Yellow Orange 615 to 575 595 D#5
Eb5
610,94 / 15,940
Δ% = 97,4
7 Ve
9 Lu
2 622,254 / 27,254
Δ% = 95,6
II ORANGE 645 to 615 630 E5 647,269 / 17,269
Δ% = 97,3
659,255 / 29,255
Δ% = 95,6
I RED 720 to 645 682,5 F5 685,757 / 3,257
Δ% = 99,5
10 Sa 1 698,456 / 15,956
Δ% = 97,7
Dark Red 750 to 720 735 F#5
Gb5
726,535 / 8,465
Δ% = 98,85
739,989 / 4,989
Δ% = 99,3
M : 98,95%
σ = 0,17
M : 97,9%
σ = 1,33

(*) numerical difference between Centres of Color Gravity and Tone Pitch. Examples : for VIOLET (A = 432) : 390 - 384,868 = 5,132 ; for VIOLET (A = 440) : 391,995 - 390 = 1,995 ; (**) percentual numerical correspondence between Centres of Color Gravity and Tone Pitch. Examples : for VIOLET (A = 432) : 100/390 x 384,868 = 98,7% ; for VIOLET (A = 440) : 100/391,995 x 390 = 99,5%. Counting the times International Pitch scores better than Verdi Pitch we get 3, whereas Verdi Pitch is 9 times closer to the heart of the hue. In International Pitch, C5 falls outside the color interval. On average, Verdi Pitch is 98,95% corresponding (with a standard deviation of 0,17), whereas International Pitch is 97,9% corresponding with a much larger standard deviation (σ = 1,33). The strength of the linear association between the two variables R (centre of color gravity & tone pitch with A = 432 Hz or A = 440 Hz) is R = 1, or strong correlation (Pearson).

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).

Both scientists & artists have disputed the validity of the alignment. However, it cannot be denied a proportionality and strong correlation is at work here. Can this be applied to the harmonic series ?


§ 5 The Harmonic Series

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 ad infinitum. For example : take a long string or tube (1:1), then the ratio of 3:2 implies a string or a tube of 2/3 length will sound a fifth higher in pitch than the longer one. At one and a half times the lenght will sound a fifth lower.

The prevalence of the octave, fifth, fourth, major & minor third in the lower part of the series contributed to the development of the concept of harmony. The intervals of the harmonic series are called "harmonic intervals" and they are all acoustically pure (beatless).

When an instrument plays a single musical tone, a whole frequency spectrum is generated. It consists of a fundamental frequency (the 1th harmonic or 1th partial) and all the overtones present together with the fundamental tone.

A "harmonic" is an integer (whole number) multiple of the fundamental frequency of a vibrating object. The noise of a drill also generates a series of overtones, but these clash and beat each other up. Musical tones satisfy the conditions of harmony and so the overtones work together with no beats. The harmonic structure of the overtones assists in understanding how one tone may move towards another tone (as in an actual harmonic progression).

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 at the same time, repeating a whole cycle at the lowest, fundamental frequency involved. Not only do these overtones define the timbre of tones (primarily by way of the relative amplitudes of the various harmonics), they are crucial to understanding the natural harmonic rules of Western tonal harmony, i.e. a harmonic logic based on the properties of the harmonic series of every musical note sounded (Hindemith).

The Harmonic Series, defining precise mathematical proportionality, is (together with the energy or amplitude of the soundwave), the foundation of every musical tone. It operates a precise & fixed mathematical relationship between : 1) all the partials of a tone (the frequency components of a tone, i.e. fundamental frequency plus overtones), 2) the frequency ratio of a tone, 3) the frequency in Hz of a tone and 4) the harmonic interval to the previous tone.

TABLE 2 below has the 16 Harmonics of C, with the corresponding colors for the partials based on the analysis above (TABLE 1).

TABLE 2 : Harmonic Series of C with Color Correspondences
Partials = 1 Fundamental + 15 Overtones Harmonic Overtone Frequency
Ratio
Frequency
A4 = 432Hz
C2 = 64,217 Hz
Tone Harmonic Interval to Previous Tone
Fundamental
Frequency
1th none 1:1 64,217 C2 fundamental
Double
Frequency
2th 1th 2:1 128,434 C3 octave
Trifold
Frequency
3th 2th 3:2 192,651 G3 fifth
Fourfold
Frequency
4th 3th 4:3 256,868 C4 fourth
Fivefold
Frequency
5th 4th 5:4 321,085 E4 major
third
Sixfold
Frequency
6th 5th 6:5 385,302 G4 minor
third
Sevenfold
Frequency
7th 6th 7:6 449,519 Bb4
-31,17c
 
Eightfold
Frequency
8th 7th 8:7 513,737 C5
Nine
Frequency
9th 8th 9:8 577,953 D5 large
whole tone
Tenfold
Frequency
10th 9th 10:9 642,17 E5 small
whole tone
Elevenfold
Frequency
11th 10th 11:10 706,387 F#5
-48,68c
 
Twelvefold
 Frequency
12th 11th 12:11 770,604 G5
Thirteenfold
Frequency
13th 12th 13:12 834,821 Ab5
+40,52
Fourteenfold
Frequency
14th 13th 14:13 899,038 Bb5
-31,17
Fifteenfold
Frequency
15th 14th 15:14 963,255 B5
Sixteenfold
Frequency
16th 15th 16:15 1027,472 C6 minor
second

The notes in bold have a large offset (in cents). Bb4, F#5 & Bb5 are lower (-) than indicated, Ab5 is higher (+). The frequencies of the overtones are calculated with the fundamental frequency as their basis (here C2). For example, the 16th Harmonic has a frequency of 1027,472 Hz or 64,217 Hz x 16. All values in Hz are based on A4 = 432 Hz, or C2 = 64,217 Hz.

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.

"... the chromatic scale can be derived just as simply from the overtone series ; that, since it makes exhaustive use the clearest overtone relations, it has an even more natural basis ; and that it is thus the most natural of all scales, and the best adapted to melodic as well as to harmonic use." - Hindemith (1942, Book 1, p.48).

The higher the harmonic, the less it is heard. Each octave in the harmonic series is divided into increasingly "smaller" and more numerous intervals. The first eight harmonics define the tone and also contain the fundamental triad (tonic) of the tonic scales erected on it, the gravity of the tone. The intervals (to the previous tone) defined by the harmonic series are "pure" & "just", i.e. in accordance with natural law, i.e. unison (1/1, frequency ratio : 1:1), octave (2/1, frequency ratio : 2:1), fifth (3/2, frequency ratio : 3:2), fourth (4/3, frequency ratio : 4:3), major third (5/4, frequency ratio : 5:4) & minor third (6/8, frequency ratio : 6:5).

The frequency ratios of the first 6 harmonics (1 - 6) define the acoustico-numerical proportions of 5 important intervals (or distances between two notes). These are the harmonic intervals or "perfect" consonances. The first 4 harmonics (first three overtones) define harmonic octave (= 2:1, C - C), harmonic fifth (= 3:2, C - G) & harmonic fourth (= 4:3, G - C). The intervals of a harmonic fifth & a harmonic fourth make an octave (3:2 + 4:3 = 12/6 = 2:1) The harmonic fifth & the harmonic fourth are relatively beatless. As the 3th harmonic has an offset of +2c, they do have a beat, but one imperceptible to the human ear (able to distinguish a difference of 5 to 6c). Only the harmonic octave is absolutely beatless. This is why the 1th harmonic is the fundamental frequency of the tone, the backbone of the harmonic series in question. Pure, perfect intervals like harmonic octave, harmonic fourth & harmonic fifth are intervals with zero beat or a beat below 5c. Harmonic octave & harmonic fifth are consonant intervals. Depending of context, the harmonic fourth can be dissonant or consonant (in fact, C - F, the fourth from the root) is not part of the harmonic series (instead F# is the 10th overtone, but with an offset of -48,68c) !

In the Harmonic Series, the fundamental with 2 overtones (1th, 2th & 3th harmonic) is followed by the harmonic major third (= 5:4, C - E) and the harmonic minor third (= 6:5, E - G). The harmonic third has two forms : major & minor. The difference in size between these two harmonic thirds is a harmonic minor second. The harmonic thirds too are beatless. The intervals of a major & minor harmonic sixth do not occur between adjacent partials. To identify them, we need to leap. The harmonic major sixth has frequency ratio 5:3, the harmonic minor sixth 8:5. Because of their mixed formats (major versus minor), these 4 intervals are all "imperfect" consonances.

The frequencies of the Harmonic Series, being integer multiples of the fundamental frequency, are naturally (bio-acoustically) related to each other by whole-numbered & small whole-numbered ratios. This is the basis of the consonance of musical intervals. The Harmonic Series diverges in a way like the Fibonacci sequence does and can be rearranged with limit points integrating the Golden Ratio (φ). The auditive (bio-acoustic) & psycho-acoustic impact of forcing the ear to adjust to a non-φ based Master Tuning (like A4 = 440 Hz) is therefore questionable. The objective, natural form of the Harmonic Series (comparable to φ) is best assisted by a standard frequency based on φ, thereby stimulating an ear constructed by way of φ. In this way, source, medium & receiver are co-relatively attuned.

TABLE 3 : Harmonic Series of the Notes of the Chromatic Scale with their Colors (A4 = 432 Hz)
Note 02th 03th 04th 05th 06th 07th 08th 09th 10th 11th 12th 13th 14th 15th 16th
 f (Hz)
C2 C3 G3 C4 E4 G4 Bb4
(-)
C5 D5 E5 F#5
(-)
G5 Ab5
(+)
Bb5
(-)
B5 C6
64,217
C#2 C#3 G#3 C#4 F4 G#4 B4
(-)
C#5 D#5 F5 G5
(-)
G#5 A5
(+)
B5
(-)
C6 C#6
68,036
D2 D3 A3 D4 F#4 A4 C5
(-)
D5 E5 F#5 G#5
(-)
A5 A#5
(+)
C6
(-)
C#6 D6
72,081
D#2 D#3 A#3 D#4 G4 A#4 C#5
(-)
D#5 F5 G5 A5
(-)
A#5 B5
(+)
C#6
(-)
D6 D#6
76,368
E2 E3 B3 E4 G#4 B4 D5
(-)
E5 F#5 G#5 A#5
(-)
B5 C6
(+)
D6
(-)
D#6 E6
80,909
F2 F3 C4 F4 A4 C5 D#5
(-)
F5 G5 A5 B5
(-)
C6 C#6
(+)
D#6
(-)
E6 F6
85,72
F#2 F#3 C#4 F#4 A#4 C#5 E5
(-)
F#5 G#5 A#5 C6
(-)
C#6 D6
(+)
E6
(-)
F6 F#6
90,817
G2 G3 D4 G4 B4 D5 F5
(-)
G5 A5 B5 C#6
(-)
D6 D#6
(+)
F6
(-)
F#6 G6
96,217
G#2 G#3 D#4 G#4 C5 D#5 F#5
(-)
G#5 A#5 C6 D6
(-)
D#6 E6
(+)
F#6
(-)
G6 G#6
101,938
A2 A3 E4 A4 C#5 E5 G5
(-)
A5 B5 C#6 D#6
(-)
E6 F6
(+)
G6
(-)
G#6 A6
108
A#2 A#3 F4 A#4 D5 F5 G#5
(-)
A#5 C6 D6 E6
(-)
F6 F#6
(+)
G#6
(-)
A6 A#6
114,422
B2 B3 F#4 B4 D#5 F#5 A5
(-)
B5 C#6 D#6 F6
(-)
F#6 G6
(+)
A6
(-)
A#6 B6
121,226
The fundamental frequency of the notes are based on A4 = 432 Hz or A2 = 108 Hz. The frequencies of the first sixteen harmonics can be calculated by multiplying the fundamental frequency with the corresponding numerator of the frequency ratio of the harmonic in question (cf. Table 2). For example : F2 (f = 85,72 Hz), so the 15th Harmonics = f.15 = 85,72 x 15 = 1285,8 Hz.  The notes with a large offset (+/- 30 cents) are always the 7th, 11th, 13th & 14th harmonics, with "+" ("-") indicating they are higher (lower) than indicated.

Harmonic unison, harmonic octave, harmonic second, major & minor harmonic third, harmonic fourth, harmonic fifth, major & minor harmonic sixth are consonant. The harmonic series also contains the harmonic minor second (16:15) and the harmonic minor seventh or "septimal minor seventh" (7:4). The latter is also an interval not occuring between adjacent partials. Together (at times) with the harmonic fourth, the minor harmonic second and the minor harmonic seventh are dissonant.

The 10 harmonic intervals at work in the harmonic series of every musical tone are the basis of what could be called a "natural" tonal harmony. However, for all kinds of reasons, musical practice deviates from this natural harmonic law, introducing a cultural harmonic practice and reinforcing it. This is fine as long as the deviation does not cause the sound to become too harsh and unmusical, hampering the experience of beauty. So although in music culture may & does alter nature, there is a limit to this. But this remains to a certain degree (inter)subjective.


§ 6 Defining Tonal Scales

Blind Harpist with Harp tuned to a Heptatonic Scale ?
Relief in Tomb Chapel of Paatonemheb (ca. 1333-1319 BCE)

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 previous note). These play a fundamental role in the elaboration of the system of 24 tonic scales defining classical (tonal) harmony. This direct relationship between tonal harmony and the Harmonic Series is often overlooked.

TABLE 4 : Harmonic Intervals
consonant dissonant
perfect imperfect (minor) second, fourtht+1
(minor) seventh
unison, fourtht, fifth, octave major & minor
thirds & sixths

When two strings are tuned an absolutely beatless interval like the octave apart, the question of how to divide the octave gives rise to scales.

A scale is an organized, attuned and characterisic "family of tones", all being frequency-wise relatives.

Historically, the first scale to be formed was the pentatonic scale.

To play on a six stringed harp, a pentatonic scale can be generated consisting of the fundamental tone (C), the octave of this tone (C) and four other notes (G, D & A). This scale of five steps : C, D, E, G, A, C (the first pentatonic mode), and all other pentatonic scales consist of tones having related frequencies with few beats. The members of each clan, family or team naturally co-operate because they continuously gravitate around each other, forming enduring dependent originations, the "solid" architectures of conventional reality.

The pentatonic group of scales (set of families) was adopted by just about every human society.

The pentatonic scales can be derived by the simple proportions given by five consecutive pitches from the circle of fifths : C, G, D, A and E. The ability to tune a harp to the pentatonic scale is the cornerstone of the development of music.

While all notes on this scale sound "in tune" together (no major discord can be gotten), they do not define any stong "harmonic pull", i.e. the clear tendency of tonet to ask for tonet+1. The latter is the functional reciprocity of a tone in terms of the type or "color" of the family of tones to which it belongs. The pentatonic scales lack semitones. Therefore, every single combinates sounds euphonious. This also explains the staticity of pentatonic music. There is no real sense of harmonic transition or progression to be construed on the basis of the selected five tones.

The circles of fifths, giving rise to the pentatonic scales, produce five pentatonic modes. The first pentatonic mode, with C as its first note or "tonic", is defined by whole note (T or two semitones) & a minor third (M or three semitones) : TTMTM. This minor interval of three semitones is the special feature of all pentatonic scales, giving this group a self-contained harmonic acoustic identity.

TABLE 5 : The Pentatonic Modes
Circle of Fifths Scale Pentatonic Mode Transposition
Rule
Steps
C C, D, E, G, A, C I TTMTM 2-2-3-2-3
G G, A, C, D, E, G II TMTTM 2-3-2-2-3
D D, E, G, A, C, D III TMTMT 2-3-2-3-2
A A, C, D, E, G, A IV MTTMT 3-2-2-3-2
E E, G, A, C, D, E V MTMTT 3-2-3-2-2

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.

This scale, with C as its first  tone or "tonic", is defined by 5 whole tones (T = two semitones) & 2 semitones (S = one semitone) : TTSTTTS. This can be done for each tone of the scale, bringing forth seven scales. These were called "modes" and given the names of the 7 provinces of Ancient Greece :

TABLE 6 : The Greek (or Ecclesiastical) Modes
Tonic Scale Mode Transposition Rule Steps
C C, D, E, F, G, A, B, C Ionian TTSTTTS 2-2-1-2-2-2-1
D D, E, F, G, A, B, C, D Dorian TSTTTST 2-1-2-2-2-1-2
E E, F, G, A, B, C, D, E Phyrgian STTTSTT 1-2-2-2-1-2-2
F F, G, A, B, C, D, E, F Lydian TTTSTTS 2-2-2-1-2-2-1
G G, A, B, C, D, E, F, G Mixolydian TTSTTST 2-2-1-2-2-1-2
A A, B, C, D, E, F, G, A Aeolian TSTTSTT 2-1-2-2-1-2-2
B B, C, D, E, F, G, A, B Locrian STTSTTT 1-2-2-1-2-2-2

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.

Of the seven Greek Modes, only two were deemed to satisfy the conditions of well-organized harmonies able to express the "major" / "minor" contrast in the form of closely related families of seven notes. The Ionian mode, with its very obvious "full stops" at the end of every "harmonic sentence" (or combinations of simultaneously sounding notes), obviously became the foundation for the scales integrating the extraverted major third, while the Aeolian mode, with less obvious definite full stops, set the standard for the scales based on the introverted minor third.

The Ionian pattern of
TTSTTTS was applied to every note of the chromatic scale. This yields 12 so-called "major" scales. The first scale of this series is C major (C, D, E, F, G, A, B, C). The Aeolian pattern of TSTTSTT applied to the chromatic scale generates 12 so-called "minor" scales. The first scale of this series is a minor (A, B, C, D, E, F, G, A). On each of the seven steps (or degrees from I to VII) of the major scales, triads could be formed.

In the major scales, the major triads are on I, IV & V, the minor triads on II, III & VI and the diminished triad on VII (the third & fifth are lowered with a semitone). In the (natural) minor scales, the minor triads are on I & IV, the major triad on V & VI , the diminished triads on II & VII, and the augmented triad on III (the fifth is raised with a semitone).

The scales directly generated by the Aeolian mode were
also called "antique" or "natural". Because these always end with an interval of a whole tone at the end (g - a in a minor), a clearer "lead tone" (one naturally moving towards the tonic) was introduced by raising the last step (g in a minor) with a semitone (to g# in a minor). This generated the harmonic minor scales. As this first adjustment brings about an augmented interval (f - g# in a minor), f was also raised to f# (in a minor). An augmented interval was preferably avoided in older music and is even hard to sing in tune today. This second adjustment generated the melodic minor scales. The change from f to f# (in a minor), i.e. the transformation of the antique minor to the melodic minor only occurs when the melody moves upwards. In a downward movement, f# is restored to f and the descending melodic minor returns (equals) the antique minor.

This set of 24 scales defines the foundation of the classical, tonal harmony of Western music.

The fundamental issue with the tonal system is the balance between the natural Harmonic Series and the degree of tempering of the natural (just, harmonic) intervals. This refers to the Tuning System, another crucial factor often overlooked in contemporary manuals on harmony. No doubt because the matter is (at first sigh) too complex and calls for a "harmonic diplomacy" difficult to teach. Indeed, the Tuning System involves "... a compromise between the natural intervals and our inability to use them - that compromise we call the tempered system, which amounts to an indefinitely extended truce." - Schoenberg (1983, p.25.)


§ 7 The Tuning System

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 tuning system, we need a unit to measure the distance (intervals) between these located tones, i.e. their frequency ratios. This unit is "cent" (c) or "one hundred", introduced by Alexander Ellis in 1875. This a logaritmic scale with 1200c to the octave. Given frequency ratio a:b, we get the following : cent = (log2 a:b) x 1200, or, given the fifth (3:2) we get : log2 3:2 =  log2 (1,5) = 0,584963 x 1200 = 701,956c. Cents are used to express the size of the microintervals (intervals smaller than one whole tone).

The fundamental question is : does the tuning system follow the acoustical laws of audition, objectively (Harmonic Series) and subjectively (structure & function of the human ear) ? Simply put, does tuning follow natural acoustic law ? Technically, this is the number of pure, harmonic intervals present in the scale. Equal Temperament has the least pure intervals present (namely one, the octave). But the octave merely delimits the scale, and does not functionally contribute to the division of the octave. So Equal Temperament is the less natural scalar definition.

The harmonic intervals are the reference of all acoustically based tuning systems, for these are beatless intervals, i.e. acoustically pure and mathematically simple (integers instead of real numbers). They are also called "just intervals". Scalar definitions based on them are untempered.

TABLE 7 : Ratios & Cents of Core Harmonic Intervals
Interval Frequency Ratio Cents
unison 1:1 0
octave 2:1 1200
fifth 3:2 701,956
fourth 4:3 498,044
major third 5:4 386,314
minor third 6:5 315,641

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.

Equal Temperament, commonly used everywhere today, has been rightly accused to "ruin harmony" (Duffin, 2007).

TABLE 8 : Equal Temperament

C C# D D# E F F# G G# A A#

 B

C
000c 100c 200c 300c 400c 500c 600c 700c 800c 900c 1000c 1100c 1200c

There is no direct relationship between Hz and c. The former positions absolute frequency, the latter proportional position (or interval distance between tones). We need to know the standard tuning (or Master Tuning) to relate both values. Indeed, as a tone increases in pitch, its location changes and so also its distance from other static tones. But the Hz value and the cent value are not measured one-to-one, since often one value increases while the other decreases. 

TABLE 9 : Frequency (Hz) / cents relationships
for A4 = 432 Hz (C3 = 128,434 Hz = 0c)

Cents C C# D D# E F F# G G# A A# B
-50 124,778 132,198 140,059 148,387 157,211 166,559 176,463 186,956 198,073 209,851 222,329 235,550
-49 124,850 132,274 140,140 148,473 157,301 166,655 176,565 187,064 198,187 209,972 222,458 235,686
-48 124,922 132,351 140,221 148,559 157,392 166,751 176,667 187,172 198,302 210,093 222,586 235,822
-47 124,995 132,427 140,302 148,644 157,483 166,848 176,769 187,280 198,416 210,215 222,715 235,958
-46 125,067 132,504 140,383 148,730 157,574 166,944 176,871 187,388 198,531 210,336 222,844 236,095
-45 125,139 132,580 140,464 148,816 157,665 167,041 176,973 187,497 198,646 210,458 222,972 236,231
-44 125,211 132,657 140,545 148,902 157,756 167,137 177,076 187,605 198,761 210,579 223,101 236,367
-43 125,284 132,733 140,626 148,988 157,847 167,234 177,178 187,713 198,875 210,701 223,230 236,504
-42 125,356 132,810 140,707 149,074 157,939 167,330 177,280 187,822 198,990 210,823 223,359 236,641
-41 125,428 132,887 140,789 149,160 158,030 167,427 177,383 187,930 199,105 210,945 223,488 236,777
Cents C C# D D# E F F# G G# A A# B
-40 125,501 132,964 140,870 149,247 158,121 167,524 177,485 188,039 199,220 211,067 223,617 236,914
-39 125,573 133,040 140,951 149,333 158,213 167,620 177,588 188,148 199,335 211,189 223,746 237,051
-38 125,646 133,117 141,033 149,419 158,304 167,717 177,690 188,256 199,451 211,311 223,876 237,188
-37 125,719 133,194 141,114 149,505 158,395 167,814 177,793 188,365 199,566 211,433 224,005 237,325
-36 125,791 133,271 141,196 149,592 158,487 167,911 177,896 188,474 199,681 211,555 224,134 237,462
-35 125,864 133,348 141,277 149,678 158,579 168,008 177,998 188,583 199,797 211,677 224,264 237,599
-34 125,937 133,425 141,359 149,765 158,670 168,105 178,101 188,692 199,912 211,799 224,394 237,737
-33 126,009 133,502 141,441 149,851 158,762 168,202 178,204 188,801 200,027 211,922 224,523 237,874
-32 126,082 133,579 141,522 149,938 158,854 168,300 178,307 188,910 200,143 212,044 224,653 238,012
-31 126,155 133,657 141,604 150,024 158,945 168,397 178,410 189,019 200,259 212,167 224,783 238,149
Cents C C# D D# E F F# G G# A A# B
-30 126,228 133,734 141,686 150,111 159,037 168,494 178,513 189,128 200,374 212,289 224,913 238,287
-29 126,301 133,811 141,768 150,198 159,129 168,591 178,616 189,237 200,490 212,412 225,043 238,424
-28 126,374 133,888 141,850 150,285 159,221 168,689 178,720 189,347 200,606 212,535 225,173 238,562
-27 126,447 133,966 141,932 150,372 159,313 168,786 178,823 189,456 200,722 212,657 225,303 238,700
-26 126,520 134,043 142,014 150,458 159,405 168,884 178,926 189,566 200,838 212,780 225,433 238,838
-25 126,593 134,121 142,096 150,545 159,497 168,981 179,030 189,675 200,954 212,903 225,563 238,976
-24 126,666 134,198 142,178 150,632 159,589 169,079 179,133 189,785 201,070 213,026 225,693 239,114
-23 126,739 134,276 142,260 150,719 159,682 169,177 179,237 189,894 201,186 213,149 225,824 239,252
-22 126,813 134,353 142,342 150,806 159,774 169,274 179,340 190,004 201,302 213,273 225,954 239,390
-21 126,886 134,431 142,425 150,894 159,866 169,372 179,444 190,114 201,419 213,396 226,085 239,529
Cents C C# D D# E F F# G G# A A# B
-20 126,959 134,509 142,507 150,981 159,959 169,470 179,547 190,224 201,535 213,519 226,216 239,667
-19 127,033 134,586 142,589 151,068 160,051 169,568 179,651 190,334 201,652 213,642 226,346 239,805
-18 127,106 134,664 142,672 151,155 160,143 169,666 179,755 190,444 201,768 213,766 226,477 239,944
-17 127,179 134,742 142,754 151,243 160,236 169,764 179,859 190,554 201,885 213,889 226,608 240,083
-16 127,253 134,820 142,836 151,330 160,329 169,862 179,963 190,664 202,001 214,013 226,739 240,221
-15 127,326 134,898 142,919 151,417 160,421 169,960 180,067 190,774 202,118 214,137 226,870 240,360
-14 127,400 134,976 143,002 151,505 160,514 170,059 180,171 190,884 202,235 214,260 227,001 240,499
-13 127,474 135,054 143,084 151,592 160,607 170,157 180,275 190,995 202,352 214,384 227,132 240,638
-12 127,547 135,132 143,167 151,680 160,699 170,255 180,379 191,105 202,469 214,508 227,263 240,777
-11 127,621 135,210 143,250 151,768 160,792 170,353 180,483 191,215 202,586 214,632 227,395 240,916
Cents C C# D D# E F F# G G# A A# B
-10 127,695 135,288 143,332 151,855 160,885 170,452 180,587 191,326 202,703 214,756 227,526 241,055
-9 127,768 135,366 143,415 151,943 160,978 170,550 180,692 191,436 202,820 214,880 227,657 241,195
-8 127,842 135,444 143,498 152,031 161,071 170,649 180,796 191,547 202,937 215,004 227,789 241,334
-7 127,916 135,522 143,581 152,119 161,164 170,748 180,901 191,658 203,054 215,128 227,921 241,473
-6 127,990 135,601 143,664 152,207 161,257 170,846 181,005 191,768 203,171 215,253 228,052 241,613
-5 128,064 135,679 143,747 152,295 161,350 170,945 181,110 191,879 203,289 215,377 228,184 241,753
-4 128,138 135,757 143,830 152,383 161,444 171,044 181,214 191,990 203,406 215,502 228,316 241,892
-3 128,212 135,836 143,913 152,471 161,537 171,142 181,319 192,101 203,524 215,626 228,448 242,032
-2 128,286 135,914 143,996 152,559 161,630 171,241 181,424 192,212 203,641 215,751 228,580 242,172
-1 128,360 135,993 144,079 152,647 161,724 171,340 181,529 192,323 203,759 215,875 228,712 242,312
Cents C C# D D# E F F# G G# A A# B
0 128,434 136,071 144,163 152,735 161,817 171,439 181,634 192,434 203,877 216,000 228,844 242,452
1 128,509 136,150 144,246 152,823 161,911 171,538 181,739 192,545 203,995 216,125 228,976 242,592
2 128,583 136,229 144,329 152,912 162,004 171,637 181,844 192,657 204,113 216,250 229,109 242,732
3 128,657 136,307 144,413 153,000 162,098 171,737 181,949 192,768 204,230 216,375 229,241 242,872
4 128,731 136,386 144,496 153,088 162,191 171,836 182,054 192,879 204,348 216,500 229,373 243,013
5 128,806 136,465 144,580 153,177 162,285 171,935 182,159 192,991 204,467 216,625 229,506 243,153
6 128,880 136,544 144,663 153,265 162,379 172,035 182,264 193,102 204,585 216,750 229,639 243,294
7 128,955 136,623 144,747 153,354 162,473 172,134 182,370 193,214 204,703 216,875 229,771 243,434
8 129,029 136,702 144,830 153,442 162,567 172,233 182,475 193,325 204,821 217,000 229,904 243,575
9 129,104 136,781 144,914 153,531 162,661 172,333 182,580 193,437 204,939 217,126 230,037 243,715
Cents C C# D D# E F F# G G# A A# B
10 129,178 136,860 144,998 153,620 162,755 172,432 182,686 193,549 205,058 217,251 230,170 243,856
11 129,253 136,939 145,082 153,709 162,849 172,532 182,791 193,661 205,176 217,377 230,303 243,997
12 129,328 137,018 145,165 153,797 162,943 172,632 182,897 193,773 205,295 217,502 230,436 244,138
13 129,402 137,097 145,249 153,886 163,037 172,732 183,003 193,885 205,414 217,628 230,569 244,279
14 129,477 137,176 145,333 153,975 163,131 172,831 183,108 193,997 205,532 217,754 230,702 244,420
15 129,552 137,256 145,417 154,064 163,225 172,931 183,214 194,109 205,651 217,880 230,835 244,562
16 129,627 137,335 145,501 154,153 163,320 173,031 183,320 194,221 205,770 218,006 230,969 244,703
17 129,702 137,414 145,585 154,242 163,414 173,131 183,426 194,333 205,889 218,131 231,102 244,844
18 129,777 137,494 145,669 154,331 163,508 173,231 183,532 194,445 206,008 218,258 231,236 244,986
19 129,852 137,573 145,754 154,421 163,603 173,331 183,638 194,558 206,127 218,384 231,369 245,127
Cents C C# D D# E F F# G G# A A# B
20 129,927 137,653 145,838 154,510 163,697 173,431 183,744 194,670 206,246 218,510 231,503 245,269
21 130,002 137,732 145,922 154,599 163,792 173,532 183,850 194,783 206,365 218,636 231,637 245,411
22 130,077 137,812 146,006 154,688 163,887 173,632 183,956 194,895 206,484 218,762 231,771 245,552
23 130,152 137,891 146,091 154,778 163,981 173,732 184,063 195,008 206,603 218,889 231,905 245,694
24 130,227 137,971 146,175 154,867 164,076 173,833 184,169 195,120 206,723 219,015 232,039 245,836
25 130,302 138,051 146,260 154,957 164,171 173,933 184,276 195,233 206,842 219,142 232,173 245,978
26 130,378 138,130 146,344 155,046 164,266 174,033 184,382 195,346 206,962 219,268 232,307 246,120
27 130,453 138,210 146,429 155,136 164,361 174,134 184,489 195,459 207,081 219,395 232,441 246,263
28 130,528 138,290 146,513 155,225 164,456 174,235 184,595 195,572 207,201 219,522 232,575 246,405
29 130,604 138,370 146,598 155,315 164,551 174,335 184,702 195,685 207,321 219,649 232,710 246,547
Cents C C# D D# E F F# G G# A A# B
30 130,679 138,450 146,683 155,405 164,646 174,436 184,809 195,798 207,441 219,776 232,844 246,690
31 130,755 138,530 146,767 155,495 164,741 174,537 184,915 195,911 207,560 219,903 232,979 246,832
32 130,830 138,610 146,852 155,584 164,836 174,638 185,022 196,024 207,680 220,030 233,113 246,975
33 130,906 138,690 146,937 155,674 164,931 174,739 185,129 196,137 207,800 220,157 233,248 247,118
34 130,982 138,770 147,022 155,764 165,027 174,840 185,236 196,251 207,920 220,284 233,383 247,260
35 131,057 138,850 147,107 155,854 165,122 174,941 185,343 196,364 208,041 220,411 233,518 247,403
36 131,133 138,931 147,192 155,944 165,217 175,042 185,450 196,478 208,161 220,539 233,653 247,546
37 131,209 139,011 147,277 156,034 165,313 175,143 185,557 196,591 208,281 220,666 233,788 247,689
38 131,285 139,091 147,362 156,125 165,408 175,244 185,664 196,705 208,401 220,794 233,923 247,832
39 131,360 139,172 147,447 156,215 165,504 175,345 185,772 196,818 208,522 220,921 234,058 247,976
Cents C C# D D# E F F# G G# A A# B
40 131,436 139,252 147,532 156,305 165,599 175,447 185,879 196,932 208,642 221,049 234,193 248,119
41 131,512 139,332 147,618 156,395 165,695 175,548 185,987 197,046 208,763 221,176 234,328 248,262
42 131,588 139,413 147,703 156,486 165,791 175,649 186,094 197,160 208,883 221,304 234,464 248,406
43 131,664 139,494 147,788 156,576 165,887 175,751 186,201 197,274 209,004 221,432 234,599 248,549
44 131,740 139,574 147,874 156,667 165,983 175,852 186,309 197,388 209,125 221,560 234,735 248,693
45 131,817 139,655 147,959 156,757 166,078 175,954 186,417 197,502 209,246 221,688 234,870 248,836
46 131,893 139,735 148,045 156,848 166,174 176,056 186,524 197,616 209,367 221,816 235,006 248,980
47 131,969 139,816 148,130 156,938 166,270 176,157 186,632 197,730 209,488 221,944 235,142 249,124
48 132,045 139,897 148,216 157,029 166,366 176,259 186,740 197,844 209,609 222,073 235,278 249,268
49 132,121 139,978 148,301 157,120 166,463 176,361 186,848 197,958 209,730 222,201 235,414 249,412
Cents C C# D D# E F F# G G# A A# B
50 132,198 140,059 148,387 157,211 166,559 176,463 186,956 198,073 209,851 222,329 235,550 249,556

Tuning Systems

A tuning system consists of three components :

(1) SCALAR DIMENSION : the number of steps dividing the octave  :

In Western classical music and also in popular music, the octave is commonly divided in 12 steps, each step being a semitone.

(2) MASTER TUNING : a standard to tune instruments or concert pitch ;

Before 1711, composers & musicians had no clear reference point to tune their instruments. Since 1939, an international consensus exists about concert pitch, with A4 = 440 Hz. This is 8 Hz sharper than Verdi pitch (A4 = 432 Hz or C3 = 128,434 Hz). Verdi pitch refers to φ (at A4 = 432 Hz, E3 = 161,817 Hz). The Golden Ratio (φ ≈ 1,618 ...) is also found in the ear as well as in the harmonic series. In the present study, concert pitch = Verdi pitch is part of the set of conditions related to the reintroduction of color & natural harmonic cycles back into music.

(3) SCALAR DEFINITION : the tuning or temperament of the scale.

Defines the interval-relationships between the tones of a scale. As the scale stays close to acoustically pure intervals (harmonic, just intervals), the frequencies of the tones (after which the notes are named) are related by ratios of two integers. The scale is "in tune" with the Harmonic Series. Historically, two tunings stand out : Pythagorean & Just Intonation. The former only uses the harmonic fifth (3:2), the latter all integer ratios, introducing limitations as to the complexity of these. So many variations of Just Intonation exist. But all these tunings generate a howling wolf interval imposing harmonic limitations. Before the 15th century, Western music was simple enough to avoid the wolf. With the Renaissance, these limitations were questioned and eventually left behind. The wolf had to be eliminated.

To enhance the harmonic capacity of the scale, i.e. its ability to allow for an increased harmonic variety, accommodate a wider compositional intent and therefore be more useful in a broader variety of musical contexts, adjustments had to be made to the tones, creating intervals modified (tempered) from their harmonic, acoustically natural format. In this way, the power of the wolf diminished or could be rendered nearly harmless. Such tempered intervals are not based on ratios of two integers. The resulting scale is then called a "temperament" (because one, some or all intervals are tempered). The later is therefore a modification of a tuning, needing radical numbers to express the ratios of one, some or indeed all of its intervals. In musical practice today, the commonly used temperament is Equal Temperament (all semitones beings 50c, a whole tone 100c and so each step of the 12 steps of the scale 100c, up to the octave above at 1200c). It is the less natural of all temperaments !

Barbour (1951), identifying over a hundred historical tunings & temperaments based around the octave, categorizes them into five broad groups : Pythagorean Tuning, Just Intonation, Meantone Temperament, Irregular Temperament(s) and Equal Temperament.

The first two are "tunings" (harmonic intervals only), the last three are "temperaments" (tempered intervals).

a) PYTHAGOREAN Tuning

SCALAR DIMENSION : 12
SCALAR DEFINITION : 3:2 (harmonic fifth) with exclusion rules for the wolf interval ;

Pythagorean Tuning defines all the notes & intervals of a scale from a series of perfect (pure) fifths with ratio of 3:2 = 701,956c, the third harmonic of the Harmonic Series (the first two being the fundamental frequency with its octave). This is easy to tune by ear.

TABLE 11 : Pythagorean Intervals
Interval Frequency Ratio Cents
unison 1:1 0
minor second 2187:2048 113,686
major second 9:8 203,910
minor third 32:27 294,136
major third 81:64 407,820
fourth 4:3 498,044
augmented fourth 729:512 611,730
fifth 3:2 701,956
minor sixth 6561:4096 815,640
major sixth 27:16 905,866
minor seventh 16:9 996,090
major seventh 243:128 1109,776
octave 2:1 1200

However, no matter how many fifths (3/2 intervals) one takes, either above or below a given note, one never arrives at an exact octave multiple of that note, but always a discrepancy larger, overshooting the target pitch, with a ratio of 129,746:128 or 1,013640625:1 instead of 1:1. There are 11 pure fifths and one narrow, out-of-tune fifth.

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 312/219 = 531441:524288 (= 1,013643265 ...). Log2 (1,013643265) = 0,01955 x 1200 = 23,46c ; the Pythagorean comma or ditonic comma.

"Now if one continues indefinitely adding fifths in an upward direction (setting up the circle of fifths), one is headed for infinity, for with the twelfth fifth a tone is reached which is also a comma higher than the octave of the original tone. (...) The ear is easily reconciled to impure fifths, if the discrepancy is very slight, as the equally tempered scale shows ; but it would not accept impure octaves." - Hindemith (1942, Book 1, p.31).

Suppose we tune in D, then we first move six times a ratio 3:2 up : D - A - E - B - F# - C# - G#. We generate the the six other notes of the chromatic scale by moving six times a ratio 3:2 down : Eb - Bb - F - C - G - (D). If we add one more note to this stack, we arrive at : Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G#. This is very similar, but not exactely identical in size to a stack of 7 octaves. Indeed, it is a Pythagorean comma larger.

Therefore, Ab and G#, when brought into the basic octave, will not coincide as expected. To get around this problem, the chromatic Pythagorean scale ignores Ab, and uses only the 12 notes from Eb to G# (C# but not Db, F# but not Gb, G# but not Ab, Bb but not A#, Eb but not D#). So only 11 perfect fifths are used to build the entire chromatic scale. The remaining fifth (from G# to Eb) is left out-of-tune ! So music combining those two notes is unplayable in this tuning. This very out-of-tune interval is the wolf interval (the 11 perfect fifths are 701,956c wide, in the exact ratio 3:2, except the solitary wolf fifth, which is only 678,495c wide = 701,956c - Pythagorean comma).

TABLE 12 : The Pythagorean Scale

C C# D Eb E F F# G G# A Bb

 B

C
1:1 2187:2048 9:8 32:27 81:64 4:3 729:512 3:2 6561:4096 27:16 16:9 243:128 2:1
0c 114 204c 294c 408c 498c 612c 702c 816 906c 998c 1110c 1200c

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.

In music not changing key very often, or not very harmonically adventurous, the wolf interval is unlikely to be a problem, as not all 12 fifths will be heard in such pieces. Because 11 fifths in Pythagorean tuning are in the simple ratio of 3:2, they sound very "smooth" and "consonant". The thirds, by contrast, with relatively complex ratios of 81:64 (for major thirds) and 32:27 (for minor thirds), sound less smooth. Pythagorean tuning is particularly well suited to music which treats fifths as consonances, and thirds as dissonances. In the West, this means music written prior to the 15th century.

b) JUST Intonation (or Just Tuning)

SCALAR DIMENSION : 12
SCALAR DEFINITION : always pure, harmonic intervals, possibly simple integers ;

Just Intonation is also based on the harmonic series, i.e. makes use of acoustically beatless intervals. Compared with Pythagorean Tuning (only using 3:2), Just Intonation arrives at other frequency ratios for the intervals.

TABLE 13 : Just Intonation (7-limit)
Interval Frequency Ratio Cents
unison 1:1 0
minor second 16:15 111,731
major second 9:8 203,910
minor third 6:5 315,641
major third 5:4 386,314
fourth 4:3 498,044
augmented fourth 7:5 582,513
fifth 3:2 701,956
minor sixth 8:5 813,686
major sixth 5:3 884,359
minor seventh 9:5 996,090
major seventh 15:8 1088,269
octave 2:1 1200

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.

TABLE 12 : Basic Just Intonation (7-limit)

C Db D Eb E F F# G Ab A Bb

 B

C
1:1 16:15 9:8 6:5 5:4 4:3 7:5 3:2 8:5 5:3 16:9 15:8 2:1
0c 112 204 316 386 498 583 702 814 884 996 1088 1200c

Wolf intervals also occur and hence the same problems as with Pythagorean Tuning persist.

c) MEANTONE Temperament

SCALAR DIMENSION : 12 and more
SCALAR DEFINITION : various attempts to make the major third pure by tempering the fifths, with no circularity ;

The first temperaments date from the 15th century, while the earliest discussions happened a century earlier. The exclusive use of the perfect fifth (Pythagorean) or other harmonic intervals (Just Intonation), causing a wolf interval, limited the harmonic possibilities of composers & musicians. The major thirds in Pythagorean were too wide and the minor thirds too narrow.

TABLE 13 : Harmonic, Pythagorean & Just Intonation Thirds
Pythagorean Major Thirds Pythagorean Minor Thirds Just Intonation Major Thirds Just Intonation Minor Thirds Harmonic Major Thirds Harmonic Minor Thirds
407,820c (81:64) 294,136c 386,314c 315,641c 386,314c (5:4) 315,641c

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.

The idea underlying Meantone Temperament was to make the major third as acoustically pure as possible. However, in a twelve-note octave, a circle of pure major thirds does not fit in (three pure major thirds = 5:4 x 5:4 x 5:4 = 125/64 or 1,953:1 instead of 2:1). In the Renaissance this was not really a problem, because composers did not use all the major thirds. Moreover, they did not notate music with more than two flats, and rarely with sharps. To make the thirds sound good, the fifths were narrowed (the harmonic fifth of 701,956c became the meantone fifth of 696,57843 in 1/4-comma or 698,3710 in 1/6-comma). This was the disadvantage of this system. But in combination with perfect euphonious thirds, these tempered fifths were tolerable.

To achieve acoustically pure major thirds, four consecutive fifths in the circle were tempered by one-quater of the difference between Pythagorian and pure major thirds. This difference of 21,506c (407,820c - 386,314), with ratio 81:80 (21,5064c) is called the syntonic comma, diatonic comma or Ptolemaic comma. This is the temperament know as "quater-comma" Meantone. Here we have eight pure thirds and four thirds which are really diminished fourths. The syntonic comma was spread over four fifths.

TABLE 13 : 1/4-comma Meantone Temperament (Piertro Aaron - 1523)

C C# D D# E F F# G G# A A#

 B

C
0c 76 193 310 5:4 503 579 697 25:16 890 1007 1083 1200

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.

In none of these systems was the wolf interval eliminated, and so the 24 keys were not all available (some chords sounding out-of-tune). But in all of them the harmonic fifth & major third were prominent.

"The strongest, most unambiguous interval, after the octave, which is unique, is the fifth, while the most beautiful is the major third, on account of the triad formed by it with its combination tones."  - Hindemith (1942, Book 1, p.88). For Hindemith, "the frequency of the combination tone is always equal to the difference between the frequencies of the directly produces tones of the interval" (p.61).

TABLE 14 : 1/6-comma Meantone Temperament (Salinas)

C C# D D# E F F# G G# A A#

 B

C
0c 89 197 305 393 502 590 698 787 895 1003 1092 1200

d) IRREGULAR Temperaments

SCALAR DIMENSION : 12
SCALAR DEFINITION : combinations of pure & tempered major thirds & fifths eliminating the wolf ;

With the discovery all fifths did not have to be tempered the same amount, a new group of temperaments emerged. These "irregular" temperaments mixed different amounts of tempering and so called for different sizes of fifths & thirds. By this adjustment, mixing harmonic & tempered fifths, the wolf interval could be driven out (closed temperaments) and more chords created. This assisted composers who intended to write chromatic lines and complexer harmonies (with sharp & more than two flat signatures), like diminished and seventh chords. The idea was to make all 24 keys around the circle of fifths become available (circular temperaments). All the notes & chords of the set of 12 major & 12 minor keys sounded "well" together ! The tirany of the wolf interval, limiting full harmonic expression, had ended. A harmonic logic with more categories & rules could be invented.

If the studies of Lehnman are correct, Johan Sebastian Bach was the first composer to musically acknowledge the crucial importance of this in his Das Wohltemperierte Klavier (BWV 846–893). The word "wohl" does not refer to "equal", but to the euphonous qualities of the 24 tonal scales in Bach's irregular temperament. Because of the variety of intervals within these irregular temperaments, each key had its own harmonic characteristics or "key color". Indeed, the flavor of the chords was slightly different in each key. Bach's temperament keeps C, D, E, F, G & A in evenly-spaced positions, at their normal place within the context of late 17th century practice. Then six remaining notes (B and the sharps F#, C#, G#, D# & A#) are put in place with adjustments making them also serve well as flats.

A staggering variety of irregular temperaments saw the light, each temperament having its own advantages and disadvantages. Some favored certain chords & keys, while others did not sound well. Or they sounded more or less all good, but were very difficult to tune. In the tuning world, some of these systems are known under the name of their inventor : Andreas Werckmeister, Johann Philipp Kirnberger, Johann Georg Neidhardt, Antonia Vallotti, Thomas Young and many others.

Two outstanding irregular temperaments : Johan Sebastian Bach and Thomas Young, both giants in their field.  According to Murray Barbour, Young's temperament is "the best of these many irregular systems" (Barbour, 1951, p.12)

TABLE 15 : Lehman - Bach Temperament

C C# D D# E F F# G G# A A#

 B

C
0c 98,045 196,09 298,045 392,180 501,955 596,090 698,045 798,045 894,135 998,045 1094,135 1200c

Irregular temperaments always leave certain fifths untempered. For example, in Young's temperament n°2, a superb example of simplicity & elegance, 6 fifths (C - G - D - A - E -  B - F#) are tempered by 1/6 of a Pythagorean comma, while the 6 others remain harmonical (pure, beatless, acoustically perfect).

Thomas Young's Well Temperament (N°2)

inside : tempering of each fifth
outside : tempering of each third
inside triangle : combined tempering of each triad

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.

TABLE 16 : Thomas Young Temperament N°2 (1/6-comma - 1807)

C C# D D# E F F# G G# A A#

 B

C
0c 256:243
90.224
195,09 32:27
294.134
392,18 4:3
498.044
1024:729 698,045 128:81
588.270
894,135 16:9
996.090
1090,225 1200c

Young N°2 defines the minor third harmonically (32:27).

"What then, is the minor triad, in reality ? I hold, following a theory which again is not entirely new, that it is a clouding of the major triad. Since one cannot even say definitely where the minor third leaves off and the major third begins, I do not believe in any polarity of the two chords. They are the high and low, the strong and weak, the light and dark, the bright and dull forms of the same sound."  - Hindemith (1942, Book 1, p.78).

"Again we may say that Young's version is an excellent irregular temperament ..."
(Barbour, 1951, p.181). The trouble with these mathematically excellent systems "is that they are too good !" (Ibidem, p.178). The step from "paper" to "practice" is difficult and involves hearing the beats of the beating fifths.

In our age, assisted by electronics, tuning & retuning has become a mere preset.

d) EQUAL Temperament

SCALAR DIMENSION : 12
SCALAR DEFINITION : tempering all intervals alike, no pure intervals, no color, no wolf ;

Already known in Antiquity (Aristoxenus in the 4th century BCE), who preferred another system, Equal Temperament became an issue around 1640, if not earlier. The father of Galileo Galilei (Vincenzo) advocated the system and composed dance suites on each of the notes of the chromatic scale (in all keys).

The octave (of 1200c) is divided into 12 of 100c and a whole tone divided in two semitones of 50c each. The scale is constructed by way of 2n/12 (n = 1 for C -unison- and n = 12 for C' -octave-). All steps are tempered by 1/12 of a Pythagorean comma, or 23.46/12 = 1.96 cents.

In the past, theorists as wel as musicians found Equal Temperament coarse, disagreeable (Thomas Smith), with a beating heightening emotion (Johann Neidhardt).

"... the ear is subject to a certain danger in being exposed only to music constructed of tempered intervals ; it accustoms itself to their clouded qualities, and like a jaded palate loses its sense of natural relations." - Hindemith (1942, Book 1, p.28).

In Equal Temperament, major thirds (400c) are far too wide compared to the harmonic major third (386,314) and minor thirds (300) too small compared to the harmonic minor third (315,641), creating an unpleasant, coarse "beat".

In all previous systems, the tone (of 9 commas) had been divided in a minor (4 commas) & a major (5 commas) semitone. This was abolished. The relationship maintained between the Harmonic Series was also eliminated. Each step of the scale is tempered. Hence, except for the octave, no acoustically pure intervals remain ! This means this temperament breaks with the underlying acoustical phenomena. This scale moves too far away from natural law to allow for color. This is the dictate of unnaturalness. The uniform intervals lack expressive variety, making the 24 keys loose their characteristic harmonic "coloration". Keys are merely shades of gray.

TABLE 17 : Equal Temperament

C C# D D# E F F# G G# A A#

 B

C
0c 100 200 300 400 500 600 700 800 900 1000 1100 1200c

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 ...

In terms of the Farben Project, tempering all intervals except the octave means the end of coloration and this is precisely the opposite of its intent. It wants to generate color.

This calls for the following tuning system :

(1) scalar dimension : the division of the octave in 12 steps ;
(2) master tuning : Verdi Pitch (A4 = 432 Hz) ;
(3) scalar definition : all tunings & temperaments are possible and should be used as a function of the music in question. Young N°2 is the best temperament. Equal Temperment is a default accommodating standardisation. It is the worse.


§ 8 Psycho-Acoustic Definitions

The impact of music on the human being, besides auditive input, is mainly through resonance, the vibration imparted on the psycho-biological system as a result of sound waves from one or more sources of sound, if possible musical instruments. These resonances happen all through the body. If these vibrations are in harmony with the laws of acoustics (in terms of the harmonic series and the φ-base of human hearing), then this impact is wellness, or beatlessness. If, due to the beating between the overtones of these sound waves, pure vibes cannot be maintained, a disturbing resonance is at hand. In louder form this becomes auditive pollution.

Psycho-acoustics aims to entrain the listener by way of sound waves most likely to trigger the natural restoring response of a system vibrating in tune with the acoustics of the harmonic series, the human ear and color.

"When a stimulus reaches the visual cortex, a visual evoked response is produced, a response that follows the frequency characteristic of the stimulus." (Evans, 2007, p.251.)

This brings the visual into the equation, opening the way for combining music & color during performance (Scriabin, the color organ). Eventually, this may lead to an individualized & audio/visually scripted entrainment, a session running HRV feedback & EEG-driven (like Mind Work Station & Bioexplorer), with parameters set to specific targets.

How to find a series of plausible Psycho-Acoustic criteria ? On the basis of my various hermeneutical studies on religious language & practice, provisional "esoteric" tables of correspondences can be put together (cf. Tabula Tabularum Esotericum, 2006).

Of these various correspondences, let us analyze the relationships proposed between the energy-wheels of the subtle energy-body (in Taoism, Hinduism & Buddhism), states of yogic consciousness (in Classical Yoga) and the symbolic language of the Qabalah (with its numerical, astrological, alchemical and psycho-dynamic semantic charts). Especially the Chakra's (energy-vortices) are interesting, for there are seven main energy-fields, each associated with a spectral color. These wheels also associate with certain functions, cognitive modes and proposed brain frequencies.

TABLE 18 : Defining Provisional Psycho-Acoustic Criteria
Wavelengths of Visible Spectrum
(in nm)
Centres of Color Gravity Tone Names
Middle C = C4
Verdi
Pitches
(in Hz)
QBL Chakra Subtle Bodies Function Cognitve
Modes
Brain frequency
VII VIOLET 400 to 380 390 G4 384,868 1 Ur crown monadic unity
selflessness
nondual Gamma
+ 90 Hz
Dark Blue 420 to 400 410 G#4
Ab4
407,754 2 Ne        
VI BLUE 450 to 420 435 A4 432 brow logoic insight
awareness
Gamma
+
50 Hz
Light Blue 475 to 450 462,5 A#4
Bb4
457,688 3 Pl throat buddhic language
speech
Gamma
+ 40 Hz
V CYAN 495 to 475 485 B4 484,904        
IV GREEN 520 to 495 507,5 C5 513,737 4 Ju
6 Su
HEART causal integration
compassion
higher self
creative Beta+
21/32 Hz
Yellow Green 565 to 520 542,5 C#5
Db5
544,286 5 Ma
8 Me
  mental   critical Beta-
13/21 Hz
formal Alpha+
10/13 Hz
III YELLOW 575 to 565 570 D5 576,651 navel thought
 will, ego
imitative Alpha-
8/10 Hz
Yellow Orange 615 to 575 595 D#5
Eb5
610,94 7 Ve
9 Lu
  astral   tribal Theta
4/8 Hz
II ORANGE 645 to 615 630 E5 647,269 sacral drives, emotions feeling affect
I RED 720 to 645 682,5 F5 685,757 10 Sa root etheric instinct, survival libidinal delta
1/4 Hz
Dark Red 750 to 720 735 F#5
Gb5
726,535   physical  

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 ?

© Wim van den Dungen, Antwerp - 2017 - philo@sofiatopia.org l SiteMap
initiated : 01 I 2014 - last update : 27 X 2014 - version n°1