Numbers and Music
Introduction
Following is an overall view regarding the nature and development of harmony and music. Harmony, in this instance, is basically another word for theory and consists of the numerical descriptions of music and its division into component parts: thus, on the one hand melody, divided into intervals, chords, and scales; and on the other hand–rhythm. Apart from gaining acquaintance with the terminology, the technical and numerical content is mostly elementary and can be related to or assessed from a single and simple formula of melody-making, the common musical scale–Do Re Mi Fa Sol La Ti do. The branch of rhythm is more elementary still.
The development of music and harmony–a necessary part of our proofs–is matched as closely as possible with the development of society, that is, from the stages when tribes developed and prevailed to the formation and stages of class society. In order to properly define harmony I added a number of new terms and had to update or clarify some already existing ones. Unless otherwise indicated, the rest of the material is common to the subject matter but organized and improved upon in accordance with the theoretical and historical outline presented here.
RHYTHM
MELODY
Science has repeatedly drawn from the lessons of harmony. By classical antiquity it figured alongside such older branches as math, geometry, and astronomy with number (quantity) providing the cornerstone of ancient philosophy. Ancient harmony uncovered a definite link between quantity and quality in nature, that is, by comparing pitches (to which a quality was assigned) according to string length (wavelength, quantity). In short, the ancients defined the numerical form of the harmonic series and eventually came to reason that sound–a single pitch or tone–formed a series of waves in the atmosphere much like the waves that form when a stone is dropped into a body of water. This basic reasoning together with its useful application in music sufficed for the continuous development of harmony and its continuous influence on other branches of science and learning up to the present day. Marin Mersenne’s discovery of the actual harmonic series in 1722, although of great importance for science, really only reinforced and de-mystified what was already known.
The wavelengths of sound have linked harmony with the subject of light and color, and, in part, inspired the periodic table of elements. It has inspired a modern-day hypothesis concerning the nature of the universe called “string theory” and other ideas regarding the possible size and shape of the universe. Radio waves captured by the radioscope led many scientists to the Big Bang hypothesis regarding the formation of the universe. Gravitational waves, hypothesized by Einstein, were discovered in 2015. It can be seen therefore that while parts of harmony are specific to the art of music, many of its aspects are part of and guided by the natural, mathematical, and historical sciences.
Just about everything in harmony refers back to the harmonic series and musical scale. The harmonic series is a natural phenomenon while the musical scale is an historical product resulting from an accumulation of experience and knowledge in melody-making, instrumentation, and acoustics. Ultimately, even the harmonic series is assessed from the historical standpoint since it is a matter of how we make use of it in actual music.
Everything can be described by numbers or numerically related systems, such as by pitch-letters. The numerical systems of even the most fundamental harmony are various. There are the whole or rational numbers which describe the harmonic series–1: 2: 3: 4: 5: 6, etc.–The harmonic series occurs when we articulate a single tone either by voice or instrument. The musical scale on the other hand provides us with 7 basic tones. These may have “syllabic names”–Do Re Mi Fa Sol La Ti do–or pitch-letter names, for example, C D E F G A B c, or numerical names following the degrees or “steps” of the scale–1 2 3 4 5 6 7 1′.
The classification of the intervals mainly derives from the degrees of the scale, for example, the interval formed at the second degree of the scale is called a “second”, at the third degree a “third”, and so forth. However, 1 to itself (1) is called unison and 1 to 1′ (one prime) is called the octave, rather than the eighth. These names are valid of course wherever the interval is found, for example, the second is found not only between the first and second degrees of the scale, but also the second and third degrees, third and fourth degrees, and so forth.
The scale enables us to transform the quantities defining the harmonic series into practical musical qualities. These are represented by the letters, hence: C: c: g: c’: e’: g’, etc., and named by their interval class, for example, C: c, the octave. Conversely, the harmonic series enables us to pinpoint the overall structure of music: partials 1:2 outlines the scale in its classic form, 2:3:4 represents its division into chords, 4:5:6 forms a harmonic chord, 6:7:8 a melodic chord, 8:9 an interval called the whole tone or major second, and so forth.
For most purposes these qualities are divisible into concords or “blending sounds” and parts and discords or “clashing sounds” and parts. Eventually, however, harmony assumes a social character, that is, just like the music it intends to define, it reflects the time and place within which it arose and developed. Such historically established social reflections, considered as musical qualities to whatever extent possible, can then be transformed back into the numbers and letters common to music.
The harmonic series leads us to consider intervals as ratios (a comparison of vibration rates–frequencies–in their lowest numerical terms), for example, the octave, 2:1, or by wave-lengths, 1:2. Numerically rational harmony traces back to the Greek philosopher Pythagoras of the 6th century BCE, although its roots are definitely much more ancient. Another form of measurement, more practical and also very ancient, establishes the interval as a “distance” between two pitches–a logarithmic number. For example, between C and D is a distance of 1 whole step (“1”). Historically, both logarithmic and rational harmony developed mainly on stringed instruments.
A ratio and distance may have an identical spelling but are never exactly equal except in the case of the octave, where 2:1 = 6 steps. Thus, the rational whole tones 9:8 and 10:9 differ from the tempered whole tone (=1), or others close to it, not only in size but also, in certain particulars, in function–yet are all written 1 and spelled the same way, for example, C 1 D.
Finally, harmony also employs positive (+) and negative numbers (-) representing the sum total of sharps (#) and flats (b), respectively– …-1, 0, +1, +2… = …1b, 0, 1#, 2#… –a series of concords known as the cycle of fifths. The fifth is an interval equal to 3:2 (e.g., G:C) or 2 1/2 tones or 3^0. Hence, these can be treated as a series of exponents as well. The first known example of cyclic tuning was described in the hieroglyphs unearthed in Ugarit (located in Syria) and dated back to about 1800 BCE, hence, well in advance of Pythagoreans.
The development of music and harmony: Tribal Society
Hunter-Gatherers. In one commonly supported scenario, modern humans emerged in Africa some 200,000 years ago, give or take a rough 50,000 years. Following the rivers and coasts they began to migrate into Asia 100,000 years ago, give or take some 25,000 years, reaching Europe and Australia somewhere around 50,000 years ago. Around 25,000 years ago, give or take about 10,000 years, Asiatic tribes crossed the Bering Land Bridge, made possible by lower sea levels at the time, and began the settlement of America. Some of the most remote regions of the world were still being settled by tribal peoples right into our own era: Madagascar, remote Pacific Islands, and so forth.
During the course of the settlement of Europe, Asia, etc., society began to transition from the middle to upper paleolithic stage. A good example of early upper paleolithic culture is found among the Eurasian tribes, well known for their cave paintings and rock carvings, figurines including the “Venus” statuettes, jewelry of ivory beads, etc., and even their music as evident by the discovery of bone and tusk (ivory) flutes, and what are possibly rasps (scrapers) and the so-called bull-roarers. Their naturalistic paintings and figurines of animals contrast quite thoroughly with their either crude or fantastic imagery of humans.
Primitive musical cultures known in modern times give us at least a rough idea of what this very early music could have been like. These are characterized by: strains and chants based on a single chord or small interval respectively; simple wind and percussive instruments, often decorated and some regarded sacred; a tendency to name instruments by the material or implement out of which it was devised; attribution of music to dream spirits or other supernatural forces–the whole of music tending to be regarded a kind of “magical talking” and hardly distinguished beyond whatever its function: ritual, dancing, narrative, signalling, amusement, etc. Much like the wearing of masks in rituals, non-sensical utterances, “coded” song texts, or the masking of the voice added to the supernatural effect.
The lack of a specific term for music, or sometimes even singing or playing (an instrument) continued well beyond this stage. Take for example the term “music”, found in very similar forms in many languages today. It is derived from the name of the Greek goddesses known as the Muses, suggesting that at one time, among the Greeks, music (the actual melody and what not) was subordinate to poetry or poetic meter, i.e., one of the branches over which one of the Muses presided. Another example is found in Old English dremen (play music) or dream (music; mirth), thought to be related to the Germanic root drumm-, tromm- (drum). If so, it can be reasoned that at some point drumming must have been common and important enough to have become associated with festivity and music in general.
Percussive instruments at this stage include the hand and foot: clapping, body slapping, stomping; clappers such as stones, sticks of bamboo, bone or wood, or spears struck together; rasps made of bone, wood, etc.–some having a basket resonator; skin drum; log drum; stomping tube, stick, or pit; various rattles, hand-held or wrapped around the ankles, wrists, etc. Wind instruments include the trumpet or horn (conch, antler, tusk); bull-roarer; reed of leaf or grass; flutes, which include such instruments as the whistle and ocarina, having 0 to 4 holes; in Australia, the didjeridu.
The Australians and many Pacific Islanders generally preserve the most archaic forms and practices, whereas the northwestern Americans, particularly those along the Pacific coast, and some Eskimos, Amazonians, southwestern Africans, and southeastern Asian tribes represent a somewhat more developed stage.
Domesticating tribes to early civilization. Altogether this stage features a neolithic or polished stone industry, improved bow and arrow, pottery, domesticated plants and animals, brick making, and metallurgy. About midway through this stage we find what are generally thought of as the world’s first civilizations, societies which created things like astronomical calendars, megalithic ceremonial architecture and writing systems, engaged in extensive trade, and so forth, but we also find the beginnings of slavery and therefore of class society… The highest stage is marked by the introduction of iron tools and weapons and leads directly to the stage of class-based civilization, the example of Greece providing us with the classic model and many details of this transition.
Overall, music is characterized by melody in two chords–a musical scale–the main type which is called anhemitonic (“without semitones”); the attribution of musical items to specific gods, epic heroes or in some cases actual individuals; further development of the instrumentation and its uses, including the formation of mixed instrumental ensembles, and the formation of instrumental chords and scales as opposed to the smaller sequences and instrumental pairings in the stage of gathering and hunting.
Musically, a higher stage more or less coinciding with the Late Bronze Age and the Iron Age which followed upon it can be differentiated in the Eastern hemisphere–what I would describe as an Age of the Tone and String–or simply the “Tone Age”. It is distinguished above all by the further development of stringed instruments; a tendency to name instruments by their tonal characteristic or engineering; development of the heptatonic or 7-tone system; and various early theoretical advances.
A scale-forming flute marks the lower stage, found, for example, among most North American tribes at the time of European colonization. Other instruments, besides those carried over from the prior stage, include the frame, water and slit drums. Indicative of the higher stage are instrumental chords and scales formed by an ensemble of the particular instrument tuned to different pitches, including those comprised into a single apparatus. Percussive scales are especially common in Sub-Saharan Africa and East Asia–the xylophone (bala; marimba), thumb piano (mbira, zanze), drum ensemble, tongue drum or ensemble, gongs and metal plates, sounding stones, and bells. A tusk ensemble is found in Liberia; in Andean America the pan-pipes; in Meso-America evidence of the double pipe and a indegenous marimba. Among the Old World tribes and peoples preparing to advance to the stage of civil society proper (classical antiquity): pan-pipes, double and triple reed pipes, harp, lyre and lute.
Vocal melody was characteristically anhemitonic in this stage and remained so in many cases well beyond. The anhemitonic stage is indicated by such things as the continued use of pentatonic scales and instrumental tunings in Africa and Asia today; by the structure of the ancient Greek system; by trace evidence of Germanic tribal music in the medieval system and music; and by “living” examples uncovered in the “backwoods” of Europe, the U.S., and other long-since heptatonic or 7-tone regions. Some of these have since become an important part of the modern national culture, or, like the blues and its derivatives, are often included among the international genres.
To be clear, it is not necessarily the number of different pitches in a given song (excluding any octaves) which determines whether or not a song is anhemitonic or diatonic, but rather the function of the different pitches.
Tone and String
The “Tone Age” is evident in the names of all kinds of instruments, the numerical and practical parts of harmony taking shape, and a variety of new associations characteristic of the epoch. These developments occurred closely alongside the development of stringed instruments which provided harmony with a decisive theoretical tool. The String established a definite, measurable relationship between its length and the corresponding pitch or tone. Etymologically there is actually very little difference between string and tone, although the possible connection has not been noted in any etymological dictionary as far as I know.
One of the simplest stringed instruments, and likely the oldest, was the single string musical bow. It is possible that the musical bow was fashioned out of or at least inspired by the hunter and warrior’s bow.
New World instruments do not appear to have advanced beyond the stage of the musical bow at the time of colonization. In fact, at least some of the New World bows have been shown to have a convincing African origin or etymology, for example, the well-known Brazilian berimbau, whereas any developed stringed instruments that are found, such as the so-called Eskimo and Apache fiddles (each having 1 or 2 strings and played with a violin-type bow), are usually considered the result of European contact. Others are native varieties of well known European, Arabian and Eastern instruments, for example, the Hawaiian ukulule, which was modelled after the small Portuguese guitar.
In the Eastern hemisphere on the other hand, the word “bow” (in forms like bent, bana, piba, etc.) sometimes survived in the name of ancient and, consequently, modern advanced stringed instruments, for example, the Sumerian pan-tur or “small bow”; Egyptian beent or vini; the classical Greek pandoura and still later the Italian mandóla. In India the vina, vana, pena or bana. According to Sachs, the Chinese pipa–in name and origin–and its linguistic derivations, the Korean bipa and Japanese biwa. The Ghanian obenta which is actually a mouth bow may also share a common origin linguistically. Among the Greeks the bow and arrow appeared as Apollo’s weapon and the lyra his favorite instrument.
The spread of “tone-type” roots is also very noticeable in much of the Eastern hemisphere. In the more obvious cases a sense of sound is conveyed, ranging from simple imitative words to corresponding mechanical and technical descriptions. In English, for example, words like thump, tap, ding, din (roar), drone (hum), and dream (Old English “mirth; music”); thunder and proper names etymologically related to or honoring the prehistoric thunder-god in northern Europe: Thursday (“Thor’s day); personal or family names, for example, Donner; and numerous place names, especially in England, Iceland, and so forth. The word threnody (lamentation) derives from Greek (possibly “tomb song”); of Latin origin tremor and tremble (shake), tempest (storm) also appear to be related.
The “tone-type” could also mean to play (to beat, strike, or sound) the instrument, as in to thrum, to strum; typically, however, it is found in the name of the instrument itself: drum, tabor, timpano, timbal, tamborine, tom-tom, etc.; tromba, trumpet, trombone, tuba (truba), drone, as an instrument; trump (the so-called Jew’s harp).
Other languages display a similar range of uses, for example, Italian tuppe tuppe (“knock-knock”), tuono (thunder), the surname Tamburrino (literally “little drum”), tammorra (tamborine), campana (bell), cupa cupa (a friction drum), etc., ancient Greek tympanon (tamborine), kymbalon (cymbals). The sense of “drum” is common in both the West and Middle East: Hebrew tof, Iranian tap, Sardinian tumbarinu, etc., but in the east the same root, sometimes the same word–or nearly so–is also regularly associated with lutes with their drum-like body, for example, the ancient Hittite tibula, the Indian tumburu or tumbaru, Russian domra, etc. Today of course most of these have a head made of wood.
A number of learned uses common to Harmony also belong to the “tone-type”: timbre or tamber (tone color, pitch), tempo (time, beat), temper (proportion), temperament (tuning), tablature (a form of notation), tenor, etc. In fact, there are many other words which may have some distant etymological connection and bearing on our subject, for examples, temple, tomb, and throne which are all associated with music. After all, at this still early stage, music, harmony, and instrumentation are still closely attached and subordinate to function rather than art and science.
The word “string” appears more or less related to Tone etymologically. Related words include stretch, strain (song), extend, tension, tendon. The Central Asiatic word for “string”–tar–can thus be understood as either a stretched-string or its tone, since tar has the same root as tone. The sitar or “3-strings (or tones)” gave rise to Homer’s kitharis, later the classical kithara, the guitar, zither, and so forth. The Greek word for “string”–chord–might also be related.
The more the twang Other instruments were simply too abherrent or indefinite in the production of tone.
Particularly in the case of the String, “tone” is further qualified by the action of tightening the string or maintaining it under tension, so as to establish a clear musical pitch. Hence, there is a similarity between
applied to the String and consequently, owing to the advantages of the String as a theoretical tool, to the numerical and practical parts of harmony taking shape.
word for string chord and related directly to the word tone or tension; the Greek term—chord–widely used in harmony proper, may also be related but is simply “string”. the latter in the 7-pitch scale or heptachord, and so forth.
The various scales and tones (tonoi) were named after the Greek and neighboring tribes. At this still early stage, nevertheless, Tone is still attached to a variety of instruments–and Music, in the case of the Greeks, to poetry and poetic meter. t. Music and the developing harmony were also intermingled with mythological imaginings. ; the
notated songs discovered in Ugarit were often signed by individual composers.
also widespread in Egypt and Italy; crotalos or clappers of shell, wood, brass; sistra (Egypt),
Class society: Ancient stage
In the West (Greece), the stage of ancient music began with Terpander and the establishment of the octave scale, and Pythagoras who defined its rational form (2:1) using the monochord. The first 4 partials of the harmonic series defined the concords–octave, fifth, and the fourth (the last already recognized by way of the tetrachord)–giving music its basic structure. The tone--either as a pitch or interval–defined the harmony and its relation to the other harmonies. Variations of the harmony were defined by the chord types: diatonic (having a “solid color”), chromatic (“soft color”), and enharmonic (“shade”). The remaining intervals were defined in relation to these and also fall under the general category of discords. Up until Aristoxenos’ time the chromatic chord, although having fallen out of popular use, was still classified alongside the diatonic as a vocal rather than instrumental genus. Aristoxenos further developed the system of intervals, chords, and scales. Unlike the Pythagoreans, he defined intervals as “distances” (logorithmically), establishing in effect a system of equal temperament. Next, the Alexandrians named the partials (ratios) used in the construction of chords, further adapting the notation. The interval 5:4 (Do Mi, a concord today) remained a type of discord for the remainder of antiquity, at least in name, as it was contrasted with the fourth. Alexandrian harmony, culminating in Ptolemy’s Harmonika (2nd cent. AD), ultimately incorporated the major accomplishments of their predecessors. Greek theory, particularly by way of Boethius’ De institutione musica (Rome, 6th cent. AD), supplied theoretical material to the polyphonic-modal system of medieval Europe. Greek harmony was also further developed and transmitted to the Middle East by way of Alfarabi’s Kitab al Musiqi (10th cent. AD) and other important works, in this way returning to the lands where many of its concepts actually originated. Western European stringed instruments usually continue kithara (e.g., guitar, zither) or pandoura (from Sumerian pantur, but introduced in classical times, mandola, bandurria, etc.). Other features of the ancient stage include the establishment of musical schools and competitions (at the Pythian games, the Olympics, the Roman Capitolia; etc.); the refinement of the aulos and kithara, etc., and invention of the organ; development of philosophy (largely inspired by harmony) and music history; the rapid development of music as an art form, and the further development of diatonic song.
Medieval or modal stage
Under the domination of the Church a new musical system began to take shape in medieval Europe, usually borrowing its definitions and concepts from the ancient Greek and Roman manuscripts. This system was based on 4 modes rather than the 7 of the ancients (each having an alternate plagal form, however) ; modes 1 and 6 were rejected being considered contrary to the doctrines of Christianity while mode 7 was rejected on the grounds that it was contrary to music. Eventually composers in Western Europe began to accompany the church songs (mostly chants) with a supporting melody, a practice they called polyphony (“many sounds”); theoretical works soon followed. Polyphony, including droning, is as old as music but having been mainly limited and spontaneous its theoretical description did not develop before this time. From the beginning all tunings were diatonic and the perfect fifth quietly assumed the role of “first concord”. These things together with an increasingly refined polyphony and the development and growing use of alteration symbols (sharps and flats) eventually transformed the church modes into the major or minor scale–the system of Western harmony or classical tonality. Thus, unlike the break between the ancient and medieval system, the medieval system evolved directly into the tonal one. This stage is also widely represented in North Africa, the Middle and Far East (the latter which possessed a pentatonic system, however). Characteristic instruments of this stage include the improved organ and the hurdy-gurdy (Europe), the Arabian lute and its many derivatives, bagpipes (parts of Europe and the Middle East), vina (Indian lute), koto (Japanese zither), etc.
Classical tonal stage
During the Renaissance the Italian musician Zarlino defined the perfect chord Do Mi Sol (the major triad) based on partials 4:5:6 thereby extending the concords from the 4th partial (the tetrad) to the 6th partial (the senario). The two “worldly” modes once rejected during the Middle Ages were admitted into the system of modes eventually becoming the major and minor scales (modes 1 and 6 respectively), while the original medieval modes appeared more and more pointless and unnatural. The elaborate polyphony became known as counterpoint. The 7th partial and 7th mode remained beyond the limits of tonality. Instruments include the harpsichord and piano, guitar, violin, mandolin, accordion, concertina, harmonica, sophisticated woodwinds such as the clarinet and oboe, and brass instruments of all kinds. By the early 18th century most instruments were tuned in equal temperament.
During the course of this period new forms of music were being brought to the attention of Western scholars. Unlike the medieval modes of the West which seemed to have no solid practical value, the pentatonic modes of the East defined a “living” music. Was the original Western scale pentatonic too? Survivals of pentatonic melodies uncovered in Scotland, Ireland, and eventually many other regions, and a re-evaluation of the ancient Greek texts suggested that this was no mere coincidence.
Alongside the question of our musical history and development, many new forms of music and harmony began to emerge in the late 19th and early 20th centuries. Some of these appeared in opposition to the “tyranny” of Western harmony, but more often than not were simply a “disguised” or expanded version of tonal music. To help evaluate these and the historical systems we need the numbers and letters and an outline of the developmental stages of society.
Modern harmony and music
Although limited by their level of musical development the basic numerical foundation in harmony was defined by the ancient Greeks. The modern system of letters (or syllables) was the work of the medieval Arabs, Indians, and Italians when they named the pitches of the octave (2:1) with the same letter, for example c:C. The tonalists further develop the system of sharps and flats ; here they act as positive and negative numbers of music.
Modern harmony, as contrasted with classical tonal and 20th century harmony, establishes the concords up to the 7th partial of the harmonic series. The name of the 7th partial is derived from the identical enharmonic interval 36:35. In fact, all prime number-partials are ultimately derived from partials 2 and 3. In the following chart, 1 = C ; the intervals by classification and size are shown on top and the distance by cents (exact only in the case of the octave–8va = 1200c) at the bottom.
The harmonic series
C C G C E G Bb C D E F# G Ab Bb B C Db D D# E F F# G G
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
It is also instructive to establish the name of the neutral intervals found in logarithmic harmony, for example, the neutral third Ep (“E-quarter tone flat”). Ep can be derived from /23 =/G =EbE.–The names of the prime number-partials and neutral intervals lead to many tuning systems (cycles), some which are already indicated, for example, septimal tuning. These systems show that the precise value (distance or ratio) represented by the letter is defined by the tuning system being employed.
Harmonic chord types were greatly increased during the 20th century to include chord clusters, chords by fourths, by seconds, and others. In addition to these we can re-introduce and expand the system of melodic chords. A part of the system of tetrachords according to ratios and spelling is shown below along with the intervallic distances by cents (approximated). It is similar to the display of chords defined by Ptolemy (in which the ratios appeared in reverse order, however) and is even more like the system of Alfarabi. Following these is a chart of the chords of classical tonality, shown for the sake of clarity.
The collection of “living” modes and the invention of new ones, including their arrangement into related groups (i.e., by modes) continues even now. Traditional tunings and scales are often adapted, if only crudely, to the equal temperament of our modern instruments. What remains lacking, often in its entirety, is the theoretical foundation : a clear and complete definition of modes and keys; “mode-notes” and key-notes ; their actual names ; their number ; general quality ; and their arrangement into the tuning cycles. Some of these features are indicated below with respect to the classical diatonic modes along with new names. The new names approximate the order in which the world was populated by the modern human race.
Like keynotes, the modes may be transferred into a cycle or tempered circle as “mode-notes”. In these forms, however, the cyclic “colors” arising from the concords, and the stepwise “colors” of the scale no longer align. The classical Greeks appear to have encountered and adressed this contradiction, at least in part, by extending the use of the prefix hypo– (“under”)–already used in the term Hypolydian, an old harmony–thus renaming the Aeolian harmony “Hypodorian”, soon after the Ionian became the “Hypophrygian” and, in some versions, the Mixolydian, became the “Hyperdorian” at the higher octave (hyper- = “over”). A whole system of keys, 7, 13, or 15 in number, was soon fashioned along these lines.
The contradiction between the scale and cycle is also clearly manifest in the difference between just intonation and Pythagorean tuning.
The arrangement below integrates the modes and keys into a single system. The numbers themselves prove that there are as many modes as there are keys and that the modes and keys are not named arbitrarily, but by their numerical class, specific size, and further, according to the tuning system being used. The names of the modes and keys need only follow the logical progression of numbers (alterations or “colors”). Here, however, the names are borrowed from various existing modes, which is also useful and instructive. The form below is tempered and subdivided by the neutral intervals. It can be shaped into the familiar circle of fifths.
Acoustics, instrumentation and song since the 20th century.
The Numbers of music
Apart from the description of instruments by their number of strings, as in Homer’s kitharis (“3-strings”), the earliest musical-numerical associations were linked closely with mythology. From these we are able to tentatively reconstruct many of the elements which later formed the comprehensive subject of harmony. These described the scale (1 4 7), or segments of it as in the formulation Dorian: Phrygian: Lydian (or in reverse). The former supply the letter-base (7) ; the first defined concord (4, the perfect fourth; therefore the first chord) ; a cycle of perfect fourths B E A. This is equal to -1, 0, +1, thus positive and negative exponents ; also, the scale divided into 2 chords. The latter define the interval as the distance between pitches ; discord (1 or 2 whole tones) ; the beginning of diatonic tuning, modes and keys and their differentiation.
The Pythagoreans broke with the old mythological conceptions, however, using the whole or rational number to describe not only music but also the essence of nature. These ideas were inspired in no small way by the discovery of the numerical form of the harmonic series and the development of other mathematical branches. They likened the nature of the universe and a person’s “soul” (being) to the tuning of the string according to certain numerical proportions. Some Greeks, such as Aristotle, firmly opposed such idealist interpretations, but by then the Pythagoreans themselves had already undermined their own (rational) number philosophy when they had discovered irrational or “nameless” numbers. In any case, many of their both scientific and idealist concepts were transmitted to the classical philosophers (especially Plato), the Alexandrians, the Romans and down through the ages. As Christianity developed some of the old classical associations were demonized while others were simply re-cast in the light of the new religion and morality. Instead of the “song of the Sirens” the scale now appears as the “choir of angels”. One of the most important of these associations, the diabolus (the “devil”) or tritone, was passed on to tonality where, together with the tonic and the concords forming the senario, defined not only tonality but some of the most fundamental aspects of melody-making.
The heptachord 1 4 7 (B E A). Our earliest knowledge of the Greek scale is the heptachord already divided into 2 melodic chords sharing a common note on the fourth string (the mese), which likely acted as a keynote. The chords represent two contrasting parts of the melody. Similarly, 4 was associated with Hermes the “trickster” and 7 with “steadfast” Apollo, two gods opposite in character and closely associated with the lyra. There is no octave (=8) in this form and it is possible that the addition of the pitch forming the octave would have created a relatively unstable harmony, the Mixolydian (mode 7), which later Greeks identified as their first mode. Therefore, contrary to the restless 7 and stable 8 (octave) of tonality, we find opposite qualities associated with these numbers in prehistoric times. In the Biblical creation story, for example, 7 appears as a day of rest. And in the opening scene of The Odyssey, as the gods sat in council at daybreak, Athena states to Zeus “Father, I have nothing more to say about Aigisthos…he ruled Mycene in its wealth for 7 years and in the 8th he met his ruin”. The ruins of the Temple of Fortuna along the banks of the Tiber in Rome reveal 4 columns front and back and 7 along the sides.–The heptachord was eventually replaced with the octave scale, but not before leaving behind for future generations the 7-tone system. For ourselves the constellation called the Lyra remains its shining monument: It was said to have been Orpheios’ lyra, washed up on the beach at Antissa where it was found by the local fishermen. From there it was passed on to the music director Terpander for examination, and then finally placed amongs the stars by Zeus. (The Lyra was also thought to have had 4 strings–a tribute then to the concord, the great discovery of the ancients and of particular importance among the Greeks, such that they generally assumed their tetrachord constituted the original scale).
The “diatonic trichord” 6:6:6. Long before the appearance of the triple sixes in the New Testament and its association with the “beast” (thought to represent Rome, capitol of the enslaving Roman Empire), it appears as a lucky combination in the story of Agamemnon’s return to Mycene after the end of the Trojan war. These same numbers can be derived from music in a very ancient association of musical forms : Dorian : Phrygian : Lydian (= 6:6:6 or C:D:E), each 6 representing an exponent, as in 3^6. The exponent is simply the number of alterations to the hexachord. These 3 modes were closely associated with the rise of diatonic melody and the differentiation between mode and key. The combination C D E, not in itself unusual in any way, can naturally be found in earlier stages of musical development. Sachs, for example, displayed a melody of the Ona of the Tierra del Fuego using precisely this combination.* In any case, the hexachord introduces another number into harmony, zero, as in 3^0, the significance of which we will get into later.
Seven. From China to the lands of the Mediterranean Sea this number had a certain mystical power attached to it as far back as we can tell. Perhaps this began in the base 10 counting system used by the ancients, where each of the first 10 numbers, except 7, were related to some other number. Eventually, however, this number was to play a major role in music. In Greek mythology it was the number associated with Apollo, the god of the 7-string lyra and founder of Greek cities. (Many non-Greek cities were also associated with this number, for example, the 7 hills of Rome). The massive stone walls of Thebes, a city with 7 gateways, were said to have placed themselves there by the music of Amphion’s lyra. Today it often appears as a lucky number just as it was in pre-Christian times, but it is also sometimes associated with evil, such as it often appears in the New Testament. In either case, however, there is often another 7 found alongside it and having an opposite attribute–for example, the 7 male youths and 7 maidens of Athens paid annually as tribute to King Minos of Crete. (As the story goes, they were forced to enter the Labyrinth where they would eventually lose their way in its intricate passageways and be devoured by the Minotaur residing within).
Unity (All) or 1. This is commonly associated with the fundamental of the harmonic series, or, what is actually not identical, the keynote of the musical scale, or the first mode and first key. Unity has been likened to many things: Fire according to Pythagoras, God according to Christianity as well as some Greek philosophers before that.
The octave 2:1. Two was a “feminine number”, the number of plurality or the contrary number of the ancients. The octave 2:1 harmonized or joined together the “contraries” (opposites), that is, it “blended” the different proportions giving rise to the fifth (3:2) and fourth (4:3). This established two disjunct fourths plus a central tone, or two overlapping fifths. The chords represented contrasting parts within the melody, creating a sense of direction. Today, the octave is usually considered the most perfect of concords (consonances), the two sounds blending completely into one another, one the higher “image” of the other. In antiquity, however, it was apparent that the octave could also lead to dissonance depending on the internal display of intervals: plurality, but in a negative sense, that is, a kind of music having more than one center, or a music that could meander into the “wrong” tonality, such as the case with the “feminine” Mixolydian harmony, the oldest octochord among the Greeks. It is for this reason that early theory and learned music generally stayed clear of the octave or “eighth string” in early Eastern and Western antiquity. Musicians who attempted to play with more than 7 strings (incorporate the octave) were slapped with fines in archaic Argos and Sparta. Ultimately, however, the radical changes taking place in Greek society–the formation of class society and the state–swept aside the old and archaic forms and concepts, allowing the arts and sciences to flourish like never before. It was in this condition that changes in the musical system, including full recognition of the octave, were able to take hold. It was the dawn of Greek civilization as reflected in the realm of music.
Three. The universal or divine number of the ancients. Like all odd numbers it was thought to be “masculine”: the Holy Trinity, the 3 Kings, the Temple of Jupiter in Rome (originally dedicated to 3 male divinities), etc., but the Greek and Italian mythology preserves a much older association : the 3 faces of the Italian goddess Diana; the 3 sisters known by Greeks as the Erinyes, and many others. The Pythagoreans defined the musical proportions by the number 3 : the arithmetic proportion or 2:3:4 which divided the octave by the fifth, and the harmonic proportion 3:4:6 which divided it by the fourth, their oldest and most important concord, hence, its name. Thus, 3 is the number which divides the octave into chords. It is also the number, however, which represents the modulations of the key or mode. Finally, as proven earlier, the names of the pitch-letters are ultimately derived from 3.
The arithmetic proportion follows the natural progression of numbers and its importance arises particularly with the use of the octave. That is, once the octave was formally introduced the ancients had to “contend” with the fifth, figure out its proper relation to everything else. Unlike later harmony in which the fifth acts as “dominant” in the Greek system the fourth remained “first concord”.
Four. This number is associated with many things: the tetrachord which is bound by the perfect fourth; the tetrad which is equal to the first 4 partials of the harmonic series and which contained all of the ancient concords.
The “fourfold” or tetraktys 6:8:9:12 (E:A:B:e). Many myths were told about Pythagoras, owing to his great wisdom and contributions to music, astronomy, math, and so forth. His community of followers claimed he was the son of Apollo, could hear the music of the heavens and perform miracles. He was said to have descended into Hades where he came across 4 blacksmiths working at their anvils and reproducing the sounds of the tetraktys with their clanging hammers. This same figure appears again in Pythagoras’ incredible proposition called the “harmony of the spheres”, which claimed that the movement of the celestial bodies (the sun, moon, known planets, and stars) actually produced the tones of the musical scale. In the later, classical version of his proposition, the tone found below the tetraktys, the Dorian key “D”, was likened to the earth around which the celestial bodies, which formed the Dorian mode on the theoretical gamut, revolved : D + E F G A B C D E. The earth does indeed seem stationary as we observe the procession of the sun, moon, planets, and stars across the sky. Hence, the earth (D) was “silent” and did not participate in the actual scale. It was simply the place from which the magnificent visible and audible harmony could be contemplated and penetrate the soul.–But can the concords of harmony actually “penetrate a person’s soul”?–Yes: in the sense that the vibrations forming the octave, fifth and fourth actually stimulate the inner ear allowing us to hear these as “correct sounds” (Hindemith). Following is one version of the celestial harmony:
Earth 1 Moon 1/2 Mercury 1 Venus 1 Sun 1 Mars 1/2 Jupiter 1 Saturn 1 Constellations
It has been wondered why the days of the week as found in ancient Greece and Rome, and consequently, in modified versions, in most of the Latin and Germanic countries today, follow a different order than that assigned by the ancients to the planets. Although there are various theories out there, I believe the real answer (one which I have not seen elsewhere) is provided by harmony since the weekdays follow the order of the cycle of fifths:
(Domenica) lunedi martedi mercoledi giovedi venerdi (sabato)
Sunday Monday Tuesday Wednesday Thursday Friday Saturday
4 1 5 2 6 3 7
The Pythagorean “L”
The Greeks called it the lambdoma, taking its name from the Greek letter “L” (^, lambda). The series 1:2:4:8 is in geometric proportion since 1/2 = 2/4 = 4/8. It represents a series of octaves, for example, C: C’: C” : C”’. The series 1:3:9:27 represented a geometric series based on the number 3 and established a cycle of perfect fourths C: F: Bb: Eb, although it would represent a cycle of perfect fifths today as shown above. In geometry each series represents the difference between the point, line, plane, and solid: x^0, x^1, x^2, x^3, while from the ancient musical standpoint a tone, chord, heptachord, and harmony (octave scale). This famous figure was thought to have been created by Pythagoras, having come to him in a vision it was said, and was first recorded by Plato.
Five. The human number.
Six. In Greek harmony it was the Phrygian number, since they were credited with adding the sixth step to the musical scale, thus forming the hexachord. The Pythagoreans defined it as a perfect (complete) number since 1 + 2 + 3 = 1 x 2 x 3.
The whole tone 9:8 or “1-step”. It is first clearly identified as a tone added to the heptachord to form the octave scale (B C D E F G A + B). Soon after it appeared as the disjunction between two melodic chords (E F G A + B C D E), and finally as a “key” (tonos) (A + B C D E F G A). The harmonies were defined by its location within the octave : the lower the mode, the higher the tone. The low, “slack” modes–therefore, high tones–were regarded feminine and suitable for certain kinds of music and purposes ; conversely, the high modes and low tones were “masculine”. The Pythagoreans likewise : A low pitch moves slowly like the planet Saturn, and is therefore the farthest from us. Likewise the names of the notes : nete, the highest note (pitch), means “lowest”, while hypate the lowest note means “highest”. Finally, the Hypolydian or “Below-the-Lydian” is located a fourth above the Lydian harmony. This reasoning allowed the Greeks to organize their tonal space on the monochord and theoretical gamuts, assigning a distinct quality (ethos) to each key and mode. In some ways the tone could be regarded a microcosm of the ancient system and perhaps for this reason they called 9 “music” and even changed the number of Muses to 9 so as to accomodate their reasoning.
The tritone /2 or (9:8)^3. The tritone defines the middle of the octave (2:1) or any interval equal or about equal to three whole tones. The /2 represents the exact middle mathematically, such as in equal temperament, whereas in other systems the middle is approximate. During the Middle Ages it was called the diabolus in musica or “devil in music”. The 7th mode was rejected in the modal system because the instability of the tritone did not make for a useful keynote (tonic). Strangely enough, by removing the “devil” from the scale (either the pitch B or F), thus, establishing the hexachord, other “devils” arise : the triple sixes and, in fact, the triple zeroes. The number zero was certainly a problem for the rational number philosophy exhibited by the ancients.
Conclusion
In spite of the various elaborations of harmony presented here and the possibility of creating music by formula, most anyone can sing and create melody without such knowledge and through these activities convey the great message of our times.
Notes
equal temperaent resolves the difference between the melodically useless Pythagorean tuning (scale) and the keys and modes.
To simply define music as diatonic is fairly meaningless whether referring to Australians, Native Americans or Babylonians, since even in Greek there is a reference to the tone (key) and not simply the diatonic scale. Also in China (the lius). From a mathematical standpoint, the diatonic scale is open to quantitative alterations, relating the whole to the part; the alterations in the tones are also the whole seen from the standpoint of opposing forces, and they are exponential and actually limited for most musical (practical) purposes.
Music Theory 