Number in Time – Music or Harmony
written 1995 Revised 2020 revised 2024
References
Guides
Classical guides to Music followed here are: Theon of Smyrna, who has a lengthy discussion of ratio and proportion in his ‘Math Useful for Understanding Plato’. Plato is represented by his dialogue of ‘Timaeus’. Thomas Taylor, the 18th century translator of Plato, who may be considered a Neo-Platonist, also has an extensive section on proportion in his ‘Theoretic Arithmetic’.
Contemporary commentators include: Lawlor, whose book, ‘Sacred Geometry’, is helpful here, on the transition of geometry into music. Lawlor is also the translator of Theon. Kadleigh, taught architecture at the Royal College of Art, London until the late 1950’s. His ‘Basic Course in Architecture’ can be considered, for our purposes here, to be a primer on the four subjects. The actual document, found by your author in the RCA library, is an unpublished, ‘homemade’ book, of mimeographed sheets, possibly lecture handouts, bound between handmade wood veneer covers. LeMee is a professor of engineering at Cooper Union, in New York. In 1988, your author attended his workshop, ‘Sacred Geometry and Music’, held at the School of Sacred Arts, in New York city. LeMee’s course was insightful, particularly on the connections of sound and space. McLain, in his book, ‘The Pythagorean Plato’, interprets Platonic symbolism from a musicological viewpoint.
Musical Consonance
Musical harmony can be said to be composed of sounds in proportional or ‘consonant’ ratios. Music’s opposite, noise, may be said to be composed of sounds that are ‘dissonant’. Consonance implies an integrated ‘action together’ whereas dissonance implies divergent or unrelated actions. The discovery of the consonant ratios which relate number, geometry and music is attributed to Pythagoras but knowledge of them, as we will discuss in later chapters, is probably much older.
Geometry may be thought of as describing visual or to a certain extent, tactile relations that require space. Geometric constructions may be thought of as still moments; like snapshots of becoming. Music is temporal. Sounds manifest through regular movement of matter in time, and sound can be described by vibrations or waves.
A vibration is like a cycle; it goes through the stages of coming into being, being, and going out of being. The cycle may be thought of as a pulse, revolution or wavelength; it can be represented graphically by a wave or, as we saw in the Geometry chapter, a point or circle, figure 1a. This may be thought of as a one dimensional movement in which directionality is undifferentiated. The revolution and the wave have a two dimensional Dyadic nature which may be represented by utilizing the qualities of the Same and the Different. In the closed cycle, the pole or point of the Same remains immobile and the point of the different moves uniformly around it. When it returns to the place of beginning a circle is enclosed, figure 1b. In the open cycle the nature of the Same assumes a quality of uniform linear directionality, while the Different continues to move around the Same. The path of the different now traces out the open wave, figure 1c.
Study of the consonant or structural patterns of vibration can thus be considered analogous to the study of being itself. Plato, in the Timaeus [47], says, “…all audible musical sound is given us for the sake of harmony, which has motions akin to the orbits in our soul and which is … a heaven sent ally in reducing to order and harmony any disharmony within us.” Or, as it has also been translated (by Bury), “…restoring order and concord within the Soul”.
Music and number.
Recalling the principle of the Monad, One can be taken as the beginning, as well as the all-encompassing, and may be represented by the circle, figure 02a; In terms of a ratio, Unity may be stated 1:1; musically, Unity might be called the fundamental. But, as we have seen, complete, fundamental unity is unmanifest, unperceivable. In order for ‘something’ to be perceivable there must be a difference, that is, a differentiation or separation, a breaking of the unity. The first step in this differentiation is for One to transform into Two. In music this relationship of 2:1 is called the octave, figure 02b. The 2:1 ratio can be musically expressed through the length of strings or pipes. Musical theory was taught at the ancient Platonic Academy with a one or two stringed instrument known as a monochord. If a string of any given length, or interval of 1, is vibrated, it produces a note called here the fundamental. A string twice as long, or an interval of 2, will vibrate at a rate that produces a sound one octave lower than the fundamental. Twoness, as we noted in the discussion of Arithmetic and Geometry, can be taken to represent its own singular quality, as well as generating Threeness by the addition of Oneness and Twoness. In music the 3:2 ratio is known as a ‘fifth’. That is, if the octaval distance is from 1 to 2, a string when plucked or struck at the 3/2 point will produce a ‘consonant’ sound five notes lower than the fundamental; a geometric method of locating this point is shown in figure 2c. The next step in the manifestation of number is Fourness; which can be considered as both a doubling of two and a unit increase from three. The ratio 4/3 is known in music as a ‘fourth’. That is, with an octaval distance of 1 to 2, a string when plucked or struck at the 4/3 point will produce a ‘consonant’ sound four notes lower than the fundamental, figure 2d.
The first four numbers can be seen to form four ratios; 1:1, 2:1, 3;2 and 4:3, these in turn define the octave and its two essential musical divisions. These first four numbers, as we saw in the Arithmetic session, can also be taken to represent the four levels of the Tetraktys and, as we saw in the Geometry session, the four stages of becoming; point, line, plane, and solid. Here we may extend the parallel, and see four ‘stages’ of musical consonance, a fundamental, its octave, its fifth consonance, and its fourth consonance.
Ratio, Proportion, Logos.
A ratio may be described as a comparative relationship between two numbers such as 2:1. A proportion, Theon says [MUUP, p49], “…is the relationship of ratios to each other, as 2 is to 1 so 8 is to 4.” This is an example of a four term, discontinuous proportion, a:b :: c:d. It is also possible to have a three term, continuous proportion such as a:b :: b:c. In such a proportion the first and last term are called the ‘extremes’ and the middle term is called the ‘mean’. There are three types of proportions used in ancient musical theory, which generate the three initial ratios found to exist within the octave.
1. In the Geometric proportion, the product of the extremes equals the square of the mean, b2 = ac, b = √ac. For example, 2:4:8, where 4 = √2×8. Or, it may be described verbally, that ‘the mean differs from each extreme by the same ratio’; in this example, the double ratio, 4 = 2×2 and 8 = 4×2.
2. In the Arithmetic proportion, the sum of the extremes is equal to twice the mean, 2b = a+c or b = a+c/2, for example, 2:3:4 where 3 = (2+4)/2. Or it can also be said that ‘the mean differs from each extreme by the same number’. In the above example 2, 3 and 4 differ from each other by 1. In this way the arithmetic mean may be said to ‘give rise’ to the ‘natural’ series, 1, 2, 3, 4… . For this reason Taylor [TA, p 97)] considers the arithmetic mean as primary. In the context of music however, the geometric mean may be considered primary because it generates the octaval relationship.
3. The Harmonic proportion can be said to combine the qualities of the other two means. The harmonic mean is equal to the product of the extremes divided by the arithmetic mean, b = 2ac/a+c, for example, 2:3:6 where 3 = 2(2×6)/(2+6) = 24/8. It can also be said to be ‘the mean which exceeds one extreme, and is exceeded by the other extreme, by the same fraction of each extreme’. In our example, 3 is greater than 2 by 1/2 of 2 and 6 is greater than 3 by 1/2 of 6.
In discussing the three means, Lawlor [SG, p81)] calls our attention to their inter-relationships. First, he notes they contain “the general formula for all our basic mathematical operations. The arithmetic proportion contains the law for addition and its inverse, subtraction, and describes the relationship which gives the natural series of cardinal numbers 1, 2, 3, 4, 5, 6, etc. The geometric proportion contains the law for multiplication and its inversion, division, and describes the relationship which gives any series of geometric progressions”.
A little later he adds, “…the arithmetic proportion shows an equality of difference, but an inequality of ratio. Thus in the arithmetic proportion 3: 5: 7, 7-5 = 5-3, but 7/5 does not equal 5/3. A geometric proportion on the other hand, is characterized by an equality of ratio but an inequality of difference. Therefore, in the geometric proportion, 2: 4: 8, 4/2 = 8/4 but 4-2 = 2 does not equal 8-4 = 4.”
“The most important and mysterious character of the harmonic proportion is the fact that the inverse of every harmonic progression is an arithmetic progression. Thus 2, 3, 4, 5, is an ascending arithmetic progression while the inverse series, 1/2, 1/3, 1/4, 1/5, is a descending harmonic progression. In music it is the insertion of the harmonic and arithmetic means between the two extremes in double ratios – representing the octave double – which gives us the progression known as the ‘musical’ proportion, that is, 1, 4/3, 3/2, 2.”
Ratio and proportion were called ‘logos’ in ancient Greece. This word had a range of meanings all having to do with comparative relationships. The basic meaning was reasoning, with or without the aid of speech.
Theon [MUUP, p 48] writes, “The relation of proportion or ratio is also called this [logos] and it is in this sense that it is said that there is a relationship of one thing to another thing. It also has the meaning of the explanation of the elements of the Universe: the recognition of things which honor and are honored, and it is in this sense that we say: taking account of something , or not taking account of it (word of honor). The calculation of bankers is also called logos, as is the discourse of Demosthenese and Lysias in their written works (speech); the definition of things which explains their essence (reason), since it is to this that it applies; the syllogism and introduction; the tales of Lybius and fables. The name logos is also given to the eulogy, to the tale and to the proverb. … But, according to Plato, the word logos is used with four different meanings: for mental thought without words; for discourse proceeding from the mind and expressed by the voice; for the explanation of the elements of the universe; and the ratio of proportion. … The ratio of proportion of two terms is a certain relationship that these terms have to each other, such as the double or triple.”
One may thus define logos as ‘proportional relationship which gives meaning’ or, to use a phrase attributed to Gregory Bateson, ‘the difference that makes a difference’. This passage also indicates a common ground, in ancient times, of proportional relationship, music, geometry, number, and “the explanation of the elements of the universe”, to be found through rational discourse.
The Octave.
Continuing with the construction of the octave, figure 3; we have seen that it defined by the 2:1 ratio. Within the interval of the octave were placed the ratios, 3:2, the interval of a fifth, and 4:3, the interval of a fourth. We now look at the interval between the fourth and the fifth. The ratio between 4/3 and 3/2, that is, 3/2 divided by 4/3, is equal to 9/8. This ratio is known in music as a ‘tone’. To complete the division of the octave, two tones are placed in the interval between 1, the ‘fundamental’, and 4/3, the fourth; and two more between 3/2, the fifth, and 2, the octave. Both of these intervals are fourths, as 2 divided by 3/2 = 4/3. The first tone interval is 1 x 9/8 = 9/8. The second tone interval is 9/8 x 9/8 = 81/64. The next interval, which is between 81/64 and 4/3, is known as the semitone. In ancient Greece this interval was called the lemma, meaning the ‘remainder’ or ‘leftover’; that interval which is leftover after the two tones are placed between 1 and 4/3. The ratio of 4/3 to 81/64 is equal to 256/243. The interval from 4/3 to 3/2 is the initial tone, that is 4/3 x 9/8 = 36/24 = 3/2. Between 3/2 and 2 are placed two more tones and one semitone. 3/2 x 9/8 = 27/16, the first tone; 27/16 x 9/8 = 243/128, the second tone; and the lemma or semitone interval is 243/128 x 256/243 = 2, thus completing the divisions of the octave. The divisions of the octave are summed up in figure 03a.
The Modular Octave
The diatonic octave is built up out of intervals or modular units, the tone and semitone. The octave has eight notes, nodes or positions, placed at seven intervals; there are five tones and two semitones. These, in turn, resolve into basic blocks of fourths, fifths, and octaves, as shown in figure 4a. Two tones and a semitone build a fourth. Three tones and a semitone, or a fourth and a tone, build a fifth. An octave can be built of a fourth and a fifth, or two fourths and a tone.
Scales and Tempering.
The progression of the octave is called a scale. This is the same term used in architecture to mean size relationship, such as the size of a building element in relation to the size of a person; or as in drawing to ‘scale’, that is, in constant size relationship, such as 1:100, or 1/8″ = 1′-0″. In music, a scale is also known as a ‘mode’. Chart below is from Kadleigh [BCA, 6th Lecture], called the ‘Pythagorean’ octave.
The arithmetic is: 9/8 x 10/9 = 5/4, 5/4 x 16/15 = 4/3, 3/2 x 10/9 = 5/3, 5/3 x 9/8 = 15/8, 15/8 x 16/15 = 2.
Construction of chords may be done with fourths, fifths, tones and octaves, as shown in figure 4a. According to Kadleigh [BCA], Greek music recognized five consonances, fourth, fifth, octave, octave & fifth, and double octave, figure 4b. If you wish to use thirds to build with, however, this Pythagorean diatonic, natural scale will have to be adjusted or ‘tempered’. This is done by slightly modifying the interval pattern. The problem is that 9/8 of 9/8 = 81/64. It is the extra 1/64 that will prevent three thirds from adding up to an octave interval. If the second and sixth steps or intervals are adjusted to be in the ratio of 10/9 instead of 9/8 the problem can be overcome to the extent that a wider range of octaves can be played on any given instrument without the sound becoming noticeably discordant. The tempered octave, as presented by Jean LeMee [SGM] is shown in figure 4c
The Pythagorean scale is made up entirely of numbers that are powers of 2 and 3. With the tempered scale powers of 2, and 3 are also multiplied by 5. Notice that the number series is the same whether the octave starts with 2 or 3, but the series is shifted by one fifth. It also might be worthy of note that all of the numbers from 1 to 10 have been employed here except seven. Although there are seven intervals, seven itself is not part of any consonance; it does not ‘consort’ with the other numbers and so remains ‘virgin’.
Music in the Timaeus.
In the dialogue of Timaeus [36], we read, regarding the construction of the world soul, that after the three components of the soul, same, other, and being, were mixed into a single whole the creator, “…went on to make appropriate subdivisions… . He began the division as follows: he first marked off [Taylor says “he received” ] a section of the whole; and then another twice the size of the first; next a third, half again as much as the second and three times the first; a fourth, twice the size of the second; a fifth, three times the third; a sixth, eight times the first; a seventh, twenty seven times the first.” The increments are 1, 2, 3, 4, 8, 9, and 27. These are traditionally arranged in the lambda pattern, as in the table below and figure 5:
Timaeus continues, “Next he filled in the double and triple intervals by cutting off further sections and placing them in the gaps, so that there were two mean terms in each interval, one exceeding one extreme and being exceeded by the other by the same fraction of the extremes; the other exceeding and being exceeded by the same numerical amount.”
The first of these is the harmonic mean (b = 2ac/a+c), the second is the arithmetic mean (b = a+c/2). The geometric mean (b = √ac) has been used to generate the initial set of intervals. Within this set of geometric mean intervals is now placed harmonic and arithmetic intervals. The two series now look like this, figure 5a:
Timaeus now takes us to the next stage: “These links produced intervals of 3/2 and 4/3 and 9/8 within the previous intervals, and he went on to fill all the 4/3 intervals with 9/8 intervals. This left over, in each case, a remainder whose terms bore the numerical ratio 256/243. And at that stage the mixture from which these sections were separated, was all used up.” In this section of the text, the two legs of the lambda are treated again as a single series. The chart below shows the lambda numbers placed in one continuous series of octaves, figure 5b.
Timaeus continues, “The Divinity then took the whole fabric and cut it down the middle into two strips, which he placed crosswise at their middle points to form a shape like the letter ‘X’; he then bent the ends round in a circle and fastened them to each other opposite the point at which the strips crossed, to make two circles, one inner and one outer. And he endowed them with uniform motion in the same place, and named the movement of the outer circle after the nature of the Same, of the inner after the nature of the Different.”
To get a sense of what this means we may refer to McClain [TPP, p 64] who points out, “… Plato’s numbers have reciprical meanings – both mathematically and tonally – it is easy to read [the] lambda as the lower half of Plato’s cross, and to reflect its meanings in the upper half.” :
The relationship of the lambda to the octave is further elaborated by McClain [TTP, p64] and Critchlow [Intro to TOA, p 14] who both refer to Nicomachus figure 5c.
The bending of the strips of soul into circles is represented by McClain [TPP, p 66], figure 6. He designates the powers of 2 which generate the octave doubling as the circle of the Different and the powers of 3 which generate fifths, and by extension, fourths and tones, as the circle of the Same. He refers to this designation as ‘traditional’ but cites no source. It seems to this author equally plausible to reverse these and call the double series the circle of the Same (because it always returns to the same place – the fundamental) and the triple series the circle of the Different (because this, when crossed with the double series, generates the different tones). In any event McClain’s point here is that both legs of the lambda represent cyclic patterns that can be represented by circles as well as lines.
Concluding this part of the dialogue, Timaeus says, “The circle of the Same he caused to revolve from left to right, and the circle of the Different from right to left on an axis inclined to it, and made the master revolution that of the Same. For he left the circle of the same whole and undivided but slit the inner circle six times to make seven unequal circles, whose intervals were double or triple, three of each; and he made these circles revolve in contrary senses relative to each other, three of them at a similar speed, and four at speeds different from each other and from that of the first three but related proportionally.” The inner circle, the circle of the Different, or we might say, the disk of the Different is again treated as an octave, figure 6a
The remainder of the passage in the Timaeus describes the combination and relative movements of the various circles, on which the heavenly bodies are placed, and brings us to the point where music merges into astronomy. A possible visualization of the circles of the Same and the Different, and the six cuts in the Different, is shown in figure 07a. Another possible visualization is shown in figure 7b. This one, interestingly, relates not only 2-d geometry with music, but relates the creation of space illusion with music.End of Notes to Music
End of Notes to Music
Music – Additional Topics
References
The Helikon
In view of the geometry of the square and the half-square used in the development of the elements of classical architecture the following description of an ancient musical teaching instrument
West writes [HHE, p240], “Ptolemy [Harm. 2.2] describes a contrivance used by some theoreticians and poetically called the Helikon, after the mountain on which the Muses dwell. It had four strings and one diagonal bridge. The proportional lengths for the concords were constructed by geometrical procedures, given a square and the midpoints of its sides, figure 09. The verticals represent the strings; the position of the third one on the horizontal axis is pinpointed by the intersection of the lines BD and AE. AE represents the bridge. The various vibrating lengths then available, 3, 4, 6, 8, 9, 12, will yield all the intervals contained in the note-series E A e a b e’.” High E = 3, A = 4, e = 6, a = 8, b = 9, and low e’ = 12. These intervals form a double octave.
Division of the Octave
The octave as noted earlier [figure 4a] is built up of units of fifths, fourths and tones. The fifth may be considered a fourth plus a tone. Ancient Greek music recognized three ‘genera’ or patterns of division of the fourth. The most familiar to us is called the ‘diatonic’ genus in which the fourth is divided into two tones plus a semi-tone and this is the pattern given in the Timaeus; next is the ‘chromatic’, in which the fourth is divided into two semi-tones plus a tone and a half, called a minor third or trihemitone; and lastly is the ‘enharmonic’ genus, in which the fourth is divided into two quarter tones plus a double tone; figure 10a.
The enharmonic may have developed from the ‘pentatonic’ scale in which the fourth is divided into only three notes and two intervals, a double tone plus a semitone. Such a division is known as a ‘trichord’. In the three principle genera the fourth is divided into three intervals and four notes called a ‘tetrachord’. There are two ways of relating the tetrachords, figure 10b; ‘conjunct’, in which the first tetrachord ends on the same note that begins the second tetrachord; and ‘disjunct’, in which the end of one tetrachord and the start of the second is separated by a whole tone.
West notes [AGM, p 164] “In the fifth century BC music of a serious character appears to have been normally of the enharmonic type. … The name ‘enharmonic’ itself implies its prime status, as it means ‘in tune’; the enharmonic genus is quite often referred to simply as ‘harmonia’. ‘the standard tuning’, whereas the chromatic genus is marked by its name as a kind of deviation, a ‘coloring’. … Some maintained that chromatic music made men soft and effeminate, whereas enharmonic music made them manly. As for diatonic, we hear that some tragic enharmonic had an admixture of diatonic, … Pure diatonic music, however, may have been typical only of certain regions of the Greek world at this period. … In the South Italian Pythagorean school it seems to have been the Diatonic genus, not the enharmonic, that enjoyed the primary status in musical theory.”
“By the late fourth century BC the enharmonic had lost a good deal of ground. Aristoxenus upheld it as being the most sophisticated and beautiful of the genera, but lamented the fact that it was now largely neglected in favor of the chromatic, which people found easier to appreciate and more ingratiating. Musical fragments from the third and second centuries BC show a mixture of chromatic and diatonic. Those from the Roman period are almost wholly diatonic.” Thus in this chapter we refer principally to the diatonic genus and it is the diatonic that is used by Timaeus in his musical cosmology
Mode
The term ‘mode’ describes patterns of which notes within the octave form the fourths and which notes within each fourth are actually played. In modern terms the mode is a combination of string tuning, key, chord pattern and even some elements of melody. Webster’s New World Dictionary, 1962 defines mode as: “a manner or way of acting,” in music: any of the various forms in which the octave was arranged in classical Greece, according to certain fixed intervals between the tones. Thus we may define mode as a complex ‘way of playing’.
West writes [AGM, p177]: “The old scales set forth by Aristides Quintilianus and identified as Dorian, Phriygian, and so on, are described as ‘harmoniai’. The word means literally ‘tunings, attunements’. It is often translated as ‘mode’, a term which in music implies above all a distinctive series of intervals in the scale, though it usually has other connotations in addition, as in Western music the major mode is generally felt to be sunnier and more extrovert than the minor.”
“In fifth and fourth century BC literature we encounter numerous references to musical ‘harmoniai’. The differences between one and another in respect to their aesthetic and emotional qualities, and their effect on the dispositions of the listener, were considered a subject of great interest and importance. The ones most often mentioned are the Dorian, Phrygian, and Lydian, but we also hear of Ionian, Aeolian, Locrian, Mixolydian, and others.”
“What did the names stand for? [Aristides] tells us that the Dorian, the Phrygian and the other harmoniai, while having some elements in common, such as the enharmonic tetrachord, differed from one another in the selection of notes used, and to some extent in ambitus [range]. Each one had its own particular set of intervalic relationships, which we are not in a position to define. From the lists of notes we can read off all the intervals that might have been used in the melody, but the individuality of the mode would depend to a considerable extent on which ones actually were played, and with what emphasis, and also on the relative frequency of the several notes. No doubt there were other differential characteristics that do not appear from the bare scales. Some types of rhythm were probably felt to go with one mode rather than another. there may have been particular melodic formulae associated with particular modes. …” West also notes that certain musical instruments were associated with certain modes, such as the aulos [reeded pipe] and the lyre with the Dorian mode; reedless pipes with the Ionian; kithera [guitar] with the Phrygian; and the citharode [zither] with the Lydian mode.
West continues, “Certainly the modes appear to have been marked in some cases by … the degree to which high or low singing was required. … Plato mentions [a tense Lydian mode] together with Mixolydian as lamentatatory and suitable for women, and he distinguishes these from ‘slack’ modes, Ionian and Lydian, which are soft and suited to the symposium. … Aristotle remarks that the tense harmoniai are not easy for old men to sing; nature offers them the slack ones instead. Clearly a ‘tense’ mode involved more high notes … It was not necessarily higher in overall pitch … it might be that [two modes] occupied the same range but that in the tense mode the melody moved more in the upper part of that octave, and in the slack mode, in the lower. … It is significant that Plato uses tenseness and slackness as a principle of classification connected with ethos [distinguishing attitude or presence of objective elements].”
West now characterizes each of the modes [p 179]: “The Dorian mode was the most widely used in the fifth century and probably earlier, and it was always well regarded. … It was used for processionals, paeans, songs of love, and in tragedy, especially for laments. Clearly it was a versatile mode … compatible with more than one mood. … it was perceived as dignified and manly. … Plato [Lach. 188d, 193d; Resp. 398e-9c, Epistle 7.336c] approves it as the finest of modes, one that mirrors the character of the man who is brave in battle, self controlled and enduring in the face of death or disaster – a model for life. According to Aristotle [Pol. 1342b12, cf. 1340b4] ‘everyone agrees that it is the steadiest and the one that most has a manly character.”
“The Phrygian seems to have been appropriate to a range of moods from cheerful piety to wild excitement or religious frenzy. Plato [Resp. 399a-c] surprisingly ignores this latter aspect … and welcomes this mode to his ideal state – the only one he does admit besides the Dorian – as expressing the character of a man who, when there is no war or crisis, conducts his affairs sensibly in an unforced way, approaching God with prayers, and with his fellow man, making and responding to reasonable requirements. Aristotle on the other hand, writes: ‘… of the modes the Phrygian … [is] exciting (orgiastika) and emotional. This is evident in practice, for all Bacchic celebration and that sort of dancing … goes most appropriately with melodies in the Phrygian mode.'” West goes on to suggest that since several important religious pieces were in this mode which “Plato and Aristotle admired as having the power to arouse and inspire, and as revealing those who stand in need of the gods …”, this may go some way to explaining Plato’s hospitality to the Phrygian.
West notes that the Lydian was ‘soft and sympotic’; but that despite its ‘slackness’ it may have had a high range as Aristotle says it was suitable for boys to learn.
West continues [p182] “The other of Plato’s slack modes, the Ionian, presents several faces. For Plato it is soft and sympotic. But Heraclides Ponticus described it as ‘not pretty or merry, but tough and severe, with a certain dignity and weight’. …” West adds in a footnote: “Late sources characterize Ionian as variegated, elegant, or intellect sharpening and spiritually uplifting.”
“Mixolydian was, besides the Dorian, the principle mode used in tragedy. … It is described as emotional, appropriate to laments and the arousal of pity.”
West [p183] goes on to mention an early Aeolian mode which he identifies as the old name of the Hypodorian. Later [p 227-8] he identifies Locrian with Hypodorian, thus suggesting an identity of the Aeolian and Locrian modes. This is interesting because Locris, as well as being a region of southern Greece is also a Pythagorean town in southern Italy, the hometown of Plato’s Timaeus. West adds [p 184] that Hypodorian is “suited to the heroic figures on the stage; Hypodorian is grand and steady (we have seen Dorian described in the same terms) …”
We may note here that the eponymous ancestors Dorus and Ion who give their names to the Dorian and Ionian modes also give their names to the Doric and Ionic orders of architecture. Interestingly the modes and the orders have similar characters. Doric and Dorian are manly, outgoing, rational; Ionic and Ionian are considerd soft, feminine, emotional.
West now moves on to describe the later history of the modes. He points out that in the attempt to integrate the modes into some sort of master scale concept, the modes were to some extent ‘reduced’ by having their names replaced by key designations. The modes were also transposed to the diatonic octave by having the interior notes of the tetrachord adjusted. It is in this Hellenistic era that the terms genus, mode, key and species become rather hopelessly intertwined.
West defines species as “the seven different series of tone and semi-tone intervals obtained by taking each note of the octave scale in turn as the starting point. The effect of the theory of keys was to reduce the old harmoniai to scales differentiated by species. Already before Aristoxenus the seven species of the Enharmonic octave had been enumerated and labeled with names borrowed from traditional modal names: Mixolydian, Lydian, Phrygian, Dorian, Hypolydian, Hypophrygian [Ionian], Locrian [Aeolian]. The names were subsequently applied to the species of the diatonic octave too.”
West [p230] gives the following translation of the modes into diatonic terms: “Mixolydian for the B species, Lydian for the C, Phrygian D, Dorian E, Hypolydian F, Ionian G, Aeolian A.”
Modes and the Planets
Godwin [HHE, p 130] is quick to point out that correspondences such as the tones of the octave with the geocentric planetary system are purely symbolic and were so understood by the various authors who give us such systems. He writes [HHE, p 125] “… just as the planetary spheres are ranked like the rungs of a ladder set up to Heaven from Earth, so each should correspond to one pitch, the totality making a musical scale (from the Latin ‘scala’ = ladder …). Plato describes them thus in the ‘Myth of Er’ (Republic, X, 617b) as eight whorls or rings rotating around the earth, on each of which stands a Siren who sings her own note, all together making a harmony.”
“The idea behind the scales of figure 11 is attributed to the school of Pythagoras: it is that the planetary spheres are spaced at intervals comparable to those stoppings of a string that produce a scale. According to the Greek writers, Pythagoras reckoned the distance from the Earth to the Moon’s sphere to be 126,000 Stades, which he represented by the interval of a whole tone. The rest of the scale, with its tones, semi-tones, etc., then shows the relative distances of the other spheres from one another. The scale as the elder Pliny gives it [figure 11a] is in the
chromatic genus of the Greek Dorian mode. It consists of two disjunct chromatic tetrachords (… marked by square brackets), plus one added note at the bottom. Censorinus and Theon of Smyrna [figure 11b], however lower the note of the Fixed Stars, making it exactly an octave from that of the earth.”
Later Godwin adds [p127], “The longer scales of figure 11d,e,f add tones above and below the planets representing the lower spheres of the four elements and the higher worlds of the Angelic Intelligences. … To illustrate and explain these scales, both the anonymous writer published by Jacques Handschin and the seventeenth century Hermeticist Robert Fludd showed the tones along the string of a monochord. …”
The monochord or polychord, figure 12, was the standard device for the teaching of music in ancient and medieval times. The calibrated scales which were slipped below or alongside the chords to mark the intervals of the notes was called the ‘kanon’ or law. The instrument itself was also known by this term.
Godwin continues, “The monochord string can equally well symbolize a celestial measuring rod that determines the distances between the spheres. Therefore it can serve as a means for translating distances directly into pitch relationships. But Fludd, at least, certainly did not consider it as astronomically correct. He uses it as a symbol for the harmonious relationships between the different spheres or levels of being.”
Godwin adds [HHE, p132], “An important symbolic element common to [these planet scales] is that the ‘mese’, the central note of the Greek system is usually assigned to the Sun … . Fludd’s two-octave scales also express the special function of the Sun in his philosophy as God’s Tabernacle in the cosmos, by placing it at the crucial midpoint of his monochord string, sounding an octave on either side, figure 13 [a similar image is shown in figure 8a]. Fludd’s many cosmic scales can run up or down, cover one, two or three octaves, but the Sun is always placed in the middle, and the scale was always in the major mode which was the norm for his time, as its inversion, the Greek Dorian, was formerly.”
Godwin next notes that another type of planetary scale ranks the planets on the basis of their relative speed against the background of the fixed stars, figure 14. The Moon is the fastest with Mercury next, then Venus, the Sun, Mars, Jupiter and Saturn, the slowest. We may note, however, that this gives the same distribution of the planets as Godwin noted in the scales of the first type.
Godwin [HHE, p132] links the late diatonic form of the modes with the planets, “The diatonic sequence of notes from which both Dorian and major modes are drawn is founded on the perfect fifth and fourth … A series of six perfect fifths or fourths in either direction produces the seven different tones of a diatonic scale: B E A D G C F (4th up or 5ths down)”
“The ancient Egyptians, according to Dio Cassius (second -third century AD) used this sequence of fifths or fourths to relate the scale to the order of days in the week. Dio is usually interpreted as intending the scale of figure 15a (identical in form to figure 14a, when he instructs us to begin with Saturn and take every fourth planet (returning on the right when we overshoot on the left). The days of the week and their planetary rulers then appear with the notes, figure 15b.”
“Thus, at least, Dio Cassius is understood by most of those (and they are not many) who have concerned themselves with this passage, such as the Abbe Roussier, Fabre d’Olivert, Edmond Bailly, Ernest Britt and the Henschels. Britt uses these tones to deduce the planetary rulerships for each of the Greek modes. He uses the earlier versions of the modes made from two conjunct tetrachords meeting on a central note, the latter being the ‘final’ or tonic. The Dorian mode thus well earns its place of honor in Greek theory and in Plato’s philosophy, for it is the only mode whose scale exactly reproduces the cosmic order, centered on the sun. Britt’s table [is shown in figure 16].”
Godwin continues [HHE, p136], “In the Renaissance, before the eight church modes had finally been reduced to the modern major and minor, there were some last ditch efforts to recapture the ethos and the marvelous effects claimed for the modes in classical times. But while keeping their Greek names the modes had changed their natures. The Dorian octave species, which in Greece had been alternatively E D C B A G F E (diatonic), E C# C B A F# F E (chromatic), or E C B+ B A F E+ E (enharmonic), was now simply D E F G A B C D. Ramis de Pareja attached the Church modes to the eight notes of the planetary scale derived from Cicero (figure 14e), and described their ethos in a few words [figure 17].“
Concluding this part of his discussion Godwin writes [p137], “Cicero’s remark that the number seven seems to be the ‘key to the universe’ is the key to understanding all these seven note [planetary scales]. … [The number seven] holds pride of place in every department of esoteric study. whether it be the system of seven Chaldaean Planets which is still the mainstay of philosophical astrology, the seven metals of Alchemy, the seven Chakras (subtle centers of the body) in Yoga, [or] the seven Heavens of Islam, … the Septenary is sure to appear whenever the human being traverses a hidden structure of time or space.”
“Now it so happens that there are seven independently moving bodies (Greek ‘planetes’, wanderers) in the visible heavens, and it is equally the case that the progression of seven fifths, and seven only, completes the diatonic scale without introducing notes of a foreign chromatic order. The mind attuned to symbolic thought sees nothing coincidental in this. Still less is it inclined to treat the Septenary as the invention of archaic peoples so awed by the seven planets that they applied the number to everything else. No: both musical scale and visible planets are understood to manifest in their respective domains a more primal Septenary. There would only be cause for surprise if this were not the case, for instance if the musical scales in use all around the world tended to have eight or eleven notes to the octave. Therefore those who assign planetary qualities to the notes of the scale are acknowledging this seven-ness as a fundamental law of the universe: a law that must manifest in all self-contained systems such as those mentioned in the preceding paragraph. …”
Although Godwin is just warming up on this subject, practical considerations prevent us from continuing farther. It is felt that enough has been presented here so that the reader may consult the primary sources directly for more detailed information on the further history of planet scales, the construction of circular tone zodiacs, Orphic musical religion, and many other such topics. The major point for us here is that music is deeply and subtly connected to arithmetic, geometry and astronomy through the agency of qualitative number.
End of Additional Notes to Music