leon crickmore : planets heptachords and the days of the week

7/25/2019 Leon Crickmore : Planets Heptachords and the Days of the Week

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PLANETS, HEPTACHORDS AND THE DAYS OF THE WEEK THE HARMONY OF THE SPHERES

Leon Crickmore < [email protected]>

Keywords

Babylonian heptachords, planets, weekdays,

1. Introduction

One of the most significant developments in recent musicology has been the transcription and interpretation of a number ofcuneiform musical texts dating from the second millennium BC. It has been established that Neo- Babylonian music, and probably Old

Babylonian music too, was based on seven diatonic heptachords. So far, eight cuneiform music texts have been published: (1) N3354

+ N3355 + N7745 + N7679; (2) UM 29-15-357 (+) N3020; (3) UET VII 74 (U7/80); (4) UET 6/3 388; (5) BM 65217 + BM

66616; (6) CBS 10996; and (7) UET VII 126 (Nabnitu Tablet 32); and (8) CBS 1766.With respect to documentary evidence

requiring musicological interpretation, four of these texts are particularly significant: UET VII 126; CBS 10996; UET VII 74 and

CBS 1766.Very little actual notated music seems to have survived. The four texts just listed contain lexicographical and mathematical

knowledge, associated with the strings of instruments, and which may also be related to the Akkadian language liturgical texts.

Primarily, therefore, these four tablets are music-theory texts.

2. The heptachords

2.1. A heptachord is a seven-note scale literally, a scale playable using seven differently tuned strings. Modern musical scales,

however, are generally considered to be eight-note octave-structures that is, ladders of eight musical pitches, the modal patterns of

which differ according to whether each scale rises or falls. However, for a credible musicological interpretation of the seven

Babylonian heptachords which is consistent with all four main cuneiform texts, it is necessary to postulate that these scales were, by

contrast, seven-note modal patterns of tones (t)and semitones (s). These patterns remained the same, regardless of the direction of

the particular scale, upwards or downwards. Thus, for example, the pattern of the heptachord isartum would be stttst, and could be

transcribed into modern letter-notation, using, for convenience, the white keys of a keyboard only, as: E, F, G, A, B, C, D (rising)and

C, B, A, G, F, E, D (falling). Table Ishows the modal patterns of each of the seven heptachordsi.

Table I

2.2. From a number of sources, we know that music-theorists in ancient Greece quantified the pitches of

their scales by means of ratios of string-length. The invention of the basic arithmetic procedures by which this was

accomplished is usually attributed to Pythagoras. Explicit documentary evidence for these procedures can be found

in Platos dialogue Timaeus. Platos so-called Timaeus Scale, with its consequential concept of the harmony of the

spheres, enormously influenced much of the speculative and cosmological thinking in Europe through subsequent

centuries, right up to the rise of modern experimental science in the seventeenth century. There is no firm evidence,

however, in any of the cuneiform tablets so far found and transcribed, that the Babylonian priest-mathematicians

quantified their heptachords. Nevertheless, the numbers required can be found in many of their mathematical

tablets, and in particular, in the standard Babylonian tables of reciprocals. Several musicologists who have

considered this matter carefully, believe, therefore, that the seven heptachords would have been quantified, using the

ratios and reciprocal ratios of sexagesimal arithmetic. Such a conclusion fits naturally within the context of the

ancient obsession with tuning systems, that can be found also in sacred texts from India and China. Table II, from

Robsonii, shows the cuneiform text MLC 1670, a typical example of the reciprocal tables found in nineteenth andeighteenth century Larsa, Ur and Nippur. The basic numbers required for the quantification of the heptachords have been highlighted.

1 2 3 4 5 6 7 String number

Modal Pattern (string intervals) Name

s t t t s t iartum

t s t t t s embbum

t t s t t t nd qablim

s t t s t t qabltum

t s t t s t kitmum

t t s t t s ptum

t t t s t t n GABA.RI


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Table II

3. The four main music texts: evidence and interpretation

3.1. UET VII 126:This tablet is a late Babylonian copy of a section of the older lexical list Nabnitu 32.The

the text is bi-lingual; it lists the names for nine strings in Sumerian and Akkadian. With regard to its musicological

interpretation, three points are of special interest: (1) the third string is described in both languages as thin; (2) the fourth string, listed

in Sumerian as string-four-small, in Akkadian is simply named Ea-creator; and (3) the

numbering of the nine strings is palindromic: 1, 2, 3, 4, 5, 4, 3, 2, 1. Furthermore, the last four of these strings are qualified by the

adjective behind that is, from the other end of the instrumentiiiThis suggests a method of tuning by reciprocal perfect fifths and

fourths from the central fifth string. Richard Dumbrillivhas ingeniously postulated that such a tuning order would have been: 5-1; 1-4;5-2; 4-3. Such a method would tune an instrument with a gamut of nine strings to the heptachord pitum (open). The heptachord,

transcribed into modern letter-notation, would be either G, A, B, C, D, E, F (rising), or A, G, F, E, D, C, B (falling). Thus the modal

pattern known aspitumcould be played in either direction rising from one end, and falling from the other - on an instrument with a

gamut of nine strings.

3.2. CBS 10996:This tablet, found near Nippur, was published by Anne Kilmer in 1960v. It is neo-Babylonian, but may possibly be a

copy of a far older text. Most musicologists accept that it is a tuning tablet. If either the minority view that it is no more than a list of

pairs of strings (dichords), or the mathematical view of paired constants was ever proved to be correct, a practical method of tuning

could still be inferred from the numbers contained in the text. For the numbers listed can be interpreted as instructions for tuning

seven heptachords, by means of perfect fifths and perfect fourths, followed by a refinement in the tuning of the thirds and sixths.


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Basic Tuning Fine-Tuning Heptachordal Name

1-5 7-5 (n GABA.RI/ nis tuhrim)

2-6 1-6 iartum

3-7 2-7 embubum

4-1 1-3 (nd qablim)

5-2 2-4 (qabltum)

6-3 3-5 (kitmum)

7-4 4-6 (ptum)

Table III

Table IIIdisplays the two columns of string-numbers in the tablet with the names of the heptachords. The dichord listed opposite

the name of each heptachord indicates the first pair of strings to be tuned to a perfect fifth. Proceeding down the column, are the

subsequent series of falling fourths and rising fifths which comprise the arithmetic algorithm for tuning. Thus to tune the heptachord

nis tuhrim/nis GABA.RI, a player would proceed as follows: 1-5, 2-6, 3-7, 4. Expressed in modern letter-notation, the resulting scale

would then be: F, G, A, B, C, D, E (rising), or B, A, G, F, E, D, C (falling - when tuned from the opposite end of the instrument). The

interval between strings 7 and 5 (as indicated in the second column of the tablet) now needs to be fine-tuned, by sweetening theintervals E-C and C-E, to bring them closer to what musicologists would describe as Just tuning.

3.3. UET VII 74:This is an Old Babylonian tablet from about 1800BC. It contains detailed instructions for the tuning/modulating

of a nine-stringed instrument, by means of tightening/sharpening (Chapter I), or loosening/flattening (Chapter II) of one of the

components of the unclear interval (la zaku),which modern musicians would call the tritone. However, as described in the text, this

cyclic procedure works for fallingscales only. Table IV(a) illustrates the tuning method described in Chapter I of the text, and Table

IV(b) the reverse procedure in Chapter IIvi.


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Table IV(a)

Table IV(b)

String Numbe 1 2 3 4 5 6 7 Tritone Re-tuning

Heptachord

c'' b' a' g' f' e' d' 5-2

Sharpen String 5 for

qablitum

s t t t s t

c'' b' a' g' f#' e' d' 1-5

Sharpen String 1 (and

8) for nis GABA.RI

s t t s t t

c#'' b' a' g' f#' e' d' 4-1Sharpen String 4 fornid qablim

t t t s t t

c#'' b' a' g#' f#' e' d' 7-4


pitum

t t s t t t

c#'' b' a' g#' f#' e' d#' 3-7


embubum

t t s t t s

c#'' b' a#' g#' f#' e' d#'6-3


kitmum

t s t t t s

c#'' b' a#' g#' f#' e#' d#' 2-6

Sharpen String 2 (and

9) for iartum (raised

by a semitone)

t s t t s t

pitum

embubum

kitmum

qablitum

nis GABA.RI

nid qablim

iartum

String Numb 1 2 3 4 5 6 7 Tritone Re-tuning

Heptachord

c'' b' a' g' f' e' d' 5-2

Flatten string 2 (and

9) for kitmum

s t t t s t

Heptachord

c'' b' a' g' f' e' d' 2-6

Flatten string 6 for

embbum

t s t t s t

Heptachord

c'' b' a' g' f' e' d' 6-3


pitum

t s t t t s

Heptachord

c'' b' a' g' f' e' d' 3-7


nd qablm

t t s t t s

Heptachord

c'' b' a' g' f' e' d' 7-4


ns GABA.RI

t t s t t t

Heptachord

c'' b' a' g' f' e' d' 4-1

Flatten string 1

(and 8) for qabltum

t t t s t t

Heptachord

c'' b' a' g' f' e' d' 1-5


iartum (lowered by

a semitone )

s t t s t t

nd qablm

ns GABA.RI

qabltum

iartum

kitmum

embbum

pitum


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3.4. CBS 1766: The date and provenance of this tablet remain uncertain. There are grounds for dating it quite late, but Fribergvii

suggests that on account of its format, which is like that of Plimpton 322,it could be an Old Babylonian text from Larsa. The text

combines the tuning numbering from CBS 10996with the first seven string-names (as far as third behind string) from UET VII 126.

At the head of the tablet stands a seven-pointed star, set in two concentric circles. The seven string-names are inscribed around the

star, each of the points of which is numbered (1-7). Waerzeggers and Siebesviiidescribe it as a possible visual-aid for tuning the seven

heptachords. Dumbrillix has suggested that it may even have been the design for a metal dial to calculate the appropriate tuning

systems for each of the heptachords. Friberg interprets it both as instructions for drawing a regular heptagon and also for tuning a

stringed instrument.

4. The heptachords: a summary

4.1.Table Vshows a transcription of all seven heptachords, both rising and falling, together with their modal patterns and some

possible tone-numbers for each of the string-pitches. The palindromic numbering of the columns has been chosen to mirror that in

UET VII 126. (For present purposes, the letters in the two columns 1, which relate to a match with CBS 1766, should be ignored.).

This thirteen-string tuning, permitting the performance of all seven heptachordal modal patterns in either direction, suggests a possible

tuning for the Queens Harp (c.2500 BC) in the British Museum. The numbers required to quantify each of these scales can be found

in four mathematical texts from the Temple Library of Nippur, CBM 11340 + 11402; 11368;11902; and 11097 (c.2200 BC)x

Table V

4.2.The distance between any two pitches is a ratio, whichrepresents the proportion by which a string has to be

lengthenedto sound the next tone of the falling scale, or shortened, using the reciprocal of the ratio, for the rising scale. These

ratios are indicated in lines 4-5 of TableV.The Babylonian priest-mathematicians seem also to have associated particular numbers

with certain of their gods. The god-numbersxirelevant to our present purpose are: 60 = Anu; 50 = Enlil/Ninurta; 40 = Ea; 30 =

Sin; 20 = Shamash; 15 = Ishtar; and 10 = Gibil/Nusku/Bel Marduk. Although these numbers do not have any direct

correspondence with the lists of heptachords and planets, they may have been used to form the harmonic ratios required to quantify

the musical pitches of the heptachordal scales. Thus the number for Anu, the first and father of all the gods is 60, the basis of

Babylonian sexagesimal arithmetic. From a musicological perspective, 60:30 or 2:1= an octave. But apparently, the Babylonians had

no word for octave in either Sumerian or Akkadian. In the Mesopotamian tonal system the octave note was more analogous to the

first day of a fresh seven-day week. On the other hand, 30:20 (Sin: Shamash) = a perfect fifth in Just Tuning; 40:30 (Ea: Sin) = a


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perfect fourth; 50:40 (Enlil: Ea) = a major third; and 60-50 (Anu: Enlil) = a minor third. And these are the four principal musical

intervals defined by the dichords in the tuning texts.

5. The planets and days of the week

5.1The origin of what we now call a week lies in religious celebrations or abstinences in various cultures on the days of new moons

and their intermediate quarters. While our seven-day structured week probably derives from the Hebrew tradition, many

Mesopotamian cultic festivals were preferably celebrated on the days of the earlier observed lunar septenary pattern.

In modern astronomy, the planets, in order of increasing distance from the sun, are: Mercury, Venus, Earth, Mars, Jupiter,Saturn, Uranus and Neptune. However in the geocentric paradigm of ancient astronomers, there were seven wandering stars

(planetes). These were normally listed in the order of their decreasing orbital cycles, namely: Saturn, Jupiter, Mars, Sun, Venus,

Mercury and Moon. Such an arrangement still survives today as the modern astrological week. In Table VIIthe decreasing cycles from

Saturn to the Moon (in black) correspond to the gradual shortening of a string. The heptachord thus sounded would be isartum (rising).

.

5.2. In many European languages, our names for the days of the week are also connected etymologically with the names of

the planets. Saturday, Sunday, and Monday are obviously related to Saturn, the Sun and the Moon. From Tuesday to Friday, the

the planetary links become clearer in French: Mardi (Mars), Mercredi (Mercury), Jeudi (Jupiter) and Vendredi

(Venus). However, when the days of the week are coupled with their respective planets, arranged in the decreasing

order of their orbital cycles, an anomaly appears: the days of the week read: Saturday, Thursday, Tuesday, Sunday,

Friday, Wednesday and Monday. To restore the normal order of the weekdays, the beginning and end of the list

have to be joined to form a ring (a zodiac, perhaps?), starting then with Monday, the reader must jump over two

subsequent weekdays to reach Tuesday, and so on for the remainder of the week. Interestingly, an analogous

pattern can be found in the order in which the heptachords emerge during the tuning procedures described in the

cuneiform music texts.

6. Correspondences with the instructions for tuning the heptachords

6.1 Lines 11-14ofCBS 10996 (Obv. Col 1) list the dichords from which the heptachords take their names in the

following order: nis tuhrim; isartum; embubum; nid qablim; qablitum; kitmum andpitum.Taking the heptachord nis tuhrim

(falling) B, A, G, F, E, D, C and applying the modulating/re-tuning instructions contained in UET VII 74, the

order in which the subsequent heptachords will be tuned is:Chapter I (tightening): nis tuhrim; nid qablim; pitum; embubum; kitmum; isartum; qablitum.

Chapter II (loosening):nis tuhrim; qablitum; isartum; kitmum; embubum; pitum; nid qablim.

Notice that, starting with nis tuhrim in the listed order of CBS 10996, ifone jumps over the next two heptachords (as we did earlier

with the days of the week), the tuning order of UET VII 74, Chapter I is produced. To bring about a match between CBS 10996 and

the tuning order of UET VII 74, Chapter II, the ends of the lists have to be joined to form a ring, and bothlists have to be read in

the reversedirection (that is, nis tuhrim, pitum ... inCBS 10996 and nis tuhrim, nid qablim... in UET VII 74, Chapter II), and then by

continually jumping over the next two heptachords in CBS 10996, the Chapter II tuning order will be discovered. In CBS 1766

(Table VII), when the planets are aligned with the points of the star, their names read from Saturn, around the circle to the leftin

black or to the right in red, occur in the order of descending sequence of their orbital periods. All this may appear somewhat

complex, and maybe even confusing, but it perhaps hints at the uncovering of a consistent set of correspondences between the planets,

the heptachords and the days of the week. Although we cannot be certain whether any of these algorithmic correspondences would

have been explicitly recognised by the Babylonian priest-mathematicians at the time the tablets were inscribed, it is reasonable tosuppose that we are dealing here with the roots from which the ancient and sophisticated concept of the harmony of the spheres

would eventually grow. The relevant strandsxii contributing to this idea are brought together in Table VI, where the heptachord

isartum, rather than nis tuhrim, has been used, since it can be taken to correspond to Saturday, the first day of the Babylonian week. The

numbers in the additional columns (i), (ii), and (iii), are explained in end-note xi.

Table VI

() () ()

( ) ( ) ()

0 0 2 1 2 1

2 0 2 2

0 0 1

0 0 1

2 20

0 1 2 1

2 10


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7. The harmony of the spheres

7.1. The harmony of the spheres is a concept from ancient analogical philosophy, which regarded the movements of the celestial

bodies as a form of music. Such music was not usually considered to be audible, but it was respected as a harmonic and religious idea

which carried cosmological implications; an idea derived from the mathematics of tuning musical strings.

7.2.The main documentary evidence for the harmony of the spheres is to be found in Platos dialogue Timaeus. In

the preface to his Sourcebook of the Pythagorean Tradition in Music, Joscelyn Godwinxiii summarizes this tradition as a many-faceted

commentary on the passage in Platos Timaeusxiv that describes how the Demiurge fashioned the World Soul. Musicologists have

identified the first octave of the so-called Timaeus scale as the Dorian octave species: E, D, C, B, A, G, F, E. In the Laches, Plato

describes the Dorian mode as the only harmony that is genuinely Greekxv. The first seven tones of this scale match the Babylonian

heptachord nid qablim. In the reverse direction (F-E), it would match nis tuhrim, which is the sihpuxviversion of nid qablim. The sihpu

form of a heptachord was its modal pattern played in the reverse direction. The modal pattern of isartumwas stttst. Thus, sehep isartum

in Nabnitu 32would indicate tsttts, which is equivalent to embubum.

7.3.When Richard Dumbrill built a playable reconstruction of the Silver Lyre of Ur (c. 2500 BC), he found that the nis tuhrim tuning

fitted the known measurements of this eleven-stringed instrument (F-B). If the gamut of such an instrument were to be increased byone further string at each end, as in the Queens Harp, the tuning would be isartum(E-C), producing a system capable of playing all

seven Babylonian heptachords in either direction. While if, alternatively, the gamut of the Silver Lyre were reduced by one string at

each end, the new nine-stringed instrument would be tuned topitum, as implied inUET VII 126. It is conceivable, therefore, that in

the third millennium BC the Babylonian heptachords were conceived as risingscales, and that it was only in the second millennium,

possibly as a result of the introduction of the new method of tuning described in UET VII 74, that falling scales became the norm.

This theoretical viewpoint persisted until the end of the Classical period in ancient Greece.

8. Conclusion

Finally, for the purpose of constructing Table VII, most of the correspondences speculated over in this article have been projected

onto the seven-pointed star at the head of the cuneiform text CBS 1766xvii. While admitting that much of the evidence cited is

circumstantial and dependent on musicological interpretation, and even allowing for the possibility that some of the assumptions of

Western musicologists about implied tonality may be partially anachronistic, the result still suggests that the harmony of the spheres,

an idea whose origin is usually attributed to Pythagoras and Plato, may have originated in Babylon, a thousand years or more earlier,

and may have arisen as part of an attempt by the Babylonian priest-mathematicians and their scribes to formulate a combined musical

and early scientific and geocentric model of their universe.


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Leon Crickmore, 28 April, 2013

Table VII

d'

1

c'

b

ag

f

e

or

10:99:10

16:1515:16

9:88:9

10:99:10

9:88:9

16:1515:16

9:88:9

t

s

t

t

t

s

t27

(or 54)1

2

3

45

6

7

CBS 1766 as a Tone Circle and Planets

Dichords in CBS 10996

Initial Tuning (5ths and 4ths)

Fine Tuning (3rds and 6ths )

The tritonic tuning procedures of UET 74

can be applied to the falling scales

t = Tone

s = Semitone

c' = Middle c

Figures in red indicate reciprocal (inverse)

scales.

Notes and Key :-

(d)

n tuhrim

Jupite

r

Jupite

r

Moon


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9 Notes and References

iIn this and some of the subsequent Tables, nis GABA.RI, the pseudo ideogram as transcribed by Anne Kilmer, should be read as nistuhrim, see S. Mirelman and Theo Krispijn, The Old Babylonian Tuning Text UET VI/3 899, Iraq 71 (2009), 43-52.

iiRobson, E. Words and Pictures: New Light on Plimpton 322,American Mathematical Monthly, 109 (2002): 105-120

iiiIn the neo-Babylonian text YBC 11381, the nine strings are listed from 1-9; each is followed by the name of a deity and a wish,formulated with a precative verbal form. Payne, Elizabeth E A New Addition to the Musical Corpus, Opening the Tablet Box,Vol. 42,Brill, Leiden (2010): 291-300

ivDumbrill, R. J. Evidence and Inference in Texts of Theory in the Ancient Near East, in Proceedings of ICONEA 2008, publishedand printed by Lulu (2010): 105-115

vOrientalia, Pontificium Institutum Biblicum Rome, Vol. 29 (1960): 278-281

viFor convenience of presentation, strings 8 and 9, which musically merely duplicate the pitches of strings 1 and 2, have been omitted.

viiFriberg, J. Seven-Sided Star Figures and Tuning Algorithms in Mesopotamian, Greek and Islamic Texts,Archiv fur Orientforschung 52(2013) : 121-155. See also Crickmore L, A Musical and Mathematical Context for CBS 1766,Music Theory Spectrum 30/2 (2008): 327-338

viiiWaerzeggers C. and Siebes R. (2007), An Alternative Interpretation of the Seven-Pointed Star on CBS 1766,Nouvelles AssyriologiquesBreves et Utilitaires 20, 2:43-45

ixDumbrill, R. J. Is the Heptachord in CBS 1766 a Dial?,Arane,Vol.1 (2008): 47-49

xHilprecht, H. V. The Babylonian Expedition of the University of Philadelphia Series A: Cuneiform Texts, (Hilprecht, Ed.) VolumeXX, Part I, Plates 10-14; transliteration of restored texts on p. 21; Published by the Department of Archaeology, University of Pennsylvania (1906)Hilprecht interprets these texts as Tables of Divisors of 604(12,960,000). The divisors 24-48 (all sexagesimal numbers in the form

2p

3q

5r

), omitting 25, and interpreted as proportional string lengths, would sound the first octave of Platos Timaeus scale, though inthe ratios of Just rather than Pythagorean tuning. 24-80 (or 81 using the 9:8 tone) are required to accommodate all seven Babylonianheptachords, also in Just tuning.

xiLivingstone, A.Mystical and Mythological Explanatory Works of Assyrian and Babylonian Scholars, Clarendon Press, Oxford, (1986): 30-33

xiiDays are listed in the assumed order of the Babylonian week;planetsby etymological association; heptachordsin the tuning orderof UET VII 74, Chapter I.There are two columns of tone-numbers. The first defines the falling-scaleproduced by the increasinglengthening of the activated string; and the second column the rising-scalesounded as that string is increasingly shortened. Thegod-numbers are simply listed in descending order.On the right side of Table VI, column (i)indicates the initial tuning-dichords from CBS 10996 (Table III). Column (ii)arranges the

integers 1-7in the order of the jump-over-two-rule. The same pattern of numbers, but starting with 7 and reading them in thereverse direction, form the tuning algorithm for the tuning ofpitum in CBS 10996 and CBS 1766. Column (iii)lists the originalnumbers around the seven-pointed star on CBS 1766,opposite which the name of the respective fallingheptachord is shown in blackin Table VII.For the risingheptachords and the planets listed in red, the points of the star would need to be numbered in theopposite direction.

xiiiGodwin, J. The Harmony of the Spheres: A Sourcebook of the Pythagorean Tradition in Music,Inner Traditions International, Vermont, (1993)

xivPlato, Timaeus, 35c-36d. See also Crickmore, L. A possible Mesopotamian origin for Platos World Soul, Hermathena 186, (2009): 5-23, especially Appendix I

xvPlato, Laches, 188d

xviCrickmore, L. A Musicological Interpretation of the Akkadian Term Sihpu, JCS 64 (2012): 59-66

xviiTable VIIhas been drawn on the assumption that the falling heptachords move around the circle to the rightto correspond with

the numbering to the right on the cuneiform tablet. To create a match between CBS 1766and CBS 10996, as originally interpreted byAnne Kilmer, and to produce her rising scales, point 6 (f)of the star has to be rotated to the 12 oclock position, becoming the newpoint 1. The first number of each dichord should then be read to the right, while the rising fifths have to be counted to the left,in thedirection of the risingscales named in red.

leon crickmore : planets heptachords and the days of the week

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