ON PENTATONY, IN A GRUNDRISSE

 

B. Lukács

 

President of the Matter Evolution Subcommittee of the

Geonomy Scientific Committee of HAS

 

CRIP RMKI H-1525 Bp. 114. Pf. 49., Budapest, Hungary

 

 

ABSTRACT

            Following Jeans a zeroth approach to pentatony is given, independently of Greek heptatony called generally diatony. Such a treatment is necessary to avoid the usual Europocentric and mathematically naďve idea that pentatony would be simply a rudimentary/truncated heptatony.

 

1. INTRODUCTION

            For more or less a century there exists a "movement" for emphasizing the pentatonic roots of Magyar folk music. (Beware: Magyar & Hungarian are not synonyms. Magyar is a language and ethnic originally from Southwestern Siberia. Hungarian is the state and its subjects without indicating language/ethnicity. Magyar is the self-name of Magyars and the root is the same as in Manyshi, a small nation along Lower Ob; Hungarian comes through Hungarus from the self-name of a Bulgarian Turk nation, in the Carpathian Basin in 8-9th c., the Onogurs, where on ogur is 10 arrows/tribes.) For Hungarian laymen it is met by complete bafflement & uninterest on one side while eager lip service on the other. So a person finishing high school with good scores can tell that yes, our folk music is pentatonic and that is great, but, of course, that person consumes only 7- and 12-tone "Western" music, which, according to personal preferences, may be either Bach or Mahler, or Freddy Mercury or Sex Pistols. And if you ask the bright person what is pentatony, the answer will be either "I have forgotten", or "pentatony uses only 5 sounds instead of 7". The second answer is true, but tells only that pentatony is poorer, and then why to care for it at all?

            I can use myself as an example. Through school years I was getting always the highest scores from Music. I could determine the actual musical scale by looking at the page by counting the flats and sharps at the beginning. Even now I remember that the solmisation names come from a tune of Guido of Arezzo sometimes in Xth century, starting as "Ut Remi..." or something such, but later somebody else changed "Ut" into "Do", although I was never taught, why, and I did not ask. I was also told that the solmisation system is "completely different" from the CDEFGAHC scale (the teacher did not tell what is the difference; now I think she did not know either), and that Kodály regarded solmisation as very important. Also, we learnt one or two pentatonic songs, and I knew the first line of the first recorded heptatonic soldiers' tune, which, according to the words, must have been composed by privates in the 7 Years War.

            So far so good. But I positively remember that I knew only that pentatony is heptatony minus 2 sounds (and I definitely did not care). In the following decades I consumed mainly heptatonic and dodekaphonic music if any, I was not too interested in the details, but as a physicist I got more or more insight into the problem.

            I think, the absurd Hungarian status of knowledge gets two different sources. The first of them has nothing to do with either musicology or with physics, but it is so absurd that I deal with it in a few sentences.

            Until the end of the First World War Hungary was a multiethnic country. Roughly half of the population spoke Magyar, a Western Siberian language; the other half were different Europeans. Surely, the Western Siberians were originally pentatonic (or not), but the other half was heptatonic. Above folk song level everything was heptatonic, from the music of the Medieval royal court through Church music to operettas and couplets. This heptatonic music was the common culture of Hungary.

            Now about 1900 a movement started. Its theses were, very roughly: common people live in very bad circumstances because they are oppressed; they are oppressed by great landowners, Austrians & Hungarian traitors licking the feet of Austrians; the people should be enlightened  and made self-conscious; let us go into little villages, collect folk songs and later city people will cease to consume international music. Pentatony of some Magyar folk songs were discovered by Bartók & Kodály c. 1910; the first formal publication seems to be [1] in 1917. (And note that the Magyar term "népdal" is not exactly "folk song" or "Volkslied", even if the translations are mirror ones. Magyar "nép" is not exactly "folk"; the term is an Ugric dvandva fom Magyar "nő+fi", Manyshi "ne+pigh" [2] meaning originally the woman and children of the family/house, so the less dominant ones. Note that between 1949 & 1989 Hungary was not a Republic but a People's Republic, in Magyar "Népköztársaság". This explicitly expressed that the new state was the society of the underprivileged, while the previous burgeois and landowner upper layers were ousted from politics. When musicologists went for collecting "magyar népdalokat", "Magyar folk songs", they looked for unadultered songs of the simple, honest and untricky "nép", not of the nation whose upper layers were influenced by heptatony.

            The irony is that they may have been right.

            Of course, the programme was absurd in a Hungary whose half never encountered pentatony for at least 1500 years. After the First World War Hungary was truncated, became dominantly Magyar-speaking, and became led by governments using more nationalistic slogans than previously, but city people continued to consume operettas & couplets and in villages the great majority of the folk songs remained heptatonic. (My background is an example. My father was city dweller and tone-deaf, so he sang infrequently, dissonantly, and mainly Italian songs. My mother came from a pure Magyar village in present Slovakia, she sang a lot when working, and her repertoire was couplets, operettas and exclusively Slovakian folk song. She learnt the folk songs in a Slovakian middle school; her village already gave up folk music.)

            Between the World Wars the leaders of the Movement were Bartók & Kodály. Bartók was quite internationally interested, but "always from clear sources", which meant "only folk music". Kodály was interested in Magyar music, so Magyar folk songs, pentatonic if possible. Both were mildly in opposition to leading politicians. Then Bartók went to USA in 1939, surely to avoid the Second World War, but his emigration has been completely mystified by politicians so we know only legends. He died there in 1945. Kodály, on the other hand, remained in Hungary, and was very mildly in opposition after the war during extreme leftist governments. (He spoke always about "poor and simple people" which sounded very near to the leftist slogans.) Also, a Kodály Method had been worked out to teach music, which, I am told, is successful in Japan and Canada. Bartók was handled as a martyr of progression or such.

            Still the overwhelming majority was not interested in Bartók. Kodály was not so refused but people showed not the least will to sing pentatonic songs. However, people learnt that Bartók & Kodály were politically preferred, so nobody told anything against them.

            Except for some highly intellectual Budapest people in the last decade of the leftist regime who told that the Kodály Method should not get central role is music teaching in ground schools. There were no repercussions (surely, Party leaders did not know more about pentatony, solmisation, Guido of Arezzo &c. than laymen did) but musicologists from the countryside started a campaign against the Budapest traitors not honouring Kodály's lifework, our pentatonic inheritance &c. The argumentation still is at this status. However, I repeat, I got always good scores from Music, but I never knew what is the Kodály Method or really what is solmisation, apart from Guido of Arezzo.

            Of course politics taught us not to be interested too much in Kodály, who was as left-sided as the ruling party even if not exactly the same way, but this is only one reason of the garbled status of knowledge about pentatony in Hungary. The main reason is terminology. In Europe, the Americas & Australia the terminology is purely heptatonic, while high music is now 12-tonic. So Old Magyar pentatony is generally formulated in heptatonic terms, which is as proper as formulating Celestial Mechanics in geocentric terms.

            Let us see first a few examples for the inherently heptatonic language of Modern Music. Generally not even the term heptatonic is used: the usual term is diatonic. Heptatony means hepta tonoi, so 7 tones in the fundamental scale before everything restarts an octave higher, but octave is again a heptatonic term from Latin octava=eighth, meaning that the eighth sound is "very similar" to the first (see later). Diatonic comes from dia tonon, "through of (all) sounds", so the scale is of 7 because that is all sounds. Octave is diapason in Greek: though all (sounds), so in Greek this word does not imply the heptatony.

            A very important step in the scale is the fifth. It means the consonant nature between the first and fifth sounds of the heptatonic scale, say C & G. In Central Europe terms coming either directly from Latin or from Italian are used, all going back to Latin quinta=fifth. Greeks of Classical Antiquity (outside the original Pythagoreans) used dia pente for the same steps, so "through 5", meaning "from first to fifth". Now, a "fifth step" in proper pentatony would be "fourth" (later) and diatony should mean the complete pentatonic scale. However then very serious ambiguities would arise when speaking about scales, steps &c. Instead I cease to use the term diatony from this moment, until the physical foundation at the end of Chapter 3 I put the ordinal number terms into quotation marks and there I will define the step names for the remaining part, and in general I try to use a language not preferring any scale to others.

            Anyways, in heptatonic language it is unavoidable to get the idea that pentatony is heptatony - 2 sounds, so a primitive heptatony (and then why to learn it?). We shall see that it is not; it is another scale. And what I am going to tell goes back to Jeans [3] who was, of course, a physicist. Only I am coming from pentatonic tradition (you already can see that not a strong tradition), so I am offended by a superstition that Music would be inherently heptatonic. Jeans was not offended (being English, not Scottish).

            In the further Chapters I will use a 5-stage evolutionary scheme, which was historically certain only for 7-tone and 12-tone scales. According to Jeans [3] the steps will be:

            0) Raw data (before musical theory)

            1) Ratios of small integers

            2) Just intonation

            3) Mean tone

            4) Tempered scale

although some stages may have been absent for some scales.

            In Chap. 2 we recapitulate the history of the favoured European heptatonic scale called generally diatonic, meaning "that of all tones" as if anything would single out the heptatonic system. (The 7-tonic is indeed better than many others. But not the best one.) Chap. 3 gives a very simplified but physical approach to consonance/dissonance. Clearly consonance/dissonance is behind preferring some scales and dispreferring others; and while dissonance is partly cultural, the Laws of Physics and the structure of the inner ear are the same for any recent human population. Chap. 4 tries to find the preferences in extending the scale for more sounds. While the set of possible choices is infinite, it seems that the path told in Chap. 4 leads to maximal consonance, and it was the path chosen by Europeans including more and more diapasons into music, and definitely by organbuilders. In this way we arrive at dodekaphony. Chapter 5 deals with the possibilities beyond dodekaphony from mathematical viewpoint.

            Until this point I more or less will follow Jeans [3]. But then Chapter 6 is about pentatonic scales. Jeans  spent only a few sentences for pentatony; but he was neither a Hungarian, nor a Scot. In Hungary the topic is often mentioned but the various kinds of pentatony are seldom distinguished and if they are then that happens almost entirely without mathematics. Chapter 7 is some sociological discussion of evolution of pentatony beyond 5 sounds. Chapter 8 deals with the Scale of Future: the 53-tone scales. While Jeans recognised the potentials of 53-tone scales, again he discussed them only in a few sentences. Finally, Chapter 9 is a short note for further tasks and possibilities of Hungarian musicology about pentatony.

 

2. HISTORY: THE HEPTATONIC SCALE OF GREEKS

            Stage 0: Raw Scale

            We do not know too much about this stage. However the Greek heptatonic/diatonic system surely had been well established at the middle of VIth century BC. So lyrists and flutists seem to have partitioned the octave into 7 musical sounds; the 8th being the octave in complex harmonia with the first, but lyras generally had only 7 strings. For simplicity we use the sequence CDEFGAHC1; of course the different "tunes" started at different sounds. Surely each instrumental musician used his own system of tuning the strings &c.

            Stage 1: Ratios of Small Integers

            Pythagoras, c. 550 BC, suggested a system for the seven musical sounds and suggested the lyra to be completed with one more string. Some completed it, some not. The system is simple, symmetric, elegant, and has a clear physical meaning. Surely his positive opinion about "number magic" helped him to find this formulation. He used the notion of lengths; if he meant lengths of solid metal bars, then the lengths are the inverses of the eigenfrequencies, and it is easier to tell the system in terms of frequencies.

            His suggested frequencies are:

 

Sound

Relative frequency

Rel. to prev.

C

1

-

D

9/8

9/8

E

81/64

9/8

F

4/3

256/243

G

3/2

9/8

A

27/16

9/8

H

243/128

9/8

C1

2

256/243

 

Table 1: The frequencies of the Pythagorean scale.

 

(Our frequencies are inverses of Pythagoras' lengths.) Pythagoras called a 9/8 step tonos, and a 256/243 one hemitonos, so tone and halftone.

            Now you can tune your lyre to 8 sounds, but you have a liberty to start on any of the CDEFGAH scale and end at the octave of the startpoint. This gives 7 different scales, where the halftones are at different places. They are the scales of strange names as Phrygian, Mixolyd &c. Some scales were believed "better" than others. Also, you can build scales of 5 tones and 2 hemitones in more ways than 7. We shall see in the next Section what principle singles out some scales.

            While ratios as 256/243 or 81/64 are not exactly those of small integers, the system is a good compromise (we are going to see, compromise of what).

            Stage 2: Just intonation

            It is a more evolved variant of the Pythagorean idea. The scale is as follows (for comparison see the Pythagorean scale again):

 

Sound

Step

Pythagorean step

C

-

-

D

9/8

9/8

E

10/9

9/8

F

16/15

256/243

G

9/8

9/8

A

10/9

9/8

H

9/8

9/8

C1

16/15

256/243

 

Table 2: The steps of heptatonic Just Intonation.

 

and then the frequencies are:

C

1

D

9/8

E

5/4

F

4/3

G

3/2

A

5/3

H

15/8

C1

2

 

Table 3: The frequencies of heptatonic Just Intonation.

 

            The ratios are now "simpler"; the importance will be cleared in the next Chapter.

 

Stage 3:Mean tone

            The idea was introduced first by Arnold Schlick [4] and later it was refined by F. Salinas, in XVIth c. [5]. It is the logically last stage of the evolution of heptatony. Since the next Chapter would be necessary for details, now we only note that the scale/system is a compromise of two natural but incompatible requirements, that i) the scale, extended to “octaves”, contain as many “fifths” as possible but ii) still the ratios be those of small integers. Especially the system now tries to compromise between the step C-E2 = 5.00 and still many “fifths” as good as possible.

            In Just Intonation E2’s frequency is exactly 5, and the result is consonant. Using pure “fifths”, after 4 steps we are at E2, and the ratio is 81/16=5.0625. Then we "correct the fifths" to steps of 51/4=1.49535... and then there will be as many almost perfect fifths as possible, and still the E/C ratio is 5/4.

            As we shall later see, this scheme of construction points out from heptatony. The logical path was indeed followed later. This path will be shown in Chap. 4.

 

            Stage 4: Tempered scale

            Ancient Greeks, being awkward about the very existence of irrational numbers, were not sensitive about root numbers. In Early Modern Ages this was not so anymore, therefore the idea of equal steps within an octave, based on the ratio 21/12=1.0594..., became possible. Maybe Glareanus [4] made the last step still within the heptatonic/diatonic system.

            Literally a dodekachordon is something of 12 strings. Such an instrument would seem to be able to play 7-tonic tunes in 12 scales (and Glareanus built up 12 scales). Observe e.g. that 6 of the 7 original "Pythagorean" scales use the sounds

C,Desz,D,Esz,E,F,Fisz,G,Asz,A,B(=Hesz),H(,C1)

so 12. The exceptional Pythagorean scale is the Mixolydian; the Mixo- showing that already the Pythagoreans regarded it as "exotic". In the Middle Ages the Church Canons called this scale Locrian; and the Locrian society was quite exotic for Ionians, indeed. (E.g. Ionians were the most patriarchal, and for them Locrians seemed the most matriarchal.) If you have 12 strings, then you may play 12 scales; but still not Locrian.

            However now you can analyse the 12 strings or sounds; and then you see that the Pythagorean halftones are not halves of the tones. In no sound algebraic sense is 256/243 a half of 9/8. A scale where you would have exactly tones and halftones, would contain 5 steps of 21/6 (tones) and 2 of 21/12 (halftones). Of course, then the only rational ratio would be the octave; but in some scales the sounds would be near to the Pythagorean one. E.g. in the "tempered Lydian"

 

Sound

Untempered

Tempered

C

1

1

D

1.1250

1.1225

E

1.2656

1.2600

F

1.3333

1.3348

G

1.5

1.4983

A

1.6875

1.6818

H

1.8984

1.8877

 

Table 4: Frequencies of the tempered scale vs. untempered Lydian.

 

With a dodekachordon permanently tuned in 12 steps of 21/12 you could play lots of "Pythagorean" scales almost perfectly.

 

3. CONSONANCES, DISSONANCES AND RESONANCES

            Two sounds, heard simultaneously or just one after another, may cause dissonant feeling; or they may not. In the latter case we may call them consonant.

            Obviously our subjective feeling may depend on lots of circumstances; and also, a sound may have many characteristics. However for monochromatic sounds, sine waves of single frequencies ν, observations are qualitatively conform, namely

            If ν1 and ν2 are near (but farther than a distinction threshold), then they are dissonant;

            This dissonancy is generally decreasing with distance;

            The two sounds are definitely not dissonant if ν1/ ν2=m/n, not too large (or: small) integers.

Maybe the first observation can be explained via the modulation with frequency ν1-ν2 for the sum; this modulation is less and less effective as the difference grows; but the third observation must have physical and physiological explanations even if we are still unsure about some details.

             Now comes the physics of strings and hairs of the inner ear. With strong simplifications both the production and the acceptance of a musical sound can be modelled with an ideal string, whose differential equation is:

              md2x/dt2 + kdx/dt + Dx = f                                                                             (1)

where f is a force; m is the mass of the string, k is the damping parameter and D is the parameter of the elastic force. Let us first take a string which got a kick but then is undisturbed.

            Let us take first the k=0 limit. Then:

              x(t)=Xsin((D/m)1/2t+C)                                                                                               (2)

where the constants X and C depend on the initial kick. The ideal string performs a pure sinusoideal with frequency (D/m)1/2 uniquely determined by the physical constants of the string. Of course, k=0 means no output, so you do not hear anything.

            If k is not 0 but not too large, then the oscillation is damped: the solution will get the form

              x(t) = Xexp{-kt/2m}sin{(mD-k2/4)1/2t/m + C)                         (3)

Again, the frequency is unique.

            However, for some musical instruments as e.g. the violin, we continuously excite the string. Then a free string could oscillate with any frequency. However a string of length L fixed at its ends can oscillate only with wavelengths

              l = L/2n                                                                                                                      (4)

because the fixed points cannot move; n is an integer. Very probably the oscillation will be the sum of several permitted modes. The connection between the wavelength and the frequency is determined by the sound velocity along the string.

            Now consider a situation when the string is not fixed, but is excited by a stationary sinusoideal force f of amplitude a and frequency w. Then the general solution of eq. (1) is:

              x(t) = Xexp{-kt/2m}sin{(mD-k2/4)1/2t/m + C) +                                 (5)

                        + {a/((w2-D/m)2+kw2/m)1/2}sin(wt+Q)

where Q is a phase shift whose form is irrelevant now. After some time the damping first term is negligible.

            So the string reacts to the stationary external oscillation, oscillates with the same frequency, but the amplitude will be generally small enough, except for the neighbourhood of the eigenfrequency given in eq. (3). So if the damping constant k is small, the string reacts only on oscillations very near to its eigenfrequency.

            Finally consider a string which is not damped but not totally linear. Without external forces its differential equation is then, say

              d2x/dt2 + W2x + Qx3 = 0                                                                                (6)

Now, this differential equation leads to elliptic integrals, which cannot be expressed analytically. However, let us take the limit when Q is very small. Then there is first a zeroth order approximation with Q=0:

              x0(t) = Asin(Wt+Z)                                                                                         (7)

where Z is an unimportant phase. Now the first order approximation is that we write the zero order one into the cubic term which we write to the rhs:

              d2x1/dt2 + W2x = -Qx13                                                                                               (8)

Now, surely, the first approximation will be the zeroth one + a small term proportional to Q:

              x1(t) = x0(t) + Qy(t)                                                                                        (9)

But then, until to terms linear in Q, we get

              d2y/dt2 + W2y = A3sin3(Wt+Z)                                                                        (10)

This we can solve analytically, but you see that powers of sin(Wt+Z) appear, which then can be expressed via sines and cosines of n(Wt+Z), where the n's are integers. Then again, the first approximation can be put back to the differential equation & so on. So a nonideal (not quite linearly elastic) string will produce its ground frequency W plus its upper harmonics nW; and also a string reacts to oscillations with its lower harmonics.

            Also, not quite one-dimensional membranes, strips &c. may produce harmonics and may react to them.

            But this means first that a musical instrument generally does not produce a "pure sound" but some combination of the ground frequency and at least upper harmonics. And similarly the "hairs" in the Corti organ of the human inner ear may react to harmonics as well. (And the inner ears of all mammals are similar.) Therefore we cannot be surprised if the “octave” of a sound seems similar to the original sound and does not cause dissonant feeling if they are produced simultaneously. Henceforth the 2:1 frequency ratio will be called diapason, through all sound (of the scale). This term is independent of heptatony.

            As for not 2:1 ratios, observe that if the second sound has 3/2 frequency of the first, then its first upper harmonics is the second upper harmonics of the first one. Also that nonlinearities in the Corti organ may lead to almost anything. And also observe the empty zones in the Asteroid Belt and similarly in Saturn's ring [7].

            Asteroids are most strongly perturbed by great Jupiter. Now at solar distances where the orbital frequency would be 5/2, 7/3, 2/1 or 5/3 the Belt is (totally or almost) empty, while there are local maxima at 4/3 and 1/1. The phenomenon can be more or less reproduced in computer simulations.

            So resonances do work. Also the good/bad feeling when two sounds are simultaneous or just follow each other can come from: cultural traditions; imprint; ancient insticts from Palaeolithe; and so on. Henceforth we simply accept the observations that sounds of frequencies in the ratio of small integers cause good feeling (so consonant), while those definitely not in simple relations are more or less dissonant.

 

4. FORWARD TO SCALES WITH LOTS OF CONSONANCIES

            The simplest consonance is the 2/1 ratio, and henceforth we will call this relation diapason. It is octave only in heptatony. However this is a terminus technicus of heptatonic music: in heptatonic scales the octave is the eighth sound, so the octava.

            3/1 is also a simple ratio, but then comes 3/2. That is quite consonant, so it would be good if the scale contained as many 3/2 ratios as possible. We will call it either {3/2} or the major consonance. The Pythagorean gamut does indeed contain the G-C step which is 3/2. Also, A-D is 3/2 and H-E and C1-F too.

            However the 3/2 ratios fare badly between diapasons, because the diapason is a replica by multiplying with 2 and not 3. So for organists playing in many diapasons the music was not sufficiently consonant.

            Now, of course, one might built up a scale based on {3/2}’s and only secondarily using diapasons. And indeed, this idea worked and gave fine organs. Jeans describes the idea simply [3]: take a first sound of unit frequency: that is C. Take its major consonance, in physicist's language a second sound with 3/2 frequency and call it G. Then again such a step: of course you are out of the original diapason because 9/4>2, so it will be D1. However a sound and its diapason are very similar, so by halving the frequency we get D. And so on.

            The scale would be very consonant but infinite, because 2m/3n cannot be 2 for any integer nonzero m and n. Some dozens of frequencies are given in Table A.1 of the Appendix. In the left column we step upwards, and after almost a cycle sharps will appear, the right column goes downward and after 2 steps flats will come. This problem must be solved somehow. The solution is generally truncation+approximation.

            But also do not forget that number 5 is also not too big, so maybe ratios of 5/3 and 5/4 would be nice too.

            In the first step we got G of the Pythagorean scale. In the second and third D and A of the next diapason. The fourth step we again go to the next diapason; and to a Pythagorean sound since

(3/2)4 = 5.0625 = 4*1.265625

so we are at the Pythagorean E2; but not at the "just intonation" scale's E2, and anyways 81/64 is far from being ratio of small integers. The next step still would reach the Pythagorean H3, but that is the last Pythagorean sound anymore.

            At this point came the Mean tone into use [4], [5]. 5.0625 is almost 5. If (3/2)4 were 5, everything would be nice. So let us use 51/4 steps instead of 3/2 and one diapason will be a tiny bit sloppy, but things will improve between diapasons. Mean tone helps for many diapasons. Now we are at Step 3.

            The next step was the tempered scale. There is no algebraic ratio except diapasons and such, so every consonance is imperfect, but not much. This is Stage 4. We return to this problem at the end of this Chapter.

            Another possibility is that you include newer and newer sounds. One diapason can be filled up with 3/2 steps, always halved the sound when you would end up in the next octave. The process is of course infinite, but you can stop when you have "already enough" sounds in your octave. For the first 71=1+35+35 Mean tone sounds see Tables A.2 & 3. E.g. you may say that you stop when you are sufficiently near to 2, where "sufficiently near" is a small enough absolute value in the ratio. With pure 3/2 steps in the heptatonic scale the error is 6.9%. For all scales of sounds <81 Table A.4 gives the error if it is <0.2. For larger errors the construction is hopeless.

            Now, scales formed with 8, 9, 10 and 11 {3/2} steps are much worse. But for 12 the error goes down to 1.4%. The next improvement happens at 41 steps, to 1.1%, and then at 53 steps to 0.2%. I stopped with the calculation at 80 steps, but Jeans claims that 306 steps would be excellent.

            However 5 steps result in 5.1% error, so pentatony is a scale more consonant than heptatony! Maybe this is behind its worldwide existence.

            Now, sometimes in the XVIIth century some instrument makers and instrument players observed that making 12 “fifths” and then stopping the process is much better that 11, 13 or 7. Their way was first to go from C upwards (and halving the frequencies if necessary) in 8 steps, and then going downward and doubling if necessary) 3 steps; and then completing the system with the diapason of C. Some sounds are just the Pythagorean or "clear" sounds, but some are not. These are somewhat higher ("sharp") or lower ("flat") than the Pythagoreans. Then the dodekatonic scale will be:

C,C-sharp,D,D-sharp,E-flat,E,F,F-sharp,G,G-sharp,A,B,H,C1

Note that in this scale D-sharp and E-flat are quite different: the respective frequencies are 1.168 and 1.1.196. In physical sense there is no such definite thing that D-sharp. Also, observe that B is H-flat in the Lydian Pythagorean scale, so in some sense B-flat = H-doubleflat with pure 3/2 steps

            And now comes the tempered scale. In strictly heptatonic times this idea was absent, roots being not true numbers. (There is an Ancient Greek proof to the conjecture that 21/2 is not a number; meaning in modern language that it is not a rational number.) After Glareanus' Dodekachordon [6] a tempering with 21/12 "half-tones" could arise as an idea, and indeed, it occurred. Finally, Bach wrote some equally tempered piano music.

 

5. DODEKAPHONY: IS IT ENOUGH?

            Above it has been told that dodekaphony is a great improvement. Here goes a Table for scales produced by less than 81 3/2 steps which are not worse than the heptatonic one:

 

Number of sounds

Error

53

0.002

41

0.011

12

0.014

65

0.016

29

0.025

24

0.027

76

0.030

70

0.036

36

0.041

58

0.049

5

0.051

48

0.056

46

0.062

7

0.069

 

Table 5: Relative discrepancies between Major consonance (3/2) and diapason (2) steps of scales of different numbers of sounds; only scales not worse than heptatonic are listed. For more, see the Appendix.

 

            We already have the dodekaphonic system, the next improvement at 41 steps would be a marginal one, so until a 53-tonic system we may remain with 12.

            However this is true only for the tempered 12-tonic scale. To be sure, stages 0, 1 and 2 never existed for the 12-scale. But the Mean tone did exist. Now, in this scale a "wolf-fifth" exists between the last upward and last downward sounds: G-sharp and E-flat. The respective frequencies are 1.563 and 1.198. The ratio is 1.221 while we should expect c. 1.33. (And then, why just a “fifth”?)

            The two sounds have been proven to be dissonant. So in Mean tone schemes it is inconvenient to stop here. Some guys made a few more steps and e.g. put two black keys between D and E &c. It is told that the organ of Händel in the London orphanery used such tricks. With a few more black keys the performance is much better; and then Stage 3 is ready for 12 sounds in the diapason.

            Stage 4 is represented by all modern pianos. Some people tell that the tempered performance is not good for consonance. I do not know: I am not listening on too tricky music.

            I think that dodekaphony was helped by Late Medieval mathematical confusions too. Namely, Late Ancient texts about Pythagorean musicology (the original texts from centuries earlier are almost completely lost) tell interesting stories how the Pythagoreans discovered laws of consonance. Some of them are impossible, and therefore late guesses. For example, [8] believes that Boethius c. 520 AD follows mainly Nicomachus of Gerasa, IInd c. AD in De institutione musica writing that once Pythagoras passed a smithy were smiths hammered iron. He heard consonances, so went in, measured the hammers and they were of 12, 9, 8 and 6 librae, respectively. Now we know that the frequency does depend on the weight of the hammer, but not linearly. However Gaudentius (IVth c. AD) describes a possible experiment, namely that Pythagoras and his followers stretched a string above a canon (c. measuring rod) divided into 12 equal parts. Now, they heard diapason consonance when using the 12 long and 6 long strings, “fourth” consonance if using 12 and 9 and “fifth” consonance if 12 and 8. (The Greek terms were diapason or harmonia, dia tessaron and dia pente; the second and third ones containing the numerals for 4 and 5 as the Latin.) You can read directly Nicomachus & Gaudentius in [9].

            Now the experiment described by Gaudentius is possible even if caution is needed. A string must be cut half and then both must be weighted with equal weights and finally tricky supports must be put to 12, 9, 8 & 6. But then, and only then, the frequency will be inversely proportional with the length.

            And now we can easily see why the Pythagoreans used a canon scaled to 12: 12 has dividers 2, 3, 4 and 6, so by integers we can realize 2/1, 3/2 and 4/3 too. With anything else up to 23 this would be impossible. But this has to do nothing with the fact that 12 {3/2}’s can be quite well approximated with diapasons while 11 or 13 not.

            I wonder if Glareanus could clearly distinguish the two 12's when working on his book about the 12 strings.

 

6. ON PENTATONY

            Many slightly different musical systems are called pentatony. Here I choose a formal definition, an observation of Jeans and a parallel to heptatony to limit the topic. So: pentatony is the use of music scales with 5 different sounds in a diapason; and [3] pentatonic tunes can be played restricting ourselves to only the black keys of the piano.

            Pentatony is very widespread on Earth; China was originally pentatonic, Japan was purely that 150 years ago, Celtic music was pentatonic, the oldest stratum of Magyar folk music is still pentatonic, pentatonic music is repeatedly reported from Eastern Europe and Siberia (Finns (?), Ugors, Turks, Mongols) & so on. It is generally believed that pentatony was the just the stage of musical evolution before heptatony.

            This may be true; lots of peoples reached pentatony but not heptatony. However surely the reason was not intellectual laziness. As shown above, 5 {3/2}’s approximate better 3 diapasons than 7 {3/2}’s 4 diapasons. So until we are interested in good major consonances, pentatony is better than heptatony; and this is even more true for the Pythagorean scale of heptatony. So the only argument for heptatony in a community using pentatony is varietas delectat; there are more combinations with more sounds, but the average degree of dissonancy will increase too.

            Of course the mechanistic idea of betterment, development & such of XIXth and XXth centuries tried to classify musical development together with that of society. As an example I can cite [10] from 1966.

            It states that first was matriarchy and a music of 2 sounds: the ground and its fifth. (You can see the inherent Mediterranocentricity of musical terms. Of course there are no fifths in a 2-grade scale. But "fifth" is a heptatonic term, trying to mean 3/2 frequency ratio to the first sound.) This represented the human nature of two sexes (if I understand correctly at all.) Later, still in matriarchy they halved the fifth as step, maybe with a third, and this represented mother, father, child. The last stage reached in matriarchy was the feminine harmonic anhemiton penta-music, but matriarchy was impossible for more. And then, about 1200 BC [rather 1100] the Chou dynasty in China introduced the Lydian scale, which is already heptatonic. But the Chous were patriarchal shepherds, and the Lydian scale is obviously the proper music for bullhorns (or ramhorns).

            Well, of course it may be that the Chous were more animal-oriented than the Shang-Yins, although the Shang-Yins were more involved in hunting than the Chous. But the Chou cultural hero, well before 1200 BC is the Millet Prince/Princess (Chinese is a genderless language), millet is a plant, and its caretaking is a more feminine activity than that of cows & bulls is.

            The literature about worldwide pentatony is tremendous, but of course I know  only a tiny part of it; which is still not written in a way digestible to physicists. There are many kinds you can arrange 5 sounds within a diapason; and I have reasons to suspect that most collectors did not explicitly recognised this problem. However, surely, because of the logic of Greek language, there may be hemitonic and anhemitonic pentatonic scales. Even with this classification the whole problems remain, since clearly an anhemitonic scale does not contain half-tones, but what is a half-tone in pentatony? Surely the proper pentatonic half-step cannot be that of the tempered 12-tone scale.

            We should keep this in mind. And indeed, there are quite different pentatonic scales. If Magyar pentatony is anhemitonic (which may be even true), still some Japanese ones are reported to have steps seeming half-tone steps in heptatony.

            Until the end of Stage 4 we will give only frequencies, not names, because names depend on conventions and pentatony has no dominant convention in Europe, America & Australia. After Stage 4 we will discuss the more or less viable "heptatonic" names.

 

Stage 0: Raw Scale:

            Surely for millennia before the advent of algebra &c. pentatony was world's highest music and lots of scales must have existed. We do not know anything certain about the scales of world's first fanatic algebrists conveniently called Old Babylonians bw. 2000 and 1600 BC; they obviously did not recognise the direct relation between lengths and sounds (as did it Pythagoras), so no mathematical tablet is extant about tuning. They used lyras & harps, but who knows how they were tuned? In the 30's the Assyrologists believed that they had a tablet ("A-A-ME-ME...") with a recorded music in 3 diapasons, but later the great Landsberg showed that the tablet is a sociological catalog [11]. Later Draffkorn-Kilmer reconstructed the music of a hymn on a tablet in the royal archive of Ugarit [12], c. 1500 BC. The text was Hurrian, but the musical parts were written in bad Akkadian. According to Draffkorn-Kilmer 7 sounds were used but not in Pythagoran relationship. Everybody may think anything about the Ugarit scale in this moment.

            In Old Egypt lots of musical instruments were in use, but it is difficult to know tuning. During the Old Kingdom harps of 6 or 7 strings were in use, which might suggest heptatony. (See e.g. the fresco in the tomb of Nenheftkai at Saqqara, Vth Dynasty.) However a well-known wall picture from the grave of Naft (Thebes West Bank, 52th grave, XVIIIth Dynasty) depicts a woman playing on a harp. The harp has 13 keys plus a few short strings not ending on keys. We cannot be sure that the painting is correct in every detail. However the picture indicates that neither 7 nor 12 were magic numbers for a musical scale then. As for Old China the highly allegoric way of formulating texts prevent us to understand some details. Very probably standard scales did exist; but which ones?

Old Chinese musicological studies are highly ethical but tell almost nothing about the sounds. In 1974, while looking for early poetical studies. F. Tőkei found one from Vth c. AD which told something about musical sounds as well [13]. The author is Yo Shen (441-513). Unfortunately, taking everything literally, the text is self-contradicting. Namely it states that

            1) there are 8 musical sounds; but

            2) they are of the scale of yang (low?) and yin (high?) ones; but

            3) the musical sounds are kung & yue ones.

You can find this in [14].

            Now even Tőkei did not want to explain the contradiction of 1) & 2) (we now know 6 yang and 6 yin sounds, and that is 12 altogether, not 8). However for 2) & 3) he told that kung and yue are the lowest and highest of the "wu yin", 5 sounds, of Chinese pentatony. The complete list is

kung→shang→kio→che→yue

Then statements 2) and 3) are not in contradiction, but speak in Vth c.  about a practice analogous to the modern one in India: maybe for a piece of music they selected 5 sounds from among the canonical 8 or 12 or any number.

            Unfortunately we know not too much about the actual yang/yin sounds 1500 years ago. In some time their number had been fixed to 12, 6 yang and 6 yin sounds, and then traditionalist Chinese historiography told that it was one of the oldest Emperors in 3rd millennium BC, Huang-ti, who wanted to canonize tuning. So he sent his minister to look for sounds. (If you are interested in the chronology of tradition, see [15], [16], [17] or simply the Appendix). The minister observed a pair of phoenix birds in the mountains: each of them uttered 6 sounds.

            Nothing indicates that these 12 sounds would have been the 12 sounds of European dodekaphony, even if Europeans tend to assume this. Some Chinese musical instruments had much more than 12 strings, tubes &c. And finally: even if we knew the 12, how should one select the actual 5 from amongst?

            As for recent Chinese, Japanese &c. pentatonic music, if they are recorded in European conventions the problem is equivalence of Eastern and European sounds. We have already seen that e.g. European mean tone and tempered 12 sounds are not exactly the same. Now, a Japanese ritual song is a part of a traditional scale maybe 1300 years old and originally completely independent from European Pythagoric heptatony. Maybe a sound is about our G-sharp; maybe it is rather A-flat. Magyar musicologists generally seem to believe that Chinese pentatony is anhemitonic, but Japanese is often not. [18] tells that in Japan various scales of 5 sounds appeared about 1300 from China or maybe from Korea, with rather unequal spacings.

            As for Magyar pentatony, the discovery is late, in 1905 or 1917 [1]. The singers were influenced for centuries by heptatonic scales and the first recorders a century ago used completely European conventions. So we do not know the details of the scales. That seems more or less sure that the remnants of the Hungarian pentatonic matter are composed in scales not containing anything similar to a Pythagorean half-tone step. Careful musicologists tell that the Magyar pentatonic scale contains tones and one-and-half tones. This will be algebraically clarified immediately. But this means that I rather restrict myself from referring to raw data. They surely exist on magnetic tapes and the proper way to transcribe them would need a proper selection of a transcription system. Algebra could help in that.

 

Stage 1: Ratios of small integers

            Of course, Pythagoras did not suggest any pentatonic scale. However we can do it now. Of course, there is no scale for only one kind of rational steps, because 21/5 is irrational. Still there is a scale of almost equal steps, where the sounds are in ratios of small integers:

1,8/7,4/3,3/2,7/4,2

The scale has even the preferred 4/3 and 3/2 steps. Now Birket-Smith tells [19] that the Javanese use 5 sounds equally spacing the diapason. I do not think that they could elaborate exactly equal irrational steps without convoluted measuring apparatuses, but they could produce approximately equal rational steps, as e.g. above.

            There is no good rational scale even with 4 equal and 1 different steps. The reason is that the unique step should contain the quartic power of the others. Even mathematically there is no such solution under the condition that there be a sound at 3/2 frequency of the first.

            However there are simple “Pythagorean” scales with two different steps, one occurring 3 times, the another 2 times. If we restrict ourselves to powers of the smallest integers, namely 2 and 3 (as Pythagoras did), the step occurring thrice is 9/8, and the other two are 32/27. We may call them half-tones and tones, but of course we may call them, of course, tones and one-and-half tones as well. In the second convention the scale is anhemitonic; and then?

            You can distribute the 32/27 steps amongst the 9/8 ones in different ways. There always will be two sounds with 3/2 or 4/3 frequency ratios, but not always from C.

           

            The most "symmetric" and "familiar" two scales are

1,5/4,4/3,3/2,15/8,2

and

1,5/4,4/3,3/2,8/5,2

The ratios are indeed of those of small integers, cca. as small as in the just intonation heptatonic scale.

 

Stage 2: Just Intonation

            Lots of scales can be found with more than 2 kinds of steps. Here, tentatively, I suggest a very symmetric one with 2 equal tones of 5/4, one hemitone of 9/8 (OK, now the pentatonic hemitone is as big as a heptatonic tone; and then?), and 2 quartertones of 16/15. The ratios are much more those of small integers than in the used heptatonic scales, except the heptatonic just intonation scale which is similar.

Now by permuting the 2 5/4, two 16/15 and one 9/8 steps different scales are obtained. Some of them differ only in the starting sound. However compare two scales with steps 5/4, 5/4, 9/8, 16/15, 16/15 and 5/4, 9/8, 5/4, 16/15, 16/15, respectively. Then, until the startpoint of the second diapason we get the frequencies

1,5/4,25/16,225/128,15/8,2 and 1,5/4,45/32,225/128,15/8,2

respectively. Now, the first scale does not contain "very consonant" 4/3 or 3/2 steps, so they can appear only between diapasons, while in the second scale the 3/2 ratio appears between the fifth and second sounds and 4/3 between the fifth and third ones. Some of the permutations are even more “familiar”.

 

Stage 3: Mean tone

            As with either Mean tone or pure 3/2 steps, you, for any case, can repeat the process mentioned in Chapter 4 to produce nice resonances/consonances. Going upwards with 3/2 steps until 5 sounds are obtained we get

1.,1.125,1.265625,1.5,1.6875

which, with the diapason of the starting sound added, is the simplest very consonant pentatonic scale, by construction. Again, if higher diapasons are routinely used (so in instrumental music with zithern, cymbals and whatnot of many strings, tubes, plates &c.), the mean tone trick is needed and then we get the simplest mean tone pentatony as

1.,1.1180,1.25,1.4953,1.6719,2

and so on in higher diapasons.

 

Stage 4: Tempered Scale

            For a Tempered Pentatone scale the step is 21/5=1.148698.... So the tempered scale is:

1,1.1487,1.3195,1.5157,1.7411,2

This is near to the Heptatonic Mean tone sounds

C,E-doubleflat,F,G,A-flat,C1

but is more near to the “almost equal” rational steps (1,8/7,4/3,3/2,7/4,2) told above. Of course only with 5 sounds in the scale tempering may be somewhat rough.

I will use henceforth D-sharp and D# as synonyms, and similarly D-flat and D&, where & stands for "b" for technical reasons.

            Now let us see the scales on heptatonic/dodekaphonic language. I give the nearest "European" sound in two systems: mean tone and tempered 12-tone ones.

            First let us go back to Point 1 with purely rational ratios. There is a scale "simplest with almost equal steps" and 10 with 3 equal greater and 2 equal smaller steps. However Scales 2-11 are in relations similar to the different old Pythagorean scales, so it well be enough to take one of them. So

 

Almost equally separated scale

Freq.

Nearest meantone

Nearest tempered 12

1

C

C

1.1429

E&&

D

1.3333

F

F

1.5

G

G

1.75

A#

B

2

C1

C1

 

Table 6: The almost equal pentatonic scale in European terms.

 

            While the steps are almost equal, a belief in dodekaphonic language would suggest 3 half-tone distances at D-F and B-C1. However while both D and B are the nearest 12-tempered sounds, they are not the same.

 

"Pythagorean"scale

            I choose here "the most familiar" of the 10 scales.

 

Freq.

Nearest meantone

Nearest tempered 12

1

C

C

1.125

D

D

1.2656

F&

E

1.5

G

G

1.6875

A

A

2

C1

C1

 

Table 7: The pentatonic “Pythagorean” scale in European terms.

 

            Now the D of meantone perceptibly differs from the pentatonic D. The third and fifth steps are bigger. Observe that in tempered dodekaphonic language this would be the system suggested by dodekaphonically schooled recorders for Magyar pentatony. This scale is indeed anhemitonic in dodekaphonic sense, because it does not contain either heptatonic Pythagorean hemitones (256/243 steps) or tempered 12 "half-steps" (21/12). But compared to the bigger 32/27 steps the smaller 9/8 ones are not much larger than half-steps.

 

Just intonation

            The two "most familiar" just intonation scales differ only in the fifth sound. So

 

Freq.

Nearest meantone

Nearest tempered 12

1

C

C

1.25

E

E

1.3333

F

F

1.5

G

G

1.6/1.875

A&/H

G#/H

2

C1

C1

 

Table 8: Pentatonic “just intonation” scale in European terms.

 

In the tempered 12 "translation" the first alternative scale would suggest a dodekaphonic half-tone. Indeed, the step is 16/15, smaller than a heptatonic tone although bigger than a heptatonic half-tone.

            As for Mean Tone, observe that the (minimal) pentatonic Mean tone scale is a subset of the heptatonic one by construction. In the common Mean tone the pentatonic scale is

C,D,E,G,A,C1

Using many diapasons simultaneously, something analogous the "wolf-fifths" may appear, but pentatonic societies did not develop organs. The simplest scale is obviously anhemitonic.

            As for Stage 4, the tempered 5-tone frequencies given above can be transcribed into Mean tone letters as

C,E&&,F,G,A#,C1

and in 12-tone tempered language it is

C,D,F,G,A#,C1

But the scale is very near to the "almost equally separated" rational one above, so see Table 6.

           

7. PENTATONY IN LAYERED SOCIETIES

            Singers generally remain within 2 diapasons because of anatomy, and people who are not professional singers do not much leave a single diapason. Even simple musical instruments remain within one; flutes, oboas & such have a few holes and no more and lyras of the Greeks had 4 or 7 strings. But in seriously layered societies  rich/mighty peoples employed professional musicians, who, if for nothing more serious than prestige, asked for complicated instruments (or built them), and of course having exceptional musicians was prestige for the princes as well.

            In Mesopotamia we enter into this stage c. 2500 BC. Leonard Wooley in the first half of last century excavated the "Royal Cemetery of Ur". The oldest rulers buried there are still not dated, Wooley's guess for 3500 BC is impossible, but the cemetery goes back before Mesannipadda, founder of the Ur I Dynasty (according to the Sumerian King List from c. 1700 BC). Mesannipadda can be dated quite satisfactorily as starting his reign in 2643±25 [20]. Wooley found many harps and the reconstructions gave 8, 11 and 15 strings. The last had 15 keys, so indeed it may have had 15 strings.

            If you do not believe in reconstructions, look at the "Mosaic Standard of Ur" again from the Royal Cemetery of Ur, now in the British Museum, and reproduced widely in various history books. There a harp player uses a harp of 11 strings. Again, we can see a harp of 11 strings on a diorite relief from Girsu (Lagash) [21], now in the Louvre; it is dated to 2130 BC. Neo-Assyrian reliefs sometimes show even more than 15 strings. Now we do not know if the Mesopotamian harps remained in one diapason, but it is not probable. A 11-tone scale has the error >0.2; for 10 (and the 11th is the double of the first) it is 0.099 (see Table A.4), not bad but worse than either 5 or 7, and for 14 & 15 the error is even higher. Probably the biggest harps already were beyond one diapason, even if the strings seen on the Girsu relief do not show too much difference in lengths.

            Old Egypt was quite centralised and extremely so during the Old Kingdom. Pharaoh Kheops had tremendous prestige independently of the number of strings of court harps; and nobody should exceed him in anything. No surprise that the Old Kingdom harp mentioned in the previous Chapter has no more than 7 strings. As for tuning we cannot know anything; but the ratio of the lengths of the longest and shortest strings is cca. 4:3, so it is improbable that the harp would have gone beyond 1 diapason.

            Then came 2 Transition periods and a Middle Kingdom; and when the New Kingdom arrives, prestige competitions are well known. No surprise that the New Kingdom fresco shows a hap with at least 15 strings. Of them, 13 is keyed; the lengths of the first and thirteenth are cca. in 13:5 ratio, and the fresco shows at least two nonkeyed ones, even shorter. This indicates a range well beyond 1 diapason, even if it would be dangerous to guess more. (The "method" can be "checked" on a Mesopotamian harp, on a relief "industrially" produced in copies c. 1800 BC [21]; copies exist in the Iraq Museum, Baghdad, the Louvre and the Oriental Institute Museum, Chicago. A woman plays on a harp of six strings; clearly not a royal musician. Now the lengths of the first and sixth strings are in cca. 2:1 ratio; so it is possible that the sixth string plays the diapason of the first.)

            In Mycenian Greece the wanakes were sacred to some degree and autocrats but tradition did not keep the memory of big and/or convoluted ensembles; and for any case such ones were unknown in Dark, Archaic and Classic ages, but then Greece had not too layered societies. In the Archaic Ages kings were abundant and heroes too, but they employed bards only transiently, and these bards were singing, with a lyra as instrument. They were not beyond the diapason, because the kings did not need/support that.

            Our solid knowledge about Chinese instrumental music does not go before Greece, in spite of Chinese pseudohistory about Huang-ti. Roughly in Classical Antiquity the schools of Konfucius and Mo-ti debate whether music is good or bad for society; but these texts tell nothing about the details of instruments or their tunings. Surely lyras, zitherns, flutes, cymbals and gongs abounded, but we would like to know the numbers of strings, holes, tubes and plates. A later text (c. from Vth c. AD [14]) mentions a Nan-kuo (or Tung-kuo) who was clumsy with an instrument of 36 tubes. Surely this instrument had at least 3 diapasons, so the almost cyclic 3/2 steps could have started; maybe even a mean tone construction as well?

            From the Vth c. AD the steppe was dominated by Turkish tribal alliances. (The Seven Magyars in the Völkerwanderung seem culturally indistinguishable from the Turks, and in 952 Emperor Constantine VIII Porphyrogenetus terms the Magyars already in the Carpathian Basin as "Turks".) Now, Turkish music is generally considered pentatonic and especially anhemitonic pentatonic. As we saw in the previous Chapter, this is expected if the range of sounds is smaller than 2 diapasons. Now, Turkish khagans had prestige, even wealth; but their courts were smaller and simpler than that of a Chinese Emperor or a Pharaoh of Egypt. Turkish society was layered but not very much. The common warrior, his wife and daughters simply sang; and as far as we can guess, chiefs and the khagan were satisfied with a few musicians at most with simple instruments.

            In the same time in Japan the Imperial Court became more and more convoluted, and China evolved really rich court ceremonies. E.g. not later than XIIth c. Japan imported the Chinese kin zithern from which the Japanese developed the koto [18]; that has 13 strings. And so on. This inevitably leads to looking for more and more sounds in nice resonances/consonances, either using pure 3/2 steps, or correcting them by a mean tone-type trick.

            Now, the 3/2 steps will not stop at 7 sounds. They can go infinitely; see Table 9, for 53. As told in Chapter 4 (and also see Table A.4) with 7 sounds in the diapason the consonance is worse than with 5. However in a big and rich court the generation of new sounds can easily reach 12 sounds, can go slightly beyond and then the musicians can experience that

1)      with 12 sounds consonance is better that with 5 (the error being 1.4% instead of 5.1%; the difference can be heard), but

2)      beyond 12 the error is again higher. Then they stop at 12, even if they continue to use only 5 in one piece of music.

 

0

1.

C

+19

1.020417

H##

-12

1.024

D&&

+7

1.044907

C#

-24

1.048575

E&&&&

+26

1.066241

H###

-5

1.069984

D&

+14

1.09183

C##

-17

1.095664

E&&&

+2

1.118034

D

+21

1.140861

C###

-10

1.144867

E&&

+9

1.168241

D#

-22

1.172343

F&&&

-3

1.196279

E&

+16

1.220704

D##

-15

1.224989

F&&

+4

1.25

E

+23

1.275522

D###

-8

1.28

F&

+11

1.306134

E#

-20

1.310719

G&&&

-1

1.337481

F

+18

1.364788

E##

-13

1.36958

G&&

+6

1.397543

F#

-25

1.402449

A&&&&

+25

1.426076

E###

-6

1.431083

G&

+13

1.460302

F##

-18

1.465429

A&&&

+1

1.495349

G

+20

1.52588

F###

-11

1.531237

A&&

+8

1.5625

G#

-23

1.567986

B&&&

-4

1.6

A&

+15

1.632667

G##

-16

1.638399

B&&

+3

1.671851

A

+22

1.705985

G###

-9

1.711975

B&

+10

1.746928

A#

-21

1.753062

C&&&

-2

1.788854

B

+17

1.825378

A##

-14

1.831786

C&&

+5

1.869186

H

-26

1.875749

D&&&&

+24

1.90735

A###

-7

1.914046

C&

+12

1.953125

H#

-19

1.959983

D&&&

 

Table 9: Mean tone sounds according to frequency

 

            And then see: Chinese have 12 sounds. (And Yo Shen in Vth c. did really have only 8??) Now on Old Chinese/Taoist grounds they may call them yang and yin sounds. But it seems that the yang-yin (or phoenix cock-hen) subscales are not in lower-higher relation. If you produce the first 12 sounds by 3/2 steps (the Mean tone alternative can be selected from Table A.3), and put the odd members to the yang series and the evens to the yin one, then you get:

 

Step

Cock/Hen (Odd/Even)

Freq.

Tempered 12 sign

1

+

1.

C

2

-

1.5

G

3

+

1.125

D

4

-

1.6875

A

5

+

1.2656

E

6

-

1.8984

H

7

+

1.4238

F#

8

-

1.0679

C#

9

+

1.6018

G#

10

-

1.2014

D#

11

+

1.8020

A#

12

-

1.3515

F

 

Table 10: The first 12 sounds with 3/2 steps and the hypothetical pair of phoenix birds.

 

(maybe they were originally answering to each other), and arranging the two subseries into ascending frequencies the results are:

Cock: C,D,E,F#,G#,A#; Hen: C#,D#,F,G,A,H

just the list I saw. Of course it is some mishmash of conventions, but that always occurs when one listens pentatonic music with heptatonic background.

            Now, the tribal alliance of the Seven Magyars surely cultivated pentatony, but First Chief/Duke Árpád in 896, when Magyars came into the Carpathian Basin, was an average Turkish Khagan. So some court music surely was established, but with simple instruments, surely not in 2 diapasons range. So surely it was the scale in Stage 3 of Chapter 6 ("anhemitonic"). Then, for 69 years, the might of the leader of the alliance was declining, bringing him down to the level of the other 6 chiefs. So no instruments of 36 tubes or 15 strings are expected. After 955 the central power started again to ascend, but from 972 (Grand Duke Géza) Christian influence is already strong because Géza is negotiating, and the Great Duchess Sarolta (Sara Oldu, White Ermin/Lady, in Bulgarian Turkish), daughter of a tribal chief, is Christian. (From contemporary German record we know her Elbe Slav (!) name Belekenigi = White Lady, again). So Gregorian tunes appear in court.

            Géza's son, Vajk, in 1000 is crowned as Stephen I,, according to Western Christian rites. Henceforth Gregorian tunes are very frequent in the Hungarian court, although Magyar, Kabar &c. people surely continued to sing pentatonic while drinking. Pentatony surely survived for a while even at high social levels at parts of the country convenient for horses; but we do not know, how long. It may have even started to evolve into "hemitonic pentatony", as in Japan & China, but we do not know anything about this, because the pentatonic movement of Bartók & Kodály was interested only in the ground level of society.

            The selection effect can be demonstrated, at least as a caricature, by a nice pentatonic song (Magyarlapád, Fejér County, RN 114, 151; TSz 53) whose text is in my clumsy translation: Do not care, darling, that I am filthy; (repeat);Because I am a maidservant; (repeat). Since human females are quite fanatic to wash themselves, great landowners do not like filthy females near to them, and at poor landowners the well was available, the heroine of the song simply does not seem to have too much self-respect.

            With such selection methods & principles early influence of Magyar pentatony on  heptatonic Western music at courts or vice versa cannot be seen.

 

8. TOWARDS BIGGER, RICHER AND MORE CONSONANT SCALES

            Table 5 shows that a 53-tone scale, generated by 3/2 steps, would be almost error-free. The discrepancy between the 53rd step and 31 steps of 2 is only 0.21 % relative, some 1/7th of the dodekaphonic scale. The first suggestion for a 53-tone scale on this ground came from the famous mathematician Mercator. (See a very brief discussion of the scale in [3].)

            Jeans tells us that in the 1850's two harmoniums were built with 53-tone scales, in London and in Springfield, Illinois. I never heard about the sound of them; and it seems from his words that Jeans did not either. The fact that the two harmoniums did not create any revelation, can maybe regarded as a negative Raw Date; I think I can explain, why the great success did not come. Stages 1 & 2 can be passed: the scale first was suggested not from ratios of small integers but from higher mathematics. However Stages 3 & 4 are straightforward.

 

Stage 3: Mean tone

            Again we start upward and downward from C by steps not exactly 3/2 but with 51/4. Say, we go 26 steps upward and 26 downward. The resulting scale is just Table 9. Table A.2 shows a few more too. For any steps, the heptatonic-based language still may be used with multiple-flats and -sharps, but is strongly forced. You can see, e.g., that C### is now higher than E&&&.

 

            If a wolf-step may appear at the end of construction, then we go a few more steps. Indeed, do not forget that 553/4 ≠(3/2)53. Therefore the best Mean tone scale will not be just the 53-tone one; that has a wolf-step, see Tables A.2 & A.3 in the Appendix.

            I think people sometimes do not clearly distinguish scales of 3/2 and Mean tone steps, because 51/4 is too near to 3/2. But after 53 steps of 51/4 there is an accumulated discrepancy realised in a wolf-step.

            So if we want to follow the ways of organ builders, then the calculation must be repeated with 51/4 steps. Table 11 gives the errors, but only the Mean tone scales not worse than the 12-tone one, up to N=80.

 

N of scale

Rel. error

12

0.0234

19

0.0204

50

0.0168

62

0.0070

31

0.0035

 

 

Table 11: The best N<81 Mean tone scales

 

 

            As for the sounds in the scale, they are parts of Table 9. As for the best, 31-tone, scale, the simplest way is to take the sounds of Table 9 until ±15, they are just 31. You can detect that the +16th sound is almost the same as the -15th (D## vs. F&&), and similarly for the -16th and the +15th (B&& vs. G##). So there is no wolf-howl; at most a cat-miaow.

            If the 53-scale harmoniums were tuned by somebody following Mean tone conventions, then there was a wolf-step. If he used pure 3/2 steps, then ratios containing 5 were absent. Maybe the nontrivial task of optimal tuning was not approached at the London and Springfield harmoniums. Mean tone tradition and 31-tone scale is an easier target; and nothing more exotic than double-flats and double-sharps appear.

            Here we can learn that Pure Mathematics cannot give an answer to any question which is not Pure Mathematics. If we know exactly what do we want, then we can exactly calculate what to do. If we have not formulated what do we want, and we apply Mathematics, surprises may occur...

 

Stage 4: Tempered Scale

            Now, contrary to Mean tone, indeed N=53 is the best (up to N=306, to be sure). The tempered scale is that of a scale of steps 21/53=1.01316.... This scale contains almost exact consonances, e.g. 1.2009 instead of 6/5, 1.2490 instead of 5/4, 1.3334 instead of 4/3 or 1.4999 instead of 3/2. So pianos would operate very well; of course the strings would give a work and the keyboard would be long. But electronic solutions will help. Here goes the scale as Table 12:

 

  1

            1.

1

  2

            1.013164

81/80=1.0125

  3

            1.026502

41/40=1.025

  4

            1.040015

26/25=1.04

  5

            1.053706

19/18=1.0555…

  6

            1.067577

16/15=1.0667…

  7

            1.081631

13/12=1.0833…

  8

            1.095869

12/11=1.0909…

  9

            1.110296

10/9=1.1111…

 10

            1.124912

9/8=1.125

 11

            1.13972

8/7=1.1429…

 12

            1.154724

22/19=1.1579…

 13

            1.169925

7/6=1.1667…

 14

            1.185326

13/11=1.1818…

 15

            1.200929

6/5=1.20

 16

            1.216739

17/14=1.2143…

 17

            1.232756

16/13=1.2308…

 18

            1.248984

5/4=1.25

 19

            1.265426

19/15=1.2667…

 20

            1.282084

23/18=1.2778…

 21

            1.298962

22/17=1.2941…

 22

            1.316062

25/19=1.3158…

 23

            1.333386

4/3=1.3333…

 24

            1.350939

23/17=1.3529…

 25

            1.368723

26/19=1.3684…

 26

            1.386741

18/13=1.3846…

 27

            1.404997

7/5=1.4

 28

            1.423492

27/19=1.4211…

 29

            1.442231

13/9=1.4444…

 30

            1.461217

19/13=1.4615…

 31

            1.480453

34/23=1.4783…

 32

            1.499942

3/2=1.5

 33

            1.519687

35/23=1.5217…

 34

            1.539693

20/13=1.5385…

 35

            1.559962

14/9=1.5555…

 36

            1.580497

19/12=1.5833…

 37

            1.601303

8/5=1.6

 38

            1.622383

13/8=1.625

 39

            1.64374

23/14=1.6429…

 40

            1.665379

5/3=1.6667…

 41

            1.687302

27/16=1.6875

 42

            1.709514

12/7=1.7143…

 43

            1.732018

19/11=1.7272…

 44

            1.754819

7/4=1.75

 45

            1.777919

16/9=1.7778…

 46

            1.801324

9/5=1.8

 47

            1.825037

31/17=1.8235…

 48

            1.849062

24/13=1.8462…

 49

            1.873404

15/8=1.875

 50

            1.898065

36/19=1.8947…

 51

            1.923052

25/13=1.9231…

 52

            1.948367

37/19=1.9474…

 53

            1.974016

79/40=1.975

 54 ≡ 1

            1.000000

1

 

Table 12: The tempered 53-tone scale.

 

 

The third column gives a "ratio of integers" near (nearer than 1/200 for italics, than 1/250 for regular and than 1/500 for bold numbers) to the actual tempered sound. Sometimes the integers are really small, sometimes not really, but never so big than at Pythagorean E or H. Lots of them contain 5, 7 or 11 in the nominator. Of course, no wolf-step can appear.

 

9. A VERY BRIEF OUTLOOK

            I wanted here to demonstrate that pentatony is not truncated heptatony. In Hungary the observation of the difference does not seem hopeless a priori even now, although hepta- and 12-music for centuries may have being diluted it.

            Namely, consider e.g. the stage of Ratio of Small Integers or Just Intonation.

            Pythagoras formulated his heptatonic system, and there the tones had 9/8 steps before them and the halftones 256/243's. Now, pentatony is not simply to drop sounds F & H. There is no rational scale with equal steps, so you cannot build up a scale with {C,D,E,G,A,C1} with equal steps. The
Pythagorean E is at 1.2656 while the analogous D of Chap. 6 is at 1.25. The Pythagorean A is at 1.6667, while the pentatonic analogy is at either 1.6 or 1.875. Such a difference (the pentatonic A would be mean tone A&) should be heard on phonograph or tape records, available.

            Of course, the singers may use other pentatonic scales; but a few such is listed in this work. And there was a region of Hungary where the dilution of pentatony must have been minimal.

            The Terra Siculorum, the Szekler Lands, was not organised into counties until 1872; the local people governed themselves in Sedes. At the beginning of XXth century Bartók & Kodály collected there pentatonic music, and for older women singers, not having been in military service, the sources were not too much influenced by non-Szekler music, except perhaps the Gregorians in Catholic villages. So the collected material should differ in the frequencies of C & A (or do & la) from those in heptatonic Europe. Now: is it?

 

ACKNOWLEDGEMENTS

            First I must express my thanks to long-deceased colleague, J. Jeans, astrophysicist, for completely formulating the problems of musical scales in mathematical and physical language. It is contained by Ref. [3]. Second, I would like to thank comments of two members of the Matter Evolution Subcommittee of the Geonomy Scientific Committee of HAS, Drs. Katalin Barlai & Emília Madarász, who can feel music much better than I. However all the responsibility is mine.

 

REFERENCES

 [1]       Z. Kodály: Ötfokú hangsor a magyar népzenében. Zenei szemle I, Temesvár, 1917

 [2]       Kálmán B.: Chrestomathia Vogulica. Tankönyvkiadó, Budapest, 1976

 [3]       J. H. Jeans: Science and Music. Dover Publ., 1968

 [4]       A. Schlick: Spiegel der Orgelmacher und Organisten, P. Drach, Speyer, 1511

 [5]       F. Salinas: De Musica Libri Septem, M. Gastius, Salamanca, 1577

 [6]       H. L. Glareanus: Dodekachordon. H. Petri, Basel, 1547

 [7]       T. Encrenaz, J.-P. Bibring & M. Blanc: The Solar System. Springer, Berlin, 1990

 [8]       Á. Szabó & Z. Kádár: Antik természettudomány. Gondolat, Budapest, 1984

 [9]       K. von Jan: Musici scriptores Graeci. Aristoteles, Euclides, Nicomachus, Bacchius, Gaudentius, Alypius et melodiarum veterum quidquid exstat. Teubner, Leipzig, 1895

[10]      W. Danckert: Tonreich und Symbolzahl in Hochkulturen und in der Primitivenwelt. Bouvier, Bonn, 1966

[11]      Anne Draffkorn-Kilmer: The Discovery of an Ancient Mesopotamian Theory of Music. Proc. Amer. Phil. Soc. 115, 131 (1971).

[12]      Anne Draffkorn-Kilmer: The Cult Song with Music from Ancient Ugarit. Rev. Assyr. Arch. Orient. 68, 1 (1974)

[13]      F. Tőkei: Sinológiai műhely. Gondolat, Budapest, 1974, p. 231

[14]      E. von Zach (ed.): Die Chinesische Anthologie. Harvard University Press, Cambridge Mass., 1958, Vol. 2, p. 938

[15]      P. Hoang: De calendario sinico variae notiones. Calendarii sinici et europaei concordantia. Zi-Ka-Wei, 1885

[16]      H. A. Giles:Religions of Ancient China. Constable & Co., London, 1906

[17]      Katalin Barlai & B. Lukács: The Firestar in China. Archaeoastronomy in Archaeology and Ethnography, BAR, Gordon, 2007, p.. 7

[18]      L. Frédéric: La vie quotidienne au Japon a l'époque des Samouraď 1185-1603. Hachette, Paris, 1968

[19]      K. Birket-Smith: The Paths of Culture. University of Wisconsin Press, Madison & Milwaukee, 1965

[20]      B. Lukács & L. Végső: The Chronology of the "Sumerian King List". Altorient. Forsch. 2, 25 (1975)

[21]      G. Komoróczy: Fénylő ölednek édes örömében. Európa, Budapest, 1970

 

APPENDIX: MORE TABLES

            Table A.1 gives the first 35 3/2 steps up and down always brought between 1 and 2 by dividing/multiplying by 2, ordered finally ascended way. The first column is the number of steps; C is at Step 0. Upward and downward steps are not distinguished in the first column but # indicates upward steps while & downward ones in Column 3. When neither appears, F and B are in the downward string, others in the upward one.

 

 0

     1.     

     C          

 12

     1.013643

     H#

 29

     1.025329

     F&&&&

 24

     1.027473

     A###

 17

     1.039318

     E&&&

 5

     1.053498

     D&

 7

     1.067871

     C#

 34

     1.080182

     G&&&&&

 19

     1.08244

     H##

 22

     1.09492

     F&&&

 31

     1.097209

     A####

 10

     1.109858

     E&&

 2

     1.125  

     D

 14

     1.140349

     C##

 27

     1.153495

     G&&&&

 26

     1.155907

     H###

 15

     1.169233

     F&&

 3

     1.185185

     E&

 9

     1.201355

     D#

 32

     1.215205

     A&&&&&

 21

     1.217745

     C###

 20

     1.231785

     G&&&

 33

     1.23436

     H####

 8

     1.24859

     F&

 4

     1.265625

     E

 16

     1.282892

     D##

 25

     1.297682

     A&&&&

 28

     1.300395

     C####

 13

     1.315387

     G&&

 1

     1.333333

     F

 11

     1.351524

     E#

 30

     1.367106

     B&&&&

 23

     1.369964

     D###

 18

     1.385758

     A&&&

 35

     1.388655

     C#####

 6

     1.404664

     G&

 6

     1.423828

     F#

 35

     1.440243

     C&&&&&

 18

     1.443254

     E##

 23

     1.459893

     B&&&

 30

     1.462945

     D####

 11

     1.479811

     A&&

 1

     1.5    

     G

 13

     1.520465

     F##

 28

     1.537994

     C&&&&

 25

     1.541209

     E###

 16

     1.558977

     B&&

 4

     1.580247

     A&

 8

     1.601807

     G#

 33

     1.620273

     D&&&&&

 20

     1.623661

     F###

 21

     1.642379

     C&&&

 32

     1.645813

     E####

 9

     1.664787

     B&

 3

     1.6875 

     A

 15

     1.710523

     G##

 26

     1.730243

     D&&&&

 27

     1.73386

     F####

 14

     1.75385

     C&&

 2

     1.777778

     B

 10     

     1.0802033

     A#

 31

     1.822807

     E&&&&&

 22

     1.826618

     G###

 19

     1.847677

     D&&&

 34

     1.851539

     F#####

 7

      1.872885

     C&

 5

      1.898438

     H

 17

      1.924338

     A##

 24

      1.946524

     E&&&&

 29

      1.950593

     G####

 12

      1.973081

     D&&

 

Table A.1: Pure 3/2 steps for 53 sounds + some others in the possible “wolf-fifth” .

 

            Table A.2 is the first 35 steps in the mean tone construction, so as Table A.1, but the step is 51/4 instead of 3/2. The first column  is the step number, the second is the frequancy in the upward string, the third is the same in the downward one:

 

N°

Up

Down

 0

1.

1.

 1

1.495349

1.337481

 2

1.118034

1.788854

 3

1.671851

1.196279

 4

1.25

1.6

 5

1.869186

1.069984

 6

1.397543

1.431083

 7

1.044907

1.914046

 8

1.5625

1.28

 9

1.168241

1.711975

 10

1.746928

1.144867

 11

1.306134

1.531237

 12

1.953125

1.024

 13

1.460302

1.36958

 14

1.09183

1.831786

 15

1.632667

1.224989

 16

1.220704

1.638399

 17

1.825378

1.095664

 18

1.364788

1.465429

 19

1.020417

1.959983

 20

1.52588

1.310719

 21

1.140861

1.753062

 22

1.705985

1.172343

 23

1.275522

1.567986

 24

1.90735

1.048575

 25

1.426076

1.402449

 26

1.066241

1.875749

 27

1.594402

1.254389

 28

1.192093

1.677721

 29

1.782595

1.121959

 30

1.332801

1.500599

 31

1.993002

1.003511

 32

1.490117

1.342176

 33

1.114122

1.795135

 34

1.666001

1.200479

 35

1.245627

1.605617

 

Table A.2: Mean Tone: 53 sounds and some more against possible wolf-steps. Observe that now the cycle does not almost ends after 2*26 steps, because the step is smaller than 3/2, but it is almost at the start again in 2*31 steps.

 

 

            Table A.3 contains the same frequencies as Table A.2, but now the sound names are given too.

 

 

 

 0 

           1.     

       C

+1

           1.495349         

       G

+2

           1.118034         

       D

+3

           1.671851         

       A

+4

           1.25     

       E

+5

           1.869186         

       H

+6

           1.397543         

       F#

+7

           1.044907         

       C#

+8

           1.5625 

       G#

+9

           1.168241         

        D#

+10

           1.746928         

        A#

+11

           1.306134         

        E#

+12

           1.953125         

        H#

+13

           1.460302         

        F##

+14

           1.09183           

        C##

+15

           1.632667         

        G##

+16

           1.220704         

        D##

+17

           1.825378         

        A##

+18

           1.364788         

        E##

+19

           1.020417         

        H##

+20

           1.52588           

        F###

+21

           1.140861         

        C###

+22

           1.705985         

        G###

+23

           1.275522         

        D###

+24

           1.90735           

        A###

+25

           1.426076         

        E###

+26

           1.066241         

        H###

+27

           1.594402         

        F####

+28

           1.192093         

        C####

+29

           1.782595         

        G####

+30

           1.332801         

        D####

+31

           1.993002         

        A####

+32

           1.490117         

        E####

+33

           1.114122         

        H####

+34

           1.666001         

        F#####

+35

           1.245627         

        C#####

-1 

           1.337481         

        F

-2 

           1.788854         

        B

-3 

           1.196279         

        E&

-4 

           1.6           

        A&

-5  

           1.069984         

        D&

-6 

           1.431083         

        G&

-7 

           1.914046         

        C&

-8 

           1.28                 

        F&

-9 

           1.711975         

        B&

-10

           1.144867         

        E&&

-11

           1.531237         

        A&&

-12

           1.024               

        D&&

-13

           1.36958           

        G&&

-14

           1.831786         

        C&&

-15

           1.224989         

        F&&

-16

           1.638399         

        B&&

-17

           1.095664         

        E&&&

-18

           1.465429         

        A&&&

-19

           1.959983         

        D&&&

-20

           1.310719         

        G&&&

-21

           1.753062         

        C&&&

-22

           1.172343         

        F&&&

-23

           1.567986         

        B&&&

-24

           1.048575         

        E&&&&

-25

           1.402449         

        A&&&&

-26

           1.875749         

        D&&&&

-27

           1.254389         

        G&&&&

-28

           1.677721         

        C&&&&

-29

           1.121959         

        F&&&&

-30

           1.500599         

        B&&&&

-31

           1.003511         

        E&&&&&

-32

           1.342176         

        A&&&&&

-33

           1.795135         

        D&&&&&

-34

           1.200479         

        G&&&&&

-35

           1.605617         

        C&&&&&

 

Table A.3: Mean tone sounds according to construction

 

            Table A.4 shows the error by which N exactly 3/2 steps approximate pure harmonias (doubling). The error is by construction a number between 0 and 0.5, the minimum of the absolute value of (3/2)N/2p, where p runs. The Table goes only until N=80, and for simplicity only the cases are given when the error is <0.2. The first column is N, the second is the best p and the third is the error.

            You can see that

            N=53 the best until N=80 (really until 305);

            This "cycle" is almost closed in 84 doublings;

            N=12 is the third best and N=41 is better only marginally;

            The next 7 "best scales" involve much more sounds than 12;

            Pentatony is the 11th best, but better than anything up to N=11;

            Pentatony can be less dissonant than the best heptatony.

It seems that the potentials of 12-tone scales are "algebraic accidents" in the sense that nobody could predict them without really good algebraic procedures. You can see that there was really not a serious reason to enlarge the scale to heptatony, and that the usual beliefs for the origin of heptatony are rather unsatisfactory. Finally, Table A.4 demonstrates that pentatony can be quite consonant and versatile.

 

 

 

53,

84,

0.002

41,

65,

0.011

12,

19,

0.014

65,

103,

0.016

29,

46,

0.025

24,

38,

0.027

76,

121,

0.030

70,

111,

0.036

36,

57,

0.041

58,

92,

0.049

5,

8,

0.051

48,

76,

0.056

46,

73,

0.062

7,

11,

0.069

60,

95,

0.070

34,

54,

0.074

72,

114,

0.082

75,

119,

0.085

22,

35,

0.087

17,

27,

0.088

19,

30,

0.088

31,

49,

0.097

63,

100,

0.097

10,

16,

0.099

51,

81,

0.109

43,

68,

0.112

39,

62,

0.121

2,

3,

0.125

55,

87,

0.127

27,

43,

0.133

14,

22,

0.140

67,

106,

0.143

68,

108,

0.143

15,

24,

0.145

56,

89,

0.154

3,

5,

0.156

26,

41,

0.156

79,

125,

0.158

44,

70,

0.166

38,

60,

0.172

32,

51,

0.177

73,

116,

0.186

20,

32,

0.188

50,

79,

0.188

61,

97,

0.197

8,

13,

0.199

 

 

Table A.4:  As Table 5, but up to discrepancy 0.2.

 

            Finally, a Table for Chinese traditional chronology after massive European contact but before discarding the tradition by the historians. The Table is from [17]; as for the sources see [15] for a Chinese scholar and [16] for an European one, both more than a century ago.

            Chinese tradition knows about c. 10 Ancient Rulers, kinds of Cultural Heroes. The first started ruling when Humanity was very ignorant, and the last nominated his son as the next ruler, so establishing Ruling Dynasties (actually the Xia).

 

Ruler

Date of reign

Fu Xi

2953-2838

Shen Nong

2838-2698

Huangdi

2698-2598

Shao Hao

2598-2514

Zhuan Xu

2514-2436

Diku

2436-2366

Gaoxin

2366-2357

Yao

2357-2255

Shun

2255-2205

Yü

2205-2197

 

Table A.5: The traditional Ancient Rulers of China

 

The transcription of Chinese ideograms vary, but Huangdi is Huang-ti in Chap. 6.

            Well, his time is the middle 3rd millennium BC. True, in canonical Chinese tradition he established not Music, but the Orders of officials. A tradition may have variants, but when in 3rd c. BC the victorious Qin leader established the Empire, he took Huang-ti as part of his ruling name; surely rather for establishing Orders of officials, than for canonising Music. In the canonical list Music was established not by Huangdi but by his first successor, Shao Hao. Not a great difference.

            The chronology, of course, cannot be accepted. Not because of the slightly too long ruling periods; those could be intelligently revised. (The first Ancient Ruler reigned 115 years; the first Japanese Emperor, Jimmu, has 120 years, not a big difference.) The problem is that the first ruler teaches the people to use nets for hunting & fishing, and the second introduces Agriculture. So Fu Xi represents the whole Mesolithe, and Shen Nong stands at the beginning of Neolithe. In Denmark the first would be Maglemosian, say, 9000 BC and the second would be c. 7000 BC. Since the periods were triggered by the Global Warming at the end of Würm III, the Chinese ages cannot differ too much.

            So the Chinese chronology is too short. Still, mid-3rd millennium is not impossible for some fundamental musical innovations; we have seen similar data for Sumerian and Egyptian musics. But of course an introduction of a scale is impossible in 2500 BC because there was then no United Chinese State. If a Shao Hao existed at all, he might have been a local chieftain at most.

            Some musical instruments surely existed in 2500. However tuned and reproducible ones, as e.g. gongs, bells &c. could not be older than the Widespread" use of bronze. C. 1500 BC the Bronze Age Shang-Yin Dynasty may have introduced (or may not) standard bells (as the tradition remembers but as a work of Huangdi/Shao Hao), or standard bronze disks or anything similar. However these instruments are not extant.