Sixteen Tones

Sixteen tones

I owe my soul to the tympano score...

The Well-Tempered Clavier (WTC) is a set of two books, each one containing 24 pairs of preludes and fugues in all major and minor keys. This gives 96 numbers of bars, providing a wide range of speculations about an eventual architecture of the books. I had the idea to study the golden ratio in this set.

Here is the table of all numbers of bars in both books, with in blue the totals in each key, and in red the computed golden ratio, rounded to the nearest integer.

Table A The Well-Tempered Clavier : Books 1 & 2

	P1	F1	PF1	φ		P2	F2	PF2	φ		PF12	φ
1	35	27	62	38	C	34	83	117	72	1	179	111
2	38	31	69	43	c	28	28	56	35	2	125	77
3	104	55	159	98	Cis	50	35	85	53	3	244	151
4	39	115	154	95	cis	62	71	133	82	4	287	177
5	35	27	62	38	D	56	50	106	66	5	168	104
6	26	44	70	43	d	61	27	88	54	6	158	98
7	70	37	107	66	Es	71	70	141	87	7	248	153
8	40	87	127	78	dis	36	46	82	51	8	209	129
9	24	29	53	33	E	54	43	97	60	9	150	93
10	41	42	83	51	e	108	86	194	120	10	277	171
11	18	72	90	56	F	72	99	171	106	11	261	161
12	22	58	80	49	f	70	85	155	96	12	235	145
13	30	35	65	40	Fis	75	84	159	98	13	224	138
14	24	40	64	40	fis	43	70	113	70	14	177	109
15	19	86	105	65	G	48	72	120	74	15	225	139
16	19	34	53	33	g	21	84	105	65	16	158	98
17	44	35	79	49	As	77	50	127	78	17	206	127
18	29	41	70	43	gis	50	143	193	119	18	263	163
19	24	54	78	48	A	33	29	62	38	19	140	87
20	28	87	115	71	a	32	28	60	37	20	175	108
21	20	48	68	42	B	87	93	180	111	21	248	153
22	24	75	99	61	b	83	101	184	114	22	283	175
23	19	34	53	33	H	46	104	150	93	23	203	125
24	47	76	123	76	h	66	100	166	103	24	289	179
	819	1269	2088			1363	1681	3044			5132

The two books were originally published in 1722 and 1742. The same numbers of bars can be found in all current editions, although some authors use some tricks to fit their theories.

The only point that can be discussed is the choice not to count the repeats of some preludes (the 11 numbers which are underlined). The table stands for the written music, while the only other choice would be to count all the repeats.

φ or Phi = 1.6180339875

Table A gives 4 golden relations between the 24 sums PF12, 4 red numbers which are too blue numbers. Table B shows these results, following a first peculiarity: the 3 strongest keys are there, with 289-287-283 bars, and the fourth relation includes the lightest key, with 125 bars.

Key	N°	P1	F1	P2	F2	bars
h	24	47	76	66	100	289
cis	4	*39	#115	*62	#71	287
b	22	24	75	83	101	283
sums P & F		P2 à	F2 à	321	538	859
C	1	35	27	34	83	179
fis	14	24	40	43	70	177
a	20	28	87	32	28	175
sums P & F		P2 à	F2 à	196	335	531

H	23	19	34	46	104	203
c	2	38	31	28	28	125

There are several other peculiarities:

– The 6 keys concerned by the strong relations are b-a-C-h, and cis-fis more signifying by their ranks, 4 and 14. The ranks of BACH's 4 letters name add up to 14. It’s widely admitted that Bach used in his music this number 14 as a signature, notably in Contrapunctus 14 ending the Art of Fugue, where for the first time he used notes BACH as the beginning of a thema.

– Within each of the keys 4-14, the 4 numbers of bars for every individual piece are 2 by 2 in golden harmony, these are the only cases among all 24 keys.

– The strongest harmony (289/179) includes first and last key (24/1, h/C), while the lightest one (203/125) concerns second and penultimate (23/2, H/c).

– The 3 strong numbers of bars (289-287-283) add up to 859, while their relatives 179-177-175 make 531, quite near the calculated golden ratio of 859 (530.9).

– The light harmony is again between Bach keys, H-c. A Fibonacci type sequence built on their numbers gives:

125-203-328-531-859-1390...

531-859-1390 are the numbers of the joined 3 strong relations, and the keys allow this kind of signature, taking in each case the keys by decreasing numbers of bars:

h-cis-b / C-fis-a = 859 / 531

C-fis-a / H-c = 531 / 328

H-c / H = 328 / 203

H / c = 203 / 125

The surprises are not over. The individual harmonies within keys 4-14 suggest a closer study, and I found that:

Within the 3 strong relations, adding up together strong preludes, strong fugues, light preludes, and light fugues, gives 4 numbers 321-538-196-335.

We already know that additions 321 + 538 and 196 + 335 lead to 531-859-(1390)

(538+321) + (335+196) = 859+531 = 1390

but substracting light preludes and fugues from strong ones, 321 – 196 and 538 – 335, leads again to 125-203-(328), the numbers of the light harmony.

(538–335) + (321–196) = 203+125 = 328

This was so beautiful that I saw in it an achievement, and failed looking any further for a time. Later I felt upset to have 6 of the 8 keys BbAaCcHh involved in the big combination, and wondered if there was a way to include the 2 missing keys, A and B (B is B flat in the German system, where H is English B natural). I was puzzled when I found how easy it was.

Just the 6 keys baCcHh totalize 1254 bars, 1254/Phi = 775.015. Half of this is 387.51, and nearer integer is 388, which is the number of bars of BA keys. Another way to put it is to start with the total number for 8 keys,

1642/Phi =1014.8…, so its golden sharing leads to integers 1015 and 627, and the difference between these integers is 388. We can write too

1642/Phi³ ≈ 388, while in the other relation between strong and weak keys we had

1390/Phi³ ≈ 328.

This is an interesting result, but it would be quite more beautiful if there was an harmonious sharing of the 6 keys baCcHh in two halves, to obtain a complete harmony 627-388-627. This is not possible with the 6 numbers of the whole keys, so I tried with the numbers in each book of the WTC. I will call tone a key in one book, in order to fit the title of this study, Sixteen tones.

So we have 12 tones b_1-2a_1-2C_1-2c_1-2H_1-2h_1-2, and the computer gives 4 sharings 6-6, only one of them looking meaningful, and it’s meaningful beyond any hope :

b₂ a₁ c_1-2 H_1-2 = 627 on one side, with the whole keys c-H (the lightest pair in golden ratio).

b₁ a₂ C_1-2 h_1-2 = 627 on the other side, with the whole extreme keys C-h (the strongest pair in golden ratio).

So it would have been sufficient to part the middle keys ba, when my purpose was to find a golden harmony including keys BA. It’s a chiasma between ba tones, which are highly different from one book to the other, that allows this perfect equality, in which two groups show the same tones, confusing major and minor :

bacchh = 627

Table C shows these 16 tones bach, with golden relations in blue/red (203/125, 289/179, [184+99]/[115+60] et 627/388).

BABA	bacHcH		b₂	a₁	c_1-2	H_1-2
	627	=	184	115	125	203
388			(+)	(+)
	627	=	99	60	179	289
(BABA)	baChCh		b₁	a₂	C_1-2	h_1-2

The distribution of minor tones baba is surprising.

To equilibrate the 328 and 468 (difference 140) of relations cH and Ch, 299 and 159 were needed, and

299/Phi = 184.79, rounded to nearest integer 185, while b₂ is 184.

159/Phi = 98.27, rounded to nearest integer 98, while b₁ is 99.

Actually, as golden relation between integers are always approximations, the factual numbers compensate the approximations of the other horizontal relations:

h/C = 289/179 < Phi, and 99/60 > Phi ; together we have the ideal distribution of 627, 388/239.

H/c = 203/125 > Phi, and 184/115 < Phi ; together we have the distribution 387/240, not so bad as 627 belongs to the numbers of which golden section is near an integer plus a half (387.507).

If ideal distributions 185-114 and 98-61 had been effective, they would too have led to distributions 387-240 and 388-239 for the two groups bachch.

Fractions 184/115 and 99/60 are equal to 8/5 and 33/20. These simplifications correspond to ideal golden sharings of 13 and 53.

I come now to major keys A and B.

Before I found the previous relations, I was a bit annoyed with their number of bars, 140 and 248, instead of 148 and 240, perfect golden distribution. Now I see that 140 is the difference between keys C-h and c-H, 468 – 328 = 140, compensated in the double equilibrium « 627 » by 299 – 159 = 140, (b₂+a₁) – (b₁+a₂).

The distribution between the two books doesn’t seem to give anything, and it’s necessary to study the distribution between Preludes and Fugues on table H to see something, and that’s again something astonishing.

	P₁	F₁	P₂	F₂
A	24	54	33	29
B	20	48	87	93

It looks like what happened in fis (golden relations F/P in each book, see table B) and cis (golden relations P/P and F/F) was a preparation to this, in which all numbers have to be mixed up to see a perfect pattern.

So we have here two golden relations, 87/54 (ideal sharing of 141) for P₂B/F₁A and 33/20 (ideal sharing of 53) for P₂A/P₁B. The other numbers in each key give again the sums 141 (48+93) and 53 (24+29).

This is quite near what happens in the 627 equilibrium, where the separation of tones a and b within the WTC books leads to 8 numbers, so we can write :

627 = 184+115+125+203 = (99+60) + (289+179)

with 99/60 and 289/179 being complementary approximations of Phi, so (99+289)/(60+179) = 388/239 is the ideal sharing of 627.

Now the 8 numbers obtained by considering prelude and fugue distribution within tones A and B lead to a similar equality, from 388 golden section of 627 :

194 = 24+29+48+93 =(33+20) + (87+54)

with 33/20 and 87/54 being complementary approximations of Phi, so (33+87)/(20+54) = 120/74 is the ideal sharing of 194, half of 388.

It’s quite striking to find the same fraction 99/60 or 33/20 in both cases, in chiasma forms b₁/a₂ et A₂/B₁.

Another striking feature is that golden relations 87/54 and 33/20 are cascading, i.e. 33 is the golden section of 54, hence 20-33-54-87 is a golden progression, reminding of what happened in table B, with the progression 125-203-328-531-859.

Another consequence is that the 140 bars of A are ideally distributed in 87 and 53, by switching Preludes 1 and 2 for example.

Last Table Bach shows the cascading numbers adding to 194 with another set of pieces adding to 194 bars. I indicated for Table A that I didn’t consider the repeats of 11 Preludes, but it might be time to notice there are 4 Preludes with repeats among the 16 tones bach, and that they correspond precisely to tones B-a-c-h, more exactly to reversed order hcaB, only possibility allowed by the 2 books of WTC. A 3^rd book would have been necessary to have direct order Bach.

87	54	33	20	= 194
93 + 48		29 + 24		= 194
B₂	a₂	c₂	h₁	= Bach
87	32	28	47	= 194

This is a case allowing an easy probabilistic evaluation. If it is given that there are 4 tones with repeats among the 16 tones bach (but this necessary condition is already a strange fact), there is only one choice, among 1820, allowing to have Bach in reversed order. Some other facts can be pointed to :

– There’s no other repeats Prelude before h₁, the only one in Book 1;

– There’s no other repeats Prelude after B₂;

– It has been wondered why Bach whose creativity seemed without limits did write a second set of Preludes of Fugues in all keys.

Now just a few words about what I think about all that. I don’t feel easy to explain it, but I don’t think Bach had any consciousness of golden ration when writing his WTC, yet this study is just a little bit of my golden work on Bach, and many scholars studies, over a hundred, already exist about golden ratio in Bach’s work, including a whole book by Guy Marchand, from his doctorate thesis. Actually it’s because the golden architecture of WTC seems too perfect that I can’t see in it a conscious project, which would be very far of anything known. All I can add is :

– Golden Ratio propaganda assumes it’s a natural harmony, then artists might use it without knowing;

– The relations might be side effects of a bigger something I have no clear idea about;

– Some points seem to fit wiser theories about Bach’s signatures, as the repeats Preludes in B-a-c-h keys.

I’ll end with something almost incredible, which might be for me part of an out of time mystery. The title of my study comes from the wellknown song Sixteen Tons. I felt like beginning with something near the chorus last line, I owe my soul to the company store. I found score was here appropriate, and arrived to tympano score, an anagram of company store.

After this choice I noticed there was 12 letters in company store, and that my two pairs of interverted letters had something alike the intervertions of tones a₁a₂ and b₁b₂ among the 12 tones bachCH, in order to compensate the difference between h₁h₂C₁C₂ and H₁H₂c₁c₂. I was quite puzzled to discover that compensatory is a perfect anagram of company store ! (with similarly 4 moved letters !)

Remi.Schulz at club-internet.fr, 04/21/2010