Mathematics seductively swims below the surface of music.  . Il faut toujours savoir pourquoi. … Et comment  (*) (*)   (One always has to know why.     …And how.)

Prof. Eugene E. Helm (1928-) . Paul Valéry (1871-1945) My mother

This paper is dedicated to all classical musicians, among those Johan Philipp Kirnberger (1721-1783), as an homage to their musicality and refined musical ears.
From the seventeenth century on already, they were capable, based on aural observation only, to develop musical well-temperaments, some of those being almost identical to a mathematically defined optimum, disclosed below.
 

Abstract:

This paper discusses the development of an objective definition of Bach- and Well-Temperaments based on the translation of very simple musical criteria to very simple arithmetics and the optimisation of these criteria using a large number of manual arithmetic iterations in spreadsheets. All results appear to be reproduceable analytically as well.

A "quality"-measuring method is proposed that allows for very simple and objective identification of musical temperaments.
 
 

1 Foreword

1.1 Motivation

Bach- and Well-Temperaments for Western Classical Music ("Wohltemperiert") receive large attention in musical literature. A definition of Well Temperament usually includes requirements that are hardly open for immediate objectification, that concern harmonious and melodious sounds and playability of a key instrument in all musical tonalities, all of this within an historical context.

We refer for example to the definition given by Andreas Werckmeister (*):

1.2 Preceding Attempts

Preceding to this paper, around 2002, an attempt was made to characterise Bach-temperaments by assigning "weights" to all major and minor thirds and to fifths.
Weights where assigned so that the temperament that shows the minimum weighted RMS value of deviations of purity of major and minor thirds and of fifths, should also show the least possible unweighted RMS (*) difference with "Bach"-temperaments (Kirnberger III, Kirnberger III unequal, Kellner, Billeter, Kelletat: see further fig. 2).

(*)   RMS: Root Mean Square.

It appeared afterwards that the used method was disclosed in 1973 already, but used in a different way, by Donald Hall (*): Donald Hall assigned weights according to frequency of occurence of chords in music scores.

(*)   “The Objective Measurement of Goodness-of-Fit for Tunings and Temperaments”, Donald Hall. Journal of Music Theory, Vol. 17, No. 2. (Autumn, 1973), pp. 274-290.

In both cases it was felt that it is not evident to make a good objective assignment of weights. See Donald Hall, page 281: "…The central idea…etc... Unfortunately, any other choice would be highly arbitrary or intuitive, so I have stayed with raw occurence data for any weights…".
The weights we obtained ourselves, to come to the closiest approach to the referred Bach temperaments, lead to an overwhelming high weight for the major third on C: a weight of 16 (16-square = 256 !) was obtained, which has to be compared to weights of 1 only for almost all other intervals.
It is this observed overwhelming importance of the major third on C that has lead to this alternative derivation of the objective characteristics of the Bach- and Well-temperaments, as disclosed below.
 

2 Proposed definition

The books "Zur musikalische Temperatur", part I, II and III, prof. H. Kelletat, Merseburger, publish extensive data on ratios of intervals and on pitches of notes, for many temperaments, especially also for well temperaments.
Moreover, in "Zur musikalische Temperatur", part II, graphs worked out by Rudolf Streich can be found, representing the characteristics of thirds (major and minor) and fifths for a number of well temperaments, and also for some so called "Bach"-temperaments: see the copied figure 1 and 2 below, (table 8 and 9 in the book):
 

Figure 1

Figure 2

.
On the graphs it can be seen that all temperaments called "Bach"-Stimmungen" have following characteristics in common:

• A pure or almost pure major third on C, and one with a somewhat lesser quality on G. Away from C a gradual transition to pythagoric major thirds, 
• A large number of pure fifths (this creates pythagoric thirds)
• An almost pure minor third on E, and away from E a gradual transition to pythagoric minor thirds

Bach- and Well-Tempering objectively strive for an optimal compromise between:

Criterium 1:	Maximalisation of the purity of the most important major thirds
Criterium 2:	Maximalisation of the purity of the minor third on E
Criterium 3:	Maximalisation of the number of pure fifths
Criterium 4:	A gradual transition of pure intervals to deviating intervals, if notes are classified according to their sequence on the circle of fifths.

Possible compliance with the above criteria will be analysed and controlled below.

2.1 Aproximation based on calculations with the PBP (Procentual Beating Pitch)

In this aproximation the purity of thirds (major and minor) and fifths will be measured by direct calculation of the procentual beating pitch (PBP) of the concerned interval, as defined below.
If p₂/p₁ stands for the pitch-ratio of concerned thirds or fifths, then the beating pitches due to an impurity of the interval can be calculated with the following formulas:
 

Fifths:		formula 1
Major thirds:		formula 2
Minor thirds:		formula 3

 Therefrom following formulas for the respective PBP (procentual beating pitch) can be derived:
 

Fifths:		formula 4
Major thirds:		formula 5
Minor thirds:		formula 6

Reason for using the PBP for purity measurements of intervals instead of the deviation of purity expressed in cents, is that from a musical point of view it is clear that beating pitches of impure intervals are directly proportional to the PBP, and can be heared directly by the ear.
In measuring purity of intervals this way we use an effective, concrete musical quality parameter; cents on the other hand are very abstract numbers for many people, only usable for standard musical measurements in general, while not being a direct quality parameter directly related to a physical (oral) observation.

2.1.1 1^st step: Maximalisation of the purity of the most important major thirds

• Criterium 1: In a first approach we set the major third on C to the pure ratio = 1,25 (= 5/4), because of its detected overwhelming importance (see par. 1.2).
• Criterium 4: Gradual transitions.

To comply with criterium 4 we use an iteration procedure in a spreadsheet that leads to minimalisation of the sum of the differences in impurity of intervals on adjacent notes, when notes are classified according to their sequence on the circle of fifths:
• the differences in impurity of intervals on adjacent notes, when notes are classified according to their sequence on the circle of fifths, are totalised
• through manual iterations in the spreadsheet this total is minimalised

• the fifths on C, G, D, A (value 1,4953 = ), with a PBP of about minus 0.93

• the fifths on E, B, Fis, Cis, Gis, Es, Bes, F, (value 1,4997.. = )

Details on the pitch ratios are given in table 1, and are visualised in figures 3 and 4.

C	Cis	D	Es	E	F	Fis	G	Gis	A	Bes	B
1.0000	1,0542	1.1180	1.1857	1.2500	1.3335	1.4059	1.4953	1.5811	1.6719	1.7783	1.8747

2.1.2 2^nd Step: Improvement of the minor third on E
 

Criterium 2:	Maximalisation of the purity of the minor third on E
Criterium 4:	Gradual transitions

On the circle of fifths it can be easily observed that an increase in purity of the minor third on E will necessarily lead to a decrease in purity of the fifths G, D and A. Therefor, an objective limit for the increase in purity of the minor third on E is the point at which the presently higher PBP of the minor third on E will be lowered to a value equal to the PBP of the worst fifth G, D or A, having associated increasing impurity: further increase of the purity of E will necessarily lead to fifth(s) with a PBP larger than the PBP of the minor third.
The same procedure as under 2.1.1 is applied in a spreadsheet, and leads to:

• equal fifths on G, D and A with a PBP of about minus 1.06
• a small third on E with the same PBP of about minus 1.06

Note:

Thanks to the observed equal value for the fifths on G, D, and A, and equal impurities for the fifths on G, D, and A and the minor third on E, the obtained values for these fifths and minor third can also be calculated analytically: 

• The ratio obtained by the third power of the ratio of the fifths on G, D or A, reduced to the initial octave, has to be equal to the ratio of the minor third on E. With "f" standing for the ratio of the fifths, and "t" for the ratio of the minor third, we can express this condition by the following equation:

formula 7

•  The PBP of the fifths on G, D, and A have to be equal to the PBP of the minor third on G. This can be expressed by the equation formula 8:

(derived from formulas 4 and 6)

formula 8

formula 9  (*)

(*)   See also Herbert A. Kellner: “Eine Rekonstruktion der wohltemperierten Stimmung von Johann Sebastian Bach”, in “Das Musikinstrument” 1/1977, p. 34 (see also in “Zur musikalischen Temperatur II, quote 102 p. 56). Comparable criteria and equations (f4 + 2f – 8 = 0 and 6f4 – 3f3 –20 = 0) were elaborated by Kellner, but by involving the MAJOR third on C (instead of the MINOR third on E used here) and fifths. Any criterium for an acceptable minor third on E necessarily leads to fifths that are worse than the fifths obtained by Kellner (1.495953506…) which cause limitations on the purity of the major third on C (1.25172…).
We therefor prefer to set this type of criterium on the minor third on E instead of on the major third on C.

2.1.4 4th Step : Improvement of the major third on G

Addition of a pure fifth on F, leads again to further inprovement of the major third on G (PBP will become = 0.32). This major third is so close to purity, that it induces the option to transfer some of its impurity to the major third on C, so that both have a same extreme low impurity. This is obtained by slight modification of the fifths on C and E, leading to an equal PBP of only 0.16 for the major thirds on C and G (their ratio = 1.2504…).
These values of the equal fifths on C and E can also be calculated analytically, because they remain as the only two unknown fifths and have equal value and tempering:

• The fifths on G, D and A were already calculated analytically: see 2.1.2
• The fifths on B, Fis, Cis, Gis, Es, Bes, and F are pure.

This "temperament will be given the name "PBP Temperament", furtheron in this paper.

It can be observed in fig. 9 that the three curves are perfectly symmetrical.
All the discussed intervals for thirds and fifths have unique values that can be calculated analytically.
From the obtained results it is clear as well that gradual transitions from pure intervals to deviating intervals (= criterium 4) are a (natural) consequence of compliance with the criteria 1 to 3, and said property can therefor no longuer be sustained as a separate criterium.

Note:
It is a very well known and proven mathematical property, that the sum of variables having a constant product will be minimal when all variables have an equal value.
The obtained set of equal deviation values for the major thirds on C and G, and for the fifths on G, D, A with minor third on E, is therefor also an analitically proven mathematical optimum.

2.2 Aproximation based on calculations with cents

For the sake of complying with more common practice, we want to work also using more "standard" calculations made in cents. 
Execution of procedures equivalent to those of 2.1.1, 2.1.2, 2.1.3 and 2.1.4 lead to same conclusions, but for the approximation with cents we can also take advantage from the conclusions of par. 2.1 to shorten the procedure: we can work immediatly the analytical way, as worked out below:

• Seven pure fifths (B, Fis, Cis, Gis, Es, Bes, F): 701.96… cents
• An even distribution of the syntonic comma expressed in cents, over three fifths G, D and A, and the minor third on E leads to following equations (with f = cent value of fifths and t = cent value of the minor third, and D = cent value of the deviation of purity):

 


	(same criterium as formula 7)	formula 10
	(pure minor third)	formula 11
	(pure fifth)	formula 12

• An even tempering of the thirds on C an G, what is equivalent with an even tempering of the fifths on C and E:...........f = 698.29… (= a ratio of 1.4968…)

This "temperament" will be given the name "Cent Temperament", furtheron in this paper.

2.3 Aproximation based on calculations with Beating Pitches

One might argue further that actual beating pitches should be used as quality parameter, instead of procentual beating pitches or cents: this is indeed "THE" real thing that is DIRECTLY heard by the ear.
From the previous approaches we can accept, that we can stay with:

• Seven pure fifths (B, Fis, Cis, Gis, Es, Bes, F)
• A distribution of the syntonic comma over three fifths G, D and A, and the minor third on E, so that all four intervals have the same absolute beating pitch. With f_A,D,E,G= pitch of notes A, D, E, G and f_D = beating pitch, we can put forward:

 


formula 13




with the outcome:

• An equal beating pitch for the thirds on C an G.

With f_B,C,E,G= pitch of notes B, C, E, G and f_d = beating pitch, we can put forward:  

	(equal beating of major thirds)	formula 14
and we know already:	(7 pure fifths)	formula 15
and:	(outcome of formula 13)	formula 16

We obtain rather complex outcomes:

It can be verified that an equal beating pitch for the thirds on C and G does not lead to an equal beating pitch for the fifths on C and E. This is different from the findings for the aproximations with PBP or cents.

This "temperament" will be given the name "Beat Temperament", furtheron in this paper.
 

3 Comparison and Identification of Musical Temperaments

We will compare the obtained "temperaments" among each other, and comparison with historical temperaments is even more important, because this should demonstrate evidence that the obtained temperaments are lying very and most close to Bach-temperaments.
With the aim of making objective comparisons we need some kind of arithmetic measurement of difference between temperaments.

We propose to calculate the unweighted RMS value of the differences in impurity of all thirds, major and minor, and all fifths, of the temperaments that have to be compared.

The above proposal can be seen as yet another and specific kind of application of the technique disclosed by Donald Hall (*). It has been worked out in a spread sheet (file Identification_of_Temperament.xls) containing a very large set of historical temperaments.

(*)  “The Objective Measurement of Goodness-of-Fit for Tunings and Temperaments”, Donald Hall. Journal of Music Theory, Vol. 17, No. 2. (Autumn, 1973), pp. 274-290, espescially p. 278 and 280.

3.1 Immediate comparison of beating pitches of the obtained temperaments

In order to compare and work with more current procedures as well, as used for example in instructions to set temperaments by ear on musical instruments, we start with table 7 below, comparing beating pitches of the fifths and of the most important thirds:
 

Beating Pitches	Fifths						Major Thirds				Fl.Th.
	B, Fis, Cis, Gis, Es, Bes, F	C	G	D	A	E	F	C	G	D	E
PBP Temp.	0	-1.14	-4.19	-3.13	-4.68	-1.43	6.80	0.42	0.64	5.71	-3.50
Cent Temp.	0	-1.67	-3.66	-2.71	-4.09	-2.09	7.19	1.30	1.95	6.04	-6.12
Beat Temp.	0	-1.15	-3.83	-3.83	-3.83	-1.59	5.96	0.64	0.64	5.96	-3.83

Table 7 displays the very high similarity of the three "temperaments", suggesting there will probably as well be minor differences only in the comparisons of said "temperaments" with historical Bach-Temperaments.

Note:

At this point it should be made very clear, we insist on this, that it has not at all been the aim of this paper to introduce yet other new temperaments. The ""new" temperaments" in this paper have been developed only to have the disposal of objective models for the evaluation of the proposed objective definition of Bach- and Well-Temperaments.

3.2 Comparison with historical temperaments

Application of the technique proposed at the begin of paragraph 3 leads to table 8 below (*):
 

Temperament	RMS of D PBP's	Temperament	RMS of D cents	Temperament	RMS of D beating
PBP Temperament	0,00	Cent Temperament	0,0	Beat Temperament	0,00
Beat Temperament	0,12	SIEVERS	0,6	PBP Temperament	0,42
Kelletat	0,15	PBP Temperament	0,7	Kelletat	0,78
Cent Temperament	0,21	Beat Temperament	0,9	Cent Temperament	0,79
Kirnberger III unequal	0,38	Kelletat	1,0	Kirnberger III unequal	1,25
STANHOPE	0,38	Kirnberger III unequal	1,4	SIEVERS	1,26
SIEVERS	0,39	Billeter	1,7	STANHOPE	1,37
Billeter	0,56	Kirnberger III	2,0	Billeter	1,89
Kirnberger III	0,64	STANHOPE	2,0	Kirnberger III	2,15
NEIDHARDT-4	0,66	Kellner	2,0	NEIDHARDT-4	2,23
Kellner	0,71	NEIDHARDT-4	2,1	d'ALEMBERT / ROUSSEAU	2,48
VOGEL (STADE)	0,74	VALLOTTI - TARTINI	2,2	VOGEL (STADE)	2,53
d'ALEMBERT / ROUSSEAU	0,79	YOUNG / Van BIEZEN	2,2	de BETHISY	2,63
VALLOTTI - TARTINI	0,83	YOUNG	2,4	VALLOTTI - TARTINI	2,63
YOUNG / Van BIEZEN	0,83	d'ALEMBERT / ROUSSEAU	2,5	Kellner	2,66
de BETHISY	0,86	VOGEL (STADE)	2,5	YOUNG / Van BIEZEN	2,71
YOUNG	0,90	MERCADIER	2,5	YOUNG	2,97
NEIDHARDT-1	0,90	BARCA	2,5	NEIDHARDT-1	2,98
MERCADIER	0,91	NEIDHARDT-1	2,6	BARCA	2,99
BARCA	0,92	BARCA - interpr. DEVIE	2,7	BARCA - interpr. DEVIE	3,02
BARCA - interpr. DEVIE	0,97	de BETHISY	2,7	MERCADIER	3,06
Kirnberger II	1,01	LAMBERT	2,8	Kirnberger II	3,34
LAMBERT	1,02	BARNES	3,0	LAMBERT	3,38
BARNES	1,03	WERCKMEISTER III	3,3	BARNES	3,55
WERCKMEISTER III	1,09	ASSELIN	3,6	RAMEAU volg. KLOP	3,57
LEGROS (2 R.T.)	1,14	WEINGARTEN	3,7	LEGROS (2 R.T.)	3,58
Von WIESE-II-b	1,15	SORGE 1744	3,7	Von WIESE-II-b	3,85
RAMEAU volg. KLOP	1,24	Kirnberger II	3,8	Kirnberger I	3,96
WEINGARTEN	1,26	NEIDHARDT-2	4,0	WERCKMEISTER III	3,97
ASSELIN	1,27	LEGROS (2 R.T.)	4,0	WEINGARTEN	4,01
Kirnberger I	1,29	NEIDHARDT-3	4,2	SORGE 1744	4,04
…	…	…	…	…	…
…	…	…	…	…	…
The full table in the spreadsheet contains 85 temperaments
…	…	…	…	…	…
…	…	…	…	…	…
MARPURG (1) ZARLINO	4,90	FOGLIANO	16,9	FOGLIANO	16,12
FOGLIANO	4,90	MARPURG (1) ZARLINO	16,9	MIDDENTOON -2/7 C	16,70
MIDDENTOON -2/7 C	5,15	MIDDENTOON -2/7 C	17,3	MARPURG (1) ZARLINO	17,60
MARPURG (4) LOULIE	5,95	MARPURG (4) LOULIE	20,0	MARPURG (4) LOULIE	20,39
SALINAS	6,57	SALINAS	22,1	SALINAS	21,25

(*)   Data of most temperaments issue from “Stemtoon & Stemmingsstelsels”, Jos De Bie, 4-de uitgave, 2001, and some from Kelletat. But also Barbour, J. Murray “Tuning and Temperament”, 1951, could easily be used as reference, although we estimate it leads to an excessive list of temperaments for comparison, with inclusion of too many hypothetical or theoretical temperaments, not used in musical practice.

• The three developed "temperaments" (PBP, cent, Beat) are very much comparable, not only when comparing their beat pitches (see 3.1), but also if compared by "RMS D impurity" measurements.
• All "Bach"-temperaments lie at the head of the table, very closely joining the three "temperaments" that were developed in this paper for the purpose to be used as reference for an objective definition of Bach- and Well-Temperaments
• At least Stanhope, Sievers and Neidhardt 4 (Große Stadt / Large City) have to be considered as being "Bach"-temperament as well, because they meet the same quality requirements as the temperaments considered as such by Kelletat
• Because Werckmeister III is recognised all round as a well temperament, it is very probable that all temperaments preceeding Werckmeister III in the table deserve the same estimation
• All well temperaments lye very far ahead of temperaments at the bottom of the table, at least by a multiplication FACTOR 6 (!), in rounded numbers.

4 Conclusion

From the calculations in this paper we have learned:

• The important major thirds are C and G
• The important minor third is E
• Gradual transitions (criterium 4) are inherent to the criteria 1 to 3, and therefor this property that was thought to be a criterium should no longuer be considered as such
• These findings are not significantly affected by the applied impurity measurement method

Herewith joined:
 

Objective comparison of temperaments against Bach- or Well-temperaments, and identification of temperaments can be done by measuring the RMS value of the differences in impurity of thirds, major and minor, and fifths.

It is clear that the above definition and measuring method do not cover some (not yet objectivised) musical criteria such as playability "around the circle of fifths", or difference in character of the tonalities, good chromatic properties, etc…, but it must be accepted as an evident fact, that these additional and essential musical properties come out as a natural and inherent lucky emanation of the criteria specified by the proposed definition.
 
 

Musicians developed Bach- and Well-Temperaments by aural observation with sensitive and trained musical ears, optimising the integral musical characteristics of temperaments over all tonalities. They very often prefer these in musical practice.
The musical definition of these temperaments, as given for example by Werckmeister, telling us WHAT is wanted, therefor remains the essential and basic defintion.

The objective musical definition proposed in this paper can only be seen as an ultimate mathematical extraction of the essential characteristics of the Bach- and Well-Temperaments. It informs us only about the HOW of their structure.
 

This definition also leads to a very simple but explicit figure that symbolically and without ambiguity displays an optimum outcome of the definition, if one discards all pure fifths of the circle of fifths, and takes into account that the marks on the lines signifie that the intervals represented by the lines with a same mark are intervals with an identical deviation of purity.
 
 

Nieuwelei, 52 B 2640 Mortsel Belgium

Acknowledgments

My first thankings go to the Music Academy of Mortsel, Belgium, where I acquired my first musical education. I joined this academy, in the hope I should get any explanation WHY a musical octave contains the asthonishing number of 53 commas! Normally one goes to an academy to learn to play music, but not so for me.
I want to express my major thankings the Department of Musicology of the State University of Ghent, Belgium, (RUG: Rijksuniversiteit Gent) offering me opportunities for interesting talkings and exchanges of ideas, leading to implicit encouragement to pursue my search for objective criteria for Bach- and Well-temperaments.
I want to thank as well the Catholic University of Leuven, Belgium, and its Association of Engineers (VILV: Vereniging van Burgerlijk Ingenieurs uit de Katholieke Universiteit Leuven), providing me free access to the Library of the University and consequently also to relevant and professional music literature worldwide. It has offered me confidence concerning the facts that this publication is meaningful and original.

back to Home page

Bibliography

Kelletat Herbert, professor