Bach s Tempered Meantone

Bach’s Tempered Meantone 1 Bach’s Tempered Meantone Related to “Das wohltemperirte Clavier” Abstract : Fifths and major thirds beat rate characteristics of famous historical temperaments are analysed. It appears that beat rate characteristics might be the actual determining factors for Baroque temperaments, mainly because beat rates are of main importance to interpreting musicians regarding harmony and possible musical affects, and to auditory tuners because of quality and ease of tuning. It is, on the other hand, not always clear whether published ratios, cents or comma’s are deduced from theoretic calculations or from concrete results on monochord measurements or settings. The revealed reality and importance of beat rate characteristics of temperaments raises additional arguments for acceptability of the Jobin proposal concerning a probable Bach temperament, or for almost identical beat rate alternatives. A novel hypothesis is proposed concerning the spirals drawn on top of the title page of “Das wohltemperirte Clavier” of Johan Sebastian Bach. Keywords Baroque ; well temperament ; meantone ; interval ; comma ; beat ; harmonic ; ratio ; cent ; Bach 1 Preamble The commonly published and dominating factors with discussions on musical temperaments are probably the investigations on purity deviations of musical intervals, measured in ratios, cents or commas. And still, musical interval beats and their beating rates are probably more affecting musical factors to interpreting musicians and auditory tuners of keyboard musical instruments. More attention might therefore have to be paid to those characteristics : beats are undesired and directly observable. Approximate auditory beat rate evaluations do not require any tool nor calculation. Impurity measurements in ratios, cents or commas on the other hand, are often nothing more but rather abstract concepts to many musicians, not of direct use or interest when playing music and also not for auditory tuning. This paper is an attempt to confirm and elucidate the importance and practical applicability of beat rate evaluations in the determination of musical temperaments, especially some Baroque ones. 2 The auditory music keyboard tuning The elementary basic concepts of musical temperaments, seen from the point of view of the interpreting musician and the auditory music keyboard tuner are discussed in this paragraph. There is of course much more that can be written on this subject, see for example : “Le Clavier Bien Obtempéré”, A. Calvet, 2020. 2021_01_28 Bach s Tempered Meantone.docx J. Broekaert 2020-11-09 Bach’s Tempered Meantone 2 Pure Musical Intervals (just and perfect intervals) 2.1 Music consists of ordained periodic sounds. Purity of coincident musical sounds is usually desired. Coincident sounds are considered pure, if no beats occur. Beats can occur due to the interference of harmonics of differing periodic sounds. Any periodic sound can mathematically be simulated by a periodic function F(t). J. Fourier (1768-1830) developed mathematical evidence that any periodic function F(t) consists of a sum of sine waves, − the harmonics −, whereby the sine wave frequencies are INTEGER multiples of a basic frequency. = sin 2 / =2 + sin 2 / cos 2 ! = ℕ ; and with : with / =2 and cos 2 / ! A small musical interval impurity leads to a beating sound, because of the summation of mutual note harmonics, of almost equal frequency. The sum of two sine waves is worked out in the formula below : sin 2 " + sin 2 # =$ + +2 cos 2 Hence, this sum corresponds to a single sine wave : • of median frequency " + # ⁄2 • with amplitude modulation from − to + • with low influence phase modulation ( = tan Figure 1 displays the effect. For example : the beating of an imperfect fifth − ratio ≈ 3/2 −, mainly results from the sum of the second harmonic of the upper note with the third harmonic of the lower note, −but also from any higher harmonic 2n of the upper note with any mutual higher harmonic 3n of the lower note−. The lowest beat rate of a fifth can therefore be set as : AB CDCEF =2 GHHIJ KLEI Fig,.1 −3 " − # " × cos '2 + 2 , at modulation frequency ";# :" # cot <2 = > ?@ # − () " − # Beating of two sine wave sounds NLOIJ KLEI =p −p This impurity measurement has been applied already by A. Kellner (1977), and is applicable for other intervals too, applying appropriate integer numbers for p1 and p2. All of the above demonstrates why important musical intervals have ratios of (low) integer numbers. 2021_01_28 Bach s Tempered Meantone.docx J. Broekaert 2020-11-09 Bach’s Tempered Meantone 3 Besides the absence of beats, there is a reinforcement of the common harmonics, leading to a pleasant and rich new sound : a beautiful “consonance” (= harmony) and also a “resonance”. 2.2 Beat rate calculation of fifths and major thirds Auditory tuning of a musical keyboard usually starts at the F3 – F4 scale (Calvet 2020). Beat rates within this scale can be calculated based on the equations tables 1 and 2. The qNote and pNote symbols stand for the beat rates of fifths (or corresponding fourths) and major thirds (or corresponding sixths). 3 3 − 2P4 + RS = 0 −2[4 + R\ = −3Z3 3P#4 − 4U#3 + RV# = 0 3P4 − 4U3 + RV = 0 3[4 − 4A3 + R] = 0 3U#3 − 2[ 4 + RX# = 0 3U3 − 2W4 + RX = 0 3A3 − 4 #3 + R_ = 0 3[ 4 − 4A 3 + R]# = 0 3W4 + RY = 4Z3 3 #3 − 2P#4 + RS# = 0 3A 3 − 4 3 + R_# = 0 5 3 − 4Z3 + aS = 0 5P4 − 4[4 + aV = 0 5U3 − 4A3 + aX = 0 5W4 − 8 #3 + aY = 0 Table 1 : calculation of fifths beating rate within the F3 – F4 scale −4P#4 + a\ = −5Z3 5P#4 − 8 3 + aV# = 0 5[4 − 8U#3 + a] = 0 5U#3 − 2P4 + aX# = 0 5A3 − 4[ 4 + a_ = 0 5[ 4 − 8U3 + a]# = 0 Table 2 : calculation of major thirds beating rate within the scale F3 – F4 5 #3 − 4A 3 + aS# = 0 5A 3 − 4W4 + a_# = 0 Quite some historical temperaments can be defined, based on the above formulas, if reliable, objective and historical data is available, regarding tuning instructions. See the Appendix A tables. All calculations were worked out based on a A4 = 415 diapason, that was quite common in Germany at Baroque time (Kammerton). A remarkable similarity can be observed between the normally and generally published figures based on ratios, cents or commas, and the recalculated ones based on beat rates. Therefore, doubts can raise on which figures are the most valid ones, taking into account that tuning at Baroque time was auditory mainly, using nothing more but a tuning fork. Measuring precision by means of a monochord is quite limited. A deviation of 1 mm only, of a chord of 1 m length corresponds to a deviation of 1200 x log2 (1001/1000) ≈ 2 cents. The cents calculations were not yet commonly applied, and were at least quite laborious, while made by hand. 3 A tempered Meantone 3.1 The natural C-major notes The auditory meantone tuning can start by setting four equal beat rate fifths C – G, G – D, D – A, A – E, in a way to obtain a just C – E third, such as was done for Appendix A. The following step includes setting the just major thirds F – A and G – B. The obtained pitches are the solution of following equations, derived from tables 1 and 2 : −R = 3P4 − 4U3 = 3U3 − 2W4 = 3W4 − 4Z3 = 3Z3 − 2W[4 concerning the involved fifths, and 0 = −a = 5P4 − 4[4 = 5 3 − 4Z3 = 5U3 − 4A3 concerning the involved major thirds of C=major. 2021_01_28 Bach s Tempered Meantone.docx J. Broekaert 2020-11-09 Bach’s Tempered Meantone 4 This way, all meantone C – major natural notes are defined (see results at table A, Appendix A). Three just major thirds are obtained, combined with six slightly diminished fifths of acceptable quality, because of their equality to those of the accepted meantone fifths. It can be verified that the F and B pitches lead to a fifths mean ratio equal to 1.502974…, slightly exceeding the pure fifths ratio, for the fifths B – F#, F# – C#, C# – G#(= Ab), Ab – Eb, Eb – Bb, Bb – F, that still have to be tuned. This condition requires that at least one of those fifths must be augmented. 3.2 The altered notes The aim for a well temperament does not allow for extension towards a chromatic scale, by adding altered notes for just major thirds to the above obtained scale, such as for the “classic” meantone. The addition of whichever just major third to the three already exiting ones, leads to one or more diminished fifths in the group of six fifths that are still to be defined. This in turn leads to an increase of the required augmentation of the remaining fifths. And this in turn again leads to undesired meantone characteristics : excessive augmentation of fifths and harsh major thirds. The “classic” meantone holds five more just major thirds, D – F#, A – C#, E – G#, Bb – D, Eb – G, on top of the three already defined ones. For this reason the aim to come to a well temperament, might correspond to the obtention of a best possible purity on those major thirds, in combination with a minimal quality loss for the remaining fifths. The latter requirement leads to the condition that additional fifths should be perfect or slightly augmented. With six fifths to be defined still, one can think of 63 (= 26 – 1) possible combinations of one to six augmented fifths (this is also zero to five just fifths). The corresponding scales can be calculated : the note pitches are the solution of the equations below, derived from table 1 : 0 = 3A3 − 4 #3 + R_ = 3 #3 − 2P#4 + RS# = 3P#4 − 4U#3 + RV# = 3U#3 − 2[ 4 + RX# = 3[ 4 − 4A 3 + R]# = 3A 3 − 4 3 + R_# In order to obtain an optimal fifths purity, the calculation is performed for equal values of qBb, qEb, qAb, qC#, qF#, qB, all of those being equal to « n x q » , with n = 1 or n = 0. Multiple analyses can be done on the obtained scales. An intriguing result among others, is the 1 – 1 – 1 – 0 – 0 – 0 combination for differing “ n “ values for the qNote. This combination corresponds with an absolute minimum (minimum minimorum) of the total of the impurity of one fifth plus the D4 – F#3, A3 – C#4, E4 – G#3 major thirds (sixths) impurities that are normally pure for the meantone : this is the sum q + pD4 + pA3 + pE4. This combination corresponds to a preference for the permitted meantone tonalities holding sharps (G –, D –, and A – major). The obtained pitches are on display in table 3. Bach =FCG= Pitches C C# D Eb E F F# G G# A Bb B 248,17 261,00 277,36 294,21 310,21 332,00 348,01 371,21 391,51 415,00 441,89 464,01 Table 3 : tempered meantone, holding optimised major thirds for the admitted temperaments holding sharps (#) The obtained beat rates are on display in table 4. 2021_01_28 Bach s Tempered Meantone.docx J. Broekaert 2020-11-09 Bach’s Tempered Meantone F3 Bb3 Eb3 Ab3 C#4 Maj. Th. 0,00 4,71 13,78 13,90 22,98 fifths – 1,67 1,16 1,16 1,16 0,00 min. th. – 17,23 – 20,66 – 25,23 – 14,50 – 14,99 Table 4 : beat rates for the table 3 scale F#3 B3 13,77 0,00 – 6,52 16,82 0,00 – 5,21 E4 14,99 – 2,61 – 5,21 5 A3 D4 G3 6,52 – 2,09 – 4,17 5,21 – 2,09 – 4,17 0,00 0,00 0,00 – 2,09 – 2,09 – 3,34 – 8,88 – 17,95 – 34,47 C4 F4 See also further, fig. 3 : display of the impurities courses of table 4 (thin lines). This paragraph is also discussed in depth in a more extensive downloadable paper : “Bach’s tempered Meantone, extensive version”, at : http://users.telenet.be/broekaert-devriendt/Index.html or http://home.deds.nl/~broekaert/Index.html 4 Bach’s Tempered Meantone The spirals drawn on top of the title page of J. S. Bach’s “Das wohltemperirte Clavier” can be compared to the fifths impurity course of the scale at par. 3.2, table 4. See fig. 2 (Amiot, 2005 : for the part containing spirals –taken from the J. S. Bach course). Figure 2 : Fifths characteristics One can observe a remarkable parallelism between the position of fifths on the impurity course, and the position of fifths on the spirals holding diverse characteristics. A door wide open to possible musicological and historical speculations… ? In line with Jobin’s hypothesis, the spirals on the figure suggest indeed an equal fifths impurity for those on C4, G3, D4, A3, E4. This fifths impurities course is very close to this hypothesis, but an absolute impurities equality for those five fifths, in association with two just major thirds (Jobin), is not possible if the fifths impurities are measured by beat rates. The obtained result also holds some disequilibrium : it is common to pay more attention to pure fifths rather than to just thirds, for the fifths impurities are easier to observe. The obtained scale has opposite characteristics. A possible alternative could strive for equal impurity of fifths and thirds. In line with the concepts of this paper, this means to strive for best equality of fifths and thirds beat rate. 2021_01_28 Bach s Tempered Meantone.docx J. Broekaert 2020-11-09 Bach’s Tempered Meantone 4.1 6 Alternative scale, with best equality of beat rate impurities, for fifths on C4, G3, D4, A3, E4 and major thirds on F3, C4, G3. It is possible to achieve equal beat rate for fifths on C, G, D, A and the major third on C. It leads indeed to five linear equations holding five variables : the variables C, G, D, E and “Beat 1”, the equal beat rate of the concerned intervals. The required equations for equal beat rate are (see tables 2 and 3) : AB 1 = 3U3 − 2P4 = 3W4 − 4U3 = 3Z3 − 2W4 = 3[4 − 4Z3 = 5P4 − 4[4 It is possible to add a sixth condition leading to a fifth unknown note B3, based on the same beat rate for the fifth on E : Beat 1 = 3B 3 – 2E 4 It can be verified, that a mathematical hazard makes that we thus obtain, NOT ADDING ANY MORE CONDITION, a major third on G3 with equal beat also, so that a seventh equation to satisfy this condition is not necessary. Applying the obtained solutions, the F3 note can finally be calculated, on the basis of a major third, taking in turn, and again, the same beat : Beat 1 = 5F 3 – 4A 3 The fifth on F will not satisfy the equal beat rate, but is of good quality, it can therefore remain unchanged. All above requirements lead to : AB 1 = W4 [4 5× 3 U3 Z3 2 × A3 P4 = = = = = = 451 101 113 253 135 151 169 The F#3, C#4, G#3/Ab3 notes can now be calculated, based on perfect fifths : 0 = 3A3 − 4 #3 = 3 #3 − 2P#4 = 3P#4 − 4U#3 The Eb4 and Bb3 notes can be calculated based on equal beat rate for fifths on Ab3, Eb4, Bb3 : AB 2 = 3Z 3 − 2[ 4 = 3[ 4 − 4A 3 = 3A 3 − 4 3 The obtained note pitches are : C4 247,90 C#4 261,33 D4 277,28 Eb4 294,18 E4 310,33 F3 331,28 F#3 348,44 G3 370,94 G#3 391,99 La3 415,00 Sib3 441,46 Si4 464,58 Table 6: BACH–3T ≈5Q≈ scale ; note pitches leading to almost equal beat rates on concerned fifths and major thirds Obtained beat rates : F3 Bb3 Eb4 G#3 C#4 F#3 B3 E4 A3 D4 G32 C4 F4 Maj. Thirds 1,84 5,51 12,85 11,62 18,43 11,80 15,25 16,30 7,80 7,35 1,84 1,84 3,37 Fifths – 1,10 0,37 0,37 0,37 0,00 0,00 0,00 – 1,84 – 1,84 – 1,84 – 1,84 – 1,84 – 2,20 min. thirds – 13,82 – 17,70 – 22,87 – 14,52 – 16,30 – 7,80 – 7,35 – 7,35 – 5,51 – 7,35 – 9,18 – 16,52 – 27,65 Table 7 : BACH–3T ≈5Q scale ; beat rates It can be observed there is a PERFECTLY identical beat rate for the fifths on C4, G3, D4, A3, E4 and the major thirds on F3, C4, G3. A remarkable equality : 1,8362831858… beats / sec. An enormous and lucky hazard. Remarkable indeed. This observation might explain the long-held confusion between “well temperament” and “equal temperament”. This confusion becomes even better explained based on 2021_01_28 Bach s Tempered Meantone.docx J. Broekaert 2020-11-09 Bach’s Tempered Meantone 7 the German terms “wohltemperiert” and “gleichschwebend” or “gleichstufig” (=”well tempered” and “with equal beat rate” or “with equal steps”) ? Here we have indeed, an equal beat rate for three thirds and five fifths of the diatonic C–major, rather than a beat rate equality for all fifths of the circle of fifths. Kellner (1977) was among the firsts to work out equal beat rates for fifths and thirds. Figure 2 illustrates the course of fifths (bold lines), related to the position of fifths on the Bach spirals. Figure 3 displays the interval beat rate impurity courses (bold lines). Figure 3 : Bach =FCG= : par. 3.2, table 3, see thin lines Bach–3T≈5Q≈ : par. 4.1, table 6, see bold lines Dotted lines : undiscussed version, holding an equal impurity distribution on all fifths from B3 to Bb3 Important note : The above comments are pure theory. The physical reality is more complex, and has to do with cords or pipes that hold inevitable imperfections : suspension points, stiffness and inhomogeneity of cords, diameter of pipes, etc… The produced sound therefore, is slightly anharmonic. This anharmonicity is normally very low for the middle scales, such as F3 to F4 for example (≤ 0,1 % ≈ 2 cent). A Bach – 3T ≈ 6Q model that contains three major thirds and SIX fifths with almost equal beat rates, is almost identical to the Bach – 3T ≈ 5Q model. Details of the calculation of this model are given in Appendix B – B6. The tuning tables for these two models are available in appendix E. The above concept was observed following calculation and comparison of a number of alternatives : 2021_01_28 Bach s Tempered Meantone.docx J. Broekaert 2020-11-09 Bach’s Tempered Meantone Jobin C dC dE 2TP / 5Q≈ 2T ≈ 5Q 2T = 5Q / cent =FCG= 3TP / 5Q≈ 3TP / 6Q≈ 3T ≈ 5Q 3T ≈ 6Q minimum 3T / 5Q≈/cent 3TP /cent 3T = 6Q/cent 2 JMT (just major third) : five equal fifths (calculated in cent) 1 JMT : on C ; as for the meantone 2 JMT : on C et G ; fifth on C differing 2 JMT : on C et G ; fifth on E differing (=meantone) 2 JMT : on C et G ; five fifths as equal as possible Equal impurities of 5 fifths and 2 thirds Equal impurities of 5 fifths and 2 thirds (in cent) 3 JMT : on F, C, G ; as for the au meantone 3 JMT : on F, C, G ; five fifths as equal as possible 3 JMT : on F, C, G ; six fifths as equal as possible Impurities of 5 fifths and 3 thirds as equal as possible Impurities of 6 fifths and 3 thirds as equal as possible Impurities of 6 fifths and 3 thirds as minimal as possible 3 JMT : on F, C, G ; and 5 fifths of equal impurity (in cent), s Equal impurities of 5 fifths and three thirds (in cent) Impurities of 6 fifths and 3 thirds as equal as possible (in cent) Impurities de 6 fifths and 3 thirds as minimal as possible Minimum / cent Table 7 : Overview of all calculated Bach alternatives (all names to be preceded by BACH–) 8 Jobin Par, 3,1 Par, 3,2 Par, 3,2 Appendix B–B1 Appendix B–B2 Appendix C–C1 Par, 4,1 Appendix B–B3 Appendix B–B4 Appendix B–B5 Appendix B–B6 Appendix B–B7 Appendix C–C2 Appendix C–C3 Appendix C–C4 Appendix C–C5 The above alternatives are calculated and discussed in depth in a downloadable more extensive text: “Bach’s tempered Meantone ; extensive text”. Downloadable at : http://users.telenet.be/broekaert-devriendt/Index.html or http://home.deds.nl/~broekaert/Index.html 4.2 Conclusion concerning the possible differing « Bach » versions The differing Bach versions can be defined, as suggested by the spirals on fig. 2, as temperaments that are evidently derived from the meantone, –all diatonic notes of the C–major are equal or almost equal indeed, and contain three just or almost just major thirds, all of this in combination with an optimisation of allowed meantone tonalities D –, A –, E – major, holding three major thirds that were just (pure) before, in the meantone, and an optimisation of three residual fifths, in favour of the Eb – and Bb – major tonalities. The above definition holds as non determining consequence, that an almost equal beat rate can be observed on the diatonic fifths, and a perfectly equal beat rate of the three remaining non perfect fifths on Bb3, Eb4, Ab3, such as displayed on the “Bach”-spirals (fig. 2). 5 Postamble Famous historical temperaments, were defined based on interval beat rates instead of the more common application of interval ratios, or cent or comma deviations. It has become clear, that all those temperaments : • are almost identical to their counterpart, commonly based on ratios, cents or comma’s. • require only a diapason, sometimes maybe a metronome, for easy auditory tuning It is not unreasonable to assume that Baroque temperaments were conceived, based on auditory observation by interpreting musicians and auditory tuners, and could therefore have been based on optimisation of interval beat rates. Post fact executed measurements by monochord at Baroque time, might at that time have lead to the commonly published definitions, based on cents, ratios or comma’s, slightly differing from the actual installed temperament based on beat rates, because of unobserved very minor measuring 2021_01_28 Bach s Tempered Meantone.docx J. Broekaert 2020-11-09 Bach’s Tempered Meantone 9 errors (unobservable at that time ! ). Very minor measuring errors in cents are almost unavoidable, and the slight anharmonic structure and pitch sliding of the real physical vibrators of musical instruments also contributes to measuring uncertainties. All above observations are valid in particular, for the Bach tuning according the Jobin proposal and the new hypotheses proposed in this paper. The exceptional dexterity and speed with which Bach could auditory tune a clavichord (Kelletat, 1981, p. 52-53; Forkel, p. 17) allows to assume that he only tuned by the ear, probably based on observable beat rates. Bach was not at all interested in calculating interval ratios (Forkel, p. 39) Based on all above arguments, it seems reasonable to assume that the spirals on the score of “Das wohltemperirte Clavier” may be associated to purity requirements concerning beats and beat rates of fifths, indeed, associated also to just or almost just major thirds on C and G, and F too. This is in support still, of Jobin’s hypothesis based on the significance of those fifths measured by commonly applied equality significance of ratios, but still in support also of the here elaborated alternative beat rate Bach temperaments. IT IS NOT THE STRICT MATHEMATICAL EXACTITUDE OF FIFTHS THAT PREVAILS,… but that what prevails in this indeed, is the “AUDITORY EQUALITY JUDGMENT” of the interpreting musicians and auditory tuners, their “brainear” (“Cervoreille” ; Calvet 2020) leading to JUST or EXCELLENT and ALMOST JUST MAJOR THIRDS on C and G, and also F. Conclusion • • • • It are the beat rates that import to performing musicians and auditory tuners Beat rate characteristics of historical temperaments are probably their determining factor Beat rate characteristics should probably deserve much more attention in musicology, and calculation of temperaments, especially for the Baroque period Taking into account the achievable auditory tuning precisions, it must be seen as a quasi certainty that Jobin and the proposed and almost equivalent beat rate Bach temperaments are valid hypotheses. 2021_01_28 Bach s Tempered Meantone.docx J. Broekaert 2020-11-09 Bach’s Tempered Meantone 10 Dedication This paper is dedicated to all classic musicians and auditory tuners. Their sensitive musical ears offer to our world all the best of the most universal and most beautiful of all languages : MUSIC. Acknowledgment I want to express my most sincere feelings of gratitude towards Amiot E., Calvet A, Jobin E and Paintoux T. Their open attitude allowed for quite intense exchange of ideas, that enabled further understanding and evolvement of insights in musical temperament and tuning matters, enabling the development of the ideas and concepts expressed in this paper. Thanks to my daughter Hilde : she drew my attention to investigate on what musicians want (C – major, diatonic interval purity) and not on what might be someone’s preferred musical temperament. Johan Broekaert Master Electro-Mechanical Engineering, option Electronics (Catholic University of Leuven, 1967 Nieuwelei, 52 2640 Mortsel [email protected] Belgium tel. 00 32 3 455 09 85 2021_01_28 Bach s Tempered Meantone.docx J. Broekaert 2020-11-09 Bach’s Tempered Meantone 11 References (internet links may change ! ) This list holds all references of the extended version of this text. Amiot E. 2008 : “Discrete Fourier Transform and Bach’s Good Temperament” https://mtosmt.org/issues/mto.09.15.2/mto.09.15.2.amiot.html Barbour J. M. 1951 : “Tuning and Temperament: A Historical Survey” Bosanquet H. 1876 : “An elementary treatise on musical intervals and temperament” https://en.xen.wiki/images/a/a7/Bosanquet_-_An_elementary_treatise_on_musical_intervals.pdf Calvet A. 2020: "Le Clavier Bien Obtempéré" http://www.andrecalvet.com/v3/index.php De Bie J. 2001 : “Stemtoon en stemmingsstelsels”, private edition, most data originate from Barbour J. 1951: “Tuning and Temperament: A Historical Survey” Devie D. 1990 : "Le Tempérament Musical : philosophie, histoire, théorie et pratique“ Forkel J. N, 1802 : "Ueber Johann Sebastian Bachs Leben, Kunst und Kunstwerke." https://reader.digitale-sammlungen.de/de/fs1/object/display/bsb10528130_00009.html Fritz B. 1756 : "Anweisung, wie man Claviere ... in allen zwölf Tönen gleich rein stimmen könne …“ https://gdz.sub.uni-goettingen.de/id/PPN630630391 Jedrzejewski F. 2002 : "Mathématiques des systèmes acoustiques, Tempéraments et modèles contemporains." L’Harmattan, Paris, 2002. Jobin E. 2005 : "BACH et le Clavier bien Tempéré"; https://www.clavecin-en-france.org/spip.php?article52. Kelletat H. 1957 : “Ein Beitrag zur Orgelbeweging. Vom Klangerlebnis in nichtgleichswebenden Temperaturen“ Kelletat H. 1960 : “Zur musikalischen Temperatur” Kelletat H. 1981 : “Zur musikalischen Temperatur; Band I. Johann Sebastian Bach und seine Zeit“ Kelletat H. 1982 : “Zur musikalischen Temperatur; Band II. Wiener Klassik“ Kelletat H. 1993 : “Zur musikalischen Temperatur; Band III. Franz Schubert“ Kellner H. 1977 : “Eine Rekonstruktion der wohltemperierten Stimmung von Johann Sebastian Bach“ Kroesbergen W. 2013 : “18th Century Quotes on J.S. Bach’s Temperament” see: https://www.academia.edu/5210832/18th_Century_Quotations_Relating_to_J.S._Bach_s_Temperament Lehman B. 2005: “Bach’s extraordinary temperament: our Rosetta Stone – 1; – 2”; Early Music Marpurg :1776: “Versuch über die musikalische Temperatur“ http://www.deutschestextarchiv.de/book/view/marpurg_versuch_1776?p=5 Napier J. 1614 : “Mirifici logarithmorum canonis descriptio" Norback J. 2002 : “A Passable and Good Temperament; A New Methodology for Studying Tuning and Temperament in Organ Music”, https://core.ac.uk/download/pdf/16320601.pdf Pitiscus B. 1603 : "Thesaurus mathematicus" Railsback O. 1938: “Scale temperament as applied to piano tuning” J. Acoust. Soc. Am. 9(3), p. 274 (1938) Rameau J.– P. 1726 : Nouveau système de musique théorique Rheticus G. 1542 : "De lateribus et angulis triangulorum (with Copernicus; 1542)" Salinas F. 1577 : “De musica libra septem” https://reader.digitale-sammlungen.de//resolve/display/bsb10138098.html Sparschuh A. 1999: “Stimm-Arithmetic des wohltemperierten Klaviers von J. S. Bach (TU Darmstadt) Stevin S. ca. 1586 : “De Thiende” (The Tenth) Stevin S. ca. 1605 : “Vande Spiegheling der Singconst” (Considerations on the art of singing) https://adcs.home.xs3all.nl/stevin/singconst/singconst.html Werckmeister A. 1681 : “Orgelprobe” Werckmeister A. 1686 : “Musicae Hodegus Curiosus” http://digitale.bibliothek.uni-halle.de/vd17/content/titleinfo/5173512 Werckmeister A. 1689 : “Orgelprobe” https://reader.digitale-sammlungen.de/de/fs1/object/display/bsb10527831_00007.html Werckmeister A. 1691 : “Musicalische Temperatur” https://www.deutsche-digitale-bibliothek.de/item/VPHYMD3QYBZNQL2UAK3Q2O35GBSVAIEL Zapf 2001: informal publication 2021_01_28 Bach s Tempered Meantone.docx J. Broekaert 2020-11-09 Bach’s Tempered Meantone Appendix A 12 Beat rate recalculation of some historic temperaments See also : Calvet 2020 Table A1 displays the obtained note pitches. Table A2 displays the applied comma divisions (q/n) and the obtained beat rates. It also displays the RMS−∆−cent of the pitch deviations from the “classic” temperament. C C# D Eb E F F# G G# A 246.76 261.43 276.98 293.45 310.90 329.39 348.97 369.72 391.71 415.00 246.73 261.43 276.92 293.45 310.87 329.48 349.08 369.72 391.77 415.00 248.23 259.38 277.53 296.95 310.28 332.00 346.91 371.19 387.86 415.00 Méantone 248.17 259.38 277.36 296.96 310.21 332.00 346.70 371.21 387.76 415.00 248.44 261.73 277.61 294.45 311.25 331.25 348.97 371.40 392.59 415.00 Werckmeister III 248.45 261.74 277.45 294.45 311.25 331.26 348.98 371.50 392.61 415.00 247.60 261.43 277.29 294.11 310.55 330.88 348.58 370.56 392.15 415.00 Vallotti 247.53 261.56 277.17 294.26 310.50 331.04 348.75 370.55 392.35 415.00 248.23 261.51 277.53 294.20 310.28 330.97 349.07 371.19 392.26 415.00 Kirnberger III / Kirnb. III "ungleich" / 248.15 261.43 277.77 294.11 310.19 330.87 348.96 371.43 392.14 415.00 “beat rate” Kirnb. III 248.17 261.44 277.36 294.12 310.21 330.89 348.98 371.21 392.16 415.00 247.60 261.73 276.67 293.45 311.25 330.13 348.97 370.14 392.59 415.00 Bendeler III 1690 247.71 261.97 276.67 293.58 311.25 330.27 349.30 370.41 392.96 415.00 247.93 261.20 277.42 293.85 310.41 330.58 348.26 370.89 391.80 415.00 Kellner 1976 247.89 261.16 277.28 293.80 310.33 330.53 348.21 370.93 391.73 415.00 247.60 262.02 277.29 294.11 310.55 330.88 349.37 370.56 392.59 415.00 Lehman 2005 247.27 262.09 277.09 294.30 310.62 349.45 349.45 370.28 392.82 415.00 247.60 261.36 277.29 293.78 310.55 330.50 348.78 370.56 391.71 415.00 Mercadier 1788 247.52 261.34 277.16 293.80 310.51 330.53 348.83 370.54 391.74 415.00 247.60 261.43 277.29 293.78 310.55 330.13 348.58 370.56 392.15 415.00 Neidhardt 1 1732 247.61 261.44 277.19 293.73 310.47 330.15 348.59 370.63 392.16 415.00 247.60 261.73 277.29 294.11 310.90 330.50 349.37 370.56 392.15 415.00 Neidhardt 2 1732 247.50 261.67 277.16 294.10 310.88 330.49 349.38 370.52 392.13 415.00 247.60 261.73 277.29 294.11 310.90 330.13 349.37 370.56 392.15 415.00 Neidhardt 3 1723 247.58 261.62 277.18 294.03 310.86 330.11 349.34 370.60 392.04 415.00 247.60 261.43 277.61 293.78 310.20 330.13 348.58 370.98 391.71 415.00 Neidhardt 4 1732 247.55 261.35 277.44 293.74 310.10 350.33 348.47 370.94 391.65 415.00 247.60 261.43 277.29 294.11 311.25 330.13 348.97 370.56 392.15 415.00 Sorge 1744 247.65 261.46 277.20 294.14 311.25 330.21 349.15 370.68 392.19 415.00 247.60 261.73 277.29 294.11 310.90 330.13 349.37 370.56 392.59 415.00 Sorge 1758 247.58 261.62 277.18 294.03 310.86 330.11 349.34 370.60 392.04 415.00 247.97 261.38 277.81 294.06 309.96 330.63 348.51 371.96 392.08 415.00 Stanhope 1806 247.92 261.32 277.56 293.98 309.90 330.56 348.42 371.88 391.97 415.00 248.05 261.97 277.67 293.98 310.48 331.55 348.42 371.15 391.97 415.00 Vogel 1975 248.04 262.22 277.32 294.26 310.27 332.02 348.31 371.07 392.35 415.00 247.84 261.10 276.13 294.30 311.25 330.45 349.48 371.04 391.64 415.00 Werckmeister VI 247.91 261.12 276.30 294.30 311.25 330.54 349.61 371.32 391.67 415.00 Table A1 : Comparison of temperament pitches : the lower rows are the beat rate calculated versions. 12TET / Fritz Bb B 439.68 439.80 444.04 443.78 441.67 441.68 441.17 441.39 441.29 441.16 441.18 440.17 440.36 440.77 440.70 440.67 441.13 440.67 440.70 440.17 440.20 441.17 441.15 440.67 440.66 440.67 440.60 440.67 440.81 440.67 440.66 440.84 440.75 440.97 441.39 441.45 441.45 465.82 465.94 463.98 464.01 466.88 466.88 464.77 465.00 465.43 465.28 465.31 465.30 465.73 465.61 465.50 465.82 465.93 465.43 465.48 465.30 465.31 466.35 466.32 466.35 466.30 465.30 465.14 465.82 466.07 466.35 466.30 464.95 464.85 464.56 464.42 466.88 466.88 Probably many more temperaments could be redefined or recalculated (see for example : Calvet, Jedrzejewski F. 2002). 2021_01_28 Bach s Tempered Meantone.docx J. Broekaert 2020-11-09 Bach’s Tempered Meantone Auditory tuning instructions (figures in beats/sec.) q = – 0.75 on all fifths q = – 2.09 on C, G, D, A | p = 0 on G, D, A, E, F, Bb, Eb q/4 = – 2.35 on C, D, G, B q = – 1.50 on F, C, G, D, A, E q = – 2.09 on C, G, D, A | q = 0 on E, B, F, Bb, Eb, Ab, C# q/3= − 1.14 on C, G, E, G# q/5= − 0.92 on C, G, D, A, B q/6= − 0.63 on F, C, G, D, A | q/12= − 0.31 on C#, G#, Eb, Bd q/16= − 0.28 on E, B, F#, C# | q/12= − 0.37 on F | q/6= − 0.74 on C, G, D q/6= − 0.78 on C, G, D, A | q/12= − 0.39 on E, B, G#, Eb q/6= − 0.73 on C, G, D | q/12= − 0.37 on F, A, B, F#, C#, Bb q/6= − 0.77 on C, G, D | q/12= − 0.39 on A, B, F#, C#, Eb, Bb q/3= − 1.15 on D, A | q/6= − 0.77 on G | q/12= − 0.38 on C, B, Bb, C# q/6= − 0.81 on C, G, D, E | q/12= − 0.40 on B, F#, Eb, Bb q/6= − 0.77 on C, G, D | q/12= − 0.39 on A, B, F#, C#, Eb, Bb (synt. comma)/3= − 1.35 on G, D, A | (schism. comma)= − 0.22 on B, Eb q/7= − 0.98 on F, C, G, A, D, E, C# | q/7=0.98 on F#, Bb 4q/7= − 2.18 on G | 2q/7= − 1.09 on F# | q/7= − 0.55 on C, B, Bb | Werckmeister VI 1691 q/7= + 0.55 on D, G# Table A2 : Tuning instructions and RMS ∆-cent of note pitch deviations from conventional values (*) : in comparison with the “classic” Kirnberger III / Kirnberger III ungleich Fritz Meantone Werckmeister III Vallotti Kirnberger III (*) Bendeler III 1690 Kellner 1976 Lehman 2005 Mercadier 1788 Neidhardt 1 1732 Neidhardt 2 1732 Neidhardt 3 1723 Neidhardt 4 1732 Sorge 1744 Sorge 1758 Stanhope 1806 Vogel 1975 Appendix B 13 RMS ∆-cent 0.32 0.56 0.31 0.72 0.48 / 0.80 1.09 0.37 1.22 0.31 0.27 0.34 0.37 0.48 0.50 0.78 0.57 1.41 0.56 Minimisation of the differences of beat rate interval impurities The “Brainear” (“Cervoreille” ; Calvet 2020) An optimisation for best possible equality of interval beat rates can in general be realised by any professional auditory keyboard tuner. However, a strict equality of interval beat rate impurities, such as calculated for a number of historical temperaments, is not always possible or easy. This problem occurs for example, if tuning a group of five fifths holding more than one just major third, as was encountered for certain Bach hypotheses. A perfect mathematical equality was not possible in those cases. The experience of a professional tuner is of fundamental importance for those peculiar cases. Besides the fine musical ear of the tuner or musician, intervenes fundamentally also his (musical) mental evaluation of the interval, his “Brainear”. An optimisation of the interval beat rate impurities, can be calculated by determination of the minimum of impurity differences of those intervals, rather than by calculation of an absolute and strict mathematical equality (often impossible on top). The mean beat rate of m major thirds and n fifths, calculated based on their absolute values (hence : positive) equals (qNote and pNote : see tables 1 et 2) : AB −R + ak ; the interval impurity difference on a note therefore equals : m+ −R + ak = Ropql + for the fifths ; and m+ −R + ak = −aopql + for the major thirds m+ kl" ΔRopql Δaopql = To be minimised therefore is : r = ΔRopql 2021_01_28 Bach s Tempered Meantone.docx + Δaopql J. Broekaert 2020-11-09 Bach’s Tempered Meantone 14 If fifths only have to be optimized, the development of this expression shows that this sum is proportional to : r ∝ −1 2 k Rk − "t# ";# → R" R# This expression can be minimised, setting its partial derivatives to zero, followed by the solution of the obtained equations. If fifths and thirds have to be minimised, it is required to calculate in complete detail. It is in any case required that all impurities are independent from one another. Therefore all impurities have to be substituted by expressions depending on the note pitches, notes that are indeed independent from one another (see tables 2 and 4). Appendix B – B6 Optimisation of beat rate impurities of the major thirds on F, C, G, and the fifths on F, C, G, D, A, E The worked out sum of squares of impurities on fifths on F3, C4, G3, D4, A3, E4 and major thirds on F3, C4, G3 corresponds to : 81 × ∆ = 2718 + 2934Py + 3726Ux + 1044Wy + 3240Zx + 2124[y + 2592Ax x −1116 x Py − 216 x Ux + 36 x Wy − 3132 x Zx + 180 x [y −2376Py Ux + 72Py Wy + 216Py Zx − 2880Py [y −864Ux Wy + 324Ux Zx + 540Ux [y − 3240Ux Ax − 1242Zx [y − 1944[y Ax −1998Wy Zx − 90Wy [y The partial derivatives set to zero lead to the equations, table B15 : F3 C4 G3 D4 E4 B3 F3 5436 – 1116 – 216 36 180 0 C4 – 1116 5868 – 2376 72 -2880 0 G3 – 216 – 2376 7452 – 864 540 – 3240 D4 36 72 – 864 2088 -90 0 E4 B3 180 – 2880 540 – 90 4248 – 1944 0 0 – 3240 0 -1944 5184 = = = = = = = A3 3132 – 216 – 324 1998 1242 0 Table B15: definition of diatonic notes s Strong simplification of the table 5 rows is possible. Based on the obtained solutions, the notes F#3, C#4, G#3/Ab3, can be calculated, setting perfect fifths : 0 = 3A3 − 4 #3 = 3 #3 − 2P#4 = 3P#4 − 4U#3 The notes Eb4 and Bb3 are obtained, setting an equal beat rate for the fifths on Ab3, Eb4 and Bb3 : AB = 3Z3 − 2[ 4 = 3[ 4 − 4A 3 = 3A 3 − 4 3 The obtained pitches are : C4 247,82 C#4 261,26 D4 277,26 Eb4 294,17 E4 310,24 F4 331,38 F#4 348,35 G4 370,86 G#4 391,89 A4 415,00 Bb4 441,50 B4 464,47 Table B16: BACH–3T ≈6Q≈ scale; note pitches for optimised inequality of concerned fifths and major thirds 2021_01_28 Bach s Tempered Meantone.docx J. Broekaert 2020-11-09 Bach’s Tempered Meantone 15 Obtained beat rates : F3 Bb3 Eb4 G#3 C#4 F#3 B3 E4 A3 D4 G3 C4 F4 Major Thirds 1,56 5,27 12,60 11,54 19,20 12,13 15,51 16,39 7,55 7,12 1,78 1,86 3,11 Fifths – 1,43 0,50 0,50 0,50 0,00 0,00 0,00 – 1,78 – 2,03 – 1,77 – 1,78 – 1,73 – 2,86 Minor thirds – 14,40 – 18,20 – 23,27 – 14,51 – 16,39 – 7,55 – 7,12 – 7,12 – 5,92 – 6,65 – 8,82 – 16,06 – 28,80 Table B17 : BACH–3T ≈6Q scale ; beat rates Appendix E A3=220 Partition F3 F4 A4 A3 A3 D4 D4 G3 G3 C4 C4 F3 F3 F4 F4 A#3 A#3 D#4 D#4 G#3 G#3 C#4 C#4 F#3 F#3 B3 B3 E4 E4 A3 E4 A4 Tuning table (for A4 = 440 ; hence: not for 415 ! ) 3 thirds and 5 fifths equal Beats Major thirds Beats 0.0 1.9 -1.9 1.9 -1.2 0.0 0.4 -0.4 0.4 0.0 0.0 0.0 1.9 -1.9 -3.9 F3 A3 F#3 A#3 G3 B3 G#3 C4 A3 C4 A#3 D4 B3 D#4 C4 E4 C#4 F4 F4 A4 1.9 12.5 1.9 12.3 8.3 5.8 16.2 1.9 19.5 3.9 2021_01_28 Bach s Tempered Meantone.docx A3=220 Partition F3 F4 A4 A3 A3 D4 D4 G3 G3 C4 C4 F3 F3 F4 F4 A#3 A#3 D#4 D#4 G#3 G#3 C#4 C#4 F#3 F#3 B3 B3 E4 E4 A3 E4 A4 J. Broekaert 3 thirds and 6 fifths almost equal Beats Major thirds Beats 0.0 1.9 -1.9 1.8 -1.5 0.0 0.5 -0.5 0.5 0.0 0.0 0.0 1.9 -2.1 -4.3 F3 A3 F#3 A#3 G3 B3 G#3 C4 A3 C4 A#3 D4 B3 D#4 C4 E4 C#4 F4 F4 A4 1.7 12.9 1.9 12.2 8.0 5.6 16.4 2.0 20.4 3.3 2020-11-09