Well Tempering based on the Werckmeister Definition

Transcription

1 Well Tempering based on the Werckmeister Definition Broekaert Johan M. Electronic Engineering, Catholic University of Leuven (KUL), Leuven, Belgium Nieuwelei, 52 B 2640 Mortsel Belgium (0)

2 Well Tempering based on the Werckmeister Definition Abstract : Based on the definition of well temperaments published by Werckmeister (1681), a number of logic mathematical steps are developed, in order to obtain mathematically optimised well tempered model- temperaments. This allows to work out some kind of objective mathematical classification of historical Well Temperaments. Historical temperaments that fit best with the models, are the same as those that are very often installed on organs, with highest established musical appreciations and notoriety. The mathematical appreciation that is worked out and the musical appreciation of historical temperaments seem to be in good accordance. All historical Well Temperaments have RMSimpurities between an absolute minimum of model- temperaments and the absolute maximum of the equal temperament. There is close affiliation of historical temperaments with said limits, and the list is well filled, hence, it makes no sense to develop new temperaments further on. The developed RMS computing modules can be used also for an accurate recognition of historical temperaments. Keywords : temperament; extended; just; intonation; musical; Pythagorean; natural harmonic; equal temperament; well temperament; optimal; model; objective; root mean square; RMS; interval; pure; third; fifth; diatonic; tune; wohltemperiert; Werckmeister 1 Issue 1.1 Musical Temperaments Differing musical schools exist since Antiquity already: we had for instance the canonical school of Pythagoras and the harmonic one of Aristoxenus. Over centuries, up to the present, musical requirements led to more than one hundred [ 1 ] historical musical temperaments. An extended survey was published by Barbour (1951 [ 2 ]). Still recently, new temperaments were developed by Kelletat (1966 [ 3, 4, 5 ]), Vogel (1975 and 1985 [ 6 ]), Michelin (1976 [ 6 ]), Kellner (1976 [ 7 ]), Barnes (1979 [ 6 ]), Billeter (1979 [ 4 ]), Asselin (1985 [ 6 ]), Lehman (2005 [ 8 ]),... Lehman also refers to 21 temperaments [ 9 ],

3 claimed to be Bach-temperaments. This evolution demonstrates there exists no universally accepted temperament: the choice of a temperament belongs to the artistic freedom of the musician. An objective, rational appreciation or ranking of temperaments does probably not exist, but temperaments can be grouped in differing families. A particular group of temperaments embraces Well Temperaments (WT). This paper investigates fundamental Well Temperament properties. 1.2 Well Temperament (WT) Werckmeister published a widely accepted WT definition (see [ 3 ] page 9) : Well temperament means a mathematical-acoustic and musical-practical organisation of the tone system within the twelve steps of an octave, so that impeccable performance in all tonalities is enabled, based on the extended just intonation (naturalharmonic tone system), while striving to keep the diatonic intervals as pure as possible. This temperament acts, while tied to given pitch ratios, as a thriftily tempered smoothing and extension of the meantone, as unequally beating half tones and as equal (equally beating) temperament. (Orgelprobe, 1681). This WT definition is still generally accepted and referred to nowadays (see [ 10 ], pages 25, 26) : The wording is the same Werckmeister uses in the title to his Musicalisches Temperatur (1691), implying that one can modulate around the circle of fifths :.. 2 Mathematical elaboration of the Well Temperament 2.1 The Equal Temperament (ET) (...and as equal temperament...)

4 The ET can be seen as an optimisation of the Pythagorean tuning. The Pythagorean tuning includes eleven pure fifths, and a reduced fifth, that can be fixed on G#-D#( Eb) for example (Claas Douwes 1699), as displayed in fig. 1, but there are in fact twelve alternatives, such as for example, that of Henri Arnaut Fig.1: Pythagorean temperament van Zwolle (±1450; small fifth on B-F#). This structure induces eight Pythagorean major thirds (the full lines in fig. 1), with a ratio of 81/64 instead of the pure ratio of 5/4. The remaining four major thirds (the dotted lines in fig. 1), which embrace the reduced fifth in their bow on the circle of fifths, are very close to purity, but a little bit small. WT requires the best possible purity for fifths and thirds :...keep the diatonic intervals as pure as possible. The even division of the Pythagorean comma over the twelve fifths, leads to fifths that all become very slightly reduced only, and quite a gain in purity for major thirds that were Pythagorean before, so that the over-all impurity, the over-all beating thus, has been reduced to a minimum The classic Equal Temperament The classic ET is obtained as mathematical optimum, when the RMS value of impurities of fifths is optimised while measured in cents, or in percentage beat pitch (PBP): Cents = 1200.log(fifth x 2/3) / log2 PBP = 100.(2.fifth - 3) impurity measured in cents impurity measured in percentage beat pitch Optimisation is achieved by working out the minimum value for:

5 2.1.2 An Equal Beat temperament A slightly different temperament is obtained as mathematical optimum, when the RMS value of impurities is optimised while measured in beats per second. Optimisation leads to an equal beat pitch for all fifths. The beat pitch on a fifth is given by: Beat = 2.f 2-3.f 1 impurity measured in beat pitch (while the second harmonic of f 2 beats with the third harmonic of f 1 ) Optimisation is achieved by working out the minimum value for: The solution can also be found by setting Beat as an unknown, in twelve equations for all twelve notes. This system of twelve linear equations with twelve unknowns is easy to solve (see footnote [ 11 ]). The unknowns are the pitches of eleven notes, this are all the notes except A, and the pitch of the beat. The solution of the system leads to : Beat = A.( ) / ( ) Beat = (for A = 440) The corresponding model- temperament is named Equal Beat in this paper. Equal Beat C C# D Eb E F F# G G# A Bb B pitch Table 1 : pitches for equal beat on fifths The above temperament was probably first devised by Barthold Fritz, in 1756 [ 12 ] Remarks: Required tuning precision: the tuning of Equal Beat by the ear should result in an equal beat for the twelfth fifth also, at the last tuning step, so that the circle of fifths becomes closed properly. The least required precision can therefore be estimated at : ± / 12 = ± 0.1 cps. The required corresponding PBP precision is : ± 0,2 %. The

6 required corresponding cents precision is : ± 0,35 cents. Unfortunately, many tuning data sources publish precisions limited to the cent only. The opinion that aural tuning of the Equal Beat or the ET is very difficult, if not impossible, is quite common. Pythagorean thirds: Pythagorean thirds are generally accepted as the highest tolerated deviation from purity of major thirds. This is probably a consequence of Pythagorean tuning over ancient centuries and habituation to these thirds since Antiquity already. Maybe therefore, Pythagorean thirds also have been implicitly accepted as such, by Werckmeister : the best possible purity...? 2.2 The Just Extended System (...based on the extended just intonation...) Table 2 displays the pitch ratios for the just extended system (Aristoxenus 335 BC). Notes within the diatonic C-major with a pure major third are marked by Mt (= 5 / 4); pure minor thirds are marked by mt (= 6 / 5); and pure fifths are marked by f (= 3 / 2). Possible alternate ratio values are : F# = 25 / 18, Gb = 64 / 45, Bb = 16 / 9. fraction 1 25/24 16/15 9/8 75/64 6/5 5/4 125/96 4/3 45/32 36/25 3/2 25/16 8/5 5/3 225/128 9/5 15/8 125/64 2/1 ratio 1, purity Mt, f mt, f Mt, f Mt, f mt, f mt Mt, f C C# Db D D# Eb E E# F F# Gb G G# Ab A A# Bb B B# C Table 2: The Just Extended System: ratios; and pure intervals within C-major Some shortcomings are clear, only within C-major already : an impure fifth D-A and third D-F; the ratio of B-F furthermore requires that at least some of the fifths B-F#, F#- C#, C#-G#, G#-D#( Eb), Eb-Bb en Bb-F must be larger than pure, inducing major thirds in turn, that are larger than Pythagorean thirds, and according remarks under 2.1.3, thirds of this type are hardly acceptable in a well temperament.

7 This system has an extraordinary tight structure: see fig. 2, displaying an arrow for any pure interval: because of purity requirements, there is no possibility to distord the triangles in the figure, because they all have a fixed size for any of their sides. The diatonic C-major scale has no arrow for fifths B-F and D-A, and third D-F, because of their impurity. Most other tonalities have a structure different from C-major, leading to different musical affects, characters: see D-major for example. Fig. 2 : The Just Extended System : pure intervals Because of all above comments, the structure of the Just Extended System needs some adjustments, so that enharmonic notes can be combined into a single note only, and whereby all tonalities obtain an acceptable harmony (purity). 2.3 Optimisation to WT of the Just Diatonic Scale in C-major The optimisation of the Just Diatonic Scale in C-major to a WT is worked out within a number of model- temperaments, by mathematical optimisation of the global purity of its thirteen relevant intervals : the impure third D-F and fifth D-A, and the existing pure intervals : this are the major thirds on F, C, G; the minor thirds on A, E, B, and all fifths (those on F, C, G, D, A and E) except the one on B. The fifth on B is not

8 evaluated, because it leads to F#, and this is not a diatonic note within C-major. The impurity of the named intervals is measured, and used to compute purity optimisations of the C-major scale. Measurement of impurities of major thirds is done with the same formulas as for fifths (see 2.1), but with the factors 2 and 3 replaced by 4 and 5 (because of the beat between the 4-th and 5-th harmonics); for minor thirds the factors 5 and 6 should be used (because of the beat between the 5-th and 6-th harmonics). Impurity measurements are possible with PBP, cents, or beat units. The optimisation of the diatonic C-major towards model- temperaments, is worked out by elaborating the minimum of the root mean square (RMS-value, according the formula below) of the thirteen relevant impurities. This elaboration of the minimum RMS is done by looping through iterations in a spread-sheet. In the mean time, the optimisation of the complete temperament is elaborated also, by a simultaneous but separate RMS-impurity optimisation of the separate group of fifths on notes outside the diatonic C-major range; the fifths on B, F#, C#, G#, Eb, Bb (according same formula as above, but with a division by six). Evaluation of temperaments, by means of RMS calculations was done earlier already, by Hall Donald [ 13 ] Elaboration of a model with minimum RMS The minimum of the RMS of Impurities is worked out numerically, looping through iterations. The numerous iterations are very simple to elaborate, but tedious. Results are displayed in table 3. These models have a quite close fit with the Marpurg temperament (see further table 11).

9 C C# D Eb E F F# G G# A Bb B PBP-Pyth Cent-Pyth Beat-Pyth Table 3 : Pitches for model- temperaments with Pythagorean thirds RMS_of_Impurity values: 1.02 % PBP, 4.50 cents, 3.49 beats/sec. The extreme purity of the optimized diatonic intervals is striking. They all have a PBP of approximately - 1 %, and this also applies for the major thirds on F, C and G, which therefore, surprisingly, are slightly smaller than pure major thirds. In this first approach it appears furthermore that the major thirds on B, F# and C# are larger than Pythagorean thirds. Consequently, the obtained models are not well tempered. A new optimisation round is necessary, avoiding the formation of major thirds larger than Pythagorean thirds Elaboration of a model with minimum RMS, avoiding major thirds larger than Pythagorean thirds Results are displayed in table 4. Those models have a close fit with Kirnberger III unequal (see further table 11). C C# D Eb E F F# G G# A Bb B PBP Cent Beat Table 4 : Pitches for model- temperaments without Pythagorean thirds RMS_of_Impurity values: 1.48 % PBP, 5.51 cents, 5.27 beats/sec. Specific for these models is the inclusion of six pure fifths : those on B, F#, C#, G#, Eb and Bb. This specific characteristic will furthermore persist in the following developments of models. The models also show minor thirds having better purity than some fifths, but the ear is much more sensitive to impurities on fifths than those on minor thirds.

10 Therefore further improvements can be thought of, whereby the best purity of minor thirds is limited to the worse impurity of fifths Elaboration of models with minimum RMS, with limitation on the purity of minor thirds Results are displayed in table 5. These models have a very close fit with Kirnberger III (see also further table 11). C C# D Eb E F F# G G# A Bb B PBP-min.th Cent-min.th Beat-min.th Table 5 : Pitches for model- temperaments with minor thirds with limitation RMS_of_Impurity values: 1.54 % PBP, 5.54 cents, 5.62 beats/sec. These models show major thirds having better purity than some fifths. For the same reason as for minor thirds, a further step can be considered leading to major thirds having a best purity limited to the worse impurity of fifths Elaboration of models with minimum RMS, with limitation on the purity of major thirds Results are displayed in table 6. These models have a very close fit with Vallotti-Tartini (see also further table 11) C C# D Eb E F F# G G# A Bb B PBP-maj.th Cent-maj.th Beat-maj.th Table 6 : Pitches for model- temperaments with major thirds with limitation RMS_of_Impurity values: 1.87 % PBP, 6.20 cents, 7.04 beats/sec.

11 2.3.5 Discussion The general evolution of the characteristics of the four types of PBP models (2.3.1 to 2.3.4) becomes clear by observing the graphic impurities displays in fig.3. The limitation of the quality of the major thirds has quite some influence on the quality of minor thirds. This appears very clear by observing the shift of the PBP-maj.th line of the minor thirds. Cent and Beat models are not displayed, but are almost identical, with very similar curves and evolution. Fig. 3 : Purity of PBP models At a first glance, no more grounds for further improvements can be thought of. 2.4 The Meantone ( and extension of the meantone ) The classic quarter comma meantone (Zarlino ) is characterised by an enlarged fifth, most often G#-D#( Eb), to such an extent that there are eight pure major thirds (see the full lines in fig. 1). The four remaining major thirds (see the dotted lines in fig. 1), embracing the enlarged fifth are consequently lager than Pythagorean thirds. The first and least requirement therefore, in a move from meantone to well temperament, is a reduction to Pythagorean thirds, of the four major thirds that are

12 much too large : this requires pure fifths on B, F#, C#, G#, Eb, Bb and F. This way six out of those seven fifths become equal to the six pure fifths obtained during optimisation of the just diatonic in C-major (see par ). The required further optimisation of remaining relevant thirds and fifths - the major thirds F, C, G; the minor thirds D, A, E, B; and fifths on F, C, G, D, A and E, with exception of B - therefore leads to the same optimisation process as in par to 2.3.4, resulting in equal models. 3 Comparison of the obtained models with Historical Temperaments The position of the developed model temperaments and the ET, if compared with historical temperaments is of much interest. It might indicate what musicians were or are implicitly looking for, regarding musical temperaments. Data on historical temperaments was obtained from [ 3 ] and [ 4 ] and [ 6 ]; or was self computed, as for Kirnberger III unequal, for example (from data from [ 4 ] fig.12). Comparison with historical temperaments can be worked out with any of the three measuring methods (PBP, cent, beat). The comparisons below are worked out with measurements of the RMS PBP impurities, for relevant intervals of the diatonic scale in C-major (see first paragraph under 2.3). Measurements in cents or beat units lead to minor classification leaps only, in comparison with PBP measurements,. 3.1 Blind Comparison An obtained comparison shortlist including all models worked out under to is displayed in table 7. The total list contains over one hundred temperaments. As expected, models of par with major thirds larger than Pythagorean thirds stand on top of the list, due to the unrestricted optimal purity conditions under which these were created. This allows for the presumption that no errors were made during the

13 elaboration of their minimum RMS. The other models have a less favourable classification. Index RMS RMS Temperament Index Temperament PBP PBP 1 PBP-Pyth 1,02 * 22 VOGEL 1,58 2 MEANTONE -2/7 C 1,02 * 23 Beat 1,62 3 RAMEAU acc. TLA 1,06 * 24 Beat-min.th 1,63 4 Beat-Pyth 1,07 * 25 Cent 1,64 5 Cent_Pyth 1,08 * 26 Cent-min.th 1,65 6 LAMBERT / CHAUMONT 1,20 * 27 GEIB / NEU-BAMBERG 1,74 * 7 ANON. De CAEN 1,21 * 28 Kirnberger III unequal 1,80 8 RAMEAU 1,21 * 29 d'alembert / ROUSSEAU 1,80 * 9 MARPURG 1,21 * 30 Kirnberger III 1,83 10 RAMEAU in F 1,21 * 31 LOUET 1,83 * 11 RAMEAU acc. KLOP 1,21 * 32 ROSSI / SAUVEUR 1,83 * Meantone 12 SALINAS 1,21 * 33 Kelletat 1,84 13 Meantone 1,21 * 34 de BETHISY 1,85 * 14 CORRETTE 1,21 * 35 SIEVERS 1,86 * 15 Meantone 1,30 * 36 PBP-maj.th 1,87 16 PBP 1,48 37 GABLER 1,90 * 17 PBP-min.th 1,54 38 STANHOPE 1,91 18 RAMEAU-MERCADIER 1,54 * 39 Cent-maj.th 1,93 19 LEGROS (3 R.T.) 1,54 * 40 VINCENT 1,97 * 20 LEGROS (2 R.T.) 1,54 * 41 Beat-maj.th 2,04 21 VOGEL (STADE) 1,58 * 42 VALLOTTI - TARTINI 2,06 * Table 7: RMS Impurity of C-major Temperaments containing major thirds larger than a Pythagorean third are marked with an asterisk (*), and form a vast majority. Although Pythagorean thirds were not explicitly rejected by Werckmeister, these will further on be considered again as the allowed maximum value for major thirds (as was already done for par to 2.3.4), because of the already mentioned habituation and acceptance (see Pythagorean thirds). 3.2 Classification of temperaments, with exclusion of those with major thirds lager than Pythagorean thirds Exclusion of temperaments with major thirds larger than Pythagorean thirds, leads to a much shorter list, displaying a very different image.

14 Index RMS RMS Temperament Index Temperament PBP PBP 1 PBP MERCADIER PBP-min.th Pythagorean (*) 2.38 Mm 3 VOGEL (STADE) 1.58 M 22 NEIDHARDT Beat BARCA Beat-min.th Kirnberger II 2.56 Mm 6 Cent Lehman_1_6_Synth Cent-min.th NEIDHARDT Kirnberger III unequal 1.80 M 27 SORGE Kirnberger III 1.83 M 28 NEIDHARDT Kelletat 1.84 Mm 29 SORGE PBP-maj.th BENDELER-III STANHOPE 1.91 Mm 31 MEANTONE -1/8 C Cent-maj.th GOTHEL / NIEDERBOBRITZSCH Beat-maj.th WERCKMEISTER-VI 3.13 m 15 Kellner 2.16 M 34 MEANTONE-1/9 C Billeter 2.21 M 35 Equal beat Kirnberger I 2.30 Mm 36 Equal temperament NEIDHARDT [14] Pure fifths on F,C,G,D,A,E Lehman_1_6_Pyth 2.34 Table 8: RMS impurity of C-major (*) Pythagorean temperament: this temperament has a small fifth on D (the fifth D-A). Temperaments marked with M or m have major (M) or minor (m) third(s) with better purity than the worse purity of fifths (see also and 2.3.4). Discussion : Most remaining models stand in very favorable positions on top of the list. The very best possible RMS-PBP is fixed by the PBP model. The highest possible RMS-PBP is also fixed : by the ET. See footnote [ 14 ] concerning position 37 in the table. The list contains temperaments, such as Kirnberger I and Kirnberger II, that are not accepted as being a WT, because of the impurity of some fifths (see [ 4 ] the chapter concerning Kirnberger temperaments). This led in the past to the publication of Kirnberger III. Temperaments containing a fifth smaller than the smallest fifth of Kirnberger II, will therefore be filtered out. The allowed minimum value of fifths is therefore set at the ratio This deviation of fifths induces an allowed maximum upwards one also, with ratio 1.509, because higher ratios lead to the same kind of

15 beating, as with ratios lower than Temperaments with inadequate fifths have to be eliminated from the list [ 15 ]. Most sources publish temperament data values rounded to the cent. As a matter of fact, this precision appears to be insufficient: see par Emmanuel Amiot warns (see [ 16 ] par. [2.1.5]) : I hope that through the process of chain-quotation, the exact values of these tunings have been preserved..... To provide for some compensation regarding possible imprecision on calculation of thirds, the allowed maximum value for Pythagorean thirds has to be increased to (81/64) x A new classification with adapted limits for fifths and major thirds has been worked out in following paragraph. 3.3 Classification of temperaments, with adapted limits for fifths and major thirds Index Temperament RMS RMS Index Temperament PBP PBP 1 PBP YOUNG / Van BIEZEN PBP-min.th MERCADIER VOGEL (STADE) 1.58 M 26 NEIDHARDT Beat-min.th LAMBERT Cent WEINGARTEN Cent-min.th BARCA Kirnberger III unequal 1.80 M 30 Lehman_1_6_Synth Kirnberger III 1.83 M 31 MEANTONE -1/7 C Kelletat 1.84 Mm 32 BENDELER-II SIEVERS 1.86 M 33 NEIDHARDT PBP-maj.th ASSELIN STANHOPE 1.91 Mm 35 SORGE Cent-maj.th NEIDHARDT Beat-maj.th SORGE VALLOTTI - TARTINI BENDELER-III Kellner 2.16 M 39 WERCKMEISTER V Billeter 2.21 M 40 BENDELER-I WERCKMEISTER III MEANTONE -1/8 C BARCA - acc. DEVIE "GOTHEL / 3.06 NIEDERBOBRITZSCH" 20 YOUNG MEANTONE -1/9 C NEIDHARDT equal beat Lehman_1_6_Pyth Equal temperament BARNES 2.34 Table 9: RMS impurity of C-major

16 Discussion : The developed models remain on top of the list : this confirms the presumption that, here also, no errors were made during the elaboration of the minimum RMS. At this point the limits of WT are clear and finally set, by the PBP model and by the ET : at the top as well as at the bottom of the list there is a close affiliation with historical temperaments, and the list is well filled. It therefore makes no sense to work on further development of new WT. It is rationally hard to understand why so many still feel the urge to do so (see also 1.1 and [ 1 ]). Notice : Vogel (Stade) has very peculiar characteristics: the fifths on F# and Bb have an isolated ratio larger than 1.5 (=1.504), surrounded by small fifths, and therefore this does exceptionally not lead to major thirds larger than Pythagorean major thirds. The purest thirds go with temperaments on top of the list, and the purest fifths go with temperaments on the bottom : see fig. 4, displaying RMS purity values of the diatonic C-major, and separate RMS impurity values for fifths and major and minor thirds, plotted in function of their temperament classification in table 5. It is possible to observe Fig. 4: Course of Impurities some peaks in the descending course of the fifths, and a sudden rise on the right on the ascending course of the major thirds. The corresponding temperaments have some disadvantage if compared with their immediate neighbors, and are those with index 12, 32, and 38 until 45 of table 5.

17 The above leads to a simple selection rule for WT: Maximal purity for the Diatonic C-Major is desired, but with best possible harmony for ALL other tonalities : a historic WT, with low index number in table 9 should be selected. Best possible purity for all tonalities, with no preference for a specific tonality : select the ET. Fifths will never pain the ear, but at slight cost of some harmony. The elementary selection rules above, are confirmed in table 10. It displays RMS values for all discussed temperaments elaborated in paragraph 2. The table clarifies that the adjustment of the Diatonic C-major to an acceptable WT leads to some jump in the obtained RMS values. The RMS value increase for temperaments to is gradual and moderate, but there is quite some difference between those temperaments and the Equal Beat and ET temperaments. Model- Temperament PBP ( % ) Cents Beat (cps) Thirds larger than Pythagorean allowed (= no WT) No thirds larger than Pythagorean (=WT) Minor thirds never better than fifths (=WT) Major thirds never better than fifths (=WT) Equal Beat (=WT) Equal Temperament (=WT) Table 10 : RMS_impurity values for the Diatonic C-Major scale 3.4 Conclusions The findings in paragraph 3.1 to 3.3, allow to propose that WT comply with following characteristics : Well Temperaments are temperaments wherefore the diatonic scale of C-major has an impurity lying between an assignable absolute mathematical minimum, and a maximum corresponding to the value for the Equal Temperament, whereby no major thirds larger than Pythagorean thirds are allowed, and fifths must have a ratio between and

18 4 Recognition of temperaments The recognition of temperaments requires an evaluation including all notes of a temperament, instead of an evaluation of the diatonic notes of C-major only, as was done in par. 2 and Used Recognition Procedure A comparison between temperaments can be made, by determination of the RMS value of the purity-differences of reciprocal intervals in one octave, for all thirds, minor and major, and all fifths, according formula below. The better the fit between temperaments, the smaller the resulting RMS will be. The recognition of a temperament can be done by comparing the unknown temperament with historical temperaments, followed by a selection and sorting for a lowest RMS of the historical temperaments. A best approximation of the PBP-models, calculated this way, is displayed in the shortlist of table 11. RMS RMS RMS RMS Temperament Temperament Temperament Temperament PBP PBP PBP PBP PBP-pyth 0.00 PBP 0.00 PBP-min.th 0.00 PBP-maj.th Kirnberger III Kirnberger III Vallotti-Tartini Marpurg M 0.43 M 0.28 Mm unequal Kirnberger III Kirnberger III Rameau M 0.67 M 0.47 M Young unequal 1779 Rameau acc. TLA 1.13 M Kelletat 1966 Vogel M Stanhope Mm 0.88 Mm Sievers M Barca acc. Devie Vogel (Stade) 0.72 M Barnes Vogel (Stade) 1.42 M Vogel (Stade) 0.93 M Vallotti-Tartini Mercadier Sievers M Sievers Kellner M Lambert Louet Kellner M Young Neidhardt Vallotti-Tartini 1750 Kelletat 1966 Kirnberger III Mm 1.57 M Werckmeister III 1681 Neidhardt Vallotti-Tartini Neidhardt Werckmeister III Neidhardt Mercadier Werckmeister III 1681 Weingarten 1750 Table 11: comparison of historical temperaments with the developed models

19 Alternate non-historical models that were worked out in this paper (with cent or beat measurements) were excluded from the list. Some temperaments not complying with set purity conditions for thirds compared with fifths (according and 2.3.4) were filtered out from the PBP-min.th column (the temperaments marked with m) and PBPmaj.th column (the temperaments marked with m and M). 4.2 Evaluation of the results It can be assessed that the above classification of temperaments, obtained by objective mathematical criteria, corresponds surprisingly well with widely prevailing musical appreciations. Out of 248 organs in the Netherlands [ 17 ], for example, more than two out of three organs are well tempered, according Werckmeister (64), Neidhardt (60), Young (9), Vallotti (12), Kirnberger (11). Also by scouting for famous musical temperament on internet, quite some good accordance can be assessed between temperaments referred to on hit pages and those in the three most right columns of table 11. Remark: wider scientifically established statistics are required, concerning installed temperaments for Western music on organs or keyboards of professional classic musicians, if an irrefutable objective confirmation or rejection of the above is wanted. 4.3 Graphical evaluation A set of ratios of twelve semitones, or quarters, or fifths or sevenths, is sufficient to determinate all notes within one octave; a ratio set for other intervals, among those the thirds, is not sufficient for unambiguous determination of all notes. It became clear though, during investigations for this paper, that major third impurity graphs are the most typical elements characterising a temperament. The impurity characteristics of

20 major thirds and fifths of some important historical temperaments are therefore combined for the display in fig. 5. A very high degree of similarity in the courses of major thirds can be observed. The figure makes clear as well that the ratios of the fifths of the displayed temperaments is much better than the allowed limit of (see par. 3.2). Fig. 5 : PBP impurity of major thirds and fifths for some temperaments of table 11 5 Conclusions Historical temperaments were mathematically evaluated, based upon the Werckmeister definition, starting from different musical requirements formulated by Werckmeister. Based upon mathematical analysis and findings in this text, it is the hope that underlying conclusions would become acceptable for the concerned scientific community :

21 5.1 Main Conclusion WELL TEMPERAMENTS ARE TEMPERAMENTS WHEREFORE THE DIATONIC SCALE OF C-MAJOR HAS AN IMPURITY LYING BETWEEN AN ASSIGNABLE ABSOLUTE MATHEMATICAL MINIMUM, AND A MAXIMUM CORRESPONDING TO THE VALUE FOR THE EQUAL TEMPERAMENT, WHEREBY NO MAJOR THIRDS LARGER THAN PYTHAGOREAN THIRDS ARE ALLOWED, AND FIFTHS MUST HAVE A RATIO BETWEEN AND Auxiliary Conclusions : Optimal models and temperaments were determined, and characterised. Widely prevailing musical appreciations of historical temperaments, and their obtained objective mathematical ranking, seem to be in good accordance. It makes no sense to attempt further development of better temperaments. An accurate objective identification of an unknown temperament is possible by applying the developed purity quality measuring methods. The way musicians were able to develop temperaments that are almost identical to the disclosed optimums, by aural means only, and this in the far past already, compel for utmost appreciation and respect. The aural optimisation of intervals, without use of any type of calculation or measuring aid, demonstrates the high refinement of the human senses in general, especially the trained musical ear, and the great intuitive power of human brains.

22 Dedication : This paper is dedicated to all classical musicians and aural tuners, as homage to their musicality and refined musical ears. Acknowledgments : thanks to my daughter Hilde, whose critical sense has lead to major improvements and very meaningful fundamental revisions, and has triggered the present approach of the problem, working out developments of models based on the just diatonic. 1 On nov. 12-th 2013, a pitch data base containing more than 500 temperaments was accessible on internet page 2 Barbour James: Tuning and Temperament: A Historical Survey. Michigan State College Press Kelletat Herbert, professor; Zur musikalischen Temperatur. Band 1. Johann Sebastian Bach und seine Zeit, 1981, ISBN Pag. 9: Wohltemperierung heißt mathematisch-akustische und praktisch-musikalischen Einrichtung von Tonmaterial innerhalb der zwölfstufigen Oktavskala zum einwandfreien Gebrauch in allen Tonarten auf der Grundlage des natürlich-harmonischen Systems mit Bestreben möglichster Reinerhaltung der diatonische Intervalle. Sie tritt auf als proportionsgebundene, sparsam temperierende Lockerung und Dehnung des mitteltönigen Systems, als ungleichschwebende Semitonik und als gleichschwebende Temperatur. (Orgelprobe, 1681). 4 Kelletat Herbert, professor; Zur musikalischen Temperatur. Band 2. Wiener Klassik, 1982, ISBN Kelletat Herbert, professor; Zur musikalischen Temperatur. Band 3. Franz Schubert, 1994, ISBN De Bie Jos: Stemtoon & Stemmingsstelsels, 4-de uitgave, 2001 (Private edition) 7 Kellner Anton, Darmstadt: Eine Rekonstruktion der wohltemperierten Stimmung von Johann Sebastian Bach, Das Musikinstrument, Heft 1/77 8 Lehman B.: Bach s extraordinary temperament: our Rosetta Stone, Early Music, 2005, Vol. XXXIII, No. 1, p. 3-23, No.2, p Oxford University Press. 9 See on oct. 30-th Mentioned temperaments are : Kelletat, Kellner, Barnes, Billeter, Lindley, Sparschuh, Jira, Zapf, Lehman, Francis, Jencka, Jobin, Maunder, Mobbs/MacKenzie, Lucktenberg, Interbartolo, Lindley/Ortgies, O'Donnell, Spanyi's Kirnberger II, Williams (retune the instrument per composition), Di Veroli. Surprisingly, Kirnberger III en Kirnberger III unequal are not mentioned in the list.

23 10 Johan Norback: A Passable and Good Temperament; A New Methodology for Studying Tuning and Temperament in Organ Music. Studies from the Department of Musicology, Göteborg University, no. 70, 2002, ISBN , ISSN , pag. 25, The equations are: (A = 440) 1. Beat + 3C - 2G = 0 2. Beat + 3C# - 2G# = 0 3. Beat + 3D = 2A 4. Beat + 3Eb - 2Bb = 0 5. Beat + 3E - 2B = 0 6. Beat - 4C + 3F = 0 7. Beat - 4C# + 3F# = 0 8. Beat - 4D + 3G = 0 9. Beat - 4Eb + 3G# = Beat - 4E = 3A 11. Beat - 4F + 3Bb = Beat - 4F# + 3B = 0 12 According information obtained on sept. 2-nd 2013 from Willem Kroesbergen, harpsichord manufacturer, the Netherlands. 13 Hall Donald: The Objective Measurement of Goodness-of-Fit for Tunings and Temperaments. Journal of Music Theory, Vol. 17, No. 2 (Autumn 1973), pp With exclusion of major thirds larger than Pythagorean thirds, an ultimate maximum of RMS value of temperaments was determined as well, besides the determination of minimum values. This maximum always reaches the same value, for any temperament having pure fifths on all diatonic notes of C-major, except B. Under this condition, all thirds of C- major are Pythagorean. All temperaments of this type have an RMS-PBP of These systematically have the lowest position, lower than the ET. These characteristics can only be perceived on some Pythagorean temperaments (those with a small fifth on B, of F#, of C#, of G#, of Eb, of Bb), that are therefore not well tempered. There are no other historical temperaments within this category. It therefore makes no sense keeping this type of temperament within the list. 15 Eliminated temperaments are: Kirnberger I and II, Werckmeister VI, Von Wiese II, Agricola, Ramis de Pareja, Schiassi, Pythagorean. 16 Emmanuel Amiot: Discrete Fourier Transform and Bach s Good Temperament. Society for Music Theory; Music Theory Online, Volume 15, Number 2, June Statistics calculated from extractions of the on-line data-base of the Huygens-Fokker Foundation, the Netherlands