DISCRETE FOURIER TRANSFORM AND BACH’S GOOD TEMPERAMENT EMMANUEL AMIOT, CPGE, PERPIGNAN FRANCE

Introduction In two memorable papers ([?, ?]) in 2005, Bradley Lehman introduced the view that the recipe for the long lost temperament of Johann Sebastian Bach, the Graal of all musicologists, had in fact been lying all the while for all to see – not unlike Poe’s Purloined Letter – as a scribbling on the front page of the autograph edition of Das wohltemperirte Clavier. As his own learned comparisons with a huge number of previously known tunings make clear, this is a vexed question, and Lehman’s ‘discovery’ attracted a number of refutals or denials, on various grounds.

Figure 1. Bach’s diagram on the first page of WTC As it is well known in the question of tunings, there is no ideal solution since a number of different properties are desirable, competing and vying with one another – the first aporia, recognized in antiquity, being the construction of a tuning with pure fifths only. This implies that the musicological debate on the quality of Lehman’s tuning (henceforth called LT) might go on for ever, as tenants of any other tunings may well put forward (in perfect good faith) different qualities1. Usually such arguments revolve around some specific intervals. Others question small pieces of scribbling on the autograph – is there a ‘c’ written on the last fifth and what to make of it, or is it irrelevant?. . . It would be better and more conclusive to put forward some global quality adressing the whole collection of major (and minor) scales, as the real question of temperaments in Bach’s time can arguably be phrased as the possibility of having all the scales sounding pleasantly, or, in other words, the possibility of arbitrary modulations in the composition and play. This is certainly what Bach was proud of having achieved, considering the sequence of 24 tonalities in both volumes of WTC. As it happens, there is an objective, geometrical quality of tunings, measured with a single number (the Musical Sameness of Scales, or MSS) that is easily computed by way of a Discrete Fourier Transform (DFT for short), and the comparison of values of MSS for the different tunings in competition so far does single out LT as a clear winner, primus inter pares. Needless to say, Bach himself knew nothing of Fourier transforms, though the MSS arguably puts forward a musical quality that Bach would certainly have found desirable. Its discovery was purely serendipitous and has nothing to do with Bach and WTC. I was investigating this DFT for theoretical reasons but endeavoured to try it on unequal temperaments. Browsing the Internet for values of tunings, I came across Lehman’s story. I was stunned when I realized, and confirmed by further computations, that LT was way above other current tunings (except Werckmeister’s IV or septenarius2) in regard of this DFT quality. 1

For instance, many pages have been written on the single third E-G# in LT versus other tunings; but the number of pure ‘tertiam minorem’, as Bach calls them, might be considered of equal or greater importance than the same for major thirds, or the size of whole tones, or any other remotely sensible feature of a given temperament. 2 In the rectified version, assuming the value 176/196 in the division of the monochord should be read as 175/196, as many have corrected. 1

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EMMANUEL AMIOT, CPGE, PERPIGNAN FRANCE

Of course, this has to be put alongside with all musicological arguments, as the perfect solution to a mathematical problem needs not be musically relevant; still the conjunction of Lehman’s arguments (including the pleasant sound of the recordings in LT), and of MSS, makes a more convincing case for Lehman’s hypothesis. But arguments, like a razor3, cut both ways: I have not tested all known temperaments, being no specialist of the question, and some lesser known temperament might achieve a better MSS than LT. Indeed, equal temperament gives an infinite value for MSS, but nowadays it seems currently accepted that equal temperament was as abhorrent for Bach as it was to be for Kirnberger and presumably for most readers of this journal. Anyway, the readers are strongly invited to test their pet tunings against LT with the formula for MSS given in section I, which is easy with a pocket calculator. I do look forward to the results of MSS testing for the alternative interpretations of Bach’s scribble (I began exploration along these lines). On the other hand, even if it is found one day that some other tuning supersedes the MSS value of LT, the fact will remain that all temperaments that had currently been used in Bach’s time have much lesser MSS, and this must forever remain a significant hallmark of LT.

1. Sameness of Scales in a Temperament I feel selfconscious about the quantity of non trivial maths that follows. Much of it, I am afraid, cannot be assumed as common knowledge, except for specialists. But I am afraid it is impossible to appreciate the import of the MSS value without at least some attempt at grabbing the maths behind. Without this understanding my paper would not make sense. I have made use of footnotes and annexes in order to lighten the reading as much as possible. In plain words, and roughly speaking, the DFT defined below enables to appraise how close a scale (meaning a sequence of notes in a given temperament) is to the mathemetical paradigm of a regular polygon. This makes great musical sense, because the biggest DFT values characterize the major scales. As it happens, the distribution of these values calculated for the 12 major scales exhibits a very special feature in the case of LT. 1.1. Discrete Fourier Transform of a Scale. Research on DFT of scales originated in the preparation of the John Clough Memorial Days in Chicago (July 2005), organized by David Clampitt and Richard Cohn, when several researchers had their interest about DFT aroused by Ian Quinn’s ground-breaking dissertation [?], wherein he rekindled an original idea of late David Lewin [?]. Fresh developments may be found in [?]. But it is a slightly different track, initiated a few months later by Dr Thomas Noll, that led to the present indicator of sameness of scales. Definition 1. Let all notes inside an octave be given by a real number between 0 and 1; this means chosing a reference note (say C) and measure all intervals from there in cents/1200. Then the Discrete Fourier Transform of a scale A = {a1 , a2 , . . . an } ⊂ [0, 1] is the map n

n

1 X 2iπ 1 X 2iπak −2iπk t/n e e = e FA : t 7→ n n k=1




ak −k t/n

k=1

(it is the Fourier Transform of the map k 7→ e2iπak ). The values FA (0), FA (1) . . . FA (n−1) are the Fourier coefficients of scale A. For instance, the tempered C major scale would be CM = {0, 1/6, 1/3, 5/12, 7/12, 3/4, 11/12} and its DFT is  2iπt 4iπt 6iπt 8iπt 10iπt 12iπt 1 t 7→ 1 + e− 7 + e− 7 + e− 7 + e− 7 + e− 7 + e− 7 7 It is perhaps useful to remind some readers that the Fourier Transform is in general a tool for checking periodicities of a given phenomenon. Here, it is easily seen that the map FA is defined from the cyclic group Zn to the field C of complex numbers4. The notes can be visualised as the points e2iπak on the unit circle, which is the quotient structure of frequencies modulo octave: see fig. ??. 3

Occam’s ? See [?].

4adding n to t does not change the value of F (t) A

DISCRETE FOURIER TRANSFORM AND BACH’S GOOD TEMPERAMENT

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1.2. DFT and Maximal Evenness. Lemma 1. |FA (1)| = 1 ⇐⇒ A is a ‘regular polygon’, i.e. a2 − a1 = a3 − a2 = · · · = a1 − an =

1 n

mod 1

For any other scale, for any t, |FA (t)| ≤ 1. This is the case when A is a whole tone scale (in equal temperament), or augmented fifth, aso. But of course a seven note scales in a (decent) twelve note temperament cannot be a regular polygon, as 12 cannot be divided into 7 equal integral parts (see again fig. ??). Still, best approximations to that are musically interesting scales: Theorem 1. Let S be the set of scales of n notes chosen in some equal temperament with m notes (m > n), meaning ∀A ∈ S, ∀a ∈ A, m a ∈ N Then the scales in S with biggest value of |FA (1)| are the Maximally Even Sets. What are Maximally Even Sets ? The ME Sets were defined in [?], extended in [?]. A recent and thorough paper on their features is [?] and the connection with DFT, discovered in [?], is investigated in [?]. The proofs of the above lemma and theorem are quarantined in the annex, together with technical definitions. For the moment let it suffice to mention that (1) Informally, a ME set is the ‘most regular’ repartition of n notes in a given temperament. (2) ME sets in (say) equal temperament are famous scales: for instance the pentatonic, whole tone, major and octatonic scales all are ME sets. (3) In several cases, including the major (and pentatonic) scales, this Maximal Evenness is somehow related to the ‘generated’ quality, e.g. the major scale is generated by consecutive fifths. This means informally that in a given context (here an equal tempered chromatic ambient universe) the size of |FA (1)| measures the regularity of the scale, i.e. its closeness to a regular polygon. The above theorem, as the next proposition, can bear some degree of approximation, and hence still stand (in this informal phrasing) in all common tunings, which are close to equal temperament. Anyway, we only need the case of 7 notes ME sets in 12 notes temperaments: Proposition 1. In 12 note equal temperament, the Maximally Even Scales with 7 notes are precisely the 12 major scales. Now we can see that |FA (1)| really measures the closeness of 7 notes-scale A to the theoretical regular heptagon, which appears as a common goal, a platonic model, that major scales strive to approximate as best they can: this is the meaning of the last proposition5. From now on we need to remember just that. For instance if scale A is a regular heptagon (say a ‘whole tone scale’ in 14 notes equal temperament) then |FA (1)| = 1; for any major scale A in 12 note-equal temperament, |FA (1)| ≈ 0.988846. This is really a strong feature of major scales and not a mere curio: Parseval’s formula P states that t=0...n−1 |FA (t)|2 = 1 = 100%. Hence, when |FA (1)| evaluates around 0.98 for a major scale, it means that almost all the energy of the scale is concentrated in this coefficient, as all other Fourier coefficients are negligible. In the case of other, random, seven note scales, the energy is spread between the coefficients: for a chromatic succession of 7 notes, |FA (1)| is around 0.74, see fig. ??. Also the next best scales for|FA (1)| are very different (e.g. B, C, C], D, E, G], A]). From there we can define the sameness coefficient of scales (MSS) inside a given temperament. 1.3. Sameness Coefficient. A temperament is simply a collection of 12 notes, i.e. a subset T of [0, 1] with 12 elements. Henceforth we will only consider scales with seven notes, that is to say ordered sequences of seven elements of T . Let us assume that the elements of T are given in order: Definition 2. A temperament, or tuning, is an ordered sequence of twelve different notes: 0 ≤ t0 < t1 < t2 < . . . t11 < 1 5Obviously this appears as a mathematical feature, not a musical one. There is much relevance to tonal functions in the fact that the major scale is Maximally Even, though. This is discussed in the litterature on ME sets and the reader is invited to consult the references given in the bibliography.

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EMMANUEL AMIOT, CPGE, PERPIGNAN FRANCE

C Cð

B

Að

D

C major

A

Dð

Gð

E

G

F Fð

Figure 2. Major scales are best approximations of regular heptagons 1

0.5

1

2

3

4

5

6

Figure 3. DFT of a major scale and a chromatic 7 notes scale Definition 3. A major scale in T is a scale of the form Aα = (a0 , . . . a6 )

with

ai = t(ki +α

mod 12)

where α is a constant and the ki ’s are the indexes of the standard C major scale: (k0 , k1 . . . k6 ) = (0, 2, 4, 5, 7, 9, 11) Example: have α = 5, we get the notes ai with i = 5, 7, 9, 10, 12 = 0, 14 = 2, 16 = 4, i.e. F major. This enables to compute |FAα (1)| for all α = 0 . . . 11, i.e. for the 12 major scales ⊂ T . For instance, taking for T the so-called pythagorean tuning with the ‘wolf fifth’ between A# and F, we get the following values for al

DISCRETE FOURIER TRANSFORM AND BACH’S GOOD TEMPERAMENT

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major scales (in semi-tone order): 0.9891, 0.9891, 0.9856, 0.9927, 0.9856, 0.9915, 0.9856, 0.9856, 0.9915, 0.9856, 0.9927, 0.9856 Notice that all scales that do not contain the wolf’s fifth are isometrical, and hence share the same absolute value of the Fourier coefficient. When I first computed these values, I was thrilled by their closeness to 1, which reflects the characterization of ME sets in Thm. 1. But as it happens, it was not the most important feature in a given temperament: the striking fact I stumbled upon was that these values are exceptionally close to one another in the case of LT. In order to visualize this phenomenon more easily, we define Definition 4. The major scale sameness (MSS) of temperament T is the inverse of the biggest discrepancy between values of coefficients |FAα (1)| for all twelve major scales in T : 1 M SS(T ) = maxα (|FAα (1)|) − minα (|FAα (1)|) This quantity is highest when all values of |FAα (1)| (for all 12 major scales) are the closest, i.e. when all major scales are almost equally similar to the ideal (theoretical) model of the regular heptagon. For instance for Pythagorean tuning, we get a maximum (resp. minimum) value of 0.9927 (resp. 0.9856) and hence 1 1 = ≈ 140. M M S(pyth) = 0.9927 − 0.9856 0.0071 For LT, no major scale comes higher than |FAα (1)| = 0.991, but none comes lower than 0.987, so the MSS is high (260). 1.4. Musical relevance: playing in all tonalities. Most analyses that I have seen, including Lehman’s, are focused on some particular intervals, with a strange predilection for E-G#. The MSS coefficient is in a nutshell a measure of the sameness of all major scales in T . This is well in tune with Bach’s own project in WTC ; though he does mention major and minor thirds on the front page (not only E-F#), plus tones and semi tones (see ??), all subsequent pages are devoted to the capacity of playing with the same tuning throughout all 24 tonalities. A tuning with high MSS means that ALL major scales are similarly close to the theoretical heptagon, i.e. similarly ‘good’. Or should one say, ‘wohl’ ? 2. Comparison of MSS for different tunings 2.1. Tables. Let us begin with a computer (or pocket calculator) program for the computation of MSS on a given T . It is convenient to put the values of the 12 elements of T in some table, as we have done: T = {t0 , t1 , . . . t11 } where the ai ’s are the values in cents, divided by 1200 (so that octave is 1). This will be the input given to the function that computes the MSS. Now we loop through the twelve major scales: For α = 0 to 11, do compute the scale Aα and its first Fourier coefficient: 6 1 X Aα = {ai = t(ki +α) mod 12 , i = 0 . . . 6} cα = e2iπ(ak −k/7) 7 k=0

Lastly, find the minimum from the maximum in those 12 values; subtract the one from the other, and take the inverse6 of the result: 1 M SS(T ) = maxα (cα ) − minα (cα ) R Fig. ?? gives this algorithm in Mathematica , for scales with any number n of notes. In the Table ??, consecutive fifths are given in cents from the origin; so in the program ??, the variable gamme had to be divided by 1200. All origins (E[) have been trimmed to the same value, 0. Of course, different origins can be given for each tuning, but this is equivalent to some rotation on the circle of notes and the quantity MSS, being geometrical in essence, is invariant under such transformations 7. I selected only a few elements in the family of meantone scales. The Neidhardt tuning used here is the 1732 one, quoted by Lehmann in his answer to Lindley and Ortgies’s acerbic refutation [?]. Lindley’s tunings are tabulated from the same source: the first two are built respectively with sevenths of Pythagorean and syntactic comma, 6This is for better visualization. 7Incidentally, it is also invariant under inversion, i.e. reversing the lists, as the inverse of any major scale is also a major scale

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EMMANUEL AMIOT, CPGE, PERPIGNAN FRANCE

fou1 Hgamme_L := ModuleB8n = Length@gammeD