MUZZULINI'S article combines a study of historical/philological nature with actual computational research and offers novel results thanks to this very combination. I appreciate its publication in Empirical Musicology Review as I find it fascinating and thoughtprovoking to interpret the process of theory creation, i.e. the exploratory investigations of a theorist, as an instance of empirical work. The fascination is amplified by the circumstance that the investigation is dedicated to the traces of musictheoretical explorations of none other than Isaac Newton. The overall subject of Newton's notebook entries and – consequently – Muzzulini's study is the search for good approximations of a selection of musical intervals within regular divisions of the octave.
MUSICAL INTERVALS AND THE PITCH HEIGHT OF JUST INTONATION
In this commentary I would like to highlight one particular aspect of musictheoretical and algebraic nature. With the choice of his title Muzzulini tacitly locates genuine musical structures such as the diatonic and chromatic scales and the hexachords within the medium of pitch height. As an alternative to the squeezing of these musical entities directly into a linear pitch height space I would like to advertise a more abstract concept of musical interval space, which supports the investigation of a variety of pitch height interpretations in terms of elements of the associated dual space.
Herein I follow Eric Regener's (1973) approach to the analysis of musical pitch notation as well as Guerino Mazzola's (1990) proposal to model the concept of pitch height in terms of linear forms on an underlying linear interval space. This approach splits the linearity of a single pitch height space into two components: (1) The operation of interval concatenation is modeled in terms of vector addition. This operation is of tremendous musical importance. (2) The mapping of musical intervals into a onedimensional „medium" of pitch height is accomplished in terms of a linear pitch height form. Its linearity guaratees that interval concatenation is thereby faithfully preserved.
I will argue that this maneuver is more than an ontological sophistry, particularly in connection with Newton's explorations and Muzzini's refinements thereof. To that end I will paraphrase some of their findings along the lines of this approach.
To begin with, we regard three intervals, the perfect octave P8, the perfect fifth P5 and the major third M3 as three linearly independent generators of a 3dimensional vector space E of musical intervals. The associated limit 5 tuning pitch height can then be given by the linear form h: E > R with h(P8) = 1, h(P5) = log_{2}(3/2), h(M3) = log_{2}(5/4). Among other possible bases for this musical interval space E we choose Newtons three chromatic step intervals a = P8  P5  M3, b = 2 P8 + 3 P5 + M3, c = P5 + 2 M3 with the transformation matrix:
The columns of this matrix show the chromatic step coordinates of the perfect octave, perfect fifth and major third, respectively. The chromatic basis a, b, c serves as the main reference for the investigation. The transformation M goes hand in hand with its dual map M^{*} which converts pitch height forms, given in chromatic coordinates into their associated octavefifththird coordinates. Hence, the inverse dual map (M^{*})^{1} sends the covector u = (1, log_{2}(3/2), log_{2}(5/4)) to its associated representation in chromatic coordinates v = (h(a), h(b), h(c)) = (log_{2}(16/15), log_{2}(135/128), log_{2}(25/24)).
MODIFYING THE PITCH HEIGHT FORM
Looking through the lens of this approach, we will see that central findings in Newton's and Muzzulini's explorations are closely related to modifications of this covector v: for every natural number n > 0 we obtain a linear form h_{n} by replacing the coordinates of the augmented covector n.v by the nearest integercoordinates, respectively. More explicitly, if r[x] denotes the integer closest to a real number x, then h_{n} is represented by the covector v_{n }= (r[n log_{2}(16/15)], r[n log_{2}(135/128)], r[n log_{2}(25/24)]).
The empirical question behind this definition is how one can measure the musical suitability of such a modified linear form h_{n} as an alternative pitch height interpretation of the interval space E. And the answer should grasp relevant apects of Daniel Muzzulini's evaluation of chromatic nEDO scales, which he explicates in terms of (in)consistencylevels and a "logcompatibility"criterion.
The following definition draws upon a set C of musical intervals chosen a priori. It is meant to represent the core of the musical interval system: The pitch height form h_{n} is said to be consistent with respect to the interval set C, if the equation h_{n}(i) = r[n.h(i)] is satisfied for all intervals i in C. This consistency condition means that the values h_{n}(i) under the modified linear form should coincide with the isolated roundings of their original pitch heights, when augmented by the factor n.
If we choose C = {7a+3b+2c} to be just the singleton set containing the octave, the definition is satisfied for all those h_{n} which satisfy h_{n}(7a+3b+2c) = n. Comparing this with Muzzulini's terms we may state — for the time being —, that this matches the condition to be "consistent and logcompatible with respect to the three chromatic intervals a, b, c". Between n = 1 and n = 5000 there are 711 linear forms, which are consistent in this sense. 2
The core collection of musically prominent notes from Newton's notebook is the chromatic scale G – A_{flat }– A – B_{flat }– B – C – C_{sharp }– D – E_{flat }– E – F – F_{sharp }– G' with the step interval pattern (a, b, a, c, a, b, a, a, c, a, b, a). Translating the scale notes to intervals anchored in the first note G, we would consider in the role of the core set C the vector sums of the twelve prefixes of this chromatic step interval pattern, i.e., the intervals a = (1, 0, 0), a+b = (1, 1, 0), 2a+b = (2, 1, 0), 2a+b+c = (2, 1, 1), etc. till 7a+3b+2c = (7, 3, 2). In this case the definition matches Muzzulini's "logcompatibility" criterion for the entire scale. Experimentally, I found to my surprise that all the 14,275 octaveconsistent instances among the linear forms h_{n} (n < 100000) turn out to be consistent with respect to these 12 chromatic intervals as well. Does plain octave consistency imply scale consistency for Newton's scale? This would be an interesting property of this scale, because for random permutations of the pattern this does not hold anymore.
As a musically reasonable strengthening one may consider the vector sums of all factors of the cyclically conceived chromatic step interval pattern in the role of the core set C. In this case the equation h_{n}(i) = r[n.h(i)] should be satisfied for all internal intervals of Newton's chromatic scale. Among the 711 consistent linear forms I found 421 satisfying this stronger condition. 3
INSPECTING THE MUZZULINI TRANSFORM
In his advanced continuation of Newton's investigation Muzzulini offers an elegant method for the control of the variety of consistent and "logcompatible" chromatic nEDO scales. After the familiar 12equaltempered chromatic scale he identifies the three cases n = 53, 118 and 612 as the best approximations of Newton's chromatic scale with respect to the measurement of the deviation from just intonation, each of them better than the previous ones. What he then does with these three extraordinary cases, can be rephrased as follows:
The three chromatic delta functions a^{*}, b^{*} and c^{*} with a^{*}(a) = 1, a^{*}(b) = a^{*}(c) = 0, b^{*}(b) = 1, b^{*}(a) = b^{*}(c) = 0, c^{*}(c) = 1, c^{*}(a) = c^{*}(a) = 0 constitute a basis in E^{*}, dual to the basis {a, b, c} in E. The three linear forms h_{612, }h_{118} and h_{53} form another basis of the dual space E^{*}: the Muzzulini basis. The columns of the unimodular transformation matrix T^{*} show the chromatic coordinates of the linear forms h_{612, }h_{118} and h_{53}, respectively. Analogously, the columns of the inverse matrix T^{*1}show the Muzzulinicoordinates of the chromatic delta functions h_{a}, h_{b}, h_{c}.
The rows of the matrix T^{*1} (i.e., the columns of the transposed matrix T^{*1* = }T^{1}) also represent the chromatic coordinates of the dual Muzzulini basis. And we get a nice explication of an observation in the article, namely that Newton's pitch height form h_{612} is dual to the interval a+2bc, whose pitch height is also known as the schisma. It is the difference h(a + 2b  c) between the Pythagorean comma h(a + 3 b  2 c) and the syntonic comma h(b  c) and, likewise, it is the difference between the syntonic comma h(b  c) and the diaschisma h(ab). In oldfashioned frequency ratios this can be written as 32,805/32,768 = (531,441/524,288)/(81/80) = (81/80)/(2,048/2,025). The other two intervals of this basis need getting used to: 4a3b+5c maps to the microchromatic step interval in 53EDO, but vanishes in 118EDO and 612EDO, while 7a9b+3c maps to the microchromatic step interval in 118EDO, but vanishes in 53EDO and 612EDO. The behavior of the latter two intervals thus illustrates interesting side effects of the rounding procedure.
Figure 1 displays the 5,000 linear forms h_{1}, … h_{5000}, and illustrates the location of the 711 consistent ones among them as well as the 241 linear forms which are consistent with respect to the internal intervals of Newton's chromatic scale. They are shown as points in Muzzulini coordinates within a cuboid. One can think of these points as vectors in E (acting on E via scalar product). The convex closure of each of the three point clouds is shown to illustrate the effectivity of this basis. We note though, that these convex closures contain other integral points as well.
Thus, we see that the height forms h_{n} up to some limit (such as n = 5,000) are nicely clustered within a bounded region in E^{*}. Those which are octaveconsistent, are concentrated in a smaller subregion, and those which are furthermore consistent with all the chromatic intervals occupy a still smaller subregion of the former. In order to explain the effect of the transformation T^{*} (turning Muzzulini coordinates into chromatic coordinates) geometrically, it is useful to inspect its eigen covectors. T^{*} is a Pisot transformation with the large real eigenvalue 69.2025 and the corresponding normalized eigencovector (0.6943, 0.5711, 0.4379). This eigencovector is quite close to the normalized coordinate representation (0.6933, 0.5719, 0.4385) of the pitch height form h. As a consequence of this property, the transformation maps small balllike point clouds into elongated tubelike point clouds along this eigendirection, which almost coincides with the pitch height direction.
PONDERING ABOUT NEWTON'S CHROMATIC MODE
Apart from the issue of nEDO approximations, the publication of Muzzulini's article in EMR is thoughtprovoking in view of the open empirical status of Newton's chromatic scale itself in terms of musical relevance, be it with respect to the music of the 17th century or beyond. To some extent was Newton indeed committed to this particular step interval pattern (a, b, a, c, a, b, a, a, c, a, b, a). After all, with respect to the search for embeddings in suitable nEDOs he was in the first instance interested in the representation of the pattern and only in the second instance in good numerical approximation. In figure 9b of the article Muzzulini reproduces a table with nEDO representations, where one row sticks out: The sequence (0, 4, 2, 6, 1, 5, 3, 7, 11, 6, 10, 8, 12) is not monotonously growing like the pitches of a scale. Butwhatever the numbers meant for Newton, they still exemplify the pattern with 4 representing a, 2 representing b and 5 representing c.
The question is: shall such 12letter patterns (like Mercator's, Newton's, Holder's or Euler's) be investigated as chromatic refinements of the diatonic modes? What are the conditions of fulfillment for an affirmative answer? Over the centuries music theorists have paid considerable attention to the classification of the step interval patterns of the diatonic modes. And since the newera theorists have also studied tone repertories, intervals and chords in the threedimensional space, generated by intervals of the major and minor triads. But chromatic modes do not seem to have gained a strong interest so far. Does this mean that there exists a broad consensus that their study is irrelevant? Or is it a potentially interesting unexplored territory? Which kind of research questions would have to be raised?
In music cognition and in music theory there is a renewed interest in the study scale degree qualia. This concept attributes different qualities to the diatonic scale degrees apart from their pitch height. Guido explained the different characters of the notes of a mode (propriatas sonorum) in terms of their different step interval neighborhoods. Would it make sense to extend this concept to a level of chromatic modes?
The theoretical path from the concept of (pseudoclassical) mode to Newton's chromatic "mode" combines two different kinds of refinement, namely: (1) the chromatic extension of a diatonic mode on the level of notes as such as (a+b, a+b, a, b+a, a+b, a+b, a), and (2) the refinement (for example) of the authentic Ionian mode TTST  TTS through Zarlino's syntonicdiatonic mode TtST  tTS. Newton's pattern accounts for a combination of both procedures. But the "refinement" also entails obstructions for a seamless extension of the diatonic theory. Guido's affinities of the hexachord TTSTT (its double periodicity with respect to the prefixes TTST and TTS) does not hold anymore for Descartes' and Newton's syntonic hexachords tTSTt and TtSTt. Nonetheless, it could be worthwhile to explore the potential meanings of chromatic scale degree qualia.
At present there are two different approaches available. Huron (2006) sees the different profiles of transition probabilities from a given scale degree to the others as a clue for statistical learning and interprets scale degree qualia as a side effect of statistical cognition. In Noll (2018a, 2018b) I constructed exomodes, i.e., parametrizations of different musical qualities of one the same interval or interval combination. Geometrically, these exomodes are located in the kernel (the zerospace) of a suitably chosen pitch height form. Noll (2018a) is dedicated to diatonic and chromatic exomodes in Reger's (1973) twodimensional note interval space, while Noll (2018b) is dedicated to the study of the exopartners of Zarlino's syntonicdiatonic modes in the threedimensional interval space E. As far as I see, nothing should be in the way of applying the approach taken in Noll (2018b) to Newton's chromatic mode.
This perspective of achieving a parametrization of possible tone qualia in the orthogonal complements of the gradients of pitch height forms adds to my motivation to object against the squeezing of musical scales and related entities into the onedimensional pitch height space.
ACKNOWLEDGEMENTS
This article was copyedited by Matthew Moore and layout edited by Diana Kayser.
NOTES

thomas.mamuth@gmail.com
Return to Text 
The sequence starts with 12, 19, 22, 31, 34, 41, 43, 46, 53, 65, 75, 77, 84, 87, 96, 99, 106, 111, 118, 130, 140, 149, 152, 159, 164, 171, 183, 193, 205, 212, 214, 217, 224, 236, 246, 248, 258, 270, 277, 280, 282, 289, 301, 311, 323, 330, 335, 342, 345, 354, 364, 366, 376, 388, 395, 398, 400, 407, 419, 429, 441, 448, 453, 460, 463, 472, 482, 484, 494, 506, 513, 516, 518, 525, 528, 537, 547, 559, 566, 571, 578, 581, 590, 593, 600, 612, …
Return to Text 
The sequence starts with 12, 34, 41, 53, 65, 87, 106, 118, 130, 140, 152, 159, 171, 183, 193, 205, 217, 224, 236, 248, 258, 270, 277, 289, 301, 311, 323, 335, 342, 354, 376, 388, 407, 419, 429, 441, 453, 460, 472, 482, 494, 506, 525, 537, 547, 559, 571, 600, 612, …
Return to Text
REFERENCES
 Huron, D. (2006). Sweet Anticipation. Music and the Psychology of Expectation. Cambridge, MA: MIT Press. https://doi.org/10.7551/mitpress/6575.001.0001
 Mazzola, G. (1990). Die Geometrie der Töne. Birkhäuser. Basel. https://doi.org/10.1007/9783034874274
 Noll, T. (2018a). One Note Samba: Navigating Notes and their Meanings within Modes and ExoModes. In M. Montiel & R. Peck (Eds.), Mathematical Music Theory: Algebraic, Geometric, Combinatorial, Topological and Applied Approaches to Understanding Musical Phenomena. https://doi.org/10.1142/9789813235311_0006
 Noll, T. (2018b). Dual latticepath transformations and the dynamics of the major and minor exomodes. Journal of Mathematics and Music 12 (3), 212232. https://doi.org/10.1080/17459737.2018.1548035
 Regener, E. (1973). Pitch Notation and Equal Temperament: A Formal Study. University of California Press, Berkeley.