ISAAC Newton (16431727) is known for his outstanding contributions to mathematics and physics. In the 1660s he invented the mirror telescope and developed the basics of the theory of ray optics published in Opticks (1704). His early contributions to music theory are less known and were not published during his lifetime. A college notebook from 1665 contains the draft of a music theoretical treatise, which has been recently edited critically by Benjamin Wardhaugh 2. Newton's manuscript contains two circular diagrams which are inspired by Descartes's Compendium Musicae (1650). Newton's diagrams interpret and generalize Descartes's just intonation hexachordal system through octave divisions into 53 and into 120 equal parts, see Figure 2. The idea to approach just intonation through 53EDO 3 was brought about by Nicolaus Mercator at about the same time – also in manuscripts, see Wardhaugh (2013, pp. 129236). Due to an early quotation by William Holder (1694/1730, pp. 7980) Mercator's treatment of the tuning problem is better known than Newton's note book. Besides the mentioned divisions Newton considered several other equal division of the octave (EDOs) with respect to their capacity to approximate just intonation tunings. These EDOs are in the focus of this article.
After a brief introduction into Pythagorean and just intonation tunings, I will give some remarks on the construction of syntonic chromatic scales in historical sources. The following close reading of Newton's sketchy note book tries to clarify why he studied certain octave divisions and neglected others.
In order to tackle Newton's optimization problem with modern means, I calculated the closest approximation to Newton's syntonic chromatic scale for all equal divisions of the octave with up to 5000 microtonal steps per octave and analysed them in terms of least square deviations and consistency. This analysis helps to assess Newton's selection of octave divisions systematically and it leads us to a general approach to the tuning problem and to an elegant description of the dependence of the best fits from the fineness of the approximating scales, i.e., their number of equal steps per octave.
The use of least square deviations to assess the quality of best fit approximation problems is a mathematical standard with many applications in data analysis, especially in statistics and probability theory. In the 19^{th} century it was applied to musical scales by Drobisch (1852, pp. 7578) and more recently by James M. Barbour (1951) and Johannes Barkowsky (2007, p. 105). Barbour's approach, which takes 12EDO as a reference scale to assess chromatic and enharmonic scales, was criticized by Donald Hall (1973).
The approximation problem studied in this article is not of a genuinely stochastic nature, because the content of the datasets generated by computer programs are uniquely determined. Nevertheless, they appear to the eye as a kind of random data. The knowledge of three excellent nEDO representations of the syntonic chromatic scale, 53EDO, 118EDO and 612EDO, explains the quasichaotic dependence of the quality of best fit on n for values of n up to 1200 very well. In other words, for EDOs with unit steps not smaller than 1 cent, the discrete interval configuration of the just intonation scale and the approximating space are both threedimensional. 4 Newton had accurate tables of (base 10) logarithms at his disposal and he used them to measure just intonation intervals with 12EDO semitones, but he had no powerful computers and software. The "empirical" approach to Newton's approximation problem given here is in line with his selection of octave divisions and it reveals general properties of equal octave division and their capacity to simulate just intonation tuning by combining them linearly.
JUST INTONATION: LIMIT3 AND LIMIT5 TUNING
Pythagorean music theory is based on the idea that musical intervals can be described with ratios of small positive integers. In antiquity these ratios refer to string lengths on a monochord. 5 The diatonic genus and the related pitch systems of the Pythagoreans, as passed down to the Middle Ages and the Renaissance, can be defined by ratios of numbers which have only 2 and 3 in their prime number factorisations. In modern terminology and notation, the socalled "Pythagorean tuning" 6 is given by rational numbers of the format 2^{k} ⋅ 3^{j} where k and j are integers (positive, zero or negative). The set of numbers L_{3}:= {2^{k} ⋅ 3^{j} k,j ∈ $\mathbb{Z}$} is a dense subset of the real numbers, which hereafter is called limit3 tuning (system). Finite subsets of L_{3} can be used to define the Pythagorean intervals, chords and scales.
Historical sources often describe pitch sets with multiterms proportions of integers and visualize them on monochord drawings. In order to describe an octave periodic scale or chord, the first and last element of the related proportion are of the ratio 2 : 1. The Pythagorean proportion
6 : 8 : 9 : 12 = (2^{1} ⋅ 3^{1}) : (2^{3} ⋅ 3^{0}) : (2^{0} ⋅ 3^{2}) : (2^{2} ⋅ 3^{1}),
for example, serves as a reference frame for the Greek diatonic, chromatic and enharmonic genera. Pitch systems from the late Renaissance and later periods are usually strictly octave periodic. However, this is not necessarily the case in the Middle Ages and antiquity. 7
Syntonic tunings or limit5 tunings are defined in the same way with the sole prime factors 2, 3 and 5, and limit7 tunings with the prime factors 2, 3, 5 and 7:
L_{5}:= {2^{k} ⋅ 3^{j} ⋅ 5^{i} k,j,i ∈ $\mathbb{Z}$}, L_{7}:= {2^{k} ⋅ 3^{j} ⋅ 5^{i} ⋅ 7^{h} k,j,i,h ∈ $\mathbb{Z}$}
Limit5 tunings can be constructed by combining octaves (2 : 1), Pythagorean fifths (3 : 2) and (just) major thirds (5 : 4). 8 Limit7 tunings also permit "just minor sevenths" (7 : 4), but normally without piling them, so that the value of the power index h in the formula for L_{7} equals 1, 0 or 1. Limit7 tunings were suggested by several theorists from the 17^{th} to the 20^{th} century, among them Christiaan Huygens (c. 1661) and Quirinius van Blankenburg (1739), see Jedrzejewski (2018). Musicians referring to "just intonation" usually mean limit5, but sometimes also limit7 tuning. In this text we frequently use the term "syntonic scales" for subsets of L_{5}. 9
Octaves, Pythagorean fifths and just major thirds are base vectors of a three dimensional "interval space" $\mathbb{Z}$^{3} = {(k,j,i)k,j,i ∈ $\mathbb{Z}$} defined by the power indices from L_{5}. Because of the uniqueness of the prime factorization there is a natural embedding:
$\mathbb{Z}$^{3} = {(k,j,i)k,j,i ∈ $\mathbb{Z}$} ≅ L_{5} ⊂ $\mathbb{Q}$ ⊂ $\mathbb{R}$
Hereby, different points in $\mathbb{Z}$^{3} correspond to different frequency ratios and arbitrary irrational numbers can be approximated arbitrarily well through limit5 ratios, which means that L_{5} is a dense subset of the real numbers $\mathbb{R}$.
Nicolas Mercator in the 17^{th} century was probably the first to represent limit5 tunings with twodimensional grids which abstract from the octave coordinate 10. JeanPhilippe Rameau (1726), Leonhard Euler (1739) and others in the 18^{th} century used the same technique to visualize scales and chords on a twodimensional lattice. The number triangles used by Boethius, Torkesey and their followers up to Robert Fludd in the early 17^{th} century are precursors of twodimensional pitch grids; however, they do not abstract from octave similarity. 11 Isaac Newton did not use grid notation in order to describe just intonation tunings, which possibly means that he did not know Mercator's manuscripts when he studied EDOs. Newton favoured combinatorics and circular arrangements which he picked up from Descartes' Compendium Musicae, see Figures 1b and 2a.
Incommensurable intervals: Euclid's algorithm and continued fractions
The mathematical reason for the quasichaotic appearance of the datasets discussed below is the mutual incommensurability of the logarithms of the primes 2, 3 and 5 (expressed in any common base, conveniently 2). This incommensurability can be summarized with the following three formulas:
 (i) log_{2}(3) ∈ $\mathbb{R}$\
 (ii) log_{2}(5) ∈ $\mathbb{R}$\$\mathbb{Q}$
 (iii) log_{2}(5) / log_{2}(3) ∈ $\mathbb{R}$\$\mathbb{Q}$
Being irrational numbers, these expressions cannot be written as integer ratios (ordinary positive fractions). Translated into music theory the first formula (i) is equivalent to the incommensurability of the just fifth and the octave so that chains of Pythagorean fifths modulo octave generate scales with potentially infinitely many pitches per octave. The second formula (ii) means that major thirds of the ratio 5 : 4 are incommensurable to the octave in the same sense. And the third formula (iii) states that just major thirds and fifths are also mutually incommensurable in terms of musical interval size. The fact that intervals of superparticular ratios are mutually incommensurable has been known since antiquity.
Historically, commensurability was tested with Euclid's algorithm, which provides a method to distinguish rational from irrational ratios of quantities: Euclid's algorithm terminates after finitely many iteration steps, if and only if the ratio of the quantities under consideration is rational. The Pythagoreans knew this fact and that the algorithm does not terminate if it is applied to the diagonal and the side of a regular pentagon. In other words, they proved that the socalled golden ratio is irrational 12. The application of Euclid's algorithm onto pairs of musical intervals viewed as continuous quantities (like distances on a straight line) produces their continued fraction expansion as a byproduct. If this expansion is finite, the intervals under consideration are commensurable, otherwise they are incommensurable. In the Sectio canonis, assigned to Euclid, it is shown that any different intervals with superparticular frequency ratios (integer ratios of the form $\frac{\mathit{n}+1}{\mathit{n}}$) are mutually incommensurable. Boethius knew this result and discussed a proof by Archytas 13. Whereas the continued fraction expansion of the golden ratio,$\frac{\sqrt{5}\mathrm{+1}}{1}$, is periodic with the simplest possible pattern [1, 1, 1, 1, …], the interval ratios related for pairs of superparticular frequency ratios have infinite continued fractions expansions which do not exhibit simple regular patterns. 14
NEWTON'S CHROMATIC SCALES AND HIS VARIOUS OCTAVE DIVISIONS
Descartes's adaption of the hexachord system to just intonation and its reduction to octave classes, as well as Newton's extension of Descartes's approach, lead us to the problem of defining justtuned diatonic and chromatic 12note scales. Differently from the Pythagorean scales of the Middle Age, which offer only few alternatives, there was no consensus about the definition of a chromatic scale in just intonation in the 17^{th} and 18^{th} centuries. Various chromatic scales were proposed, which occupy different and incongruent regions in pitch grids for just intonation. 15.
From syntonic hexachords to chromatic scales
Guido of Arezzo's Micrologus (c. 1025) constructs a tone system with seven overlapping hexachords "utremifasola", each of them of the same symmetric interval structure TTsTT, a Pythagorean diatonic semitone s (256 : 243) flanked on both sides by pairs of major tones T (9 : 8). Guido of Arezzo placed these congruent hexachords at ut = Γ, C, F, G, c, f and g, so that his system comprises two octaves plus a major sixth [Γ, ee], where Γ, the lowest note of the system, is one octave below G, and ee, the highest note of the system one octave above e. There is no Bflat in the lowest octave, but Bflat and B in the middle and upper octaves.
The Pythagorean hexachord (Pyth_{Hex}) can be completed to octachords (Pyth_{Oct}), so that the octave is portioned into five tones and two diatonic semitones, see Table 1. For this purpose, the minor third (32 : 27) between the major sixth (27 : 16) and the octave, is divided into a Pythagorean semitone and a major tone, which results in two equivalent diatonic scales a Pythagorean fifth/fourth apart. Hence, the Pythagorean hexachord Pyth_{Hex} on C together with the transpositions by a fourth (4 : 3) and by a fifth (3 : 2) generate a limit3 diatonic system of eight pitch classes, CDEFGAB♭B, if the pitches are reduced to one octave. The pitches can be arranged as a stack of fourths BEADGCFB♭ with the chromatic semitone B♭B of the ratio 2187 : 2048, which is approximately 25% larger than the diatonic semitones EF and B♭B of the ratio 256 : 243.
Pythagorean hexachord  Pythagorean octachords  

Interval structure  TTsTT  TTsTTsT TTsTTTs 
Ratio structure 
Note: The two octachords are cyclic permutations of the same interval pattern. They define the same diatonic scale.
Guido's system was quoted time and again through the late Middle Ages and Renaissance up to the Templum musicae (1618/24) by Robert Fludd (1618/24, p. 161) published immediately before René Descartes (15961650) composed the Compendium musicae dedicated to Isaac Beeckman (1619). The Compendium written 1618, the earliest complete treatise by Descartes, circulated in at least four manuscript copies during his lifetime, only 1650 immediately after his death it appeared in two Latin prints and three years later afterwards in an English edition (1653). Descartes's manuscript is lost. 16 One of the four circular pitch diagrams contained in the Compendium Descartes gives a system of three structurally identical hexachords "utremifasola" (mollis, naturalis and duris), whose references are Pythagorean fifths apart as in Guido's system, but whose symmetric interval configuration tTSTt is composed of superparticular limit5 ratios with major tones T (9 : 8), minor tones t (10 : 9) and diatonic semitones S (16 : 15) in place of Guido's limit3 intervals. Descartes's syntonic hexachord and its completions to octachords are shown in Table 2.
Symmetric syntonic hexachord  Syntonic octachords  

Interval structure  tTSTt  tTSTtST tTSTtTS 
Ratio structure 
Note: The two ways of dividing the minor third between the sixth scale degree and the octave lead to two incongruent diatonic scales, in which the order of the major and minor tones between the respective tonics and their third scale degrees are reversed. Newton looked at both types of diatonic extensions, see Figures 2c/d.
Descartes' reinterpretation of Guido of Arezzo's Pythagorean hexachords results in a diatonic just intonation system with both varieties of B (Bduris and Bmollis) and two ambiguous pitches D and G, see Figure 1a. Just intonation as well as meantone temperaments were investigated and widely debated in the 16^{th} century by Fogliano (1529), Gioseffo Zarlino (1558/62, 1588), Francisco Salinas (1577) and Vincenzo Galilei (1581). Just intonation was usually justified by the consonance of the major thirds and by referring to Ptolemy's superparticular tetrachord divisions.
The use of concentric rings to visualize hexachordal tone systems has a very early forerunner in Theinred of Dover (Ms. Bodley 842, midtwelfth or early fourteenth century), where four hexachords are represented on a circle that closes after two octaves, see Figure 2b 17 Theinred's visualization was probably not known to Descartes, and almost certainly not to Newton. Descartes, however, was at least superficially familiar with Zarlino's theories from his college education at Flèche 18. Zarlino's Sopplimenti musicali contain a diagram, which could have been a source of inspiration for one of Descartes's circular pitch diagrams, Zarlino (1588, Libro Ottavo p. 296). Newton in 1665 certainly knew either the English or more likely the Latin edition of Descartes's Compendium musicae, when he investigated musical scales.
Newton's constructions shown in Figures 2c and 2d hold five sevennote scales (utremifasollafa) instead of the three Cartesian hexachords (utremifasolla) with ut = FCGDA and four ambiguous notes D, A, E and B. The hexachords of Figure 2c are equal to Descartes's (tTSTt) whereas in Figure 2d they are of the format TtSTt with a major tone T between ut and re. The numbers around the circle refer to 53EDO and in Figure 2d also to 120EDO (semitones to 1 d.p.). The seventh grades fa, the minor sevenths with respect to ut, are indicated but not labeled in Figure 2c. Together Newton's five sevennote scales cover eleven out of twelve chromatic pitches (without G♯/A♭).
Syntonic chromatic scales with 12 pitches per octave
In his college notebook, Newton expressed the sizes of the syntonic intervals by 12tempered semitones with an accuracy of 0.0001 cent (see Figure 8) 19. There are various ways to select the pitches of a 12note chromatic scale within L_{5}. The pitch sets from the Figures 2c and 2d are inconsistent with Figures 3 and 8, because the transpositions of the syntonic hexachord/scale by multiples of Pythagorean fifths generate configurations which are contained in two rows of fifths instead of three rows as in Figure 3.
Whereas Pythagorean chromatic scales have only two kinds of semitones, syntonic chromatic scales are composed of three or four differently sized semitones (see Figures 4 and 5). Newton's chromatic scale is equivalent to a scale proposed by Marin Mersenne (1636), and it has three kinds of semitones: seven diatonic semitones a of the ratio 16 : 15, three greater chromatic semitones b of the ratio 135 : 128 and two lesser chromatic semitones c of the ratio 25 : 24, producing the semitone pattern "abacabaacaba" with reference G, see Figures 3a/3b. It also contains a Pythagorean third FA (81 : 64) defined through four Pythagorean fifths 20 so that the major triad FAC has a "mistuned" third compared with the perfect major triads on C, G and D of the G major scale. Holder's scale has five lesser chromatic semitones (25 : 24) but no greater chromatic semitones. Whereas the scales by Newton and Holder have only three different semitones, the solutions by Mercator and Euler require all four different semitones shown in Fig. 4. Euler's selection is equivalent to a configuration proposed by Athanasius Kircher (1650), and Newton's selection to a configuration by Marin Mersenne (1636). The earliest syntonic scale over three generations of fifths with precisely 12 pitch classes is by Kepler (1619). 21
612EDO
In his notebook, Newton examined the divisions of the octave into 15, 19, 20, 24, 25, 29, 36, 41, 51, 53, 59, 100, 120 and 612 equal parts in order to approximate the syntonic chromatic scale, see Figures 9a to 9d. How did he arrive at this particular selection of octave divisions, and how accurate are they? The most striking number is certainly 612, which results in a microtonal unit of little less than 2 cent. If a sound of 440 Hz 22 is increased by this tiny interval, its frequency increases by 0.5 Hz only. The algebraic key to this specific division of the octave can be found in Figure 6a.
The "numbers" in Figure 6a/6b indicate sizes of the syntonic intervals with respect to G in terms of 12EDO semitones plus/minus unit fractions of 12EDO whole tones. The fractions not highlighted red or blue in the transcription permit the calculation of the 612EDO representation of the pitch set GA♭ACDFF♯G containing optimum approximations of the Pythagorean sequence FCGDA as well as perfect minor and major triads FA♭C and DF♯A (spanning three rows of fifths), so that the overall quality of the approximation within the region of Newton's scale is made sure. Comparing the factorizations of the neighbored numbers 612, 615 and 611 with the denominators in Figure 6b (612 = 12 · 51 = 36 · 17 , 615 = 41 · 15 and 611 = 47 · 13) helps to understand why Newton preferred 612EDO having a Pythagorean comma equivalent of 12 units, see also Appendix_1 (http://hdl.handle.net/1811/92832). The three groups of pitches defined by the denominators in Figure 6b are highlighted in Figure 7 with the same colors. The denominators are symmetrically arranged about the tritonus/diminished fifth in the middle between C and D. The denominator 41 makes Newton's scale slightly asymmetrical. The number 6 instead of 6  $\frac{1}{\mathrm{41}}$ would have meant perfect symmetry, but tritonus and diminished fifth cannot be equal in just intonation. 24 Making Newton's chromatic scale mirror symmetric in limit5 tuning would require two different pitches C♯ and D♭ (6 ± $\frac{1}{\mathrm{41}}$), i.e., 13 pitches per octave. Newton addresses this point by giving two opposite scales in Figure 6a.
The intervals of Newton's chromatic scale in 612EDO are shown in Table 3. Among the equal divisions of the octave looked at by Newton the division into 612 parts provides the most accurate approximation of the chromatic scale, not primarily because 612EDO has the largest number of parts, but because the deviations of the best fit measured in unit intervals are the smallest. For example, 720EDO is less accurate than 612EDO, although it has a smaller unit interval 25.
G  A♭  A  B♭  B  C  C♯  D  E♭  E  F  F♯  G 

0  57  104  161  197  254  301  358  415  451  508  555  612 
0  56.983  103.994  160.977  197.020  254.003  301.014  357.997  414.980  451.023  508.006  555.017  612 
Note: The deviations to the decimal values (3 d.p.) in the last row, as calculated with logarithms, have a maximum of only 0.023 for B♭ and E.
Since Newton measured the deviations in unit fractions of 6EDO units (whole tones) the denominators of the obtained rationalizations are necessarily multiples of 6. Newton searched for good approximations with fewer parts than 612, and he probably wanted to make them independent of the 12 equally spaced semitones, which he used to measure the syntonic chromatic scale. He could achieve this by examining equal divisions of the octave with unit intervals which are multiples of the unit of 612EDO like 306EDO, 204EDO or 51EDO determined by the divisors of 612. The 51EDO unit is in the range of the pitch resolution of the human ear for pairs of successive tones and about the size of the Pythagorean comma. Newton's numbers for the chromatic scale in 51EDO are on top of his list, shown in Figure 9b.
Why at all should one study tone systems with so many pitches per octave, if the smallest steps are imperceptibly small? From the end of the 15^{th} century to the early 20^{th} century many different limit5 chromatic and enharmonic systems with up to 53 pitches per octave have been proposed. Among the enharmonic scales given by Athanasius Kircher (1650) there is a scale with 38 pitches per octave, and Arthur von Oettingen (1917) depicts a selection of 53 limit5 pitches. One reason for the study of fine grained "enharmonic" syntonic scales in the 16^{th} century was the wish to represent the Greek enharmonic tetrachords with their "quarter tones" in limit5, see Zarlino (1562, 162) 26.
The analysis of the intervals in historical enharmonic scales with 22 or more pitch classes shows that they are much better represented in 118EDO than in 53EDO. Since 612EDO is the coarsest tuning that accounts accurately and consistently for two differently sized commas – the Pythagorean and the syntonic with 12 and 11 units – it is the best base to study tunings which correspond to large coherent regions of the syntonic pitch grid. If the distinction between the Pythagorean and syntonic commas is irrelevant, 53EDO with "Mercator comma"units turns out to be the most convenient finite tuning for Pythagorean and syntonic scales. And two successive sounds distant by a Mercator comma are perceptibly different in pitch.To put it differently, whereas lookup tables for exponential functions and logarithms used in astronomy have many thousand entries to cover the range between 1 and 10, musical lookup tables require only 53, 118 or 612 entries to enable their user to calculate the ratios of arbitrary limit 5intervals within an octave with high precision. Jost Bürgi's exponential table (Bürgi 1620) has the base 1.0001 = 10^{1/23027.002…} = 2^{1/6931.818…} so that it requires 6,932 entries to cover the range between 1 and 2 of a musical octave. Newton needed only the values of 12 base 10 logarithms, when he determined the optimum bases for musical logarithms and he did not even need numerical values for their bases, 2^{$\frac{1}{\mathrm{53}}$} ≈ 1.01316 and 2^{$\frac{1}{\mathrm{612}}$} ≈ 1.00113.
With similar intentions Aristoxenous divided the whole tone into sixty parts in order to compare the many varieties of the Greek diatonic, chromatic and enharmonic tetrachords suggested by different authors. The corresponding metasystem is 360EDO, if the tone is interpreted as the sixth part of an octave as Vincenzo Galilei assumed (Galilei 1581, 58), so that historically Newton is not completely isolated with his finegrained division of the octave.
51EDO and 53EDO
From the 612EDO representation of the chromatic scale Newton could determine with fewer parts without extensive calculations. Taking the accuracy of 612EDO for granted, it can be used to find coarser good equal divisions of the octave by proportionally reducing the 612EDO representations. It follows from the previous paragraph that the irrational number 2^{1/612}, the frequency ratio of the 612EDO unit, can serve as the common ratio of a geometric series which approximates Newton's syntonic scale more accurately than any other equal division of the octave 27. An idea for the varying quality of the approximations can be obtained by comparing the decimal parts of the numbers in the bottom row of Table 1 with the corresponding decimal parts in the bottom lines (labeled 'direct') of the following Tables 4 and 5 28.
For finding the interval sizes of the 51EDO scale one is tempted to divide the numbers in the 612EDO scale by 12 and round the results to the next integer, see the rows labeled "Newton" and "../12" in Table 4. Thereby, a simple rescaling is performed in order to calculate the pitches in 51EDO from the 612EDO pitches.
Newton's values for 51EDO differ at two places from these rescaled values. This is not to be considered a mistake, since with Newton's values all the diatonic semitones (GA♭, AB♭, BC, C♯D, DE♭, EF, F♯G) measure 5 units, the greater chromatic semitones (A♭A, CC♯, FF♯) measure 4 and the lesser chromatic semitones (B♭B, E♭E) measure 2 units each. In other words, the two adjustments create a consistent scale, in which equal syntonic intervals are rendered with equal numbers of 51EDO units: 7 ⋅ 5 + 3 ⋅ 4 + 2 ⋅ 2 = 51.
The major tones in Newton's 51EDO scale measure 9 units so that six of them are 3 units larger than an octave. This difference, the 51EDO equivalent of the Pythagorean comma, is three times as big as the Pythagorean comma 29, so that Newton's 51EDO representation must be considered a very weak approximation to the syntonic chromatic scale.
G  A♭  A  B♭  B  C  C♯  D  E♭  E  F  F♯  G  

Newton  0  57  104  161  197  254  301  358  415  451  508  555  612 
../12  0  4.8  8.7  13.4  16.4  21.2  25.1  29.8  34.6  37.6  42.3  46.3  51 
rounded  0  5  9  13  16  21  25  30  35  38  42  46  51 
Newton  0  5  9  14  16  21  25  30  35  37  42  46  51 
direct  0  4.75  8.67  13.42  16.42  21.17  25.09  29.83  34.58  37.58  42.33  46.25  51 
Note: The pitch reference is G = 0 so that the size of the octave is either 612 or 51 units. Newton's 51EDO representation differs at two places from simple rescaling. The last row in this table shows that rescaling from 612EDO and direct logarithmic calculation result in the same integer approximations.
If instead only the 612EDO sizes of the three semitones 57, 47, 36 are divided by 12 and rounded to the next integer, ($\frac{\mathrm{57}}{\mathrm{12}}$,$\frac{\mathrm{47}}{\mathrm{12}}$,$\frac{\mathrm{36}}{\mathrm{12}}$) ≈ (4.8,3.9,3) ≈ (5,4,3), the division of the octave into 53 equal parts is obtained in a natural way because of 7 ⋅ 5 + 3 ⋅ 4 + 2 ⋅ 3 = 53 = 6 ⋅ 9  1. In this partition of the octave both the syntonic and the Pythagorean comma measure one unit. The 53EDO values of the chromatic scale can also be determined correctly by rescaling the 612EDO values with $\frac{\mathrm{53}}{\mathrm{612}}$ and rounding the results to the closest integer, see Table 5. The last row of the same table confirms that proportional rescaling gives the same numbers as the direct logarithmic computation. Furthermore, the maximum deviation in the best fit for 53EDO is only 0.068 units (compared with 0.418 in 51EDO).
G  A♭  A  B♭  B  C  C♯  D  E♭  E  F  F♯  G  

Newton  0  57  104  161  197  254  301  358  415  451  508  555  612 
..*(53/612)  0  4.9  9.0  13.9  17.1  22.0  26.1  31.0  35.9  39.1  44  48.1  53 
Newton  0  5  9  14  17  22  26  31  36  39  44  48  53 
direct  0  4.93  9.01  13.94  17.06  21.997  26.07  31.003  35.94  39.06  43.99  48.07  53 
41EDO and 29EDO
The representation of the chromatic scale in 41EDO and 29EDO can be directly obtained from the values of 53EDO by reducing the semitone sizes evenly, see Table 6. The last row shows that the same procedure cannot be continued down to 17EDO, since the related lesser chromatic semitones would have size zero.
G  A♭  A  B♭  B  C  C♯  D  E♭  E  F  F♯  G  

53EDO  0  5  9  14  17  22  26  31  36  39  44  48  53 
0  1  2  3  4  5  6  7  8  9  10  11  12  
41EDO  0  4  7  11  13  17  20  24  28  30  34  37  41 
0  1  2  3  4  5  6  7  8  9  10  11  12  
29EDO  0  3  5  8  9  12  14  17  20  21  24  26  29 
0  1  2  3  4  5  6  7  8  9  10  11  12  
17EDO  0  2  3  5  5  7  8  10  12  12  14  15  17 
Note: The semitone size numbers of 53EDO are reduced by 1 or 2 units to obtain their values in 41EDO and 29EDO. A like reduction by 3 units to obtain 17EDO would result in chromatic semitones B♭B and E♭E of size 0.
This shows that there is no consistent scale with less than 29 units which distinguishes between the three kinds of semitones and preserves their natural order ($\frac{\mathrm{25}}{\mathrm{24}}$<$\frac{\mathrm{135}}{\mathrm{128}}$<$\frac{\mathrm{16}}{\mathrm{15}}$) at the same time. Newton did not consider 65EDO, 77EDO and 89EDO, although they can be derived from 53EDO in the same way as 41EDO and 29EDO, probably because they divide the octave into more parts and give less accurate approximations than 53EDO.
Overview and classification of Newton's EDOs
The results of my analysis of the remaining scales studied by Newton are summarized in the following. Newton examined only divisions of the octave which assign fixed sizes to the three different semitones. Their sizes and the resulting minor and major tones are listed in Table 7.
nEDO  612  51  53  41  29  36  60  120  120  100  100  25  20  15  19  24 

25 : 24 (c)  36  2  3  2  1  1  3  6  8  3  1  1  2  1  3  0 
135 : 128 (b)  47  4  4  3  2  2  4  8  9  8  7  3  3  1  2  1 
16 : 15 (a)  57  5  5  4  3  4  6  12  11  10  11  2  1  2  1  3 
10 : 9 (t)  93  7  8  6  4  5  9  18  19  13  12  3  3  1  4  3 
9 : 8 (T)  104  9  9  7  5  6  10  20  20  18  18  5  4  3  3  4 
Note: The numbers highlighted in the EDOs with n = 25 or less indicate inconsistencies within the syntonic scale to be approximated.
Newton's 17 different EDOs can be classified as follows:
 612, 51, 53, 41, 29: derivation from 612EDO and 53EDO
 59: optimization of the major third or indirect derivation from 120EDO or 53EDO
 24, 36, 60, 120: refinement of 12EDO / 120: decimal division of the 12EDO semitone
 100, 25, 20: decimal division of the octave
 15, 19, 20, 24, 25: small numbers (relative semitone sizes not respected)
Only, 29EDO, 41EDO, 53EDO and 612EDO, agree fully with the isolated rounding, which shows that Newton must have determined the representations of the scale by assigning fixed sizes to three independent intervals, e.g., to the three different semitones – at various places in the notebook he used the abbreviations a, b and c for the three semitones and r, s, t for the major tone, the minor tone and the diatonic semitone. Also, the latter three intervals determine the corresponding approximation completely.
Newton did not consider the division into 118 equal parts, which is the closest approximation of his syntonic chromatic scale for octave division between 53EDO and 612EDO. He could have found this division, if he had investigated the rejected divisions into 59 parts or the division into 120 parts in greater detail. And he did not consider the accurate and consistent divisions into 65 (= 53 + 1 ⋅ 12), 77 (= 53 + 2 ⋅ 12) and 89 (= 53 + 3 ⋅ 12) parts, perhaps because they use more parts than 53EDO and are less accurate.
Without logarithms the divisions into 53 equal parts can be found by comparing the Pythagorean comma to the Pythagorean semitone, which are nearly in the size ratio 1 : 4. This approximate ratio was known by Boethius (1867, 293295: Inst. Mus. III, 14) so that the whole tone, consisting of two Pythagorean semitones ("limmae") and a Pythagorean comma gives 9 units, and the octave – six tones minus a comma – measures6 ⋅ 9  1 = 53 units, with a unit interval close to the Pythagorean comma. In fact, the octave measures 55.8 syntonic commas or 51.15 Pythagorean commas, so that Mercator's "artificial comma" is a good average between the two commas, being only marginally greater in magnitude (.475) than the true average'.This explains the accurate Pythagorean and syntonic intervals in 53EDO.
The division of the octave into 53 equal parts was studied by Nicolaus Mercator at about the same time as Newton or even earlier 30. As Benjamin Wardhaugh (personal communication, 2020) points out, it is not very likely that the two men exchanged ideas about tuning, since Newton at this time was still a student with no public profile. We do not know whether Newton, at the age of 22, was familiar with the aforementioned comparison of the Pythagorean comma to the semitone by Boethius 31.
COMPUTATIONAL ANALYSIS OF ARBITRARY EDOAPPROXIMATIONS
For the computer assisted analysis described in this section, I have created small Java programs with Processing. They generated lists of scales and key figures which I examined with spread sheets. One of these programs determined and analyzed the nEDO approximations of limit5 diatonic and chromatic scales for n < 2000, see Appendix_2 (http://hdl.handle.net/1811/92832). This analysis of the least square deviations can be related to Newton's octave divisions.
For this purpose, the diatonic and chromatic scales of just intonation are expressed with ratios to the reference notes C resp. G as follows. The syntonic diatonic scale (limit5 diatonic scale) has two alternative pitches for the second degree D, so that both the major tone and the minor tone above the key C are present:
C  D  D  E  F  G  A  B  C 
1  $\frac{\mathrm{10}}{9}$  $\frac{9}{8}$  $\frac{5}{4}$  $\frac{4}{3}$  $\frac{3}{2}$  $\frac{5}{3}$  $\frac{\mathrm{15}}{8}$  2 
The two varieties of D differ by a syntonic comma c (81 : 80). This selection is in accordance with Descartes's definition of the diatonic scale, with the symmetric pattern tctSTtTS with which Newton was familiar. The (syntonic) chromatic scale is taken from Newton's notebook (Figure 8) and defined by:
G  A♭  A  B♭  B  C  C♯  D  E♭  E  F  F♯  G 
1  $\frac{\mathrm{16}}{\mathrm{15}}$  $\frac{9}{8}$  $\frac{6}{5}$  $\frac{5}{4}$  $\frac{4}{3}$  $\frac{\mathrm{45}}{\mathrm{32}}$  $\frac{3}{2}$  $\frac{8}{5}$  $\frac{5}{3}$  $\frac{\mathrm{16}}{9}$  $\frac{\mathrm{15}}{8}$  2 
For a given value of n (i.e., for the division of the octave into n equal intervals), the program calculates the sizes of all the intervals $\frac{p}{q}$ with respect to the reference note (C resp. G) with the formula n ⋅ log_{2}$\frac{p}{q}$ and rounds them to the next integer. In the following this procedure is called isolated (logarithmic) rounding, if all the intervals of the diatonic or chromatic scale are determined in this way, see Appendix_3 (http://hdl.handle.net/1811/92832). The column headed dev in the Tables 8 and 9 contain the root mean square (rms) deviations of n ⋅ log_{2}$\frac{p}{q}$ from the rounded values. This deviation is thus expressed in units of the corresponding nEDO approximation. The last column dev_c contains the rmsdeviation expressed in cent. The cent values naturally converge against 0 for increasing values of n, whereas the deviations with respect to the size of the unit intervals do not. In the following the latter is simply called deviation, if the context is clear, and the root mean square deviation measured in cent, is called centdeviation.
n  C  D  D  E  F  G  A  B  C  dev  dev_c 

12  0  2  2  4  5  7  9  11  12  0.1135  11.35 
53  0  8  9  17  22  31  39  48  53  0.0460  1.04 
118  0  18  20  38  49  69  87  107  118  0.0373  0.38 
559  0  85  95  180  232  327  412  507  559  0.0306  0.07 
612  0  93  104  197  254  358  451  555  612  0.0166  0.03 
1783  0  271  303  574  740  1043  1314  1617  1783  0.0152  0.01 
2513  0  382  427  809  1043  1470  1852  2279  2513  0.0134  0.006 
4296  0  653  730  1383  1783  2513  3166  3896  4296  0.0030  0.0008 
Note: The values in column dev form a monotonically decreasing sequence if indexed by n. There is no longer subset of indices n from 12 to 5000 with this property.
n  G  A♭  A  B♭  B  C  C♯  D  E♭  E  F  F♯  G  dev  dev_c 

12  0  1  2  3  4  5  6  7  8  9  10  11  12  0.1076  10.76 
53  0  5  9  14  17  22  26  31  36  39  44  48  53  0.0504  1.14 
118  0  11  20  31  38  49  58  69  80  87  98  107  118  0.0325  0.33 
612  0  57  104  161  197  254  301  358  415  451  508  555  612  0.0157  0.03 
1783  0  166  303  469  574  740  877  1043  1209  1314  1480  1617  1783  0.0156  0.01 
2513  0  234  427  661  809  1043  1236  1470  1704  1852  2086  2279  2513  0.0151  0.007 
3684  0  343  626  969  1186  1529  1812  2155  2498  2715  3058  3341  3684  0.0134  0.004 
4296  0  400  730  1130  1383  1783  2113  2513  2913  3166  3566  3896  4296  0.0024  0.0007 
559  0  52  95  147  180  232  275  327  379  412  464  507  559  0.0358  0.08 
Note: The record n = 559 at the bottom is found by the optimization of the diatonic scale (Table 8), but not in the optimization of the chromatic scale (Table 9).
The Tables 8 and 9 list optimal approximations with respect to the rmsdeviation (dev) in the following sense. For all integers k less than n the deviation in kEDO is greater than the deviation in nEDO. For example, the smallest number for which the deviations of the chromatic and the diatonic scale are smaller than the deviations for 12 is 53. And between 53EDO and 118EDO there is no equal division kEDO with a deviation smaller than the deviation for 53EDO. This procedure produces maximal monotonically decreasing sequences of deviations.
The rankings created by this optimization algorithm are very similar across all sets of syntonic pitches tested, which comprised the full chromatic and diatonic scales, the major triad and the combination of the diatonic semitone with the major and minor tones 32. This is not totally unexpected, since the syntonic system (limit5 tuning) is generated by Pythagorean fifths and syntonic major thirds (modulo octave). The deviations for the diatonic scale, the chromatic scale as well as major thirds and fifths (the perfect major triad) for n = 12 to 133 are visualized in Figure 10.
Logcompatibility and consistency
Newton's syntonic chromatic scale has three different semitones, the diatonic semitone (16 : 15 of 111.73 cent), the greater chromatic semitone (135 : 128 of 92.18 cent) and the lesser chromatic semitone (25 : 24 of 70.67 cent). His scale with reference note G has seven diatonic semitones a, three greater chromatic semitones b and two lesser chromatic semitones c, and the octave is filled by the semitonepattern abacabaacaba 33.
In the following, a chromatic nEDO scale is called consistent, if the semitones of the same kind are rendered as equal intervals, and it is called strongly consistent, if it is consistent and the condition 2c > a > b > c is satisfied, so that the relative size order of the three semitones is respected and the lesser chromatic semitone c is greater than half a diatonic semitone a 34. Scales agreeing with the directly rounded logarithms are called logcompatible. The smallest positive integers (a, b, c) with 2c > a > b are (5, 4, 3). With these values for a, b and c the 7 diatonic, the 3 greater and the 2 lesser chromatic semitones add up to an octave of 53 units (53 = 7 ⋅ 5 + 3 ⋅ 4 + 2 ⋅ 3). Since the isolated roundings of Newton's syntonic chromatic scale agree with the consistent 53EDO representation, 53EDO provides the coarsest strongly consistent and logcompatible chromatic scale.
In order to classify the chromatic scales for n up to 1200 "consistency levels" as defined in Table 10 are used. Hereby, level 0 is equivalent with strong consistency. These definitions quantify the similarity of interval size relationships between the approximating EDOscales and Newton's syntonic chromatic scale. The restrictions become gradually weaker for increasing consistency levels. True syntonic chromatic scales would have consistency level 0, their approximation through 12EDO have consistency level 3, because 12EDO does not distinguish between different semitones and has a = b = c.
cnslevel  condition 

0  2c > a > b > c 
1  a > b > c and 2c ≤ a 
2  a > b = c 
3  a = b ≥ c 
4  Not a ≥ b ≥ c 
Note: The variables a, b and c are the sizes of the diatonic, the greater chromatic and the lesser chromatic semitones.
For n ≤ 1200 there are 374 divisions of the octave into n equal intervals with logcompatible chromatic scales. The lowest deviations for scales with consistency levels > 0 occurs at 12EDO 35, 34EDO and 41EDO. The maximum deviation occurs at 306EDO (306 = $\frac{1}{2}$ ⋅ 612). 53EDO and 65EDO are the only divisions of the octave into less than 100 equal parts with a relative deviation lower than that of 12EDO.
53, 118 AND 612 – A GENERATOR SYSTEM AND ITS GEOMETRIC INTERPRETATION
A close inspection of the numbers n with small deviations between the best EDO fit and the syntonic chromatic scale reveals that they are in a close relationship to 612, 118 and 53, the numbers with the smallest relative deviations. The deviations for 53EDO, 118EDO and 612EDO are visualized in Figure 11. The analysis of the data explored in this section shows that all the numbers n up to 1200 with a logcompatible syntonic nEDO scale can be written in the format n = 612x + 118y + 53z, where x, y and z are integers close to 0. The number n of the 32 scales with the lowest deviations can be written in this format with 0 ≤ x ≤ 2, 2 ≤ y ≤ 2 and 2 ≤ z ≤ 1.
Therefore 53EDO, 118EDO and 612EDO form a kind of generator system for logcompatible consistent scales. More precisely, the correct sizes of the nEDO intervals can be linearly combined from the corresponding sizes in 53EDO, 118EDO and 612EDO for all logcompatible scales with n ≤ 1200.
If a given n can be written as n = 612x + 118y + 53z, where 1 ≤ x ≤ 3, 9 ≤ y ≤ 10 and 8 ≤ z ≤ 7, the nEDO sizes of the diatonic semitone a and the chromatic semitones b and c can be calculated with the formulas a = x ⋅ 57 + y ⋅ 11 + z ⋅ 5, b = x ⋅ 47 + y ⋅ 9 + z ⋅ 4 and c = x ⋅ 36 + y ⋅ 7 + z ⋅ 3 so that the sizes of the three semitones a, b and c are uniquely determined and hence the entire chromatic scale. In other words, the logcompatible nEDO representation (for arbitrary values of n ≤ 1200) of the syntonic chromatic scale can be determined without using logarithms – just on base of the values of 53EDO, 118EDO and 612EDO. The main task for a given n is to express it as a suitable linear combination of 612, 118 and 53. Parts of the results of the corresponding calculations are shown in Table 11.
The following example uses n = 935 and shows how the nEDO representations of Newton's chromatic scale can be calculated by hand from the first four columns of Table 11: Because 935 can be written as 935 = 2 ⋅ 612  2 ⋅ 118  1 ⋅ 53 the coordinates for 935EDO (dev rank 21) are (x,y,z) = (2,2,1) By substituting them into the formulas
 a = x ⋅ 57 + y ⋅ 11 + z ⋅ 5
 b = x ⋅ 47 + y ⋅ 9 + z ⋅ 4
 c = x ⋅ 36 + y ⋅ 7 + z ⋅ 3,
the sizes of the 935EDO semitones can be obtained as follows:
 a = 2 ⋅ 57 + (2) ⋅ 11 + (1) ⋅ 5 = 87
 b = 2 ⋅ 47 + (2) ⋅ 9 + (1) ⋅ 4 = 72
 c = 2 ⋅ 36 + (2) ⋅ 7 + (1) ⋅ 3 = 55
With these values of a, b and c and the semitonepattern abacabaacaba the calculation of the 935EDO representation of Newton's chromatic scale can be calculated as a cumulative sum:
 a = 87
 a + b = 87 + 72 = 129
 a + b + a = 129 + 87 = 246
 a + b + a + c = 246 + 55 = 301
 …
n  x  y  z  rnk  a  b  c  scale  dev_u  cmpt  cns  dev_rnk 

612  1  0  0  1  57  47  36  0 57 104 161 197 254 301 358 415 451 508 555 612  0.0157  0  0  1 
1171  2  0  1  5  109  90  69  0 109 199 308 377 486 576 685 794 863 972 1062 1171  0.0227  0  0  2 
730  1  1  0  2  68  56  43  0 68 124 192 235 303 359 427 495 538 606 662 730  0.0306  0  0  3 
118  0  1  0  1  11  9  7  0 11 20 31 38 49 58 69 80 87 98 107 118  0.0325  0  0  4 
559  1  0  1  2  52  43  33  0 52 95 147 180 232 275 327 379 412 464 507 559  0.0358  0  0  5 
494  1  1  0  2  46  38  29  0 46 84 130 159 205 243 289 335 364 410 448 494  0.0408  0  0  6 
1053  2  1  1  6  98  81  62  0 98 179 277 339 437 518 616 714 776 874 955 1053  0.0462  0  0  7 
677  1  1  1  3  63  52  40  0 63 115 178 218 281 333 396 459 499 562 614 677  0.0463  0  0  8 
441  1  1  1  3  41  34  26  0 41 75 116 142 183 217 258 299 325 366 400 441  0.0503  0  0  9 
53  0  0  1  1  5  4  3  0 5 9 14 17 22 26 31 36 39 44 48 53  0.0504  0  0  10 
1106  2  1  0  5  103  85  65  0 103 188 291 356 459 544 647 750 815 918 1003 1106  0.0526  0  0  11 
171  0  1  1  2  16  13  10  0 16 29 45 55 71 84 100 116 126 142 155 171  0.0585  0  0  12 
848  1  2  0  5  79  65  50  0 79 144 223 273 352 417 496 575 625 704 769 848  0.0611  0  0  13 
65  0  1  1  2  6  5  4  0 6 11 17 21 27 32 38 44 48 54 59 65  0.0614  0  0  14 
236  0  2  0  4  22  18  14  0 22 40 62 76 98 116 138 160 174 196 214 236  0.0649  0  0  15 
665  1  0  1  2  62  51  39  0 62 113 175 214 276 327 389 451 490 552 603 665  0.0656  0  0  16 
783  1  1  1  3  73  60  46  0 73 133 206 252 325 385 458 531 577 650 710 783  0.0694  0  0  17 
795  1  2  1  6  74  61  47  0 74 135 209 256 330 391 465 539 586 660 721 795  0.0715  0  0  18 
1118  2  0  2  8  104  86  66  0 104 190 294 360 464 550 654 758 824 928 1014 1118  0.0717  0  0  19 
376  1  2  0  5  35  29  22  0 35 64 99 121 156 185 220 255 277 312 341 376  0.072  0  0  20 
935  2  2  1  9  87  72  55  0 87 159 246 301 388 460 547 634 689 776 848 935  0.0765  0  0  21 
323  1  2  1  6  30  25  19  0 30 55 85 104 134 159 189 219 238 268 293 323  0.0767  0  0  22 
547  1  1  1  3  51  42  32  0 51 93 144 176 227 269 320 371 403 454 496 547  0.0767  0  0  23 
289  0  2  1  5  27  22  17  0 27 49 76 93 120 142 169 196 213 240 262 289  0.0801  0  0  24 
1000  2  1  2  9  93  77  59  0 93 170 263 322 415 492 585 678 737 830 907 1000  0.0811  0  0  25 
988  2  2  0  8  92  76  58  0 92 168 260 318 410 486 578 670 728 820 896 988  0.0816  0  0  26 
183  0  2  1  5  17  14  11  0 17 31 48 59 76 90 107 124 135 152 166 183  0.0843  0  0  27 
506  1  0  2  5  47  39  30  0 47 86 133 163 210 249 296 343 373 420 459 506  0.0861  0  0  28 
901  1  2  1  6  84  69  53  0 84 153 237 290 374 443 527 611 664 748 817 901  0.0862  0  0  29 
624  1  1  2  6  58  48  37  0 58 106 164 201 259 307 365 423 460 518 566 624  0.0919  0  0  30 
388  1  1  2  6  36  30  23  0 36 66 102 125 161 191 227 263 286 322 352 388  0.0921  0  0  31 
1159  2  1  1  6  108  89  68  0 108 197 305 373 481 570 678 786 854 962 1051 1159  0.0922  0  0  32 
966  1  3  0  10  90  74  57  0 90 164 254 311 401 475 565 655 712 802 876 966  0.0929  0  0  33 
354  0  3  0  9  33  27  21  0 33 60 93 114 147 174 207 240 261 294 321 354  0.0974  0  0  34 
429  1  2  1  6  40  33  25  0 40 73 113 138 178 211 251 291 316 356 389 429  0.0979  0  0  35 
882  2  2  2  12  82  68  52  0 82 150 232 284 366 434 516 598 650 732 800 882  0.1006  0  0  36 
106  0  0  2  4  10  8  6  0 10 18 28 34 44 52 62 72 78 88 96 106  0.1009  0  0  37 
913  1  3  1  11  85  70  54  0 85 155 240 294 379 449 534 619 673 758 828 913  0.1009  0  0  38 
258  1  3  0  10  24  20  15  0 24 44 68 83 107 127 151 175 190 214 234 258  0.104  0  0  39 
224  0  1  2  5  21  17  13  0 21 38 59 72 93 110 131 152 165 186 203 224  0.1043  0  0  40 
205  1  3  1  11  19  16  12  0 19 35 54 66 85 101 120 139 151 170 186 205  0.1065  0  0  41 
407  0  3  1  10  38  31  24  0 38 69 107 131 169 200 238 276 300 338 369 407  0.1073  0  0  42 
12  0  1  2  5  1  1  1  0 1 2 3 4 5 6 7 8 9 10 11 12  0.1076  0  3  43 
742  1  2  2  9  69  57  44  0 69 126 195 239 308 365 434 503 547 616 673 742  0.1076  0  0  44 
270  1  2  2  9  25  21  16  0 25 46 71 87 112 133 158 183 199 224 245 270  0.108  0  0  45 
817  2  3  1  14  76  63  48  0 76 139 215 263 339 402 478 554 602 678 741 817  0.1081  0  0  46 
1019  1  3  1  11  95  78  60  0 95 173 268 328 423 501 596 691 751 846 924 1019  0.1103  0  0  47 
301  0  3  1  10  28  23  18  0 28 51 79 97 125 148 176 204 222 250 273 301  0.112  0  0  48 
1041  2  2  1  9  97  80  61  0 97 177 274 335 432 512 609 706 767 864 944 1041  0.1121  0  0  49 
870  2  3  0  13  81  67  51  0 81 148 229 280 361 428 509 590 641 722 789 870  0.1125  0  0  50 
718  1  0  2  5  67  55  42  0 67 122 189 231 298 353 420 487 529 596 651 718  0.1159  0  0  51 
342  0  2  2  8  32  26  20  0 32 58 90 110 142 168 200 232 252 284 310 342  0.1171  0  0  52 
836  1  1  2  6  78  64  49  0 78 142 220 269 347 411 489 567 616 694 758 836  0.1174  0  0  53 
1065  2  0  3  13  99  82  63  0 99 181 280 343 442 524 623 722 785 884 966 1065  0.1218  0  0  54 
130  0  2  2  8  12  10  8  0 12 22 34 42 54 64 76 88 96 108 118 130  0.1228  0  0  55 
600  1  1  2  6  56  46  35  0 56 102 158 193 249 295 351 407 442 498 544 600  0.1233  0  0  56 
311  1  3  1  11  29  24  18  0 29 53 82 100 129 153 182 211 229 258 282 311  0.1241  0  0  57 
1084  1  4  0  17  101  83  64  0 101 184 285 349 450 533 634 735 799 900 983 1084  0.1251  0  0  58 
1183  2  1  3  14  110  91  70  0 110 201 311 381 491 582 692 802 872 982 1073 1183  0.1252  0  0  59 
764  2  3  2  17  71  59  45  0 71 130 201 246 317 376 447 518 563 634 693 764  0.1256  0  0  60 
…  …  …  
1083  5  19  5  411  101  84  62  0 101 185 286 348 449 533 634 735 797 898 982 1083  0.734  6  0  1188 
83  3  18  7  382  8  7  3  0 8 15 23 26 34 41 49 57 60 68 75 83  0.7356  6  1  1189 
88  3  19  6  406  8  6  7  0 8 14 22 29 37 43 51 59 66 74 80 88  0.7359  6  4  1190 
1088  1  18  8  389  101  83  66  0 101 184 285 351 452 535 636 737 803 904 987 1088  0.7401  6  0  1191 
524  4  19  6  413  49  41  29  0 49 90 139 168 217 258 307 356 385 434 475 524  0.7479  6  0  1192 
695  4  18  7  389  65  54  39  0 65 119 184 223 288 342 407 472 511 576 630 695  0.7484  6  0  1193 
647  2  19  7  414  60  49  40  0 60 109 169 209 269 318 378 438 478 538 587 647  0.7508  6  0  1194 
476  2  18  8  392  44  36  30  0 44 80 124 154 198 234 278 322 352 396 432 476  0.753  6  0  1195 
1136  5  19  6  422  106  88  65  0 106 194 300 365 471 559 665 771 836 942 1030 1136  0.7599  6  0  1196 
35  3  19  7  419  3  2  4  0 3 5 8 12 15 17 20 23 27 30 32 35  0.7632  6  4  1197 
577  4  19  7  426  54  45  32  0 54 99 153 185 239 284 338 392 424 478 523 577  0.7757  6  0  1198 
594  2  19  8  429  55  45  37  0 55 100 155 192 247 292 347 402 439 494 539 594  0.7798  6  0  1199 
1189  5  19  7  435  111  92  68  0 111 203 314 382 493 585 696 807 875 986 1078 1189  0.7883  6  0  1200 
The numbers 53 and 118 are relatively prime. Therefore, every integer can be written in the format n = x ⋅ 53 + y ⋅ 118 36. It is important to note that the strong restrictions on the coefficients x, y, z are responsible for the feature that the intervals of the logcompatible consistent chromatic scales can be calculated as described. For example, if the condition on z were weakened to 8 ≤ z ≤ 8, the numbers 306 and 918 could be written in two ways as combinations of 612, 118 and 53:
306 = 118 + 8 ⋅ 53 = 612 + 118  8 ⋅ 53 and 918 = 612  118 + 8 ⋅ 53 = 2 ⋅ 612 + 118  8 ⋅ 53
Only the combinations with z = 8 result in the correct logcompatible semitones, if the formulas for a, b and c are applied. Newton's chromatic scale has a particularly weak approximation in 306EDO, although it is well suited to approximate Pythagorean tuning 37, and also 918EDO has very large deviations.
The coordinates highlighted grey in Table 11 are located in the smallest cuboid of Figure 12. The column "rnk" (rank) is calculated with x^{2} + y^{2} + z^{2}. The incompatibility index cmpt measures the total deviations from the scale defined through direct logapproximation in nEDO units. For the definition of the consistency levels cns, see Table 10. The integer triplets (x,y,z) correspond to points in a threedimensional space. It can be shown that within a cuboid grid of the dimensions 7 × 38 × 15 38 the representation of a number n in the format n = x ⋅ 612 + y ⋅ 118 + z ⋅ 53 is unique and that there are no larger cuboids with this property. The results of the geometrical analysis are summarized and visualized in Figure 12.
The incompatibility index (cmpt) measures the deviation of a given representation of the syntonic chromatic scale from the best logarithmic pitch values. It is defined as the sum of the individual absolute deviations in nEDO units, so that an index of 0 is perfect compatibility. There are 372 numbers n between 12 and 1200 with the incompatibility index 0, which means that the related scales agree with the direct logarithmic approximations, and 13 scales have the maximum logincompatibility index of 7.
There are 356 scales with n between 12 and 1200 which are strongly consistent and logcompatible at the same time. Only 16 compatible scales have consistency levels greater than 0 – none of these values is greater than 3. The corresponding octave dividers n are listed in Table 12.
cns  cmpt  n 

1  0  29, 41, 63, 82 
2  0  7, 19, 31, 43, 55 
3  0  10, 12, 22, 24, 34, 36, 46, 56, 58 
The lowest deviation 0.273 for a logincompatible scale occurs at n = 920 and the highest deviation of 0.3912 for a logcompatible scale at n = 306. In other words, the root mean square deviation does not fully separate compatible from incompatible scales. It is decisive only for deviations less than 0.273 and for deviations greater than 0.3912.
A simple ranking of the representations n = x ⋅ 512 + y ⋅ 118 + z ⋅ 53 is defined by the Euclidean norm, d = $\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$ the distance of the point (x,y,z) from the origin (0, 0, 0) and it corresponds to the square root of the column rnk in Table 11. Low values of d correlate to low deviations (dev) of the nEDO chromatic scales from the syntonic chromatic scale. The graphs in Figures 13a/13b show that the coordinates x, y, z of a number n can be used to estimate the quality of the fit of an nEDO system with the syntonic chromatic scale. However, there is no monotone relationship between Euclidean norms of these points and the corresponding root mean square deviations.
The two curves in Figure 14 show that the Euclidean norms d = $\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$ of n = x ⋅ 512 + y ⋅ 118 + z ⋅ 53 (norm: blue) and the deviations of their nEDO representation from the syntonic chromatic scale (rms_u: red) form very similar patterns. This means that the coordinates (x, y, z) explain the quasichaotic dependence on n of the deviations to a wide extent. Both graphs have pronounced minima at 12, 53, 53 ± 12 as well as at 118, 118 ± 12. Some of the strong maxima (indicating inaccurate approximations) occur at 47, 47 ± 12. Recall that 118 = 2 · 53 + 12 = 65 + 53 = 2 · 59 and that the poor 59 (rejected by Newton, see Figure 9a), is exactly in the middle between the pronounced minima at 53 and 65.
CONCLUSION
The controversies about syntonic pitch systems and their many ambiguities, as well as the presence of various tempered tunings in the 16^{th} century and later, demanded mathematical skills and powerful tools to compare the related scales and the sizes of the many different intervals efficiently. Accurate tables of logarithms were published in the early 17^{th} century. The use of equal divisions of the octave to measure pitch systems was advanced by Simon Stevin, Marin Mersenne, Nicolaus Mercator, Christiaan Huygens, Isaac Newton and others. The basic idea was to find fine grained "metascales" (lowest common denominators) which could express the scales under consideration as subsets very accurately. Newton found two of the three best metascales for limit5 chromatic scales among the equal divisions of the octave with less than 1783 parts.
It is unlikely that the division of the octave into 612 equal parts could have been found without using a logarithm table. However, the division into 53 parts, which is at the same time an optimum for the Pythagorean tuning, could have been found with the knowledge of Boethius. Jacobus Leodiensis in the 14^{th} century proposed a monochord of 53 commas only, a scale with 41 Pythagorean commas and 12 intervals a little smaller 40. My tentative reconstruction of Newton's implicit line of reasoning shows how his various chromatic EDOapproximations can be derived from the 612EDO representation of the chromatic scale with no need for further logarithms.
The reconstruction of Newton's approach to the tuning problem led to some more general mathematical considerations in the final part of this article. I have shown that the optimum divisions of the octave into 53, 118 and 612 equal parts can be used to generate a comprehensive collection of consistent nEDO scales for arbitrary divisions of the octave with up to n = 1200 equal parts. This collection contains the scales which agree with direct rounding as a subset. The interpretation of the related numbers n as points in a threedimensional space revealed an almost linear relationship between their distance from the origin and the quality of the related nEDO representation of the syntonic chromatic scale. The required calculations rely only on the knowledge of three optimum equal divisions, 53EDO, 118EDO and 612EDO – the key to the large data sets added to this text is hidden in Newton's analysis of the chromatic scale given in Figure 6a.
ACKNOWLEDGEMENTS
This text is based on research made for the project "Sound Colour Space – A Virtual Museum" (20152016) at Zurich University of the Arts, funded by the Swiss National Science Foundation (105216_156979). The author wishes to thank Martin Neukom, Benjamin Wardhaugh and Roman Oberholzer for their critical and helpful comments. Furthermore, the insightful comments and suggestions by the reviewers helped a lot to improve my contribution and find its final form. This article was copyedited by Matthew Moore and layout edited by Diana Kayser.
NOTES

Correspondence can be addressed to: Dr. Daniel Muzzulini, Bläsiring 42, CH4057 Basel, daniel@muzzulini.ch.
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Wardhaugh (2013, pp. 85113). The College Notebook (MS Add.4000), Cambridge University Library, Manuscript 225, is also published online <http://cudl.lib.cam.ac.uk/view/MSADD04000/1> [visited 20200428]
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EDO stands for Equal Division of the Octave; 12EDO is equal temperament with 12 pitches per octave.
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Comparable statements holding "to a great extent" are frequently given in perceptual psychology, where some (two or three) main factors are determined that would explain a complex phenomenon such as musical timbre by projecting experimental data to a low dimensional structure.
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A monochord is an experimental device (or musical instrument) with a single string that can be divided with moveable bridges in order to study the relationships between musical intervals.
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This is a misleading designation, because the Pythagorean chromatic and enharmonic genera and systems (as for example described and transmitted by Boethius) had interval ratios containing higher prime numbers. In the course of the history they disappeared and the medieval modes were usually constructed on base of proper diatonic tetrachords and hexachords only.
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The full Greek systems (systema teleion) have two different octaves; only the upper octave contains both B♭ and B.
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The combinations lead to just minor thirds (6 : 5) and two different whole tone steps (9 : 8 and 10 : 9).
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Vincenzo Galilei and Gioseffo Zarlino in the 16^{th} century used the term "Sintono di Tolomei" for the just intonation diatonic scale. Limit5 pitches of the same name differ by syntonic commas, which are a little smaller than Pythagorean commas.
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One of these grids uses three rows of fifths, see Wardhaugh (2013, p. 161, p. 227)
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The use of musical number triangles in historical sources is discussed in Muzzulini (2015, 212215)
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The continued fractions expansions are independent of the base of the underlying number system. Irrational numbers correspond with infinite continued fractions (Euclid's algorithm does not come to an end for irrational ratios), cf. Schechter (1980). The convergence of the continued fractions approximation to a given irrational number is comparably slow.
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It follows from Proposition 3 of the Euclidean "Sectio canonis" (Barker Ed. 1989, p. 195; Lindley & TurnerSmith 1993, pp. 228229).
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Continued fractions were used in music theoretical writings of the 18^{th} and 19^{th} centuries Lambert (1774, 59) and Drobisch (1852, 76), see also Carey, N., & Clampitt, D. (1989, 197198).
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See Barbour (1951), Žabka (2010; 2013), Muzzulini & Vogtenhuber (2016)
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Wardhaugh (2013, pp. 184), Descartes (1987, 47)
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I became aware of this diagram through a personal communication by Susan F. Weiss (1999). See also Snyder (1983, 1986).
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Gaukroger (1995, pp. 5559). But Descartes did not treat meantone temperaments in his "Compendium musicae" or the tuning of lutes.
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The cent is a modern unit of interval size, so that the 12edo semitone measures 100 cent and the octave therefore 1200 cent. To obtain cent values from Newton's values multiply his numbers by 100.
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There are 118 classes of chromatic scales that can be written as uninterrupted chains of Pythagorean fifths (3 : 2) and small fifths (40 : 27). See also Žabka (2013). The Pythagorean chromatic 12note scale has just two different semitones (256 : 243 and 2187 : 2048). By convention, the note names of chromatic semitones begin with the same letter and differ by a simple alteration, such as CC♯, D♭D or F♯F♯♯, whereas the notes EF, C♯D, CDd or F♯♯G♯ form diatonic semitones. In syntonic chromatic scales the chromatic and/or diatonic semitones can occur in different varieties.
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See Barbour (1951), Muzzulini, D. & Vogtenhuber (2016)
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Frequency is measured in Hertz, i.e., full periods per second. Mark Lindley looked at Newton's octave divisions more than 30 years ago and claimed the Newton favored 53EDO over all other divisions (Lindley 1987, pp. 206210).
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The curves are explained in Wardhaugh (2013, 9697).
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If they were the same, they would be equal to 6 semitones of the 12tempered tuning.
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The distinction between deviations with respect to the unit and absolute deviations (in cent) is crucial for the notion of quality of best fit used in this article.
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See http://soundcolourspace.zhdk.ch/diagrams/39
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It is even optimal for EDOs with a unit interval greater than 0.67 cent. The coarsest better approximation is 1783EDO, see Table 7.
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Later in this text the deviations are formally measured as root mean square deviations.
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According to Boethius, the interval of the Pythagorean comma is defined as six whole tones minus one octave. It has the ratio 531,441 : 524,228. Boethius (1867, 264267: Inst. Mus. II, 31)
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Mercator's music theoretical manuscripts are published in Wardhaugh (2013, 129236).
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Newton knew an interpretation of the Greek tetrachord's genera with syntonic intervals (Ptolemy, Zarlino) not looked at by Descartes.
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See Appendix_5 (http://hdl.handle.net/1811/92832) for detailed data about this point.
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Newton used the same abbreviations for the three semitones of his scale.
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If the condition 2c > a is violated, the lesser chromatic semitone is more like a quartertone than a semitone.
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Although 12EDO does not distinguish between different semitones at all, it is a consistent approximation to Newton's syntonic chromatic scales.
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This is a general property for pairs of relatively prime numbers, so that even 612 could be obtained from 53 and 118 only:
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Notice that 306EDO is related to 612EDO as 59EDO to 118EDO. Both 306EDO and 59EDO give a "very poor" chromatic scale where some pitches are to be rounded by almost half an EDO unit. Newton did not give the full representation for 59EDO, and he did not look at all at 306EDO.
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The dimensions are given as side lengths of the cuboid not as numbers of grid points.
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The two graphs are differently scaled in the ydirection.
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Jacobus Leodiensis (ca. 1260 – after 1330), suggests and depicts a monochord with 53 microintervals per octave: 41 Pythagorean commas and 12 intervals a little smaller than the Pythagorean comma (Pythagorean semitones minus three Pythagorean commas), cf. Smits van Waesberghe (ed), (1988, p. 106, p. 139, f. 51r)
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