Incommensurable intervals: Euclid's algorithm and continued fractions

The mathematical reason for the quasi-chaotic 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:

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 so-called 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 by-product. 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

From syntonic hexachords to chromatic scales

Guido of Arezzo's Micrologus (c. 1025) constructs a tone system with seven overlapping hexachords "ut-re-mi-fa-so-la", each of them of the same symmetric interval structure T-T-s-T-T, 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 B-flat in the lowest octave, but B-flat 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 limit-3 diatonic system of eight pitch classes, C-D-E-F-G-A-B♭-B, if the pitches are reduced to one octave. The pitches can be arranged as a stack of fourths B-E-A-D-G-C-F-B♭ with the chromatic semitone B♭-B of the ratio 2187 : 2048, which is approximately 25% larger than the diatonic semitones E-F and B♭-B of the ratio 256 : 243.

Table 1. Pythagorean hexachords and octachords related to Guido of Arezzo's tone system.

	Pythagorean hexachord	Pythagorean octachords
Interval structure	T-T-s-T-T	T-T-s-T-T-s-T T-T-s-T-T-T-s
Ratio structure

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 (1596-1650) 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 "ut-re-mi-fa-so-la" (mollis, naturalis and duris), whose references are Pythagorean fifths apart as in Guido's system, but whose symmetric interval configuration t-T-S-T-t is composed of superparticular limit-5 ratios with major tones T (9 : 8), minor tones t (10 : 9) and diatonic semitones S (16 : 15) in place of Guido's limit-3 intervals. Descartes's syntonic hexachord and its completions to octachords are shown in Table 2.

Table 2. Syntonic hexachords and octachords.

	Symmetric syntonic hexachord	Syntonic octachords
Interval structure	t-T-S-T-t	t-T-S-T-t-S-T t-T-S-T-t-T-S
Ratio structure

Descartes' re-interpretation of Guido of Arezzo's Pythagorean hexachords results in a diatonic just intonation system with both varieties of B (B-duris and B-mollis) 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, mid-twelfth 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 seven-note scales (ut-re-mi-fa-sol-la-fa) instead of the three Cartesian hexachords (ut-re-mi-fa-sol-la) with ut = F-C-G-D-A and four ambiguous notes D, A, E and B. The hexachords of Figure 2c are equal to Descartes's (t-T-S-T-t) whereas in Figure 2d they are of the format T-t-S-T-t with a major tone T between ut and re. The numbers around the circle refer to 53-EDO and in Figure 2d also to 120-EDO (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 seven-note scales cover eleven out of twelve chromatic pitches (without G♯/A♭).

Figure 2a. Descartes' just intonation hexachords (ut-re-mi-fa-sol-la) with ut = F, C and G result in two ambiguous notes, G and D, differing by syntonic commas (324 : 320= 80 : 81 = 486 : 480). (Descartes 1650, 35)

Figure 2b. Theinred of Dover's hexachords on a two-octave circle with four hexachords on ut = B♭, F, C, G together with an interpretation in terms of a 12 semitone note frame without F♯. The tuning is not specified, so that the diagram is compatible with Pythagorean as well as 12-EDO tuning, depending on the interpretation of the Tonus and Semitonium. (MS. Bodley 842, fol. 80v)

Figure 2c.

Figure 2d.

Figures 2c/2d. Newton's generalization of Descartes's hexachords (Newton 1665, fol. 110r, fol. 109r, MS Add. 4000: Reproduced by kind permission of the Syndics of Cambridge University Library).

Syntonic chromatic scales with 12 pitches per octave

In his college notebook, Newton expressed the sizes of the syntonic intervals by 12-tempered semitones with an accuracy of 0.0001 cent (see Figure 8) 19. There are various ways to select the pitches of a 12-note chromatic scale within L₅. 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 "a-b-a-c-a-b-a-a-c-a-b-a" with reference G, see Figures 3a/3b. It also contains a Pythagorean third F-A (81 : 64) defined through four Pythagorean fifths 20 so that the major triad F-A-C 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

Figure 3a. Newton's syntonic chromatic scale in a rectangular pitch grid as a closed vector path. The notes (pitch classes) of the C-major scale are highlighted. The G-major scale is congruent with the standard diatonic major scale as described by Mercator, see Figure 1a.

Figure 3b. The three kinds of semitones, a, b and c, of Newton's chromatic scale together with their frequency ratios. The vector b + a forms a major tone of the ratio 9 : 8, and c + a forms a minor tone of the ratio 10 : 9, see also Figure 1b.

Figure 4. Vector representations of the four kinds of semitones occurring in limit-5 tunings. The lesser and greater chromatic semitones, as well as the ordinary and large diatonic semitones, differ by syntonic commas; the lesser chromatic and ordinary diatonic semitones, as well as the greater chromatic and large diatonic semitones, differ by enharmonic changes or major diesis (128 : 125) – an octave minus three just major thirds.

Figure 5. Grid representations of four different syntonic chromatic scales from historical sources. The large diatonic semitones 27 : 25 in the scales by Mercator (c. 1660), Holder (1694) and Euler (1739) are highlighted.

612-EDO

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.

Figure 6a. The curves in the central part analyze all the intervals of the chromatic scale, whether they are greater or smaller than the corresponding intervals with respect to G. Newton (1665, 108r) 23.

Figure 6b. Transcription of the note names and numbers from Figure 6a.

The "numbers" in Figure 6a/6b indicate sizes of the syntonic intervals with respect to G in terms of 12-EDO semitones plus/minus unit fractions of 12-EDO whole tones. The fractions not highlighted red or blue in the transcription permit the calculation of the 612-EDO representation of the pitch set G-A♭-A-C-D-F-F♯-G containing optimum approximations of the Pythagorean sequence F-C-G-D-A as well as perfect minor and major triads F-A♭-C and D-F♯-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 612-EDO 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 limit-5 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.

Figure 7. The 612-EDO pitches of the notes described in Figures 6a/b correspond to pitches on and within the parallelogram F-A♭-A-F♯. The pitches D♭, E♭, G, B, C♯ fit to 615-EDO.

The intervals of Newton's chromatic scale in 612-EDO 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 612-EDO has the largest number of parts, but because the deviations of the best fit measured in unit intervals are the smallest. For example, 720-EDO is less accurate than 612-EDO, although it has a smaller unit interval 25.

Since Newton measured the deviations in unit fractions of 6-EDO 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 612-EDO like 306-EDO, 204-EDO or 51-EDO determined by the divisors of 612. The 51-EDO 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 51-EDO are on top of his list, shown in Figure 9b.

Figure 8. Newton gives base 10 logarithms of the intervals of the syntonic chromatic scale (3^rd column). The last number column expresses these intervals as multiples of 12-tempered semitones with an accuracy of 0.0001 cent. Newton (1665, 105v)

Figure 9a. The intervals of the syntonic chromatic scale approximated by multiples of unit intervals, which result from various divisions of the octave into equal parts.
Newton (1665, 108v)

Figure 9b. It is unclear how the almost regular pattern in the fourth row from the bottom (0 4 2 6 1 5 3 7 11 6 10 8 12) is related to the divisions of the octave. Newton (1665, 108r)

Figure 9c. Above: Two versions of 120-EDO with semitones of the sizes 11, 9, 8 resp. 12, 8, 6 units, the latter are equivalent to 6, 4, 3 in 60-EDO. Newton (1665, 106r). Below: A 24-EDO representation with a lesser chromatic semitone of size 0. Newton (1665, 108r).

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 limit-5 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 limit-5 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 limit-5, 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 118-EDO than in 53-EDO. Since 612-EDO 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, 53-EDO 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 look-up tables for exponential functions and logarithms used in astronomy have many thousand entries to cover the range between 1 and 10, musical look-up tables require only 53, 118 or 612 entries to enable their user to calculate the ratios of arbitrary limit 5-intervals 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 meta-system is 360-EDO, 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 fine-grained division of the octave.

51-EDO and 53-EDO

From the 612-EDO representation of the chromatic scale Newton could determine with fewer parts without extensive calculations. Taking the accuracy of 612-EDO for granted, it can be used to find coarser good equal divisions of the octave by proportionally reducing the 612-EDO representations. It follows from the previous paragraph that the irrational number 2^1/612, the frequency ratio of the 612-EDO 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 51-EDO scale one is tempted to divide the numbers in the 612-EDO 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 51-EDO from the 612-EDO pitches.

Newton's values for 51-EDO 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 (G-A♭, A-B♭, B-C, C♯-D, D-E♭, E-F, F♯-G) measure 5 units, the greater chromatic semitones (A♭-A, C-C♯, F-F♯) 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 51-EDO units: 7 ⋅ 5 + 3 ⋅ 4 + 2 ⋅ 2 = 51.

The major tones in Newton's 51-EDO scale measure 9 units so that six of them are 3 units larger than an octave. This difference, the 51-EDO equivalent of the Pythagorean comma, is three times as big as the Pythagorean comma 29, so that Newton's 51-EDO representation must be considered a very weak approximation to the syntonic chromatic scale.

If instead only the 612-EDO 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 53-EDO values of the chromatic scale can also be determined correctly by rescaling the 612-EDO 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 53-EDO is only 0.068 units (compared with 0.418 in 51-EDO).

41-EDO and 29-EDO

The representation of the chromatic scale in 41-EDO and 29-EDO can be directly obtained from the values of 53-EDO by reducing the semitone sizes evenly, see Table 6. The last row shows that the same procedure cannot be continued down to 17-EDO, since the related lesser chromatic semitones would have size zero.

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 65-EDO, 77-EDO and 89-EDO, although they can be derived from 53-EDO in the same way as 41-EDO and 29-EDO, probably because they divide the octave into more parts and give less accurate approximations than 53-EDO.

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.

Newton's 17 different EDOs can be classified as follows:

• 612, 51, 53, 41, 29: derivation from 612-EDO and 53-EDO
• 59: optimization of the major third or indirect derivation from 120-EDO or 53-EDO
• 24, 36, 60, 120: refinement of 12-EDO / 120: decimal division of the 12-EDO semitone
• 100, 25, 20: decimal division of the octave
• 15, 19, 20, 24, 25: small numbers (relative semitone sizes not respected)

Only, 29-EDO, 41-EDO, 53-EDO and 612-EDO, 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 53-EDO and 612-EDO. 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 53-EDO 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, 293-295: 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 53-EDO.

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.

Log-compatibility 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 semitone-pattern a-b-a-c-a-b-a-a-c-a-b-a 33.

In the following, a chromatic n-EDO 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 log-compatible. 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 53-EDO representation, 53-EDO provides the coarsest strongly consistent and log-compatible 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 EDO-scales 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 12-EDO have consistency level 3, because 12-EDO does not distinguish between different semitones and has a = b = c.

For n ≤ 1200 there are 374 divisions of the octave into n equal intervals with log-compatible chromatic scales. The lowest deviations for scales with consistency levels > 0 occurs at 12-EDO 35, 34-EDO and 41-EDO. The maximum deviation occurs at 306-EDO (306 = $\frac{1}{2}$ ⋅ 612). 53-EDO and 65-EDO are the only divisions of the octave into less than 100 equal parts with a relative deviation lower than that of 12-EDO.