1. Introduction

A logarithmic approximant (or approximant for short) is an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as:

• Why do certain temperaments such as 12edo provide a good approximation to 5-limit just intonation?
• Why are certain commas small, and roughly how small are they?
• Why does the 3-limit framework produce aesthetically pleasing scale structures?

The exact size, in cents, of an interval with frequency ratio r is

$\qquad J_c = 1200 \log_2{r} = 1200 \ln{r}/\ln{2}$

where for just intervals r is rational and can be written as the ratio of two integers:

$\qquad r = n/d$

When manipulating approximants it is convenient to work with a different logarithmic base, in which the interval is defined as

$\qquad J = \tfrac{1}{2} \ln{r}$

This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio e2 = 7.38906... This unit interval can conveniently be termed the dineper (dNp), being twice the size of the natural unit for logarithmic measurement, the Neper.

Comparing the two units of measurement we find

1 dineper = 2400/ln(2) = 3462.468 cents

which is about 1.4 semitones short of three octaves.

The logarithmic size of an interval with a given frequency ratio can be conveniently notated as that ratio underlined. Thus 3/2 is the perfect fifth. This can also be expressed by an explicit function: if bim(r) = (r-1)/(r+1), then r = bim(r). The inverse function can be written mib(v) = (1+v)/(1-v).

Three types of approximants are described here:

• Bimodular approximants (first order rational approximants)
• Padé approximants of order (1,2) (second order rational approximants)

2. Bimodular approximants

Definition

The bimodular approximant of an interval with frequency ratio r = n/d is

$\qquad v = \frac{r-1}{r+1}$

v can thus be expressed as

$\qquad v = \frac{n-d}{n+d} \\$

###### = (frequency difference) / (frequency sum)

###### =½ (frequency difference) / (mean frequency)

$\qquad r = \frac{1+v}{1-v}$

Properties

When r is small, v provides an approximate relative measure of the logarithmic size of the interval. This approximation was exploited by Joseph Sauveur in 1701 and later by Euler and others.

Noting that the exact size (in dineper units) of the interval with frequency ratio r is

$\qquad J = \tfrac{1}{2} \ln{r}$

the relationship between v and J can be expressed as

$\qquad v = \frac{r-1}{r+1} = \frac{e^{2J}-1}{e^{2J}+1} = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...$

which shows that vJ and provides an indication of the size and sign of the error involved in this approximation.

J can be expressed in terms of v as

$\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...$

The function v(r) is the order (1,1) Padé approximant of the function J(r) =½ ln r in the region of r = 1, which has the property of matching the function value and its first and second derivatives at this value of r. The bimodular approximant function is thus accurate to second order in r – 1.

As an example, the size of the perfect fifth (in dNp units) is

$\qquad J = \tfrac{1}{2} \ln{3/2} = 0.20273...$

The bimodular approximant for this interval (r = 3/2) is

$\qquad v = (3/2 – 1)/(3/2 + 1) = (3 – 2)/(3 + 2) = 1/5 = 0.2$

and the Taylor series indicates that the error in this value is about

$\qquad -\tfrac{1}{3}v^3 = -0.00267...$

The approximants of superparticular intervals are reciprocals of odd integers, as shown in Figure 1.

######Figure 1. Bimodular approximants for low-order superparticular intervals

If v[J] denotes the bimodular approximant of an interval J with frequency ratio r,

$\qquad v[-J] = -v[J] \\ \qquad v[J_1 +J_2] = \frac{v_1+v_2}{1+v_1 v_2}$

This last result is equivalent to the identity expressing tanh(J1 + J1) in terms of tanh(J1) and tanh(J2).

Bimodular approximants and equal temperaments

While bimodular approximants have historically been used as a means of estimating the sizes of very small intervals, they remain reasonably accurate as the interval size is increased to an octave or more. And being easily computable, they provide a quick means of comparing the relative sizes of intervals. For example:

Two perfect fourths (r = 4/3, v = 1/7) approximate a minor seventh (r = 9/5, v = 2/7)

Three major thirds (r = 5/4, v = 1/9) or two 7/5s (v = 1/6) or five 8/7s (v = 1/15) approximate an octave (r = 2/1, v = 1/3)

Bimodular approximants (abbreviated to ‘approximants’ here) also provide simple explanations for the properties of certain equal temperaments.

Tuning the perfect fourth and perfect fifth in the ratio of their approximants (1/7 : 1/5 = 5 : 7) and adjusting their sum to a pure octave yields 12edo (considered as a 3-limit temperament). This is an example of the high accuracy typically obtainable from a tempering policy which takes two intervals which are similar in size and not too large, tunes them in their approximant ratio, and normalises their sum to a pure interval.

Aspects of 12edo considered as a 5-limit temperament can be explained by noting that it tunes the major third, major sixth and octave in the ratio of their approximants (1/9 : 1/4 : 1/3 = 4 : 9 : 12). The accuracy here is lower because the octave is of a size where the approximant has a significant error, and tuning the octave pure assigns the entire error to the smaller intervals.

Tuning the major third and perfect fifth in the in the ratio of their approximants (1/9 : 1/5) and tuning the fifth pure yields Carlos Alpha.

Tuning the minor third and perfect fifth in the in the ratio of their approximants (1/11 : 1/5) and tuning the fifth pure yields Carlos Beta.

Tuning the minor third and major third in the ratio of their approximants (1/11 : 1/9) and adjusting their sum to a perfect fifth yields Carlos Gamma . This temperament has high accuracy because it conforms to the policy noted above.

Tuning the octave pure while preserving the ratios specified above yields, respectively, 31edo, 19edo and 34edo.

Tuning the intervals 9/7, 7/5 and 5/3 in the ratio of their approximants (1/8 : 1/6 : 1/4 = 3 : 4 : 6) and adjusting their sum to a perfect twelfth yields the equally tempered Bohlen-Pierce scale.

Tuning the intervals 11/9, 9/7, 3/2 and 5/3 in the ratio of their approximants (1/10 : 1/8 : 1/5 : 1/4 = 4 : 5 : 8 : 10) and adjusting their sum to a major tenth yields 88 cent equal temperament.

Relationships of this sort can be identified in all equal temperaments.

Bimodular commas

As a consequence of the near-rational interval relationships implied by approximants, any pair of source intervals can be used to define a comma.

Given two intervals J1 and J2 (with J1 < J2) and their approximants v1 and v2, we define the bimodular residue as

$\qquad b_r(J_1,J_2) = \frac{J_2}{v_2} - \frac{J_1}{v_1}$

and using the Taylor series expansion of J(v) we find

$\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (v_2^2 – v_1^2) = \tfrac{1}{3} (v_2 + v_1)(v_2 – v_1)$

The bimodular comma is obtained from the bimodular residue by means of a rational multiplier which ensures that the result (in line with the usual convention applied to commas) is a linear combination of J1 and J2 with integer coefficients sharing no common factor:

$\qquad b(J_1,J_2) ≈ b_m(J_1,J_2) b_r(J_1,J_2)$

where

$\qquad v_1 = \frac{j_1}{g_1}, v_2 = \frac{j_2}{g_2}$

and (with rare exceptions)

$\qquad b_m(J_1,J_2) ≈ \frac{LCM(j_1,j_2)}{GCD(g_1,g_2)}$

The bimodular residue is accurately estimated by

$\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (J_1+J_2)(J_2-J_1)$

and therefore

$\qquad b(J_1,J_2) ≈ \tfrac{1}{3} (J_1+J_2)(J_2-J_1) b_m$

Examples

If the source intervals are the perfect fourth (f = 4/3) and the perfect fifth (F = 3/2), then v1 = 1/7, v2 = 1/5, and b is the Pythagorean comma:

$\qquad b(F,f) = b_r(F,f) = \frac{F}{\tfrac{1}{5}} - \frac{f}{\tfrac{1}{7}} = 5F – 7f$

If the source intervals are the perfect fourth (f = 4/3) and the minor seventh (m7 = 9/5), then v1 = 1/7, v2 = 2/7, br = 2/7 and b is the syntonic comma:

$\qquad b(m_7,f) = b_r(m_7,f) = \tfrac{2}{7} \left( \frac{m_7}{\tfrac{2}{7}} - \frac{f}{\tfrac{1}{7}} \right) = m_7 – 2f$

For further examples of bimodular commas, and a fuller account of bimodular approximant theory, refer to this paper. See also Don Page comma (another name for this type of comma).

3. Padé approximants of order (1,2)

Definition

In the section on bimodular approximants it was shown than an interval of logarithmic size J (measured in dineper units) is related to its bimodular approximant by

$\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...$

where

$\qquad v = \frac{r-1}{r+1}$

and r is the interval’s frequency ratio.

Another way to express this relationship is with a continued fraction:

$\qquad J = \tanh^{-1}{v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))$

The first convergent of this continued fraction is v, the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is

$\qquad y = \frac{v}{1-v^2/3}$

Values of this rational approximant for some simple 5-limit intervals are shown in the table below.

 Interval J########### (1,2) Padé approximant y# Perfect twelfth = 3/1 6/11 Octave = 2/1 9/26 Major sixth = 5/3 12/47 Perfect fifth = 3/2 15/74 Perfect fourth = 4/3 21/146 Major third = 5/4 27/242

The denominators of these fractions rapidly get large, so this type of approximant has limited usefulness. However, when combined with bimodular approximants it has occasional value in explaining apparent numerical coincidences and the smallness of the associated commas. For example:

(3/1) / (6/5) = 6.0257 ≈ (6/11) / (1/11) = 6 (kleisma)

(3/1) / (7/4) = 1.9632 ≈ (6/11) / (3/11) = 2 (septimal diesis = 49/48)

(2/1) / (7/6) = 4.4966 ≈ (9/26) / (1/13) = 9/2 (|-11 -9 0 9> comma)

(2/1) / (27/25) = 9.0065 ≈ (9/26) / (1/26) = 9 (ennealimma)

(5/3) / (49/45) = 5.9986 ≈ (12/47) / (2/47) = 6

(5/3) / (25/22) = 3.9960 ≈ (12/47) / (3/47) = 4

(5/3) / (26/21) = 2.3918 ≈ (12/47) / (5/47) = 12/5

(5/3) / (27/20) = 1.7022 ≈ (12/47) / (7/47) = 12/7

(3/2) / (20/17) = 2.4949 ≈ (15/74) / (6/74) = 5/2

Definition

The quadratic approximant q of an interval J with frequency ratio r = n'/d is

$\qquad q(r) = \tfrac{1}{2} (r^{1/2} – r^{-1/2}) \\ \qquad = \tfrac{1}{2} (e^J - e^{-J}) \\ \qquad = \sinh{J} \\ \qquad = J + \tfrac{1}{3!} J^3 + \tfrac{1}{5!} J^5 + ...$

If this is compared with the expression for the bimodular approximant,

$\qquad v = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...$

it is apparent that q is about twice as accurate as v, with an error of opposite sign.

While v is the frequency difference divided by twice the arithmetic frequency mean, q is the frequency difference divided by twice the geometric frequency mean:

$\qquad q = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}$

$\qquad \sqrt{r} = q + \sqrt{1+q^2}$

The following are the quadratic approximants of some simple 5-limit intervals:

 Interval J##################### Quadratic approximant q ## Perfect twelfth = 3/1 1/√3 Octave = 2/1 1/2√2 Minor seventh = 9/5 2/3√5 Pythagorean minor seventh = 16/9 7/24 Major sixth = 5/3 1/√15 Minor sixth = 8/5 3/4√10 Perfect fifth = 3/2 1/2√6 Perfect fourth = 4/3 1/4√3 Major third = 5/4 1/4√5 Minor third = 6/5 1/2√30 Pythagorean minor third = 32/27 5/24√6 Large tone = 9/8 1/12√2 Small tone = 10/9 1/6√10 Diatonic semitone = 16/15 1/8√15 Chroma = 25/24 1/20√6 Syntonic comma = 81/80 1/72√5

Expressed in terms of the bimodular approximant, v = j/g,

$\qquad q = \frac{v}{\sqrt{1-v^2}} = \frac{j}{\sqrt{g^2-j^2}}$

Quadratic approximants of just intervals thus have the form q = j/√k, where j and k are integers and j2 + k = g2 is a perfect square.

The presence of a square root in the denominator of q (except where J is a double interval) means that quadratic approximants do not, on the whole, imply approximate rational ratios between just intervals or commas of the conventional type. Their interest stems from the fact that ratios involving integer square roots are expressible as repeating continued fractions.

Properties

If v[J] and q[J] denote, respectively, the bimodular and quadratic approximants of an interval J with frequency ratio r, and qn denotes q[Jn] , then

$\qquad v = \tanh{J}, q = \sinh{J}, \frac{q}{v} = \cosh{J} \\ \qquad \sqrt{r} = e^J = q(\frac{1}{v} + 1) \\ \qquad \frac{1}{\sqrt{r}} = e^{-J} = q(\frac{1}{v} - 1) \\ \qquad \frac{1}{q^2} = \frac{1}{v^2} – 1 \\ \qquad q[-J] = -q[J] \\ \qquad q[J_2 + J_1] = q_1 q_2 (\frac{1}{v_2} + \frac{1}{v_1}) \\ \qquad q[J_2 - J_1] = q_1 q_2 (\frac{1}{v_2} - \frac{1}{v_1}) \\ \qquad \frac {q[J_2 + J_1]}{q[J_2 - J_1]} = \frac{v_2+v_1}{v_2-v_1} \\ \qquad q[J_2 + J_1] q[J_2 - J_1] = q_2^2 - q_1^2 \\$

The last two expressions are rational for just intervals, and the last result is equivalent to the hyperbolic trigonometric identity

$\qquad \sinh{(J_2 + J_1)} \sinh{(J_2 - J_1)} = \sinh^2{J_2} - \sinh^2{J_1}$

Where two quadratic approximants have the same square root in the denominator their ratio is rational. This seems to suggest a new source of approximate rational interval ratios, and therefore a new source of commas, but in this situation the approximants always represent the sum and difference of a pair of just intervals, and their ratio can be derived by an alternative route using the bimodular approximants of those intervals.

For example

$\qquad \frac{octave}{large \, tone} ≈ \frac{1}{2√2} / \frac{1}{12√2} = 6$

where large tone = 9/8.

However, this can also be derived from bimodular approximants. Using

$\qquad \frac {q[J_2 + J_1]}{q[J_2 - J_1]} = \frac{v_2+v_1}{v_2-v_1}$

with J2 = F =3/2 and J1 = f = 4/3 this gives

$\qquad \frac{octave}{large \, tone} ≈ \frac{q[F+f]}{q[F-f]} \\ \qquad = \frac{v[F] + v[f]}{v[F] - v[f]} = \frac{1/5 + 1/7}{1/5 - 1/7} = 6$

The quadratic approximant q of a double interval 2J (for example, the ditone) is rational, which suggests using ½ q(r2) as a rational approximant of J (where J has frequency ratio r):

$\qquad \tfrac{1}{2} q(r^2) = \tfrac{1}{4} (r - \frac{1}{r}) = \tfrac{1}{2} \sinh{2J} = J + \tfrac{2}{3}J^3 + \tfrac{2}{15}J^5 + ...$

However, this approximant is both less accurate and more complex than the corresponding bimodular approximant, and consequently of limited value.

The most interesting approximate interval ratios derivable from quadratic approximants are irrational.

Relative sizes of intervals between 3 frequencies in arithmetic progression

Theorem

If three harmonics of a fundamental frequency form an arithmetic progression, then the ratio of the logarithmic sizes of the intervals formed between the lower and upper pairs of harmonics is close to the geometric mean of these intervals’ frequency ratios.

Remarks

If the harmonics have indices n – m, n and n + m, the two intervals have reduced frequency ratios n/(n – m) and (n + m)/n. It can be assumed that n and m have no common factor.

m is the degree of epimoricity of the intervals. When m = 1 the intervals are adjacent superparticular intervals.

The geometric mean of the frequency ratios is the frequency ratio corresponding to the arithmetic mean of the intervals.

Proof

The ratio of the intervals as estimated from their quadratic approximants is

$\qquad \tfrac{m}{2\sqrt{n(n-m)}} / \tfrac{m}{2\sqrt{(n+m)n}} = \sqrt{\frac{n+m}{n-m}}$

which is the geometric mean of their frequency ratios.

Examples

The ratio of the perfect fifth, F = 3/2, to the perfect fourth, f = 4/3, as estimated by their quadratic approximants (1/2√6 and 1/4√3) is √2, which is the frequency ratio of the arithmetic mean of these intervals (the half-octave).

F/f = 701.955/498.045 = 1.40942,

√2 = 1.41421.

The ratio of the large tone, T = 9/8, to the small tone, t = 10/9, as estimated by their quadratic approximants (1/12√2 and 1/6√10) is √5/2, which is the frequency ratio of the mean tone.

T'/t = 203.910/182.404 = 1.11790,

√5/2 = 1.11803.

Argent temperament

As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, 1.4000.

It can be shown that the error in a pair of intervals tuned in the ratio of their approximants is minimised if the sum of the intervals is normalised – in this case to a pure octave. If this is done while maintaining the √2 ratio the perfect fifth and fourth are tempered to

###Perfect fifth = 3/2 = 702.944 cents

###Perfect fourth = 4/3 = 497.056 cents

This fifth is wide by 0.989 cents, and the fourth narrow by the same amount. These errors are of about half the magnitude, and of opposite sign, as their counterparts in 12edo (where these intervals are tuned in the ratio of their bimodular approximants).

A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the silver ratio (sometimes called the silver mean), δs = √2 + 1 = 2.4142. On this basis, and by analogy with golden meantone temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘argent temperament' is proposed instead.

Argent temperament has interesting fractal properties which help to explain why 3-limit tuning forms aesthetically pleasing scales.

The continued fraction expansion of the silver ratio has a particularly simple form:

$\qquad \delta_s = √2 + 1 = 2 + 1/(2 + 1/(2 + 1/(2 + ...)))$

As a result, if two intervals L and s are tuned in the silver ratio, with s = L/δs, subtracting twice the small interval s from the large interval L leaves a remainder of size s/δs:

$\qquad L – 2s = (\delta_s – 2)s = s/\delta_s$

(since 1/δs = √2 - 1 = δs - 2) and consequently this process can be continued indefinitely to generate sequences of decreasing intervals as follows. The names are assigned according to Pythagorean conventions (the limma being the Pythagorean semitone 256/243) followed by tempered and just sizes in cents:

 Octave 1200.00 (1200.00) Perfect fourth## 497.06 (498.04) Tone 205.89 (203.91) Limma 85.28 (90.22) Pythag comma 35.32 (23.46) Perfect 11th## 1697.06 (1698.04) Perfect fifth 702.94 (701.96) Minor third## 291.17 (294.13) Apotome## 120.61 (113.69) 17-tone comma## 49.96 (66.76)

Thus for example:

###octave = 2×fourth + tone

###fourth = 2×tone + limma

###tone = 2×limma + Pythag comma

###perfect 11th (8/3) = 2×fifth + minor third

###fifth = 2×(minor third) + apotome

When picturing these relationships it makes most musical sense to place the small interval between the two larger ones, as in the ‘continued fraction jigsaw’ below.

The following relationships hold in the table, the first two being valid for the pure intervals as well as their tempered counterparts:

• Subtracting twice an interval from the interval on its left generates the interval on its right.
• An interval in the second row is the sum of the interval immediately above and the interval diagonally above and to the right.
• Adjacent horizontal pairs have ratio δs = √2 + 1.
• Adjacent vertical pairs have ratio √2.
• Extending the table to a third row yields consisting of the intervals in the first row multiplied by 2, and so on.

The regularity of this scheme, combined with the fact that the ratios between closely related intervals are of order 2, means that its intervals form orderly sequences in which successive terms are clearly differentiated but of comparable magnitude – highly desirable properties for the formation of musical scales.

In this fractal temperament, multiplying or dividing any interval by the factor δs = √2 + 1 produces another interval in the temperament. Any tempered interval J’ can be split into three parts, two of equal size J’/δs and the other of size J’/δs2.

A similar principle applies to multiplication and division by the factor √2, except that intervals in the top row of the table cannot be divided by √2 to yield another interval in the temperament. These properties means that the temperament would support compositional techniques based on novel types of intervallic augmentation and diminution.

Successive convergents of the silver ratio produce ratios involving Pell numbers.

###√2 + 1 ≈ 2, 5/2, 12/5, 29/12, 70/29…,

Other approximations to the silver ratio are provided by ratios of consecutive half Pell-Lucas numbers, which are formed by adding consecutive Pell numbers

###√2 + 1 ≈ 3, 7/3, 17/7, 41/17, 99/41…,

This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, 41edo, 70edo etc.).

The accuracy of the argent fifth means that the scheme produces workable approximations to the true sizes of the 3-limit intervals featured in the table. However, if the table is extended one further step to the right, errors of sign begin to occur (the next column containing the 29-tone comma and minus the 41-tone comma).

Figure 2 is a continued fraction jigsaw showing the sizes of the octave (o), fourth (f), tone (T), limma (sp), Pythagorean comma (p) and 29-tone comma (p29) as tempered by 41edo - an approximation to argent temperament. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.

######Figure 2. Continued fraction jigsaw for 41edo

Figure 3 is a geometrical representation of argent temperament in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = large tone, mp#= Pythagorean minor third, sp#= Pythagorean limma, Xp#= Pythagorean apotome, p = Pythagorean comma.

######Figure 3. Geometrical representation of argent temperament

Argent temperament tunes the augmented fourth (tritone) and diminished fifth (double minor third) in the ratio 3/2√2, which is also the ratio of the quadratic approximants of 10/7 and 7/5:

$\qquad \frac{q[10/7]}{q[7/5]}= \frac{ \tfrac{3} {2\sqrt{70}} } { \tfrac{2} {2\sqrt{35}} } = \tfrac{3}{2\sqrt{2}}.$

This means that in argent temperament the augmented fourth is very close to 10/7 and the diminished fifth is very close to 7/5. The discrepancy in each case is just 0.175 cents. Another way to express the first of these relationships is

$\qquad 3 (\tfrac{1}{2\sqrt{6}} – \tfrac{1}{4\sqrt{3}}) ≈ \tfrac{3}{2\sqrt{70}},$

which after squaring both sides leads to √2 ≈ 99/70, a well-known approximation which can be confirmed by noting that 99/70 = √(2 + 1/4900).

As a consequence of these relationships the tempered diatonic semitone (85.281 cents) is close to 21/20 (84.467 cents), the tempered chromatic semitone (120.606 cents) is close to 15/14 (119.443 cents), and the tempered Pythagorean comma (35.325 cents) is close to 50/49 (34.976 cents).

If these 7-limit intervals are considered to be tempered to their 3-limit counterparts argent is an example of hemifamity temperament. Hemifamity (3120/3103) is the bimodular comma formed from 10/7 and 9/8

By the Gelfond-Schneider theorem the frequency ratios of all argent intervals (r = 2√2a+b, where a and b are integers) are transcendental, with the exception of octave multiples (a = 0). The frequency ratio of the tempered perfect eleventh (8/3 = 2.6666...) is the Gelfond-Schneider constant or Hilbert number, 2√2 = 2.665144...

Golden temperaments

It has been shown in an example above that the ratio of the large tone (T = 9/8) to the small tone (t = 10/9) is closely approximated by

$\qquad T/t = \sqrt{5}/2$

It follows that

$\qquad (T + t/2)/t = (\sqrt{5}+1)/2 = \phi$

where ϕ = 1.61803... is the golden ratio.

If a Fibonacci sequence of intervals is formed from the pair of intervals Tt/2 and t, and extended in both directions, it can thus be expected that the ratios between successive intervals in this sequence will also be close to ϕ. The sequence formed in this way is Sequence 1 in the following table.

 Sequence 1:# #t/2 - 3c# #2c #t/2 - c #T - t/2 #t #T + t/2# #M + t/2# #2M Sequence 2:# #magic #diesis #chroma# #semitone# #t #mp #f - c #m6p - c# Difference: #-3σ/2 #σ #-σ/2 #σ/2 #0 #σ/2 #σ/2 #σ Seq 1 ratios: #1.6120## #1.6204 #1.6171 #1.6184# #1.6179 #1.6181 #1.6180 Seq 2 ratios: #1.3865 #1.7212 #1.5810 #1.6325 #1.6125 #1.6201 #1.6172

where f = 4/3, T = 9/8, t = 10/9, M = 5/4, magic = 3125/3072, diesis = 128/125, chroma = 25/24, semitone = 16/15, mp = 32/27, c = syntonic comma = 81/80, m6p = 128/81, σ = schisma = 32805/32768.

The ratios between successive intervals in Sequence 1 are shown in the row labelled ‘Seq 1 ratios’, and are indeed close to ϕ.

Sequence 2 is another Fibonacci sequence of intervals which differ from those in Sequence 1 by small amounts of the order of one schisma (σ), as indicated by the row marked ‘Difference’ (which is itself a Fibonacci sequence).

The ratios of consecutive pairs of intervals in Sequence 2 are shown in the row labelled ‘Seq 2 ratios’. They approximate ϕ rather less accurately.

A suitable name for 5-limit tunings in which the intervals in either Sequence 1 or Sequence 2, or both, are tempered to exactly ϕ would be ‘golden temperaments’.

Tempering the Sequence 2 ratios to ϕ while tuning the octave pure and tempering out the syntonic comma yields golden meantone temperament.

Tempering the Sequence 1 ratios to ϕ yields a range of temperaments which can be made extremely accurate by, for example, tuning the octave and fifth (and therefore all Pythagorean intervals) pure. In this temperament the errors in the intervals s, t, M and m=6/5 are all ±0.02106 cents.

Tempering out the schisma tunes Sequences 1 and 2 identically so that the ratios between consecutive intervals can be fixed at ϕ in both sequences. Normalised to a pure octave, the resulting temperament, ‘golden schismatic’, has a fifth of 701.791061 cents (error -0.163 cents) and a major third of 385.671509 cents (error -0.642 cents).

If the quadratic approximants q1, q2 and q3 of a set of three intervals J1, J2 and J3 satisfy

$\qquad q_1^2 + q_2^2 = q_3^2$

they can be said to form a Pythagorean triple.

The following are three examples. In the first and third cases, their counterparts in 12edo, J1', J2' and J3', are also Pythagorean triples:

 #J1 #J2 #J3 #q1 #q2 #q3 #J1' #J2' #J3'# #6/5# #5/4 #4/3# #1/2√30# #1/4√5 #1/4√3 #3 #4 #5 #4/3 #12/5# #5/2# #1/4√3 #7/4√15# #3/2√10# #8/5 #12/5 #8/3 #3/4√10 #7/4√15 #5/4√6 #8 #15 #17

A small 34edo comma

As Gene Ward Smith has noted, the 5-limit comma |-433 -137 280> (‘selenia’) is remarkably small at just 0.004764 cents. The minute size of this comma can be explained using quadratic approximants.

It can be shown, using a suitable comma-based lattice, that every comma tempered out by 34edo can be expressed as an integer linear combination of the //gammic// comma |-29 -11 20> (4.769 cents) and the semisuper comma (AKA vishnuzma) |23 6 -14> (3.338 cents). In particular,

###selenia = 7 gammic – 10 semisuper

So to prove that selenia is small we must show that gammic'/semisuper ≈ 10/7.

Gammic and semisuper are both bimodular commas:

###gammic = b(6/5,5/4)

###semisuper = b(25/24,4/3)

Using a result given in the section on bimodular commas, the size of b(J1,J2) can be estimated using

$\qquad b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 – J_1^2) b_m$

Estimating J2 and J1 with their quadratic approximants we then have

$\qquad b(J_1,J_2) ≈ \frac{1}{3} (q_2^2 – q_1^2) b_m$

For gammic:

###J₁= 6/5, J₂= 5/4

###v= 1/11, v= 1/9, bm = 1

###q₁² = (1/4)(1/30), q² = (1/4)(1/20)

###gammic = b(J,J) ≈ (1/12) (1/30 – 1/20) = (1/12) (1/60)

For semisuper:

###J₁= 25/24, J₂= 4/3

###v= 1/49, v= 1/7, bm = 1/7

###q₁² = (1/4)(1/600), q² = (1/4)(1/12)

###semisuper = b(J,J) ≈ (1/12) (1/12 – 1/600)(1/7) = (1/12) (7/600)

Therefore

###gammic/semisuper10/7

as required.

To estimate the size of selenia we must quantify the error in this ratio. A more accurate analysis gives

$\qquad b(J_1,J_2) ≈ \left( \tfrac{1}{3} (q_2^2 – q_1^2) – \tfrac{2}{15} (q_2^4 – q_1^4) \right) b_m \\ \qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{5} (q_1^2 + q_2^2) ) b_m$

So to improve our estimates of b(J1,J2) we should multiply them by

$\qquad f = 1 – \tfrac{2}{5} (q_1^2 + q_2^2)$

Thus a better estimate for gammic/semisuper is

$\qquad \frac{gammic}{semisuper} ≈ \frac{10 f_{gamma}} {7 f_{semisuper}}$

from which it follows that

###selenia = 7 gammic - 10 semisuper

######## ≈ 7 gammic (fgammic - fsemisuper)/fgammic

Putting in the numbers:

###fgammic = 1 – (2/5) (1/4) (1/30 + 1/20) = 1 – 1/120

###fsemisuper = 1 – (2/5)(1/4) (1/600 + 1/12) = 1 – (1/120) (51/50)

###fgammic - fsemisuper = 1/6000

Therefore

###selenia ≈ 7 gammic (1/6000) (120/119) = gammic/850 = 0.00561 cents

which is within 20% of the accurate value, 0.00476 cents. (The discrepancy is due to the influence of terms in q6, which become significant when the f values are very similar.)

In summary, the reason selenia is small (compared to gammic and semisuper) is because the quadratic approximants of gammic and semisuper are in the ratio 10/7. The reason it is very small (of order gammic/1000 rather than gammic/10) is because the fractional errors in those approximants are almost the same. That in turn is because the squares of the source intervals of these bimodular commas have nearly the same sum. Note that the quadratic approximants of three of these intervals form a Pythagorean triple:

$\qquad \left( q(\tfrac{6}{5}) \right)^2 + \left( q(\tfrac{5}{4}) \right)^2 = \left( q(\tfrac{4}{3}) \right)^2$

and (q(25/24))2 , being small in comparison to the other terms, compromises this equality only slightly.