Logarithmic approximants: Difference between revisions

Line 1:

=~~'''~~1. Introduction~~'''~~=

= 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:~~~~

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:

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

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

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

Line 20:

</math>

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.

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

Line 34:

<ul><li>Bimodular approximants (first order rational approximants)</li><li>Padé approximants of order (1,2) (second order rational approximants)</li><li>Quadratic approximants</li></ul>

=~~'''~~2. Bimodular approximants~~'''~~=

= 2. Bimodular approximants =

==~~~~Definition~~~~==

== Definition ==

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

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

<math>\qquad v = \frac{r-1}{r+1}

</math>

''~~~~v ~~~~''can thus be expressed as

''v ''can thus be expressed as

<math>\qquad v = \frac{n-d}{n+d} \\

Line 51:

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

~~~~''r'' ~~~~can be retrieved from ~~~~''v''~~~~ using the inverse relation

''r'' can be retrieved from ''v'' using the inverse relation

<math>\qquad r = \frac{1+v}{1-v}</math>

==~~~~Properties~~~~==

== 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.

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

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

<math>\qquad J = \tfrac{1}{2} \ln{r}</math>

the relationship between ~~~~''v''~~~~ and ~~~~''J''~~~~ can be expressed as

the relationship between ''v'' and ''J'' can be expressed as

<math>\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 - ...</math>

which shows that ~~~~''v'' ≈ ''J''~~~~ and provides an indication of the size and sign of the error involved in this approximation.

which shows that ''v'' ≈ ''J'' 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

''J'' can be expressed in terms of ''v'' as

<math>\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...</math>

The function ~~~~''v(r)''~~~~ is the order (1,1) [http://en.wikipedia.org/wiki/Pad%C3%A9_approximant 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~~~~.

The function ''v(r)'' is the order (1,1) [http://en.wikipedia.org/wiki/Pad%C3%A9_approximant 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

Line 78:

<math>\qquad J = \tfrac{1}{2} \ln{3/2} = 0.20273...</math>

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

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

<math>\qquad v = (3/2 – 1)/(3/2 + 1) = (3 – 2)/(3 + 2) = 1/5 = 0.2</math>

Line 92:

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

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

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

<math>\qquad v[-J] = -v[J] \\

\qquad v[J_1 +J_2] = \frac{v_1+v_2}{1+v_1 v_2}</math>

This last result is equivalent to the identity expressing ~~~~tanh(''J''~~~~1 + ~~~~''J''~~~~1~~~~)~~~~ in terms of ~~~~tanh(''J''~~~~1~~~~)~~~~ and ~~~~tanh(''J''~~~~2~~~~).~~~~

This last result is equivalent to the identity expressing tanh(''J''1 + ''J''1) in terms of tanh(''J''1) and tanh(''J''2).

==~~~~Bimodular approximants and equal temperaments~~~~==

== 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)

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)

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.

Line 126:

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

==~~~~Bimodular commas~~~~==

== 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 ~~~~''J''~~~~1~~~~ and ~~~~''J''~~~~2~~~~ (with~~~~ ''J''~~~~1~~~~ < ~~~~''J''~~~~2~~~~) and their approximants ~~~~''v''~~~~1~~~~ and ''~~~~v~~~~''~~~~2~~~~, we define the ''bimodular residue'' as

Given two intervals ''J''1 and ''J''2 (with ''J''1 < ''J''2) and their approximants ''v''1 and ''v''2, we define the ''bimodular residue'' as

<math>\qquad b_r(J_1,J_2) = \frac{J_2}{v_2} - \frac{J_1}{v_1}</math>

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

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

<math>\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)</math>

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 ~~~~''J''~~~~1~~~~ and ''~~~~J~~~~''~~~~2~~~~ with integer coefficients sharing no common factor:

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 ''J''1 and ''J''2 with integer coefficients sharing no common factor:

<math>\qquad b(J_1,J_2) ≈ b_m(J_1,J_2) b_r(J_1,J_2)</math>

Line 158:

===Examples===

If the source intervals are the perfect fourth (~~~~''f'' =~~~~ <u~~>4/3~~~~'')'' and the perfect fifth (~~~~''F'' = 3/2</u~~>), ~~~~then~~ ~~''v''1 = 1/7~~, ~~''v''2 = 1/5~~~~, and ''~~~~b~~~~'' is the Pythagorean comma:

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

<math>\qquad b(F,f) = b_r(F,f) = \frac{F}{\tfrac{1}{5}} - \frac{f}{\tfrac{1}{7}} = 5F – 7f</math>

If the source intervals are the perfect fourth (~~~~''f'' = 4/3</u~~></span~~>) and the minor seventh (~~~~''m''~~~~7 ~~~~= 9/5), ~~~~then ~~~~''v''~~~~1 ~~~~= 1/7~~~~, ~~~~''v''2 = 2/7~~~~, ''~~~~b~~~~''r ~~~~= 2/7~~~~ and ''~~~~b~~~~'' is the syntonic comma:

If the source intervals are the perfect fourth (''f'' = 4/3) and the minor seventh (''m''7 = 9/5), then ''v''1 = 1/7, ''v''2 = 2/7, ''b''r = 2/7 and ''b'' is the syntonic comma:

<math>\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</math>

Line 168:

For further examples of bimodular commas, and a fuller account of bimodular approximant theory, refer to [[:File:Bimod_Approx_2014-6-8.pdf|this paper]]. See also [[Don_Page_comma|Don Page comma]] (another name for this type of comma).

=~~'''~~3. Padé approximants of order (1,2)~~'''~~=

= 3. Padé approximants of order (1,2) =

==~~~~Definition~~~~==

== 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

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

<math>\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...</math>

Line 179:

<math>\qquad v = \frac{r-1}{r+1}</math>

and ''~~~~r~~~~'' is the interval’s frequency ratio.

and ''r'' is the interval’s frequency ratio.

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

Line 185:

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

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

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

<math>\qquad y = \frac{v}{1-v^2/3}</math>

Line 193:

{| class="wikitable"

|-

| | ''Interval ~~~~J~~~~''###########

| | ''Interval J''###########

| | ''(1,2) Padé approximant ~~~~y~~~~''#

| | ''(1,2) Padé approximant y''#

|-

| | Perfect twelfth = 3/1

Line 234:

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

=~~'''~~4. Quadratic approximants~~'''~~=

= 4. Quadratic approximants =

==~~~~Definition~~~~==

== Definition ==

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

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

<math>\qquad q(r) = \tfrac{1}{2} (r^{1/2} – r^{-1/2}) \\

Line 248:

<math>\qquad v = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...</math>

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

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:

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:

<math>\qquad q = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}</math>

''~~~~r~~~~'' can be retrieved from ''~~~~q~~~~'' using

''r'' can be retrieved from ''q'' using

<math>\qquad \sqrt{r} = q + \sqrt{1+q^2}</math>

Line 262:

{| class="wikitable"

|-

| | ''Interval'' ''~~~~J~~~~''#####################

| | ''Interval'' ''J''#####################

| | ''Quadratic approximant'' ~~~~''q''~~~~ #~~#~~

| | ''Quadratic approximant'' ''q'' ##

|-

| | ~~~~Perfect twelfth = 3/1</u~~></span~~>

| | Perfect twelfth = 3/1

| | ~~~~ 1/√3~~~~

| | 1/√3

|-

| | ~~~~ Octave = 2/1</u~~></span~~>

| | Octave = 2/1

| | ~~~~ 1/2√2~~~~

| | 1/2√2

|-

| | ~~~~ Minor seventh = 9/5</u~~></span~~>

| | Minor seventh = 9/5

| | ~~~~ 2/3√5~~~~

| | 2/3√5

|-

| | ~~~~ Pythagorean minor seventh = 16/9</u~~></span~~>

| | Pythagorean minor seventh = 16/9

| | ~~~~ 7/24~~~~

| | 7/24

|-

| | ~~~~ Major sixth = 5/3</u~~></span~~>

| | Major sixth = 5/3

| | ~~~~ 1/√15~~~~

| | 1/√15

|-

| | ~~~~ Minor sixth = 8/5</u~~></span~~>

| | Minor sixth = 8/5

| | ~~~~ 3/4√10~~~~

| | 3/4√10

|-

| | ~~~~ Perfect fifth = 3/2</u~~></span~~>

| | Perfect fifth = 3/2

| | ~~~~ 1/2√6~~~~

| | 1/2√6

|-

| | ~~~~ Perfect fourth = 4/3</u~~></span~~>

| | Perfect fourth = 4/3

| | ~~~~ 1/4√3~~~~

| | 1/4√3

|-

| | ~~~~ Major third = 5/4</u~~></span~~>

| | Major third = 5/4

| | ~~~~ 1/4√5~~~~

| | 1/4√5

|-

| | ~~~~ Minor third = 6/5</u~~></span~~>

| | Minor third = 6/5

| | ~~~~ 1/2√30~~~~

| | 1/2√30

|-

| | ~~~~ Pythagorean minor third = 32/27</u~~></span~~>

| | Pythagorean minor third = 32/27

| | ~~~~ 5/24√6~~~~

| | 5/24√6

|-

| | ~~~~ Large tone = 9/8</u~~></span~~>

| | Large tone = 9/8

| | ~~~~ 1/12√2~~~~

| | 1/12√2

|-

| | ~~~~ Small tone = 10/9</u~~></span~~>

| | Small tone = 10/9

| | ~~~~ 1/6√10~~~~

| | 1/6√10

|-

| | ~~~~ Diatonic semitone = 16/15</u~~></span~~>

| | Diatonic semitone = 16/15

| | ~~~~ 1/8√15~~~~

| | 1/8√15

|-

| | ~~~~ Chroma = 25/24</u~~></span~~>

| | Chroma = 25/24

| | ~~~~ 1/20√6~~~~

| | 1/20√6

|-

| | ~~~~ Syntonic comma = 81/80</u~~></span~~>

| | Syntonic comma = 81/80

| | ~~~~ 1/72√5~~~~

| | 1/72√5

|}

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

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

<math>\qquad q = \frac{v}{\sqrt{1-v^2}} = \frac{j}{\sqrt{g^2-j^2}}</math>

Quadratic approximants of just intervals thus have the form ''~~~~q = j/√k~~~~'', where ''~~~~j~~~~'' and ''~~~~k~~~~'' are integers and ''~~~~j~~~~''~~~~2~~~~''~~~~ + k = g~~~~''~~~~2~~~~ is a perfect square.

Quadratic approximants of just intervals thus have the form ''q = j/√k'', where ''j'' and ''k'' are integers and ''j''2'' + k = g''2 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.

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~~~~==

== Properties ==

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

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

<math>\qquad v = \tanh{J}, q = \sinh{J}, \frac{q}{v} = \cosh{J} \\

Line 351:

<math>\qquad \frac {q[J_2 + J_1]}{q[J_2 - J_1]} = \frac{v_2+v_1}{v_2-v_1}</math>

with ~~~~''J''~~~~2 ~~~~= ''F'' =3/2</u~~> and~~ ~~''J''~~~~1 ~~~~= ''f'' = 4/3</u~~></span> this gives

with ''J''2 = ''F'' =3/2 and ''J''1 = ''f'' = 4/3 this gives

<math>\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</math>

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

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

<math>\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 + ...</math>

Line 364:

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

==~~~~Relative sizes of intervals between 3 frequencies in arithmetic progression~~~~==

== Relative sizes of intervals between 3 frequencies in arithmetic progression ==

===~~~~Theorem~~~~===

=== 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~~~~===

=== 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.

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 [[Superpartient|degree of epimoricity]] of the intervals. When ''~~~~m~~~~'' = 1 the intervals are adjacent superparticular intervals.

''m'' is the [[Superpartient|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~~~~===

=== Proof ===

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

Line 383:

which is the geometric mean of their frequency ratios.

===~~~~Examples~~~~===

=== Examples ===

The ratio of the perfect fifth, ~~~~''F'' = 3/2</u~~></span~~>, to the perfect fourth, ~~~~''f'' = 4/3</u~~></span~~>, 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).

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,~~~~

''F/f'' = 701.955/498.045 = 1.40942,

~~~~√2 = 1.41421.~~~~

√2 = 1.41421.

The ratio of the large tone, ~~~~''T'' = 9/8</u~~></span~~>, to the small tone, ~~~~''t'' = 10/9</u~~></span~~>, 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.

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'<nowiki/>'''/t'''''<nowiki/>''' = 203.910/182.404 = 1.11790,'''~~~~

''T'<nowiki/>'''/t'''''<nowiki/>''' = 203.910/182.404 = 1.11790,'''

~~~~√5/2 = 1.11803.~~~~

√5/2 = 1.11803.

==~~~~Argent temperament~~~~==

== 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.

Line 407:

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 [http://en.wikipedia.org/wiki/Silver_ratio silver ratio] (sometimes called the silver mean), ''~~~~δ~~~~''~~s = ~~√2 + 1 = 2.4142~~~~. On this basis, and by analogy with [[Golden_Meantone|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.

A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the [http://en.wikipedia.org/wiki/Silver_ratio silver ratio] (sometimes called the silver mean), ''δ''√2 + 1 = 2.4142. On this basis, and by analogy with [[Golden_Meantone|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.

Line 415:

<math>\qquad \delta_s = √2 + 1 = 2 + 1/(2 + 1/(2 + 1/(2 + ...)))</math>

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

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

<math>\qquad L – 2s = (\delta_s – 2)s = s/\delta_s</math>

(since 1''/~~~~δ~~~~''~~~~s ~~~~= √2 - 1 = ''δ''~~~~s~~~~ - 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 <u~~>256/243~~) followed by tempered and just sizes in cents:

(since 1''/δ''s = √2 - 1 = ''δ''s - 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:

{| class="wikitable"

Line 491:

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

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

<ul><li>Subtracting twice an interval from the interval on its left generates the interval on its right.</li><li>An interval in the second row is the sum of the interval immediately above and the interval diagonally above and to the right.</li><li>Adjacent horizontal pairs have ratio ''δ''''s'' = √2 + 1.</li><li>Adjacent vertical pairs have ratio √2.</li><li>Extending the table to a third row yields consisting of the intervals in the first row multiplied by 2, and so on.</li></ul>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 ''~~~~δ~~~~''~~~~''s'' ~~~~= √2 + 1~~~~ produces another interval in the temperament. Any tempered interval ''~~~~J’~~~~'' can be split into three parts, two of equal size ''~~~~J’~~~~''~~/<~~/~~span>~~''~~~~δ~~~~''~~~~s~~~~ and the other of size ''~~~~J’~~~~''~~~~/~~~~''δs2''.

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

Line 509:

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 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.

[[File:Continued_fraction_jigsaw_41edo.png|alt=Continued fraction jigsaw 41edo.png|800x396px|Continued fraction jigsaw 41edo.png]]

Line 515:

######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 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, mppp#= Pythagorean apotome, p = Pythagorean comma.

[[File:Silver_temperament_graphic.png|alt=Silver temperament graphic.png|800x587px|Silver temperament graphic.png]]

Line 535:

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 [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem Gelfond-Schneider theorem] the frequency ratios of all argent intervals (~~~~''r'' = ~~2√2~~''a''+''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...</u~~></span~~>) is the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant Gelfond-Schneider constant] or Hilbert number, 2√2 = 2.665144~~~~...

By the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem Gelfond-Schneider theorem] the frequency ratios of all argent intervals (''r'' = 2√2''a''+''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 [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant 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</u~~></span~~>) to the small tone (~~~~''t'' = 10/9</u~~></span~~>) is closely approximated by

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

<math>\qquad T/t = \sqrt{5}/2</math>

Line 546:

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

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

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

If a Fibonacci sequence of intervals is formed from the pair of intervals ~~~~''T'' – ''t''/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.

If a Fibonacci sequence of intervals is formed from the pair of intervals ''T'' – ''t''/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.

{| class="wikitable"

|-

| | Sequence 1:#

| | ~~~~#~~~~''t''/2 - 3''c''~~~~#~~~~

| | #''t''/2 - 3''c''#

| | ~~~~#~~~~2''c''

| | #2''c''

| | ~~~~#~~~~''t''/2 ''- c''

| | #''t''/2 ''- c''

| | ~~~~#~~~~''T - t''/2

| | #''T - t''/2

| | ~~~~#~~~~''t''

| | #''t''

| | ~~~~#~~~~''T + t''/2~~~~#~~~~

| | #''T + t''/2#

| | ~~~~#~~~~''M + t''/2~~~~#~~~~

| | #''M + t''/2#

| | ~~~~#~~~~2''M''

| | #2''M''

|-

| | Sequence 2:#

| | ~~~~#~~~~''magic''

| | #''magic''

| | ~~~~#~~~~''diesis''

| | #''diesis''

| | ''~~~~#~~~~chroma~~~~#~~~~''

| | ''#chroma#''

| | ~~~~#~~~~''semitone''~~#~~

| | #''semitone''#

| | ~~~~#~~~~''t''

| | #''t''

| | ~~~~#~~~~''mp''

| | #''mp''

| | ~~~~#~~~~''f - c''

| | #''f - c''

| | ~~~~#~~~~''m6p - c''#

| | #''m6p - c''#

|-

| | Difference:

| | ~~~~#~~~~-3''σ''/2

| | #-3''σ''/2

| | ~~~~#~~~~''σ''

| | #''σ''

| | ~~~~#~~~~-''σ''/2

| | #-''σ''/2

| | ~~~~#~~~~''σ''/2

| | #''σ''/2

| | ~~~~#~~~~0

| | #0

| | ~~~~#~~~~''σ''/2

| | #''σ''/2

| | ~~~~#~~~~''σ''/2

| | #''σ''/2

| | ~~~~#~~~~''σ''

| | #''σ''

|-

| | Seq 1 ratios:

| |

| | ~~~~#~~~~1.6120</span~~>##~~

| | #1.6120##

| | ~~~~#~~~~1.6204

| | #1.6204

| | ~~~~#~~~~1.6171

| | #1.6171

| | ~~~~#~~~~1.6184~~~~#~~~~

| | #1.6184#

| | ~~~~#~~~~1.6179

| | #1.6179

| | ~~~~#~~~~1.6181

| | #1.6181

| | ~~~~#~~~~1.6180

| | #1.6180

|-

| | Seq 2 ratios:

| |

| | ~~~~#~~~~1.3865

| | #1.3865

| | ~~~~#~~~~1.7212

| | #1.7212

| | ~~~~#~~~~1.5810

| | #1.5810

| | ~~~~#~~~~1.6325

| | #1.6325

| | ~~~~#~~~~1.6125

| | #1.6125

| | ~~~~#~~~~1.6201

| | #1.6201

| | ~~~~#~~~~1.6172

| | #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.</u~~></span~~>

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~~~~''.

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).

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.

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’.

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|golden meantone]] temperament.

Tempering the Sequence 2 ratios to ''ϕ'' while tuning the octave pure and tempering out the syntonic comma yields [[Golden_Meantone|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</u~~></span~~> are all ±0.02106 cents.

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).

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).

==~~~~Pythagorean triples of quadratic approximants~~~~==

== Pythagorean triples of quadratic approximants ==

If the quadratic approximants ~~~~''q''1'', q''2~~~~ and ~~~~''q''3~~~~ of a set of three intervals ~~~~''J''1, ''J''2~~~~ and ''~~~~J~~~~''3 satisfy

If the quadratic approximants ''q''1'', q''2 and ''q''3 of a set of three intervals ''J''1, ''J''2 and ''J''3 satisfy

<math>\qquad q_1^2 + q_2^2 = q_3^2</math>

Line 625:

they can be said to form a [http://en.wikipedia.org/wiki/Pythagorean_triple Pythagorean triple].

The following are three examples. In the first and third cases, their counterparts in 12edo, ~~~~''J''1', ''J''2'~~~~ and ~~~~''J''3'~~~~, are also Pythagorean triples:

The following are three examples. In the first and third cases, their counterparts in 12edo, ''J''1', ''J''2' and ''J''3', are also Pythagorean triples:

{| class="wikitable"

|-

| | ~~~~#~~~~''J''1

| | #''J''1

| | ~~~~#~~~~''J''2

| | #''J''2

| | ~~~~#~~~~''J''3

| | #''J''3

| | ~~~~#~~~~''q''1

| | #''q''1

| | ~~~~#~~~~''q''2

| | #''q''2

| | ~~~~#~~~~''q''3

| | #''q''3

| | ~~~~#~~~~''J''1'

| | #''J''1'

| | ~~~~#~~~~''J''2'

| | #''J''2'

| | ~~~~#~~~~''J''3'~~#~~

| | #''J''3'#

|-

| | ~~#~~6/5#

| | #6/5#

| | ~~~~#~~~~5/4

| | #5/4

| | ~~~~#~~~~4/3~~#~~

| | #4/3#

| | ~~~~#~~~~1/2√30~~~~#~~~~

| | #1/2√30#

| | ~~~~#~~~~1/4√5

| | #1/4√5

| | ~~#~~1/4√3

| | #1/4√3

| | ~~#~~3

| | #3

| | ~~#~~4

| | #4

| | ~~~~#~~~~5

| | #5

|-

| | ~~~~#~~~~4/3

| | #4/3

| | ~~#~~12/5</u~~>#~~

| | #12/5#

| | ~~~~#~~~~5/2</u~~>#~~

| | #5/2#

| | ~~~~#~~~~1/4√3

| | #1/4√3

| | ~~#~~7/4√15~~~~#~~~~

| | #7/4√15#

| | ~~~~#~~~~3/2√10~~~~#~~~~

| | #3/2√10#

| |

|-

| | ~~~~#~~~~8/5

| | #8/5

| | ~~~~#~~~~12/5

| | #12/5

| | ~~#~~8/3

| | #8/3

| | ~~#~~3/4√10

| | #3/4√10

| | ~~~~#~~~~7/4√15

| | #7/4√15

| | ~~#~~5/4√6

| | #5/4√6

| | ~~#~~8

| | #8

| | ~~~~#~~~~15

| | #15

| | ~~~~#~~~~17

| | #17

|}

Line 675:

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

~~~~#~~##~~''selenia'' = 7 ''gammic'' – 10 ''semisuper''

###''selenia'' = 7 ''gammic'' – 10 ''semisuper''

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

Line 681:

''Gammic'' and ''semisuper'' are both bimodular commas:

~~~~##~~#~~''gammic'' = </span~~>''b''(6/5,5/4)~~

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

~~~~##~~#~~''semisuper'' = </span~~>''b''(25/24,4/3)~~

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

~~~~Using a result given in the section on bimodular commas, the size of ~~~~''b''(''J''1,''J''2)~~~~ can be estimated using

Using a result given in the section on bimodular commas, the size of ''b''(''J''1,''J''2) can be estimated using

<math>\qquad b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 – J_1^2) b_m</math>

~~Estimating ~~''J''2~~ and ~~''J''1~~~~ with their quadratic approximants we then have

''J''2''J''1 with their quadratic approximants we then have

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

~~~~For ''gammic'':~~~~

For ''gammic'':

~~~~###~~~~''J''₁= 6/5, ''J''₂= 5/4

###''J''₁= 6/5, ''J''₂= 5/4

~~~~###~~~~''v''₁ ~~~~= 1/11, ~~~~''~~~~v~~~~''₂ ~~~~= 1/9, ~~~~''~~~~b~~~~''~~~~m~~~~ = 1~~~~

###''v''₁ = 1/11, ''v''₂ = 1/9, ''b''m = 1

~~~~###~~~~''q''</span~~>₁² = ~~~~(1/4)(1/30),~~~~ ''~~~~q~~~~''~~~~₂''² ='' ~~~~(1/4)(1/20)~~

###''q''₁² = (1/4)(1/30), ''q''₂''² ='' (1/4)(1/20)

~~~~###~~~~''gammic'' = </span~~>''b''(''J''~~~~₁~~~~,''J''~~~~₂~~~~) ≈ (1/12) (1/30 – 1/20) = (1/12) (1/60)~~

###''gammic'' = ''b''(''J''₁,''J''₂) ≈ (1/12) (1/30 – 1/20) = (1/12) (1/60)

~~~~For ''semisuper:''~~~~

For ''semisuper:''

~~~~###~~~~''J''₁= 25/24, ''J''₂= 4/3

###''J''₁= 25/24, ''J''₂= 4/3

~~~~###~~~~''v''₁ ~~~~= 1/49, ~~~~''~~~~v~~~~''₂ ~~~~= 1/7, ~~~~''~~~~b~~~~''~~~~m~~~~ = 1/7~~~~

###''v''₁ = 1/49, ''v''₂ = 1/7, ''b''m = 1/7

~~~~###~~~~''q''</span~~>₁² = ~~~~(1/4)(1/600),~~~~ ''~~~~q~~~~''~~~~₂''² ='' ~~~~(1/4)(1/12)~~

###''q''₁² = (1/4)(1/600), ''q''₂''² ='' (1/4)(1/12)

~~~~###~~~~''semisuper'' = </span~~>''b''(''J''~~~~₁~~~~,''J''~~~~₂~~~~) ≈ (1/12) (1/12 – 1/600)(1/7) = (1/12) (7/600)~~

###''semisuper'' = ''b''(''J''₁,''J''₂) ≈ (1/12) (1/12 – 1/600)(1/7) = (1/12) (7/600)

~~~~Therefore~~~~

Therefore

~~###~~''gammic/semisuper'' ≈ ~~~~10/7~~~~

''gammic/semisuper'' ≈ 10/7

as required.

Line 724:

\qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{5} (q_1^2 + q_2^2) ) b_m</math>

~~So to improve our estimates of ~~''b''(''J''1,''J''2)~~~~ we should multiply them by

''b''(''J''1,''J''2) we should multiply them by

<math>\qquad f = 1 – \tfrac{2}{5} (q_1^2 + q_2^2)</math>

Line 732:

<math>\qquad \frac{gammic}{semisuper} ≈ \frac{10 f_{gamma}} {7 f_{semisuper}}</math>

~~~~from which it follows that~~~~

from which it follows that

~~~~###~~~~''selenia'' = 7 ''gammic'' - 10 ''semisuper''~~~~

###''selenia'' = 7 ''gammic'' - 10 ''semisuper''

~~~~######## ~~~~≈ 7 ''gammic'' (''f''~~~~''~~~~gammic~~ - fsemisuper~~)~~~~/~~fgammic~~''

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

~~~~Putting in the numbers:~~~~

Putting in the numbers:

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

''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)~~~~

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

~~~~###~~~~''fgammic - fsemisuper ~~''~~~~= 1/6000~~

###''fgammic - fsemisuper ''= 1/6000

Therefore

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

###''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 ''q~~~~''~~~~6~~~~'','' which become significant when the ''~~~~f~~~~'' values are very similar.)

q''6'','' 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:

Line 756:

<math>\qquad \left( q(\tfrac{6}{5}) \right)^2 + \left( q(\tfrac{5}{4}) \right)^2 = \left( q(\tfrac{4}{3}) \right)^2</math>

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

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

=Sources and acknowledgements=

@@ Line 1: / Line 1: @@
 {{todo|intro|inline=1}}
-='''<span style="font-size: 20px;">1. Introduction</span>'''=
+= 1. Introduction =
-<span style="font-family: Arial,Helvetica,sans-serif;">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:</span>
+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:
-<ul><li><span style="font-family: Arial,Helvetica,sans-serif;">Why do certain temperaments such as 12edo provide a good approximation to 5-limit just intonation?</span></li><li><span style="font-family: Arial,Helvetica,sans-serif;">Why are certain commas small, and roughly how small are they?</span></li><li><span style="font-family: Arial,Helvetica,sans-serif;">Why does the 3-limit framework produce aesthetically pleasing scale structures?</span></li></ul>
+<ul><li>Why do certain temperaments such as 12edo provide a good approximation to 5-limit just intonation?</li><li>Why are certain commas small, and roughly how small are they?</li><li>Why does the 3-limit framework produce aesthetically pleasing scale structures?</li></ul>
 The exact size, in cents, of an interval with frequency ratio ''r'' is
@@ Line 20: / Line 20: @@
 </math>
-This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio <span style="font-family: Georgia,serif; font-size: 110%;">e</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span><span style="font-family: Georgia,serif; font-size: 110%;"> = 7.38906...</span> This unit interval can conveniently be termed the dineper (dNp), being twice the size of the natural unit for logarithmic measurement, the Neper.
+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
@@ Line 34: / Line 34: @@
 <ul><li>Bimodular approximants (first order rational approximants)</li><li>Padé approximants of order (1,2) (second order rational approximants)</li><li>Quadratic approximants</li></ul>
-='''<span style="font-size: 20px;">2. Bimodular approximants</span>'''=
+= 2. Bimodular approximants =
-==<span style="font-family: Arial,Helvetica,sans-serif;">Definition</span>==
+== Definition ==
-The bimodular approximant of an interval with frequency ratio ''<span style="font-family: Georgia,serif; font-size: 110%;">r = n/d</span>'' is
+The bimodular approximant of an interval with frequency ratio ''r = n/d'' is
 <math>\qquad v = \frac{r-1}{r+1}
 </math>
-''<span style="font-family: Georgia,serif; font-size: 110%;">v </span>''can thus be expressed as
+''v ''can thus be expressed as
 <math>\qquad v = \frac{n-d}{n+d} \\
@@ Line 51: / Line 51: @@
 <span style="color: #ffffff;">######</span> =½ (frequency difference) / (mean frequency)
-<span style="font-family: Georgia,serif; font-size: 110%;">''r'' </span>can be retrieved from <span style="font-family: Georgia,serif; font-size: 110%;">''v''</span> using the inverse relation
+''r'' can be retrieved from ''v'' using the inverse relation
 <math>\qquad r = \frac{1+v}{1-v}</math>
-==<span style="font-family: Arial,Helvetica,sans-serif;">Properties</span>==
+== Properties ==
-When <span style="font-family: Georgia,serif; font-size: 110%;">''r'' </span>is small, <span style="font-family: Georgia,serif; font-size: 110%;">''v''</span> 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.
+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 <span style="font-family: Georgia,serif; font-size: 110%;">''r''</span> is
+Noting that the exact size (in dineper units) of the interval with frequency ratio ''r'' is
 <math>\qquad J = \tfrac{1}{2} \ln{r}</math>
-the relationship between <span style="font-family: Georgia,serif; font-size: 110%;">''v''</span> and <span style="font-family: Georgia,serif; font-size: 110%;">''J''</span> can be expressed as
+the relationship between ''v'' and ''J'' can be expressed as
 <math>\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 - ...</math>
-which shows that <span style="font-family: Georgia,serif; font-size: 110%;">''v'' ≈ ''J''</span> and provides an indication of the size and sign of the error involved in this approximation.
+which shows that ''v'' ≈ ''J'' and provides an indication of the size and sign of the error involved in this approximation.
-''<span style="font-family: Georgia;">J</span>'' can be expressed in terms of <span style="font-family: Georgia,serif; font-size: 110%;">''v''</span> as
+''J'' can be expressed in terms of ''v'' as
 <math>\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...</math>
-The function <span style="font-family: Georgia,serif; font-size: 110%;">''v(r)''</span> is the order (1,1) [http://en.wikipedia.org/wiki/Pad%C3%A9_approximant Padé approximant] of the function <span style="font-family: Georgia,serif; font-size: 110%;"> ''J(r) =''½ ln ''r'' </span> in the region of <span style="font-family: Georgia,serif; font-size: 110%;">''r'' = 1</span>, which has the property of matching the function value and its first and second derivatives at this value of ''<span style="font-family: Georgia,serif; font-size: 110%;">r</span>''. The bimodular approximant function is thus accurate to second order in <span style="font-family: Georgia,serif; font-size: 110%;">''r'' – 1</span>.
+The function ''v(r)'' is the order (1,1) [http://en.wikipedia.org/wiki/Pad%C3%A9_approximant 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
@@ Line 78: / Line 78: @@
 <math>\qquad J = \tfrac{1}{2} \ln{3/2} = 0.20273...</math>
-The bimodular approximant for this interval (<span style="font-family: Georgia,serif; font-size: 110%;">''r'' = 3/2</span>) is
+The bimodular approximant for this interval (''r'' = 3/2) is
 <math>\qquad v = (3/2 – 1)/(3/2 + 1) = (3 – 2)/(3 + 2) = 1/5 = 0.2</math>
@@ Line 92: / Line 92: @@
 <span style="color: #ffffff;">######</span>Figure 1. Bimodular approximants for low-order superparticular intervals
-If <span style="font-family: Georgia,serif; font-size: 110%;">''v''[''J''] </span>denotes the bimodular approximant of an interval <span style="font-family: Georgia,serif; font-size: 110%;">''J''</span> with frequency ratio ''<span style="font-family: Georgia,serif; font-size: 110%;">r</span>'',
+If ''v''[''J''] denotes the bimodular approximant of an interval ''J'' with frequency ratio ''r'',
 <math>\qquad v[-J] = -v[J] \\
 \qquad v[J_1 +J_2] = \frac{v_1+v_2}{1+v_1 v_2}</math>
-This last result is equivalent to the identity expressing <span style="font-family: Georgia,serif; font-size: 110%;">tanh(''J''</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1 + </span><span style="font-family: Georgia,serif; font-size: 110%;">''J''</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span><span style="font-family: Georgia,serif;">)</span> in terms of <span style="font-family: Georgia,serif; font-size: 110%;">tanh(''J''</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span><span style="font-family: Georgia,serif; font-size: 110%;">)</span> and <span style="font-family: Georgia,serif; font-size: 110%;">tanh(''J''</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span><span style="font-family: Georgia,serif; font-size: 110%;">).</span>
+This last result is equivalent to the identity expressing tanh(''J''1 + ''J''1) in terms of tanh(''J''1) and tanh(''J''2).
-==<span style="font-family: Arial,Helvetica,sans-serif; font-size: 15px;">Bimodular approximants and equal temperaments</span>==
+== 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, ''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>'' = 1/7) approximate a minor seventh (''r'' = 9/5, ''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>'' = 2/7)
+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, ''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>'' = 1/9) or two <u>7/5</u>s (''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>'' = 1/6) or five <u>8/7</u>s (''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>'' = 1/15) approximate an octave (''r'' = 2/1,''<span style="font-family: Georgia,serif; font-size: 110%;"> v</span>'' = 1/3)
+Three major thirds (''r'' = 5/4, ''v'' = 1/9) or two <u>7/5</u>s (''v'' = 1/6) or five <u>8/7</u>s (''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.
@@ Line 126: / Line 126: @@
 Relationships of this sort can be identified in all equal temperaments.
-==<span style="font-family: Arial,Helvetica,sans-serif;">Bimodular commas</span>==
+== 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 <span style="font-family: Georgia,serif; font-size: 110%;">''J''</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> and <span style="font-family: Georgia,serif; font-size: 110%;">''J''</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span> (with<span style="font-family: Georgia,serif; font-size: 110%;"> ''J''</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> &lt; <span style="font-family: Georgia,serif; font-size: 110%;">''J''</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span>) and their approximants <span style="font-family: Georgia,serif; font-size: 110%;">''v''</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> and ''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>''<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span>, we define the ''bimodular residue'' as
+Given two intervals ''J''1 and ''J''2 (with ''J''1 &lt; ''J''2) and their approximants ''v''1 and ''v''2, we define the ''bimodular residue'' as
 <math>\qquad b_r(J_1,J_2) = \frac{J_2}{v_2} - \frac{J_1}{v_1}</math>
-and using the Taylor series expansion of <span style="font-family: Georgia,serif; font-size: 110%;">''J''(''v'')</span> we find
+and using the Taylor series expansion of ''J''(''v'') we find
 <math>\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)</math>
-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 <span style="font-family: Georgia,serif; font-size: 110%;">''J''</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> and ''<span style="font-family: Georgia,serif; font-size: 110%;">J</span>''<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span> with integer coefficients sharing no common factor:
+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 ''J''1 and ''J''2 with integer coefficients sharing no common factor:
 <math>\qquad b(J_1,J_2) ≈ b_m(J_1,J_2) b_r(J_1,J_2)</math>
@@ Line 158: / Line 158: @@
 ===Examples===
-If the source intervals are the perfect fourth (<span style="font-family: Georgia,serif; font-size: 110%;">''f'' =</span> <u><span style="font-family: Georgia,serif; font-size: 110%;">4/3</span></u>'')'' and the perfect fifth (<span style="font-family: Georgia,serif; font-size: 110%;">''F'' = <u>3/2</u></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">), </span><span style="font-family: Arial,Helvetica,sans-serif;">then</span> <span style="font-family: Georgia,serif; font-size: 110%;">''v''1 = 1/7</span>, <span style="font-family: Georgia,serif; font-size: 110%;">''v''2 = 1/5</span>, and ''<span style="font-family: Georgia,serif; font-size: 110%;">b</span>'' is the Pythagorean comma:
+If the source intervals are the perfect fourth (''f'' = <u>4/3</u>'')'' and the perfect fifth (''F'' = <u>3/2</u>), then ''v''1 = 1/7, ''v''2 = 1/5, and ''b'' is the Pythagorean comma:
 <math>\qquad b(F,f) = b_r(F,f) = \frac{F}{\tfrac{1}{5}} - \frac{f}{\tfrac{1}{7}} = 5F – 7f</math>
-If the source intervals are the perfect fourth (<span style="font-family: Georgia,serif; font-size: 110%;">''f'' = <u>4/3</u></span>) and the minor seventh (<span style="font-family: Georgia,serif; font-size: 110%;">''m''</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">7 </span><span style="font-family: Georgia,serif; font-size: 110%;">= <u>9/5</u>), </span><span style="font-family: Arial,Helvetica,sans-serif;">then </span><span style="font-family: Georgia,serif; font-size: 110%;">''v''</span>1 <span style="font-family: Georgia,serif; font-size: 110%;">= 1/7</span>, <span style="font-family: Georgia,serif; font-size: 110%;">''v''2 = 2/7</span>, ''<span style="font-family: Georgia,serif; font-size: 110%;">b</span>''r <span style="font-family: Georgia,serif; font-size: 110%;">= 2/7</span> and ''<span style="font-family: Georgia,serif; font-size: 110%;">b</span>'' is the syntonic comma:
+If the source intervals are the perfect fourth (''f'' = <u>4/3</u>) and the minor seventh (''m''7 = <u>9/5</u>), then ''v''1 = 1/7, ''v''2 = 2/7, ''b''r = 2/7 and ''b'' is the syntonic comma:
 <math>\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</math>
@@ Line 168: / Line 168: @@
 For further examples of bimodular commas, and a fuller account of bimodular approximant theory, refer to [[:File:Bimod_Approx_2014-6-8.pdf|this paper]]. See also [[Don_Page_comma|Don Page comma]] (another name for this type of comma).
-='''<span style="font-size: 21.33px;">3. Padé approximants of order (1,2)</span>'''=
+= 3. Padé approximants of order (1,2) =
-==<span style="font-family: Arial,Helvetica,sans-serif;">Definition</span>==
+== Definition ==
-In the section on bimodular approximants it was shown than an interval of logarithmic size ''<span style="font-family: Georgia,serif; font-size: 110%;">J</span>'' (measured in dineper units) is related to its bimodular approximant by
+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
 <math>\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...</math>
@@ Line 179: / Line 179: @@
 <math>\qquad v = \frac{r-1}{r+1}</math>
-and ''<span style="font-family: Georgia,serif; font-size: 110%;">r</span>'' is the interval’s frequency ratio.
+and ''r'' is the interval’s frequency ratio.
 Another way to express this relationship is with a continued fraction:
@@ Line 185: / Line 185: @@
 <math>\qquad J = \tanh^{-1}{v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))</math>
-The first convergent of this continued fraction is ''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>'', the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is
+The first convergent of this continued fraction is ''v'', the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is
 <math>\qquad y = \frac{v}{1-v^2/3}</math>
@@ Line 193: / Line 193: @@
 {| class="wikitable"
 |-
-| | ''Interval <span style="font-family: Georgia,serif; font-size: 110%;">J</span>''<span style="color: #ffffff;">###########</span>
+| | ''Interval J''<span style="color: #ffffff;">###########</span>
-| | ''(1,2) Padé approximant <span style="font-family: Georgia,serif; font-size: 110%;">y</span>''<span style="color: #ffffff;">#</span>
+| | ''(1,2) Padé approximant y''<span style="color: #ffffff;">#</span>
 |-
 | | Perfect twelfth = <u>3/1</u>
@@ Line 234: / Line 234: @@
 (<u>3/2</u>) / (<u>20/17</u>) = 2.4949 ≈ (15/74) / (6/74) = 5/2
-='''<span style="font-size: 21.33px;">4. Quadratic approximants</span>'''=
+= 4. Quadratic approximants =
-==<span style="font-family: Arial,Helvetica,sans-serif;">Definition</span>==
+== Definition ==
-The quadratic approximant ''<span style="font-family: Georgia,serif; font-size: 110%;">q</span>'' of an interval ''<span style="font-family: Georgia,serif; font-size: 110%;">J</span>'' with frequency ratio <span style="font-family: Georgia,serif; font-size: 110%;">''r'' = ''n'<nowiki/>'''/d'''''</span>''' is'''
+The quadratic approximant ''q'' of an interval ''J'' with frequency ratio ''r'' = ''n'<nowiki/>'''/d'''''''' is'''
 <math>\qquad q(r) = \tfrac{1}{2} (r^{1/2} – r^{-1/2}) \\
@@ Line 248: / Line 248: @@
 <math>\qquad v = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...</math>
-it is apparent that ''<span style="font-family: Georgia,serif; font-size: 110%;">q</span>'' is about twice as accurate as ''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>'', with an error of opposite sign.
+it is apparent that ''q'' is about twice as accurate as ''v'', with an error of opposite sign.
-While ''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>'' is the frequency difference divided by twice the arithmetic frequency mean, ''<span style="font-family: Georgia,serif; font-size: 110%;">q</span>'' is the frequency difference divided by twice the geometric frequency mean:
+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:
 <math>\qquad q = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}</math>
-''<span style="font-family: Georgia,serif; font-size: 110%;">r</span>'' can be retrieved from ''<span style="font-family: Georgia,serif; font-size: 110%;">q</span>'' using
+''r'' can be retrieved from ''q'' using
 <math>\qquad \sqrt{r} = q + \sqrt{1+q^2}</math>
@@ Line 262: / Line 262: @@
 {| class="wikitable"
 |-
-| | ''Interval'' ''<span style="font-family: Georgia,serif; font-size: 110%;">J</span>''<span style="color: #ffffff;">##################### </span>
+| | ''Interval'' ''J''<span style="color: #ffffff;">##################### </span>
-| | ''Quadratic approximant'' <span style="font-family: Georgia,serif; font-size: 110%;">''q''</span><span style="color: #ffffff; font-family: Georgia,serif; font-size: 110%;"> ##</span>
+| | ''Quadratic approximant'' ''q'' ##
 |-
-| | <span style="font-family: Arial,Helvetica,sans-serif;">Perfect twelfth = <u>3/1</u></span>
+| | Perfect twelfth = <u>3/1</u>
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> 1/√3</span>
+| |  1/√3
 |-
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> Octave = <u>2/1</u></span>
+| |  Octave = <u>2/1</u>
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> 1/2√2</span>
+| |  1/2√2
 |-
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> Minor seventh = <u>9/5</u></span>
+| |  Minor seventh = <u>9/5</u>
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> 2/3√5</span>
+| |  2/3√5
 |-
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> Pythagorean minor seventh = <u>16/9</u></span>
+| |  Pythagorean minor seventh = <u>16/9</u>
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> 7/24</span>
+| |  7/24
 |-
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> Major sixth = <u>5/3</u></span>
+| |  Major sixth = <u>5/3</u>
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> 1/√15</span>
+| |  1/√15
 |-
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> Minor sixth = <u>8/5</u></span>
+| |  Minor sixth = <u>8/5</u>
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> 3/4√10</span>
+| |  3/4√10
 |-
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> Perfect fifth = <u>3/2</u></span>
+| |  Perfect fifth = <u>3/2</u>
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> 1/2√6</span>
+| |  1/2√6
 |-
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> Perfect fourth = <u>4/3</u></span>
+| |  Perfect fourth = <u>4/3</u>
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> 1/4√3</span>
+| |  1/4√3
 |-
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> Major third = <u>5/4</u></span>
+| |  Major third = <u>5/4</u>
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> 1/4√5</span>
+| |  1/4√5
 |-
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> Minor third = <u>6/5</u></span>
+| |  Minor third = <u>6/5</u>
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> 1/2√30</span>
+| |  1/2√30
 |-
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> Pythagorean minor third = <u>32/27</u></span>
+| |  Pythagorean minor third = <u>32/27</u>
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> 5/24√6</span>
+| |  5/24√6
 |-
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> Large tone = <u>9/8</u></span>
+| |  Large tone = <u>9/8</u>
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> 1/12√2</span>
+| |  1/12√2
 |-
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> Small tone = <u>10/9</u></span>
+| |  Small tone = <u>10/9</u>
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> 1/6√10</span>
+| |  1/6√10
 |-
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> Diatonic semitone = <u>16/15</u></span>
+| |  Diatonic semitone = <u>16/15</u>
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> 1/8√15</span>
+| |  1/8√15
 |-
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> Chroma = <u>25/24</u></span>
+| |  Chroma = <u>25/24</u>
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> 1/20√6</span>
+| |  1/20√6
 |-
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> Syntonic comma = <u>81/80</u></span>
+| |  Syntonic comma = <u>81/80</u>
-| | <span style="font-family: Arial,Helvetica,sans-serif;"> 1/72√5</span>
+| |  1/72√5
 |}
-Expressed in terms of the bimodular approximant,''<span style="font-family: Georgia,serif; font-size: 110%;"> v = j/g</span>'',
+Expressed in terms of the bimodular approximant,'' v = j/g'',
 <math>\qquad q = \frac{v}{\sqrt{1-v^2}} = \frac{j}{\sqrt{g^2-j^2}}</math>
-Quadratic approximants of just intervals thus have the form ''<span style="font-family: Georgia,serif; font-size: 110%;">q = j/√k</span>'', where ''<span style="font-family: Georgia,serif; font-size: 110%;">j</span>'' and ''<span style="font-family: Georgia,serif; font-size: 110%;">k</span>'' are integers and ''<span style="font-family: Georgia,serif; font-size: 110%;">j</span>''<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span>''<span style="font-family: Georgia,serif; font-size: 110%;"> + k = g</span>''<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span> is a perfect square.
+Quadratic approximants of just intervals thus have the form ''q = j/√k'', where ''j'' and ''k'' are integers and ''j''2'' + k = g''2 is a perfect square.
-The presence of a square root in the denominator of ''<span style="font-family: Georgia,serif; font-size: 110%;">q</span>'' (except where ''<span style="font-family: Georgia,serif; font-size: 110%;">J</span>'' 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.
+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.
-==<span style="font-family: Arial,Helvetica,sans-serif;">Properties</span>==
+== Properties ==
-If ''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>''<span style="font-family: Georgia,serif; font-size: 110%;">[''J'']</span> and <span style="font-family: Georgia,serif; font-size: 110%;">''q''[''J'']</span> denote, respectively, the bimodular and quadratic approximants of an interval ''<span style="font-family: Georgia,serif; font-size: 110%;">J</span>'' with frequency ratio ''<span style="font-family: Georgia,serif; font-size: 110%;">r</span>'', and ''<span style="font-family: Georgia,serif; font-size: 110%;">q</span>''<span style="font-family: Georgia,serif; font-size: 80%;">n</span> denotes <span style="font-family: Georgia,serif; font-size: 110%;">''q''[''J''n]</span> , then
+If ''v''[''J''] and ''q''[''J''] denote, respectively, the bimodular and quadratic approximants of an interval ''J'' with frequency ratio ''r'', and ''q''n denotes ''q''[''J''n] , then
 <math>\qquad v = \tanh{J}, q = \sinh{J}, \frac{q}{v} = \cosh{J} \\
@@ Line 351: / Line 351: @@
 <math>\qquad \frac {q[J_2 + J_1]}{q[J_2 - J_1]} = \frac{v_2+v_1}{v_2-v_1}</math>
-with <span style="font-family: Georgia,serif; font-size: 110%;">''J''</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2 </span><span style="font-family: Georgia,serif; font-size: 110%;">= ''F'' =<u>3/2</u></span> <span style="font-family: Arial,Helvetica,sans-serif;">and</span> <span style="font-family: Georgia,serif; font-size: 110%;">''J''</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1 </span><span style="font-family: Georgia,serif; font-size: 110%;">= ''f'' = <u>4/3</u></span> this gives
+with ''J''2 = ''F'' =<u>3/2</u> and ''J''1 = ''f'' = <u>4/3</u> this gives
 <math>\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</math>
-The quadratic approximant ''<span style="font-family: Georgia,serif; font-size: 110%;">q</span>'' of a double interval <span style="font-family: Georgia,serif; font-size: 110%;">2''J''</span> (for example, the ditone) is rational, which suggests using <span style="font-family: Georgia,serif; font-size: 110%;">½ ''q''(''r''</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span><span style="font-family: Georgia,serif; font-size: 110%;">)</span> as a rational approximant of ''<span style="font-family: Georgia,serif; font-size: 110%;">J</span>'' (where ''<span style="font-family: Georgia,serif; font-size: 110%;">J</span>'' has frequency ratio ''<span style="font-family: Georgia,serif; font-size: 110%;">r</span>''):
+The quadratic approximant ''q'' of a double interval 2''J'' (for example, the ditone) is rational, which suggests using ½ ''q''(''r''2) as a rational approximant of ''J'' (where ''J'' has frequency ratio ''r''):
 <math>\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 + ...</math>
@@ Line 364: / Line 364: @@
 The most interesting approximate interval ratios derivable from quadratic approximants are irrational.
-==<span style="font-family: Arial,Helvetica,sans-serif;">Relative sizes of intervals between 3 frequencies in arithmetic progression</span>==
+== Relative sizes of intervals between 3 frequencies in arithmetic progression ==
-===<span style="font-family: Arial,Helvetica,sans-serif;">Theorem</span>===
+=== 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.
-===<span style="font-family: Arial,Helvetica,sans-serif;">Remarks</span>===
+=== Remarks ===
-If the harmonics have indices ''<span style="font-family: Georgia,serif; font-size: 110%;">n – m, n</span>'' and ''<span style="font-family: Georgia,serif; font-size: 110%;">n + m</span>'', the two intervals have reduced frequency ratios ''<span style="font-family: Georgia,serif; font-size: 110%;">n/(n – m)</span>'' and ''<span style="font-family: Georgia,serif; font-size: 110%;">(n + m)/n</span>''. It can be assumed that ''<span style="font-family: Georgia,serif; font-size: 110%;">n</span>'' and ''<span style="font-family: Georgia,serif; font-size: 110%;">m</span>'' have no common factor.
+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.
-''<span style="font-family: Georgia,serif; font-size: 110%;">m</span>'' is the [[Superpartient|degree of epimoricity]] of the intervals. When ''<span style="font-family: Georgia,serif; font-size: 110%;">m</span>'' = 1 the intervals are adjacent superparticular intervals.
+''m'' is the [[Superpartient|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.
-===<span style="font-family: Arial,Helvetica,sans-serif;">Proof</span>===
+=== Proof ===
 The ratio of the intervals as estimated from their quadratic approximants is
@@ Line 383: / Line 383: @@
 which is the geometric mean of their frequency ratios.
-===<span style="font-family: Arial,Helvetica,sans-serif;">Examples</span>===
+=== Examples ===
-The ratio of the perfect fifth, <span style="font-family: Georgia,serif; font-size: 110%;">''F'' = <u>3/2</u></span>, to the perfect fourth, <span style="font-family: Georgia,serif; font-size: 110%;">''f'' = <u>4/3</u></span>, 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).
+The ratio of the perfect fifth, ''F'' = <u>3/2</u>, to the perfect fourth, ''f'' = <u>4/3</u>, 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).
-<span style="font-family: Georgia,serif; font-size: 110%;">''F/f'' = 701.955/498.045 = 1.40942,</span>
+''F/f'' = 701.955/498.045 = 1.40942,
-<span style="font-family: Georgia,serif; font-size: 110%;">√2 = 1.41421.</span>
+√2 = 1.41421.
-The ratio of the large tone, <span style="font-family: Georgia,serif; font-size: 110%;">''T'' = <u>9/8</u></span>, to the small tone, <span style="font-family: Georgia,serif; font-size: 110%;">''t'' = <u>10/9</u></span>, 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.
+The ratio of the large tone, ''T'' = <u>9/8</u>, to the small tone, ''t'' = <u>10/9</u>, 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.
-<span style="font-family: Georgia,serif; font-size: 110%;">''T'<nowiki/>'''/t'''''<nowiki/>''' = 203.910/182.404 = 1.11790,'''</span>
+''T'<nowiki/>'''/t'''''<nowiki/>''' = 203.910/182.404 = 1.11790,'''
-<span style="font-family: Georgia,serif; font-size: 110%;">√5/2 = 1.11803.</span>
+√5/2 = 1.11803.
-==<span style="font-family: Arial,Helvetica,sans-serif;">Argent temperament</span>==
+== 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.
@@ Line 407: / Line 407: @@
 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 [http://en.wikipedia.org/wiki/Silver_ratio silver ratio] (sometimes called the silver mean), ''<span style="font-family: Georgia,serif; font-size: 110%;">δ</span>''<span style="vertical-align: sub;">s </span>= <span style="font-family: Georgia,serif; font-size: 110%;">√2 + 1 = 2.4142</span>. On this basis, and by analogy with [[Golden_Meantone|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.
+A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the [http://en.wikipedia.org/wiki/Silver_ratio silver ratio] (sometimes called the silver mean), ''δ''√2 + 1 = 2.4142. On this basis, and by analogy with [[Golden_Meantone|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.
@@ Line 415: / Line 415: @@
 <math>\qquad \delta_s = √2 + 1 = 2 + 1/(2 + 1/(2 + 1/(2 + ...)))</math>
-As a result, if two intervals ''<span style="font-family: Georgia,serif; font-size: 110%;">L</span>'' and ''<span style="font-family: Georgia,serif; font-size: 110%;">s</span>'' are tuned in the silver ratio, with <span style="font-family: Georgia,serif; font-size: 110%;">''s = L/δ''</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span>, subtracting twice the small interval ''<span style="font-family: Georgia,serif; font-size: 110%;">s</span>'' from the large interval ''<span style="font-family: Georgia,serif; font-size: 110%;">L</span>'' leaves a remainder of size <span style="font-family: Georgia,serif; font-size: 110%;">''s/δ''</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span>:
+As a result, if two intervals ''L'' and ''s'' are tuned in the silver ratio, with ''s = L/δ''s, subtracting twice the small interval ''s'' from the large interval ''L'' leaves a remainder of size ''s/δ''s:
 <math>\qquad L – 2s = (\delta_s – 2)s = s/\delta_s</math>
-(since 1''/<span style="font-family: Georgia,serif; font-size: 110%;">δ</span>''<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s </span><span style="font-family: Georgia,serif; font-size: 110%;">= √2 - 1 = ''δ''</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span><span style="font-family: Georgia,serif; font-size: 110%;"> - 2</span>) 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 <u><span style="font-family: Arial,Helvetica,sans-serif;">256/243</span></u>) followed by tempered and just sizes in cents:
+(since 1''/δ''s = √2 - 1 = ''δ''s - 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 <u>256/243</u>) followed by tempered and just sizes in cents:
 {| class="wikitable"
@@ Line 491: / Line 491: @@
 The following relationships hold in the table, the first two being valid for the pure intervals as well as their tempered counterparts:
-<ul><li>Subtracting twice an interval from the interval on its left generates the interval on its right.</li><li>An interval in the second row is the sum of the interval immediately above and the interval diagonally above and to the right.</li><li>Adjacent horizontal pairs have ratio ''<span style="font-family: Georgia,serif; font-size: 110%;">δ</span>''<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">''s'' </span><span style="font-family: Georgia,serif; font-size: 110%;">= √2 + 1.</span></li><li>Adjacent vertical pairs have ratio <span style="font-family: Georgia,serif; font-size: 110%;">√2</span>.</li><li>Extending the table to a third row yields consisting of the intervals in the first row multiplied by 2, and so on.</li></ul>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.
+<ul><li>Subtracting twice an interval from the interval on its left generates the interval on its right.</li><li>An interval in the second row is the sum of the interval immediately above and the interval diagonally above and to the right.</li><li>Adjacent horizontal pairs have ratio ''δ''''s'' = √2 + 1.</li><li>Adjacent vertical pairs have ratio √2.</li><li>Extending the table to a third row yields consisting of the intervals in the first row multiplied by 2, and so on.</li></ul>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 ''<span style="font-family: Georgia,serif; font-size: 110%;">δ</span>''<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">''s'' </span><span style="font-family: Georgia,serif; font-size: 110%;">= √2 + 1</span> produces another interval in the temperament. Any tempered interval ''<span style="font-family: Georgia,serif; font-size: 110%;">J’</span>'' can be split into three parts, two of equal size ''<span style="font-family: Georgia,serif; font-size: 110%;">J’</span>''<span style="font-family: Georgia,serif; font-size: 110%;">/</span>''<span style="font-family: Georgia,serif; font-size: 110%;">δ</span>''<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span> and the other of size ''<span style="font-family: Georgia,serif; font-size: 110%;">J’</span>''<span style="font-family: Georgia,serif; font-size: 110%;">/</span>''<span style="font-family: Georgia,serif; font-size: 110%;">δ</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span>''.
+In this fractal temperament, multiplying or dividing any interval by the factor ''δ''''s'' = √2 + 1 produces another interval in the temperament. Any tempered interval ''J’'' can be split into three parts, two of equal size ''J’''/''δ''s 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.
@@ Line 509: / Line 509: @@
 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 (s<span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;">p</span>), Pythagorean comma (p) and 29-tone comma (p<span style="font-size: 60%;">29</span>) 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 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.
 [[File:Continued_fraction_jigsaw_41edo.png|alt=Continued fraction jigsaw 41edo.png|800x396px|Continued fraction jigsaw 41edo.png]]
@@ Line 515: / Line 515: @@
 <span style="color: #ffffff;">######</span>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, m<span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;">p</span><span style="color: #ffffff;">#</span>= Pythagorean minor third, s<span style="font-size: 80%; vertical-align: super;">p</span><span style="color: #ffffff;">#</span>= Pythagorean limma, X<span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;">p</span><span style="color: #ffffff;">#</span>= Pythagorean apotome, p = Pythagorean comma.
+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, mppp<span style="color: #ffffff;">#</span>= Pythagorean apotome, p = Pythagorean comma.
 [[File:Silver_temperament_graphic.png|alt=Silver temperament graphic.png|800x587px|Silver temperament graphic.png]]
@@ Line 535: / Line 535: @@
 If these 7-limit intervals are considered to be tempered to their 3-limit counterparts argent is an example of hemifamity temperament. Hemifamity (<u>3120/3103</u>) is the bimodular comma formed from <u>10/7</u> and <u>9/8</u>
-By the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem Gelfond-Schneider theorem] the frequency ratios of all argent intervals (<span style="font-family: Georgia,serif; font-size: 110%;">''r'' = 2</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">√2''a''+''b''</span>, where''<span style="font-family: Georgia,serif; font-size: 110%;"> a</span>'' and ''<span style="font-family: Georgia,serif; font-size: 110%;">b</span>'' are integers) are transcendental, with the exception of octave multiples (<span style="font-family: Georgia,serif; font-size: 110%;">''a'' = 0</span>). The frequency ratio of the tempered perfect eleventh (<span style="font-family: Georgia,serif; font-size: 110%;"><u>8/3</u> = <u>2.6666...</u></span>) is the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant Gelfond-Schneider constant] or Hilbert number, <span style="font-family: Georgia,serif; font-size: 110%;">2</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">√2</span><span style="font-family: Georgia,serif; font-size: 110%;"> = 2.665144</span>...
+By the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem Gelfond-Schneider theorem] the frequency ratios of all argent intervals (''r'' = 2√2''a''+''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 (<u>8/3</u> = <u>2.6666...</u>) is the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant 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'' <span style="font-family: Georgia,serif; font-size: 110%;">= <u>9/8</u></span>) to the small tone (<span style="font-family: Georgia,serif; font-size: 110%;">''t'' = <u>10/9</u></span>) is closely approximated by
+It has been shown in an example above that the ratio of the large tone (''T'' = <u>9/8</u>) to the small tone (''t'' = <u>10/9</u>) is closely approximated by
 <math>\qquad T/t = \sqrt{5}/2</math>
@@ Line 546: / Line 546: @@
 <math>\qquad (T + t/2)/t = (\sqrt{5}+1)/2 = \phi</math>
-where <span style="font-family: Georgia,serif; font-size: 110%;">''ϕ'' = 1.61803</span>... is the golden ratio.
+where ''ϕ'' = 1.61803... is the golden ratio.
-If a Fibonacci sequence of intervals is formed from the pair of intervals <span style="font-family: Georgia,serif; font-size: 110%;">''T'' – ''t''/2</span> and ''<span style="font-family: Georgia,serif; font-size: 110%;">t</span>'', and extended in both directions, it can thus be expected that the ratios between successive intervals in this sequence will also be close to ''<span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span>''. The sequence formed in this way is Sequence 1 in the following table.
+If a Fibonacci sequence of intervals is formed from the pair of intervals ''T'' – ''t''/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.
 {| class="wikitable"
 |-
 | | Sequence 1:<span style="color: #ffffff;">#</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff;">#</span>''t''/2 - 3''c''<span style="color: #ffffff; font-family: Georgia,serif;">#</span> </span>
+| | <span style="color: #ffffff;">#''t''/2 - 3''c''# </span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>2''c''</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#2''c''</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>''t''/2 ''- c''</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#''t''/2 ''- c''</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>''T - t''/2 </span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#''T - t''/2 </span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>''t''</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#''t''</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>''T + t''/2<span style="color: #ffffff; font-family: Georgia,serif;">#</span></span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#''T + t''/2#</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>''M + t''/2<span style="color: #ffffff; font-family: Georgia,serif;">#</span> </span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#''M + t''/2# </span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>2''M'' </span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#2''M'' </span>
 |-
 | | Sequence 2:<span style="color: #ffffff;">#</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>''magic''</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#''magic''</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>''diesis''</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#''diesis''</span>
-| | ''<span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>chroma<span style="color: #ffffff; font-family: Georgia,serif;">#</span></span>''
+| | ''<span style="color: #ffffff; font-family: Georgia,serif;">#chroma#</span>''
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>''semitone''<span style="color: #ffffff; font-family: Georgia,serif;">#</span></span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#''semitone''#</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>''t''</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#''t''</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>''mp''</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#''mp''</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>''f - c''</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#''f - c''</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>''m6p - c''</span><span style="color: #ffffff;">#</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#''m6p - c''</span><span style="color: #ffffff;">#</span>
 |-
 | | Difference:
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>-3''σ''/2</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#-3''σ''/2</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>''σ''</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#''σ''</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>-''σ''/2</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#-''σ''/2</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>''σ''/2</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#''σ''/2</span>
-| | <span style="color: #ffffff; font-family: Georgia,serif;">#</span>0
+| | #0
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>''σ''/2</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#''σ''/2</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>''σ''/2</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#''σ''/2</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>''σ'' </span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#''σ'' </span>
 |-
 | | Seq 1 ratios:
 | |
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6120</span><span style="color: #ffffff; font-family: Georgia,serif;">##</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#1.6120</span>##
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6204</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#1.6204</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6171</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#1.6171</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6184<span style="color: #ffffff; font-family: Georgia,serif;">#</span> </span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#1.6184# </span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6179</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#1.6179</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6181</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#1.6181</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6180</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#1.6180</span>
 |-
 | | Seq 2 ratios:
 | |
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.3865</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#1.3865</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.7212</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#1.7212</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.5810</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#1.5810</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6325</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#1.6325</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6125</span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#1.6125</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6201 </span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#1.6201 </span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6172 </span>
+| | <span style="color: #ffffff; font-family: Georgia,serif;">#1.6172 </span>
 |}
-where <span style="font-family: Georgia,serif; font-size: 110%;">''f'' = <u>4/3</u>, ''T'' = <u>9/8</u>, ''t'' = <u>10/9</u>, ''M'' = <u>5/4</u>, ''magic'' = <u>3125/3072</u>, ''diesis'' = <u>128/125</u>, ''chroma'' = <u>25/24</u>, ''semitone'' = <u>16/15</u>, ''mp'' = <u>32/27</u>, ''c'' = ''syntonic comma'' = <u>81/80</u>, ''m6p'' = <u>128/81</u>, ''σ'' = ''schisma'' = <u>32805/32768.</u></span>
+where ''f'' = <u>4/3</u>, ''T'' = <u>9/8</u>, ''t'' = <u>10/9</u>, ''M'' = <u>5/4</u>, ''magic'' = <u>3125/3072</u>, ''diesis'' = <u>128/125</u>, ''chroma'' = <u>25/24</u>, ''semitone'' = <u>16/15</u>, ''mp'' = <u>32/27</u>, ''c'' = ''syntonic comma'' = <u>81/80</u>, ''m6p'' = <u>128/81</u>, ''σ'' = ''schisma'' = <u>32805/32768.</u>
-The ratios between successive intervals in Sequence 1 are shown in the row labelled ‘Seq 1 ratios’, and are indeed close to ''<span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span>''.
+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 (''<span style="font-family: Georgia,serif; font-size: 110%;">σ</span>''), as indicated by the row marked ‘Difference’ (which is itself a Fibonacci sequence).
+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 ''<span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span>'' rather less accurately.
+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 ''<span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span>'' would be ‘golden temperaments’.
+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 ''<span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span>'' while tuning the octave pure and tempering out the syntonic comma yields [[Golden_Meantone|golden meantone]] temperament.
+Tempering the Sequence 2 ratios to ''ϕ'' while tuning the octave pure and tempering out the syntonic comma yields [[Golden_Meantone|golden meantone]] temperament.
-Tempering the Sequence 1 ratios to ''<span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span>'' 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 ''<span style="font-family: Georgia,serif; font-size: 110%;">s, t</span>'', ''<span style="font-family: Georgia,serif; font-size: 110%;">M</span>'' and <span style="font-family: Georgia,serif; font-size: 110%;">''m''=<u>6/5</u></span> are all ±0.02106 cents.
+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''=<u>6/5</u> 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 ''<span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span>'' 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).
+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).
-==<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%; vertical-align: sub;">Pythagorean triples of quadratic approximants</span>==
+== Pythagorean triples of quadratic approximants ==
-If the quadratic approximants <span style="font-family: Georgia,serif; font-size: 110%;">''q''1'', q''2</span> and <span style="font-family: Georgia,serif; font-size: 110%;">''q''3</span> of a set of three intervals <span style="font-family: Georgia,serif; font-size: 110%;">''J''1, ''J''2</span> and ''<span style="font-family: Georgia,serif; font-size: 110%;">J</span>''3 satisfy
+If the quadratic approximants ''q''1'', q''2 and ''q''3 of a set of three intervals ''J''1, ''J''2 and ''J''3 satisfy
 <math>\qquad q_1^2 + q_2^2 = q_3^2</math>
@@ Line 625: / Line 625: @@
 they can be said to form a [http://en.wikipedia.org/wiki/Pythagorean_triple Pythagorean triple].
-The following are three examples. In the first and third cases, their counterparts in 12edo, <span style="font-family: Georgia,serif; font-size: 110%;">''J''1', ''J''2'</span> and <span style="font-family: Georgia,serif; font-size: 110%;">''J''3'</span>, are also Pythagorean triples:
+The following are three examples. In the first and third cases, their counterparts in 12edo, ''J''1', ''J''2' and ''J''3', are also Pythagorean triples:
 {| class="wikitable"
 |-
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; line-height: 0px; overflow: hidden;">#</span>''J''1</span>
+| | <span style="color: #ffffff; line-height: 0px; overflow: hidden;">#''J''1</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>''J''2</span>
+| | <span style="color: #ffffff; font-family: Georgia;">#''J''2</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>''J''3</span>
+| | <span style="color: #ffffff; font-family: Georgia;">#''J''3</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>''q''1</span>
+| | <span style="color: #ffffff; font-family: Georgia;">#''q''1</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>''q''2</span>
+| | <span style="color: #ffffff; font-family: Georgia;">#''q''2</span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>''q''3 </span>
+| | <span style="color: #ffffff; font-family: Georgia;">#''q''3 </span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>''J''1' </span>
+| | <span style="color: #ffffff; font-family: Georgia;">#''J''1' </span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>''J''2' </span>
+| | <span style="color: #ffffff; font-family: Georgia;">#''J''2' </span>
-| | <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>''J''3'<span style="color: #ffffff; font-family: Georgia;">#</span></span>
+| | <span style="color: #ffffff; font-family: Georgia;">#''J''3'#</span>
 |-
-| | <span style="color: #ffffff; font-family: Georgia;">#</span><u>6/5</u><span style="color: #ffffff;">#</span>
+| | #<u>6/5</u><span style="color: #ffffff;">#</span>
-| | <span style="color: #ffffff; font-family: Georgia;">#</span><u>5/4</u>
+| | #<u>5/4</u>
-| | <span style="color: #ffffff; font-family: Georgia;">#</span><u>4/3</u><span style="color: #ffffff; font-family: Georgia;">#</span>
+| | #<u>4/3</u>#
-| | <span style="color: #ffffff; font-family: Georgia;">#</span>1/2√30<span style="color: #ffffff; font-family: Georgia;">#</span>
+| | #1/2√30#
-| | <span style="color: #ffffff; font-family: Georgia;">#</span>1/4√5
+| | #1/4√5
-| | <span style="color: #ffffff; font-family: Georgia;">#</span>1/4√3
+| | #1/4√3
-| | <span style="color: #ffffff; font-family: Georgia;">#</span>3
+| | #3
-| | <span style="color: #ffffff; font-family: Georgia;">#</span>4
+| | #4
-| | <span style="color: #ffffff; font-family: Georgia;">#</span>5
+| | #5
 |-
-| | <span style="color: #ffffff; font-family: Georgia;">#</span><u>4/3</u>
+| | #<u>4/3</u>
-| | <span style="color: #ffffff; font-family: Georgia;">#</span><u>12/5</u><span style="color: #ffffff; font-family: Georgia;">#</span>
+| | #<u>12/5</u>#
-| | <span style="color: #ffffff; font-family: Georgia;">#</span><u>5/2</u><span style="color: #ffffff; font-family: Georgia;">#</span>
+| | #<u>5/2</u>#
-| | <span style="color: #ffffff; font-family: Georgia;">#</span>1/4√3
+| | #1/4√3
-| | <span style="color: #ffffff; font-family: Georgia;">#</span>7/4√15<span style="color: #ffffff; font-family: Georgia;">#</span>
+| | #7/4√15#
-| | <span style="color: #ffffff; font-family: Georgia;">#</span>3/2√10<span style="color: #ffffff; font-family: Georgia;">#</span>
+| | #3/2√10#
 | |
 | |
 | |
 |-
-| | <span style="color: #ffffff; font-family: Georgia;">#</span><u>8/5</u>
+| | #<u>8/5</u>
-| | <span style="color: #ffffff; font-family: Georgia;">#</span><u>12/5</u>
+| | #<u>12/5</u>
-| | <span style="color: #ffffff; font-family: Georgia;">#</span><u>8/3</u>
+| | #<u>8/3</u>
-| | <span style="color: #ffffff; font-family: Georgia;">#</span>3/4√10
+| | #3/4√10
-| | <span style="color: #ffffff; font-family: Georgia;">#</span>7/4√15
+| | #7/4√15
-| | <span style="color: #ffffff; font-family: Georgia;">#</span>5/4√6
+| | #5/4√6
-| | <span style="color: #ffffff; font-family: Georgia;">#</span>8
+| | #8
-| | <span style="color: #ffffff; font-family: Georgia;">#</span>15
+| | #15
-| | <span style="color: #ffffff; font-family: Georgia;">#</span>17
+| | #17
 |}
@@ Line 675: / Line 675: @@
 It can be shown, using a suitable [[Comma-based_lattices|comma-based lattice]], that every comma tempered out by 34edo can be expressed as an integer linear combination of the [[Gammic_node|//gammic// comma]] |-29 -11 20&gt; (4.769 cents) and the ''semisuper'' comma (AKA ''[[vishnuzma|vishnuzma]]'') |23 6 -14&gt; (3.338 cents). In particular,
-<span style="color: #333333;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span>''selenia'' = 7 ''gammic'' – 10 ''semisuper''</span>
+###''selenia'' = 7 ''gammic'' – 10 ''semisuper''</span>
 <span style="color: #333333;">So to prove that ''selenia'' is small we must show that ''gammic'<nowiki/>'''/semisuper'''''<nowiki/>''' ≈ 10/7.'''</span>
@@ Line 681: / Line 681: @@
 <span style="color: #333333;">''Gammic'' and ''semisuper'' are both bimodular commas:</span>
-<span style="color: #333333;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span>''gammic'' = </span><span style="color: #333333; font-family: Georgia,serif; font-size: 110%;">''b''(<u>6/5</u>,<u>5/4</u>)</span>
+###''gammic'' = </span>''b''(<u>6/5</u>,<u>5/4</u>)
-<span style="color: #333333;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span>''semisuper'' = </span><span style="color: #333333; font-family: Georgia,serif; font-size: 110%;">''b''(<u>25/24</u>,<u>4/3</u>)</span>
+###''semisuper'' = </span>''b''(<u>25/24</u>,<u>4/3</u>)
-<span style="font-family: Arial,Helvetica,sans-serif;">Using a result given in the section on bimodular commas, the size of </span><span style="font-family: Georgia,serif; font-size: 110%;">''b''(''J''1,''J''2)</span><span style="color: #333333;"> can be estimated using</span>
+Using a result given in the section on bimodular commas, the size of ''b''(''J''1,''J''2)<span style="color: #333333;"> can be estimated using</span>
 <math>\qquad b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 – J_1^2) b_m</math>
-<span style="color: #333333;">Estimating </span><span style="color: #333333; font-family: Georgia,serif; font-size: 110%;">''J''2</span><span style="color: #333333;"> and </span><span style="color: #333333; font-family: Georgia,serif; font-size: 110%;">''J''1</span><span style="color: #333333;"> with their quadratic approximants we then have</span>
+''J''2''J''1<span style="color: #333333;"> with their quadratic approximants we then have</span>
 <math>\qquad b(J_1,J_2) ≈ \frac{1}{3} (q_2^2 – q_1^2) b_m</math>
-<span style="font-family: Arial,Helvetica,sans-serif;">For ''gammic'':</span>
+For ''gammic'':
-<span style="font-family: Georgia,serif; font-size: 110%;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span>''J''₁= 6/5, ''J''₂= 5/4</span>
+<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###''J''₁= 6/5, ''J''₂= 5/4</span>
-<span style="font-family: Georgia,serif; font-size: 110%;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span>''v''</span>₁ <span style="font-family: Georgia,serif; font-size: 110%;">= 1/11, </span>''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>''₂ <span style="font-family: Georgia,serif; font-size: 110%;">= 1/9, </span>''<span style="font-family: Georgia,serif; font-size: 110%;">b</span>''<span style="font-family: Georgia,serif; font-size: 60%;">m</span><span style="font-family: Georgia,serif; font-size: 110%;"> = 1</span>
+<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###''v''</span>₁ = 1/11, ''v''₂ = 1/9, ''b''m = 1
-<span style="font-family: Georgia,serif; font-size: 110%;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span>''q''</span><span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">₁² = </span><span style="font-family: Georgia,serif; font-size: 110%;">(1/4)(1/30),</span> ''<span style="font-family: Georgia,serif; font-size: 110%;">q</span>''<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">₂''² ='' </span><span style="font-family: Georgia,serif; font-size: 110%;">(1/4)(1/20)</span>
+<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###''q''</span>₁² = (1/4)(1/30), ''q''₂''² ='' (1/4)(1/20)
-<span style="font-family: Arial,Helvetica,sans-serif;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span>''gammic'' = </span><span style="font-family: Georgia,serif; font-size: 110%;">''b''(''J''</span>₁<span style="font-family: Georgia,serif; font-size: 110%;">,''J''</span>₂<span style="font-family: Georgia,serif; font-size: 110%;">) ≈ (1/12) (1/30 – 1/20) = (1/12) (1/60)</span>
+<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###''gammic'' = </span>''b''(''J''₁,''J''₂) ≈ (1/12) (1/30 – 1/20) = (1/12) (1/60)
-<span style="font-family: Arial,Helvetica,sans-serif;">For ''semisuper:''</span>
+For ''semisuper:''
-<span style="font-family: Georgia,serif; font-size: 110%;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span>''J''₁= 25/24, ''J''₂= 4/3</span>
+<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###''J''₁= 25/24, ''J''₂= 4/3</span>
-<span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span>''v''</span>₁ <span style="font-family: Georgia,serif; font-size: 110%;">= 1/49, </span>''<span style="font-family: Georgia,serif;">v</span>''₂ <span style="font-family: Georgia,serif; font-size: 110%;">= 1/7, </span>''<span style="font-family: Georgia,serif;">b</span>''<span style="font-family: Georgia,serif; font-size: 60%;">m</span><span style="font-family: Georgia,serif; font-size: 110%;"> = 1/7</span>
+<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###''v''</span>₁ = 1/49, ''v''₂ = 1/7, ''b''m = 1/7
-<span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span>''q''</span><span style="font-family: Arial,Helvetica,sans-serif;">₁² = </span><span style="font-family: Georgia,serif;">(1/4)(1/600),</span> ''<span style="font-family: Georgia,serif; font-size: 110%;">q</span>''<span style="font-family: Arial,Helvetica,sans-serif;">₂''² ='' </span><span style="font-family: Georgia,serif;">(1/4)(1/12)</span>
+<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###''q''</span>₁² = (1/4)(1/600), ''q''₂''² ='' (1/4)(1/12)
-<span style="font-family: Arial,Helvetica,sans-serif;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span>''semisuper'' = </span><span style="font-family: Georgia,serif; font-size: 110%;">''b''(''J''</span>₁<span style="font-family: Georgia,serif; font-size: 110%;">,''J''</span>₂<span style="font-family: Georgia,serif; font-size: 110%;">) ≈ (1/12) (1/12 – 1/600)(1/7) = (1/12) (7/600)</span>
+<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###''semisuper'' = </span>''b''(''J''₁,''J''₂) ≈ (1/12) (1/12 – 1/600)(1/7) = (1/12) (7/600)
-<span style="font-family: Arial,Helvetica,sans-serif;">Therefore</span>
+Therefore
-<span style="color: #ffffff;">###</span><span style="font-family: Arial,Helvetica,sans-serif;">''gammic/semisuper'' ≈ </span><span style="font-family: Georgia,serif; font-size: 110%;">10/7</span>
+''gammic/semisuper'' ≈ 10/7
 <span style="color: #333333;">as required.</span>
@@ Line 724: / Line 724: @@
 \qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{5} (q_1^2 + q_2^2) ) b_m</math>
-<span style="color: #333333;">So to improve our estimates of </span><span style="font-family: Georgia,serif; font-size: 110%;">''b''(''J''1,''J''2)</span> <span style="color: #333333;">we should multiply them by</span>
+''b''(''J''1,''J''2) <span style="color: #333333;">we should multiply them by</span>
 <math>\qquad f = 1 – \tfrac{2}{5} (q_1^2 + q_2^2)</math>
@@ Line 732: / Line 732: @@
 <math>\qquad \frac{gammic}{semisuper} ≈ \frac{10 f_{gamma}} {7 f_{semisuper}}</math>
-<span style="font-family: Arial,Helvetica,sans-serif;">from which it follows that</span>
+from which it follows that
-<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><span style="font-family: Georgia,serif; font-size: 110%;">''selenia'' = 7 ''gammic'' - 10 ''semisuper''</span>
+###''selenia'' = 7 ''gammic'' - 10 ''semisuper''
-<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">######## </span><span style="font-family: Georgia,serif; font-size: 110%;">≈ 7 ''gammic''<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"> (''f''</span></span>''<span style="font-family: Georgia,serif; font-size: 70%; vertical-align: sub;">gammic</span><span style="font-family: Georgia,serif; font-size: 110%;"> - f</span><span style="font-size: 70%; vertical-align: sub;">semisuper</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">)</span><span style="font-family: Georgia,serif; font-size: 110%;">/f</span><span style="font-size: 70%; vertical-align: sub;">gammic</span>''
+######## ≈ 7 ''gammic''<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"> (''f''</span>''gammic - fsemisuper)/fgammic''
-<span style="font-family: Arial,Helvetica,sans-serif;">Putting in the numbers:</span>
+Putting in the numbers:
-''<span style="color: #ffffff;">###</span><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">gammic </span>''<span style="font-family: Georgia,serif; font-size: 110%;">=</span> <span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1 – (2/5) (1/4) (1/30 + 1/20) = 1 – 1/120</span>
+''fgammic ''= 1 – (2/5) (1/4) (1/30 + 1/20) = 1 – 1/120
-<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span>''<span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">semisuper </span>''<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">= 1 – (2/5)(1/4) (1/600 + 1/12) = 1 – (1/120) (51/50)</span>
+###''fsemisuper ''= 1 – (2/5)(1/4) (1/600 + 1/12) = 1 – (1/120) (51/50)
-<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span>''<span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">gammic</span> <span style="font-family: Georgia,serif;">- </span><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">semisuper </span>''<span style="font-family: Georgia,serif; font-size: 110%;">= 1/6000</span>
+###''fgammic - fsemisuper ''= 1/6000
 <span style="color: #333333;">Therefore</span>
-<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><span style="font-family: Georgia,serif; font-size: 110%;">''selenia'' ≈ 7 ''gammic'' (1/6000) (120/119) = ''gammic''/850 = 0.00561</span><span style="color: #333333;"> cents</span>
+###''selenia'' ≈ 7 ''gammic'' (1/6000) (120/119) = ''gammic''/850 = 0.00561<span style="color: #333333;"> cents</span>
-<span style="color: #333333;">which </span>is within 20% of the accurate value, 0.00476 cents. (The discrepancy is due to the influence of terms in ''<span style="font-family: Georgia,serif; font-size: 110%;">q</span>''<span style="font-family: Georgia,serif; font-size: 80%; vertical-align: super;">6</span>'','' which become significant when the ''<span style="font-family: Georgia,serif; font-size: 110%;">f</span>'' values are very similar.)
+q''6'','' 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:
@@ Line 756: / Line 756: @@
 <math>\qquad \left( q(\tfrac{6}{5}) \right)^2 + \left( q(\tfrac{5}{4}) \right)^2 = \left( q(\tfrac{4}{3}) \right)^2</math>
-and <span style="font-family: Georgia,serif; font-size: 110%;">(''q''(25/24))</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span> , being small in comparison to the other terms, compromises this equality only slightly.
+and (''q''(25/24))2 , being small in comparison to the other terms, compromises this equality only slightly.
 =Sources and acknowledgements=