Logarithmic approximants: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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<h4>Original Wikitext content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=1. Introduction=

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=**1. Introduction**=

~~The term~~ //logarithmic approximant// (or //approximant// for short) ~~denotes~~ 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:

* Why do certain temperaments such as 12edo provide a good approximation to 5-limit just intonation?

* Why are certain commas small, and roughly how small are they?

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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{3v}{3-v^2}

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

[[math]]

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

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

[[math]]

where ~~//octave// = __2/1__,~~ //large tone// = __9/8__.

where //large tone// = __9/8__.

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

[[math]]

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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===

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.

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###Perfect fourth = __4/3__ = 497.056 cents

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

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

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

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

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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 = ~~\frac{~~s}{\delta_s}

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

[[math]]

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) followed by tempered and just sizes in cents:

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* Adjacent vertical pairs have ratio √2.

* Extending the table to a third row yields consisting of the intervals in the first row multiplied by 2, and so on.

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

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 have comparable magnitude – highly desirable properties for the formation of musical scales.

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

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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 silver temperament. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.

[[image:Continued fraction jigsaw 41edo.png width="800" height="396"]]

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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.

|| Sequence 1:# || #//t/2 - 3c//# || #//2////c// || #//t/2 - c// || #//T - t/2// || #//t// || #//T + t/2// || #//M + t/2// || #//2M// ||

|| Sequence 1:# || #//t///2 - 3//c//# || #2//c// || #//t///2 //- c// || #//T - t///2 || #//t// || #//T + t///2 || #//M + t///2 || #2//M// ||

|| Sequence 2:# || #magic || #diesis || #chroma# || #semitone# || #//t// || #//mp// || #//f - c// || #//m6p - c//# ||

|| Sequence 2:# || #//magic// || #//diesis// || //#chroma#// || #//semitone//# || #//t// || #//mp// || #//f - c// || #//m6p - c//# ||

|| Difference: || #-3σ/2 || #σ || #-σ/2 || #σ/2~~ ||~~ #0 </span> || #σ/2 || #σ/2 || #σ ||

|| Difference: || #-3//σ///2 || #//σ// || #-//σ///2 || #//σ///2 || #0 || #//σ///2 || #//σ///2 || #//σ// ||

|| Seq 1 ratios: || || #1.6120## || #1.6204 || #1.6171 || #1.6184# || #1.6179 || #1.6181 || #1.6180 ||

|| Seq 2 ratios: || || #1.3865 || #1.7212 || #1.5810 || #1.6325 || #1.6125 || #1.6201 || #1.6172 ||

where f = 4/3, T = 9/8, t = 10/9, M = 5/4, magic = ~~3125~~/~~3072~~, diesis = ~~128~~/~~125~~, semitone = 16/15, mp = 32/27, c = syntonic comma = 81/80, m6p = ~~128~~/81, σ = schisma = ~~32805~~/~~327__68~~.__

where //f// = __4/3__, //T// = __9/8__, //t// = __10/9__, //M// = __5/4__, //magic// = __3125/3072__, //diesis// = __128/125__, //chroma// = __25/24__, //semitone// = __16/15__, //mp// = __32/27__, //c// = //syntonic comma// = __81/80__, //m6p// = __128/81__, //σ// = //schisma// = __32805/32768.__

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

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

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

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

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

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they can be said to form a 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:

|| #//J//1 || #~~~~//J//2 || #/~~/J//3 || <~~span style="font-family: Georgia;">#</~~span>//q//1 || <~~span style="font-family: Georgia;">#</~~span>//q//2 || <~~span style="font-family: Georgia;">#</~~span>//q//3 || <~~span style="font-family: Georgia;">#</~~span>//J//1' || <~~span style="font-family: Georgia;">#</~~span>//J//2' || #~~//J//3'# ||

|| #//J//1 || #//J//2 || #//J//3 || #//q//1 || #//q//2 || #//q//3 || #//J//1' || #//J//2' || #//J//3'# ||

|| #~~<~~/~~span>6/5~~# || #</span~~>5/4 || #~~<~~/~~span>4/3~~#</span~~> || #~~1/2√30~~#</span~~> || #~~1/4√5 ||~~ #~~1/4√3 ||~~ #~~<~~/~~span>3~~ || #~~4 ||~~ #~~5 ||

|| #__6/5__# || #__5/4__ || #__4/3__# || #1/2√30# || #1/4√5 || #1/4√3 || #3 || #4 || #5 ||

|| #4/3 || #~~<~~/~~span>12/5~~#~~<~~/~~span>~~ || #5/2# || #1/4√3 || #7/4√15# || #3/2√10# || || || ||

|| #__4/3__ || #__12/5__# || #__5/2__# || #1/4√3 || #7/4√15# || #3/2√10# || || || ||

|| #8/5 || #12/5 || #8/3 || #3/4√10 || #7/4√15 || #5/4√6 || #~~~~8 || #~~~~15 || #~~~~17 ||

|| #__8/5__ || #__12/5__ || #__8/3__ || #3/4√10 || #7/4√15 || #5/4√6 || #8 || #15 || #17 ||

==A small 34edo comma==

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The minute size of this comma can be explained using quadratic approximants.

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// comma |-29 -11 20> (4.769 cents) and the //semisuper// comma |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/////semisuper// ≈ 10/7.

###//Gammic// and //semisuper// are both __bimodular commas__:

//Gammic// and //semisuper// are both __bimodular commas__:

###~~~~//gammic// = //b//(__6/5__,__5/4__)

###//gammic// = //b//(__6/5__,__5/4__)

###~~~~//semisuper// = //b//(__25/24__,__4/3__)

###//semisuper// = //b//(__25/24__,__4/3__)

Using a result given in the section on bimodular commas, the size of the bimodular comma //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

[[math]]

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

[[math]]

For //gammic//:

###~~~~//J//₁= 6/5, //J//₂= 5/4

###//J//₁= 6/5, //J//₂= 5/4

###~~~~//v//</span~~>₁ ~~~~= 1/11, //v//~~~~₂ ~~= 1/9, //b//m = 1

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

###~~~~//q//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// = //b//(//J//</span~~>₁~~~~,//J//</span~~>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://

~~and 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//</span~~>₁ ~~~~= 1/49, //v//~~~~₂ ~~= 1/7, //b//m = 1/7

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

###//semisuper// = //b//(//J//</span~~>₁~~~~,//J//</span~~>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

###//gammic/semisuper// ≈ 10/7

as required.

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

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

[[math]]

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

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

[[math]]

\qquad f = 1 – \tfrac{2}{15} (q_1^2 + q_2^2)

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

[[math]]

Thus a better estimate for //gammic/semisuper// is

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from which it follows that

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

####### ##≈ 7 //gammic// (~~1 -~~ //f////~~semisuper~~// / //f////~~gammic~~//)

######## ≈ 7 //gammic// (//f////gammic// - //f////semisuper//)///fgammic//

Putting in the numbers ~~we find~~

Putting in the numbers:

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

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

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

Therefore

###//selenia// ≈ 7 (1/6000) (120/119) ~~//gammic//~~ = //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.)

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:

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

=Source=

=Source=

This article is based on original research by Martin Gough. See [[file:Bimod Approx 2014-6-8.pdf|this paper]] for a fuller account of bimodular approximants.</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Logarithmic approximants</title></head><body><h1 id="toc0"><a name="x1. Introduction"></a>1. Introduction</h1>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Logarithmic approximants</title></head><body><h1 id="toc0"><a name="x1. Introduction"></a>1. Introduction</h1>

~~The term~~ logarithmic approximant (or approximant for short) ~~denotes~~ 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>

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

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<!-- ws:start:WikiTextMathRule:25:

[[math]]&lt;br/&gt;

\qquad y = \frac{3v}{3-v^2}&lt;br/&gt;[[math]]

\qquad y = \frac{v}{1-v^2/3}&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad y = \frac{3v}{3-v^2}</script>

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

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

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\qquad \frac{octave}{large \, tone} ≈ \frac{1}{2√2} / \frac{1}{12√2} = 6&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad \frac{octave}{large \, tone} ≈ \frac{1}{2√2} / \frac{1}{12√2} = 6</script>

where ~~octave = 2/1,~~ large tone = 9/8.

where large tone = 9/8.

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

<!-- ws:start:WikiTextMathRule:34:

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

<h2 id="toc12"> </h2>

<h2 id="toc13"><a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression"></a>Relative sizes of intervals between 3 frequencies in arithmetic progression</h2>

<h2 id="toc13"><a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression"></a>Relative sizes of intervals between 3 frequencies in arithmetic progression</h2>

<h3 id="toc14"><a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Theorem"></a>Theorem</h3>

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.

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

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

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

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

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

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<!-- ws:start:WikiTextMathRule:39:

[[math]]&lt;br/&gt;

\qquad L – 2s = (\delta_s – 2)s = ~~\frac{~~s}{\delta_s}&lt;br/&gt;[[math]]

\qquad L – 2s = (\delta_s – 2)s = s/\delta_s&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad L – 2s = (\delta_s – 2)s = ~~\frac{~~s}{\delta_s}</script>

--><script type="math/tex">\qquad L – 2s = (\delta_s – 2)s = s/\delta_s</script>

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) followed by tempered and just sizes in cents:

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

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

<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 δs = √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 have ~~similar~~ 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 δs = √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 have comparable magnitude – highly desirable properties for the formation of musical scales.

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

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

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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 silver temperament. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.

<img src="/file/view/Continued%20fraction%20jigsaw%2041edo.png/541636098/800x396/Continued%20fraction%20jigsaw%2041edo.png" alt="Continued fraction jigsaw 41edo.png" title="Continued fraction jigsaw 41edo.png" style="height: 396px; width: 800px;" />

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<td>Sequence 1:#

</td>

<td>#t/2 - 3c#

<td>#t/2 - 3c#

</td>

<td>#</span~~><em~~>2~~~~c

<td>#2c

</td>

<td>#t/2 - c

<td>#t/2 - c

</td>

<td>#T - t/2

<td>#T - t/2

</td>

<td>#t

</td>

<td>#T + t/2

<td>#T + t/2

</td>

<td>#M + t/2

<td>#M + t/2

</td>

<td>#2M

<td>#2M

</td>

</tr>

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<td>Sequence 2:#

</td>

<td>#magic

<td>#magic

</td>

<td>#diesis

<td>#diesis

</td>

<td>#chroma#

<td>#chroma#

</td>

<td>#semitone#

<td>#semitone#

</td>

<td>#t

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<td>Difference:

</td>

<td>#-3σ/2

<td>#-3σ/2

</td>

<td>#σ

<td>#σ

</td>

<td>#-σ/2

<td>#-σ/2

</td>

<td>#σ/2

</td>

<td>#</span~~>0 ~~

<td>#0

</td>

<td>#σ/2

<td>#σ/2

</td>

<td>#σ/2

<td>#σ/2

</td>

<td>#σ

<td>#σ

</td>

</tr>

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</table>

where f = 4/3, T = 9/8, t = 10/9, M = 5/4, magic = 3125/3072, diesis = 128/125, semitone = 16/15, mp = 32/27, c = syntonic comma = 81/80, m6p = 128/81, σ = schisma = ~~32805/327~~68.

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

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

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

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

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

Tempering the Sequence 1 ratios to ϕ yields a range of temperaments which can be made extremely accurate by, for example, tuning the octave and fifth (and therefore all Pythagorean intervals) pure. In this temperament the ~~error~~ in the intervals s, t~~, m~~ ~~and~~ M 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).

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<td>#J1

</td>

<td>#~~~~J2

<td>#J2

</td>

<td>#~~~~J3

<td>#J3

</td>

<td>#~~~~q1

<td>#q1

</td>

<td>#~~~~q2

<td>#q2

</td>

<td>#~~~~q3

<td>#q3

</td>

<td>#~~~~J1'

<td>#J1'

</td>

<td>#~~~~J2'

<td>#J2'

</td>

<td>#~~~~J3'#~~~~

<td>#J3'#

</td>

</tr>

<tr>

<td>#<~~/span~~>6/5#

<td>#6/5#

</td>

<td>#<~~/span~~></~~span~~>~~5/4~~

<td>#5/4

</td>

<td>#<~~/span~~></~~span~~>~~4/3~~#~~~~

<td>#4/3#

</td>

<td>#~~~~1/2√30#~~~~

<td>#1/2√30#

</td>

<td>#~~~~1/4√5

<td>#1/4√5

</td>

<td>#~~~~1/4√3

<td>#1/4√3

</td>

<td>#~~~~3

<td>#3

</td>

<td>#~~~~4

<td>#4

</td>

<td>#~~~~5

<td>#5

</td>

</tr>

<tr>

<td>#<~~/span~~>4/3

<td>#4/3

</td>

<td>#<~~/span~~></~~span~~>~~12/5~~#~~~~

<td>#12/5#

</td>

<td>#<~~/span~~></~~span~~>~~5/2~~#~~~~

<td>#5/2#

</td>

<td>#~~~~1/4√3

<td>#1/4√3

</td>

<td>#~~~~7/4√15#~~~~

<td>#7/4√15#

</td>

<td>#~~~~3/2√10#~~~~

<td>#3/2√10#

</td>

<td>

Line 1,253:

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</tr>

<tr>

<td>#<~~/span~~>8/5

<td>#8/5

</td>

<td>#<~~/span~~></~~span~~>~~12/5~~

<td>#12/5

</td>

<td>#<~~/span~~></~~span~~>~~8/3~~

<td>#8/3

</td>

<td>#~~~~3/4√10

<td>#3/4√10

</td>

<td>#~~~~7/4√15

<td>#7/4√15

</td>

<td>#~~~~5/4√6

<td>#5/4√6

</td>

<td>#~~~~8

<td>#8

</td>

<td>#~~~~15

<td>#15

</td>

<td>#~~~~17

<td>#17

</td>

</tr>

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The minute size of this comma can be explained using quadratic approximants.

It can be shown, using a suitable <a class="wiki_link" href="/Comma-based%20lattices">comma-based lattice</a>, that every comma tempered out by 34edo can be expressed as an integer linear combination of the gammic comma |-29 -11 20&gt; (4.769 cents) and the semisuper comma |23 6 -14&gt; (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/semisuper ≈ 10/7.

###Gammic and semisuper are both bimodular commas:

Gammic and semisuper are both bimodular commas:

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

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

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

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

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

<!-- ws:start:WikiTextMathRule:43:

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\qquad b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 – J_1^2) b_m&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 – J_1^2) b_m</script>

~~ ~~

Estimating J2 and J1 with their quadratic approximants we then have

<!-- ws:start:WikiTextMathRule:44:

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\qquad b(J_1,J_2) ≈ \frac{1}{3} (q_2^2 – q_1^2) b_m&lt;br/&gt;[[math]]

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

~~ ~~

For gammic:

<~~span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"~~>~~###~~</~~span~~><~~/span~~></~~span~~>J~~₁= 6/5~~, ~~&lt~~;em>J<~~/em>₂~~= ~~5/4&lt~~;~~/span> ~~

###J₁= 6/5, J₂= 5/4

~~<~~span style="font-family: Georgia,serif~~;">~~###~~<~~/span~~></span~~>&lt~~;em>v</~~span~~>₁ ~~= 1/11,~~ v<~~/em>₂ <~~span style="~~font-family: Georgia,serif; font-size: 110%;">~~= 1/9,~~ </span~~&gt~~;~~b<~~span~~ style="font-family: ~~Georgia~~,serif; font-size: 60%;">m = 1<~~br />

###v₁ = 1/11, v₂ = 1/9, bm = 1

<~~span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&~~gt;~~###~~</~~span~~></span~~&gt~~;~~&lt~~;em>q</~~span~~>~~₁² =~~ </span>(1/4)(1/30), q₂² = (1/4)(1/20)

###q₁² = (1/4)(1/30), q₂² = (1/4)(1/20)

~~&lt~~;~~span style="~~font-family: Arial,Helvetica,sans-~~serif;">###~~~~gammic = b(J</span~~>₁~~~~,J</span~~>₂~~~~) ≈ (1/12) (1/30 – 1/20) = (1/12) (1/60)~~ 

###gammic = b(J₁,J₂) ≈ (1/12) (1/30 – 1/20) = (1/12) (1/60)

For semisuper:

~~and for~~ semisuper:

###J₁= 25/24, J₂= 4/3

###~~~~J₁= 25/24, J₂= 4/3

###v₁ = 1/49, v₂ = 1/7, bm = 1/7

###v</span~~>₁ ~~~~= 1/49, v</em~~>₂ ~~~~= 1/7, bm = 1/7

###q₁² = (1/4)(1/600), q₂² = (1/4)(1/12)

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

###semisuper = b(J₁,J₂) ≈ (1/12) (1/12 – 1/600)(1/7) = (1/12) (7/600)

Therefore

###gammic/semisuper ≈ 10/7

as required.

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

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

<!-- ws:start:WikiTextMathRule:45:

[[math]]&lt;br/&gt;

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Line 1,317:

<!-- ws:start:WikiTextMathRule:46:

[[math]]&lt;br/&gt;

\qquad f = 1 – \tfrac{2}{15} (q_1^2 + q_2^2)&lt;br/&gt;[[math]]

\qquad f = 1 – \tfrac{2}{5} (q_1^2 + q_2^2)&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad f = 1 – \tfrac{2}{15} (q_1^2 + q_2^2)</script>

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

Thus a better estimate for gammic/semisuper is

<!-- ws:start:WikiTextMathRule:47:

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from which it follows that

###selenia = 7 gammic - 10 semisuper

####### ##≈ 7 gammic (~~1 -~~ f~~semisuper~~ / f~~gammic~~)

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

Putting in the numbers ~~we find~~

Putting in the numbers:

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

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

###fgammic - fsemisuper = 1/6000

Therefore

###selenia ≈ 7 ~~(1/6000) (120/119)~~ gammic = 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 q6, which become significant when the f values are very similar.)

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

Line 1,346:

Line 1,342:

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

<h1 id="toc22"><a name="Source"></a>Source</h1>

<h1 id="toc22"><a name="Source"></a>Source</h1>

This article is based on original research by Martin Gough. See <a href="/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf" onclick="ws.common.trackFileLink('/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf');">this paper</a> for a fuller account of bimodular approximants.</body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-02-21 11:17:47 UTC</tt>.<br>
+: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-02-21 12:57:21 UTC</tt>.<br>
-: The original revision id was <tt>541645916</tt>.<br>
+: The original revision id was <tt>541651356</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 <h4>Original Wikitext content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;1. Introduction&lt;/span&gt;=
+<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=**&lt;span style="font-size: 20px;"&gt;1. Introduction&lt;/span&gt;**=
-&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;The term //logarithmic approximant// (or //approximant// for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as:&lt;/span&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;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:&lt;/span&gt;
 * &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why do certain temperaments such as 12edo provide a good approximation to 5-limit just intonation?&lt;/span&gt;
 * &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why are certain commas small, and roughly how small are they?&lt;/span&gt;
@@ Line 174: / Line 174: @@
 The first convergent of this continued fraction is //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;//, the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is
 [[math]]
-\qquad y = \frac{3v}{3-v^2}
+\qquad y = \frac{v}{1-v^2/3}
 [[math]]
 Values of this rational approximant for some simple 5-limit intervals are shown in the table below.
@@ Line 268: / Line 268: @@
 \qquad \frac{octave}{large \, tone} ≈ \frac{1}{2√2} / \frac{1}{12√2} = 6
 [[math]]
-where //octave// = __2/1__, //large tone// = __9/8__.
+where //large tone// = __9/8__.
 However, this can also be derived from bimodular approximants. Using
 [[math]]
@@ Line 286: / Line 286: @@
 The most interesting approximate interval ratios derivable from quadratic approximants are irrational.
 == ==
-==&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Relative sizes of intervals between 3 frequencies in arithmetic progression&lt;/span&gt;==
+==&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Relative sizes of intervals between 3 frequencies in arithmetic progression&lt;/span&gt;==
 ===&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Theorem&lt;/span&gt;===
 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.
@@ Line 313: / Line 313: @@
 &lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;Perfect fourth = __4/3__ = 497.056 cents
 This fifth is wide by 0.989 cents, and the fourth narrow by the same amount. These errors are of about half the magnitude, and of opposite sign, as their counterparts in 12edo (where these intervals are tuned in the ratio of their bimodular approximants).
-A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the ‘silver ratio’ (sometimes called the ‘silver mean’), δS = √2 + 1 = 2.4142. On this basis, and by analogy with ‘golden meantone’ temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘silver 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 ‘silver ratio’ (sometimes called the ‘silver mean’), //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;//&lt;span style="vertical-align: sub;"&gt;s &lt;/span&gt;= &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;√2 + 1 = 2.4142&lt;/span&gt;. On this basis, and by analogy with ‘golden meantone’ temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘silver temperament’ is proposed instead.
 Silver temperament has interesting fractal properties which help to explain why 3-limit tuning forms aesthetically pleasing scales.
 The continued fraction expansion of the silver ratio has a particularly simple form:
@@ Line 321: / Line 321: @@
 As a result, if two intervals L and s are tuned in the silver ratio, with &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//s = L/δ//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s&lt;/span&gt;, subtracting twice the small interval //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;s&lt;/span&gt;// from the large interval //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;L&lt;/span&gt;// leaves a remainder of size &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//s/δ//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s&lt;/span&gt;:
 [[math]]
-\qquad L – 2s = (\delta_s – 2)s = \frac{s}{\delta_s}
+\qquad L – 2s = (\delta_s – 2)s = s/\delta_s
 [[math]]
 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) followed by tempered and just sizes in cents:
@@ Line 357: / Line 357: @@
 * Adjacent vertical pairs have ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;√2&lt;/span&gt;.
 * Extending the table to a third row yields consisting of the intervals in the first row multiplied by 2, and so on.
-The regularity of this scheme, combined with the fact that the ratios between closely related intervals are of order 2, means that its intervals form orderly sequences in which successive terms have similar magnitude – highly desirable properties for the formation of musical scales.
+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 have comparable magnitude – highly desirable properties for the formation of musical scales.
 In this fractal temperament, multiplying or dividing any interval by the factor &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= √2 + 1&lt;/span&gt; produces another interval in the temperament. Any tempered interval //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J’&lt;/span&gt;// can be split into three parts, two of equal size //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J’&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;/&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s&lt;/span&gt; and the other of size //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J’&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;/&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;2&lt;/span&gt;//.
 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 368: / Line 368: @@
 Figure 2 is a //continued fraction jigsaw// showing the sizes of the octave (o), fourth (f), tone (T), limma (s&lt;span style="vertical-align: super;"&gt;p&lt;/span&gt;), Pythagorean comma (p) and 29-tone comma (p&lt;span style="font-size: 60%;"&gt;29&lt;/span&gt;) as tempered by 41edo - an approximation to silver temperament. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.
 [[image:Continued fraction jigsaw 41edo.png width="800" height="396"]]
@@ Line 387: / Line 388: @@
 where &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//ϕ// = 1.61803&lt;/span&gt;... is the golden ratio.
 If a Fibonacci sequence of intervals is formed from the pair of intervals &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//T// – //t///2&lt;/span&gt; and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;t&lt;/span&gt;//, and extended in both directions, it can thus be expected that the ratios between successive intervals in this sequence will also be close to //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;//. The sequence formed in this way is Sequence 1 in the following table.
-|| Sequence 1:&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;//t/2 - 3c//&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt; &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//2////c//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//t/2 - c//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//T - t/2// &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//t//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//T + t/2//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//M + t/2// &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//2M// &lt;/span&gt; ||
+|| Sequence 1:&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;//t///2 - 3//c//&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt; &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;2//c//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//t///2 //- c//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//T - t///2 &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//t//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//T + t///2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//M + t///2 &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;2//M// &lt;/span&gt; ||
-|| Sequence 2:&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;magic&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;diesis&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;chroma&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;semitone&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//t//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//mp//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//f - c//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//m6p - c//&lt;/span&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt; ||
+|| Sequence 2:&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//magic//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//diesis//&lt;/span&gt; || //&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;chroma&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;/span&gt;// || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//semitone//&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//t//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//mp//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//f - c//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//m6p - c//&lt;/span&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt; ||
-|| Difference: || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;-3σ/2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;σ&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;-σ/2&lt;/span&gt; || &lt;span style="font-family: "Palatino Linotype","Book Antiqua",Palatino,serif;"&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;σ/2&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;span style="font-family: "Lucida Sans Unicode","Lucida Grande",sans-serif; font-size: 80%;"&gt;0 &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;σ/2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;σ/2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;σ &lt;/span&gt; ||
+|| Difference: || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;-3//σ///2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//σ//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;-//σ///2&lt;/span&gt; || &lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;//σ///2 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;0 || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//σ///2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//σ///2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//σ// &lt;/span&gt; ||
 || Seq 1 ratios: ||   || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6120&lt;/span&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;##&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6204&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6171&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6184&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt; &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6179&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6181&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6180&lt;/span&gt; ||
 || Seq 2 ratios: ||   || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.3865&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.7212&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.5810&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6325&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6125&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6201 &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6172 &lt;/span&gt; ||
-where &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f = 4/3, T = 9/8, t = 10/9, M = 5/4, magic = 3125/3072, diesis = 128/125, semitone = 16/15, mp = 32/27, c = syntonic comma = 81/80, m6p = 128/81, σ = schisma = 32805/327__68.__&lt;/span&gt;
+where &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//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.__&lt;/span&gt;
 The ratios between successive intervals in Sequence 1 are shown in the row labelled ‘Seq 1 ratios’, and are indeed close to //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;//.
 Sequence 2 is another Fibonacci sequence of intervals which differ from those in Sequence 1 by small amounts of the order of one schisma (//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;σ&lt;/span&gt;//), as indicated by the row marked ‘Difference’ (which is itself a Fibonacci sequence).
@@ Line 398: / Line 399: @@
 A suitable name for 5-limit tunings in which the intervals in either Sequence 1 or Sequence 2, or both, are tempered to exactly //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;// would be ‘golden temperaments’.
 Tempering the Sequence 2 ratios to //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;// while tuning the octave pure and tempering out the syntonic comma yields golden meantone temperament.
-Tempering the Sequence 1 ratios to //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;// 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 error in the intervals //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;s, t, m&lt;/span&gt;// and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;M&lt;/span&gt;// are all ±0.02106 cents.
+Tempering the Sequence 1 ratios to //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;// 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 //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;s, t&lt;/span&gt;//, //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;M&lt;/span&gt;// and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//m//=__6/5__&lt;/span&gt; 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 //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;// 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).
@@ Line 408: / Line 409: @@
 they can be said to form a Pythagorean triple.
 The following are three examples. In the first and third cases, their counterparts in 12edo, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//1', //J//2'&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//3'&lt;/span&gt; , are also Pythagorean triples:
-|| &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; line-height: 0px; overflow: hidden;"&gt;#&lt;/span&gt;//J//1&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;//J//2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;//J//3&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;//q//1&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;//q//2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;//q//3 &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;//J//1' &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;//J//2' &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;//J//3'&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; ||
+|| &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; line-height: 0px; overflow: hidden;"&gt;#&lt;/span&gt;//J//1&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;//J//2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;//J//3&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;//q//1&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;//q//2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;//q//3 &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;//J//1' &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;//J//2' &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;//J//3'&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt; ||
-|| &lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;6/5&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;5/4 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;4/3&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;1/2√30&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;1/4√5 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;1/4√3 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;3 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;4 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;5 ||
+|| &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;__6/5__&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;__5/4__ || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;__4/3__&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;1/2√30&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;1/4√5 || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;1/4√3 || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;3 || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;4 || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;5 ||
-|| &lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;4/3 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;12/5&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;5/2&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;1/4√3 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;7/4√15&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;3/2√10&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt; ||   ||   ||   ||
+|| &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;__4/3__ || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;__12/5__&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;__5/2__&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;1/4√3 || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;7/4√15&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;3/2√10&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt; ||   ||   ||   ||
-|| &lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;8/5 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;12/5 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;8/3 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;3/4√10 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;7/4√15 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;5/4√6 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;8 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;15 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;17 ||
+|| &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;__8/5__ || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;__12/5__ || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;__8/3__ || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;3/4√10 || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;7/4√15 || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;5/4√6 || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;8 || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;15 || &lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;17 ||
 ==A small 34edo comma==
@@ Line 417: / Line 418: @@
 &lt;span style="color: #333333;"&gt;The minute size of this comma can be explained using qu&lt;/span&gt;adratic approximants.
 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// comma |-29 -11 20&gt; (4.769 cents) and the //semisuper// comma |23 6 -14&gt; (3.338 cents). In particular,
-&lt;span style="color: #333333;"&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;//selenia// = 7 //gammic// – 10 //semisuper//&lt;/span&gt;
+&lt;span style="color: #333333;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;//selenia// = 7 //gammic// – 10 //semisuper//&lt;/span&gt;
 &lt;span style="color: #333333;"&gt;So to prove that //selenia// is small we must show that //gammic/////semisuper// ≈ 10/7.&lt;/span&gt;
-&lt;span style="color: #333333;"&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;//Gammic// and //semisuper// are both __bimodular commas__:&lt;/span&gt;
+&lt;span style="color: #333333;"&gt;//Gammic// and //semisuper// are both __bimodular commas__:&lt;/span&gt;
-&lt;span style="color: #333333;"&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;//gammic// = &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;//b//(__6/5__,__5/4__)&lt;/span&gt;
+&lt;span style="color: #333333;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;//gammic// = &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;//b//(__6/5__,__5/4__)&lt;/span&gt;
-&lt;span style="color: #333333;"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;//semisuper// = &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;//b//(__25/24__,__4/3__)&lt;/span&gt;
+&lt;span style="color: #333333;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;//semisuper// = &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;//b//(__25/24__,__4/3__)&lt;/span&gt;
 &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Using a result given in the section on bimodular commas, the size of the bimodular comma&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; //b//(//J//1,//J//2)&lt;/span&gt;&lt;span style="color: #333333;"&gt; can be estimated using&lt;/span&gt;
 [[math]]
 \qquad b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 – J_1^2) b_m
 [[math]]
 &lt;span style="color: #333333;"&gt;Estimating &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;//J//2&lt;/span&gt;&lt;span style="color: #333333;"&gt; and &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;//J//1&lt;/span&gt;&lt;span style="color: #333333;"&gt; with their quadratic approximants we then have&lt;/span&gt;
 [[math]]
 \qquad b(J_1,J_2) ≈ \frac{1}{3} (q_2^2 – q_1^2) b_m
 [[math]]
 &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;For //gammic//:&lt;/span&gt;
-&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;//J//₁= 6/5, //J//₂= 5/4&lt;/span&gt;
+&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;//J//₁= 6/5, //J//₂= 5/4&lt;/span&gt;
-&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;//v//&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₁ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/11, &lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;//&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₂ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/9, &lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 60%;"&gt;m&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; = 1&lt;/span&gt;
+&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;//v//&lt;/span&gt;₁ &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/11, &lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;//₂ &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/9, &lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 60%;"&gt;m&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; = 1&lt;/span&gt;
-&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;//q//&lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;₁² = &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(1/4)(1/30),&lt;/span&gt; //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;//&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;₂//² =// &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(1/4)(1/20)&lt;/span&gt;
+&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;//q//&lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;₁² = &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(1/4)(1/30),&lt;/span&gt; //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;//&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;₂//² =// &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(1/4)(1/20)&lt;/span&gt;
-&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;//gammic// = &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//b//(//J//&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₁&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;,//J//&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₂&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;) ≈ (1/12) (1/30 – 1/20) = (1/12) (1/60)&lt;/span&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;//gammic// = &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//b//(//J//&lt;/span&gt;₁&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;,//J//&lt;/span&gt;₂&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;) ≈ (1/12) (1/30 – 1/20) = (1/12) (1/60)&lt;/span&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;For //semisuper://&lt;/span&gt;
-&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;and for //semisuper://&lt;/span&gt;
+&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;//J//₁= 25/24, //J//₂= 4/3&lt;/span&gt;
-&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;//J//₁= 25/24, //J//₂= 4/3&lt;/span&gt;
+&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;//v//&lt;/span&gt;₁ &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/49, &lt;/span&gt;//&lt;span style="font-family: Georgia,serif;"&gt;v&lt;/span&gt;//₂ &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/7, &lt;/span&gt;//&lt;span style="font-family: Georgia,serif;"&gt;b&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 60%;"&gt;m&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; = 1/7&lt;/span&gt;
-&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;//v//&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₁ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/49, &lt;/span&gt;//&lt;span style="font-family: Georgia,serif;"&gt;v&lt;/span&gt;//&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₂ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/7, &lt;/span&gt;//&lt;span style="font-family: Georgia,serif;"&gt;b&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 60%;"&gt;m&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; = 1/7&lt;/span&gt;
 &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;//q//&lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;₁² = &lt;/span&gt;&lt;span style="font-family: Georgia,serif;"&gt;(1/4)(1/600),&lt;/span&gt; //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;//&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;₂//² =// &lt;/span&gt;&lt;span style="font-family: Georgia,serif;"&gt;(1/4)(1/12)&lt;/span&gt;
-&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;//semisuper// = &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//b//(//J//&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₁&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;,//J//&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₂&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;) ≈ (1/12) (1/12 – 1/600)(1/7) = (1/12) (7/600)&lt;/span&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;//semisuper// = &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//b//(//J//&lt;/span&gt;₁&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;,//J//&lt;/span&gt;₂&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;) ≈ (1/12) (1/12 – 1/600)(1/7) = (1/12) (7/600)&lt;/span&gt;
 &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Therefore&lt;/span&gt;
 &lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;//gammic/semisuper// ≈ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;10/7&lt;/span&gt;
 &lt;span style="color: #333333;"&gt;as required.&lt;/span&gt;
-&lt;span style="color: #333333;"&gt;To estimate the size of //selenia// we must quantify for the error in this ratio. A more accurate analysis gives&lt;/span&gt;
+&lt;span style="color: #333333;"&gt;To estimate the size of //selenia// we must quantify the error in this ratio. A more accurate analysis gives&lt;/span&gt;
 [[math]]
 \qquad b(J_1,J_2) ≈ \left( \tfrac{1}{3} (q_2^2 – q_1^2) – \tfrac{2}{15} (q_2^4 – q_1^4) \right) b_m \\
@@ Line 454: / Line 452: @@
 &lt;span style="color: #333333;"&gt;So to improve our estimates of &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//b//(//J//1,//J//2)&lt;/span&gt; &lt;span style="color: #333333;"&gt;we should multiply them by&lt;/span&gt;
 [[math]]
-\qquad f = 1 – \tfrac{2}{15} (q_1^2 + q_2^2)
+\qquad f = 1 – \tfrac{2}{5} (q_1^2 + q_2^2)
 [[math]]
 &lt;span style="color: #333333;"&gt;Thus a better estimate for //gammic/semisuper// is&lt;/span&gt;
@@ Line 463: / Line 461: @@
 &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;from which it follows that&lt;/span&gt;
 &lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//selenia// = 7 //gammic// - 10 //semisuper//&lt;/span&gt;
-&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;####### ##&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;≈ 7 //gammic// (1 - //f//&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;semisuper&lt;/span&gt;// &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;/ //f//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;//gammic//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;)&lt;/span&gt;
+&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;######## &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;≈ 7 //gammic//&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt; (//f//&lt;/span&gt;&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;gammic&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; - //f//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;//semisuper//)&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;/&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;gammic&lt;/span&gt;//
-&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Putting in the numbers we find&lt;/span&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Putting in the numbers:&lt;/span&gt;
 //&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;//gammic// = 1 – (2/5) (1/4) (1/30 + 1/20) = 1 – 1/120&lt;/span&gt;
 &lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;//semisuper// = 1 – (2/5)(1/4) (1/600 + 1/12) = 1 – (1/120) (51/50)&lt;/span&gt;
 &lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;gammic &lt;/span&gt;&lt;span style="font-family: Georgia,serif;"&gt;- &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;semisuper &lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/6000&lt;/span&gt;
 &lt;span style="color: #333333;"&gt;Therefore&lt;/span&gt;
-&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//selenia// ≈ 7 (1/6000) (120/119) //gammic// = //gammic///850 = 0.00561&lt;/span&gt;&lt;span style="color: #333333;"&gt; cents&lt;/span&gt;
+&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//selenia// ≈ 7 //gammic// (1/6000) (120/119) = //gammic///850 = 0.00561&lt;/span&gt;&lt;span style="color: #333333;"&gt; cents&lt;/span&gt;
 &lt;span style="color: #333333;"&gt;which &lt;/span&gt;is within 20% of the accurate value, 0.00476 cents. (The discrepancy is due to the influence of terms in //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 80%; vertical-align: super;"&gt;6&lt;/span&gt;//,// which become significant when the //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;// 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 477: / Line 475: @@
 and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(//q//(25/24))&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;2&lt;/span&gt; , being small in comparison to the other terms, compromises this equality only slightly.
-=Source=
+=&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Source&lt;/span&gt;=
 This article is based on original research by Martin Gough. See [[file:Bimod Approx 2014-6-8.pdf|this paper]] for a fuller account of bimodular approximants.</pre></div>
 <h4>Original HTML content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Logarithmic approximants&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:49:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x1. Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:49 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;1. Introduction&lt;/span&gt;&lt;/h1&gt;
+<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Logarithmic approximants&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:49:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x1. Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:49 --&gt;&lt;strong&gt;&lt;span style="font-size: 20px;"&gt;1. Introduction&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
-  &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;The term &lt;em&gt;logarithmic approximant&lt;/em&gt; (or &lt;em&gt;approximant&lt;/em&gt; for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as:&lt;/span&gt;&lt;br /&gt;
+  &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;A &lt;em&gt;logarithmic approximant&lt;/em&gt; (or &lt;em&gt;approximant&lt;/em&gt; for short) is an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as:&lt;/span&gt;&lt;br /&gt;
 &lt;ul&gt;&lt;li&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why do certain temperaments such as 12edo provide a good approximation to 5-limit just intonation?&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why are certain commas small, and roughly how small are they?&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why does the 3-limit framework produce aesthetically pleasing scale structures?&lt;/span&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
 The exact size, in cents, of an interval with frequency ratio &lt;em&gt;r&lt;/em&gt; is&lt;br /&gt;
@@ Line 674: / Line 672: @@
 &lt;!-- ws:start:WikiTextMathRule:25:
 [[math]]&amp;lt;br/&amp;gt;
-\qquad y = \frac{3v}{3-v^2}&amp;lt;br/&amp;gt;[[math]]
+\qquad y = \frac{v}{1-v^2/3}&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;\qquad y = \frac{3v}{3-v^2}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:25 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;\qquad y = \frac{v}{1-v^2/3}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:25 --&gt;&lt;br /&gt;
 Values of this rational approximant for some simple 5-limit intervals are shown in the table below.&lt;br /&gt;
@@ Line 917: / Line 915: @@
 \qquad \frac{octave}{large \, tone} ≈ \frac{1}{2√2} / \frac{1}{12√2} = 6&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\qquad \frac{octave}{large \, tone} ≈ \frac{1}{2√2} / \frac{1}{12√2} = 6&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:33 --&gt;&lt;br /&gt;
-where &lt;em&gt;octave&lt;/em&gt; = &lt;u&gt;2/1&lt;/u&gt;, &lt;em&gt;large tone&lt;/em&gt; = &lt;u&gt;9/8&lt;/u&gt;.&lt;br /&gt;
+where &lt;em&gt;large tone&lt;/em&gt; = &lt;u&gt;9/8&lt;/u&gt;.&lt;br /&gt;
 However, this can also be derived from bimodular approximants. Using&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:34:
@@ Line 939: / Line 937: @@
 The most interesting approximate interval ratios derivable from quadratic approximants are irrational.&lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:73:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;!-- ws:end:WikiTextHeadingRule:73 --&gt; &lt;/h2&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:75:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:75 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Relative sizes of intervals between 3 frequencies in arithmetic progression&lt;/span&gt;&lt;/h2&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:75:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:75 --&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Relative sizes of intervals between 3 frequencies in arithmetic progression&lt;/span&gt;&lt;/h2&gt;
   &lt;!-- ws:start:WikiTextHeadingRule:77:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Theorem"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:77 --&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Theorem&lt;/span&gt;&lt;/h3&gt;
   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.&lt;br /&gt;
@@ Line 967: / Line 965: @@
 &lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;Perfect fourth = &lt;u&gt;4/3&lt;/u&gt; = 497.056 cents&lt;br /&gt;
 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).&lt;br /&gt;
-A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the ‘silver ratio’ (sometimes called the ‘silver mean’), δS = √2 + 1 = 2.4142. On this basis, and by analogy with ‘golden meantone’ temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘silver temperament’ is proposed instead.&lt;br /&gt;
+A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the ‘silver ratio’ (sometimes called the ‘silver mean’), &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;&lt;/em&gt;&lt;span style="vertical-align: sub;"&gt;s &lt;/span&gt;= &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;√2 + 1 = 2.4142&lt;/span&gt;. On this basis, and by analogy with ‘golden meantone’ temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘silver temperament’ is proposed instead.&lt;br /&gt;
 Silver temperament has interesting fractal properties which help to explain why 3-limit tuning forms aesthetically pleasing scales.&lt;br /&gt;
 The continued fraction expansion of the silver ratio has a particularly simple form:&lt;br /&gt;
@@ Line 977: / Line 975: @@
 &lt;!-- ws:start:WikiTextMathRule:39:
 [[math]]&amp;lt;br/&amp;gt;
-\qquad L – 2s = (\delta_s – 2)s = \frac{s}{\delta_s}&amp;lt;br/&amp;gt;[[math]]
+\qquad L – 2s = (\delta_s – 2)s = s/\delta_s&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;\qquad L – 2s = (\delta_s – 2)s = \frac{s}{\delta_s}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:39 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;\qquad L – 2s = (\delta_s – 2)s = s/\delta_s&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:39 --&gt;&lt;br /&gt;
 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) followed by tempered and just sizes in cents:&lt;br /&gt;
@@ Line 1,035: / Line 1,033: @@
 When picturing these relationships it makes most musical sense to place the small interval between the two larger ones, as in the ‘continued fraction jigsaw’ below.&lt;br /&gt;
 The following relationships hold in the table, the first two being valid for the pure intervals as well as their tempered counterparts:&lt;br /&gt;
-&lt;ul&gt;&lt;li&gt;Subtracting twice an interval from the interval on its left generates the interval on its right.&lt;/li&gt;&lt;li&gt;An interval in the second row is the sum of the interval immediately above and the interval diagonally above and to the right.&lt;/li&gt;&lt;li&gt;Adjacent horizontal pairs have ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= √2 + 1.&lt;/span&gt;&lt;/li&gt;&lt;li&gt;Adjacent vertical pairs have ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;√2&lt;/span&gt;.&lt;/li&gt;&lt;li&gt;Extending the table to a third row yields consisting of the intervals in the first row multiplied by 2, and so on.&lt;/li&gt;&lt;/ul&gt;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 have similar magnitude – highly desirable properties for the formation of musical scales.&lt;br /&gt;
+&lt;ul&gt;&lt;li&gt;Subtracting twice an interval from the interval on its left generates the interval on its right.&lt;/li&gt;&lt;li&gt;An interval in the second row is the sum of the interval immediately above and the interval diagonally above and to the right.&lt;/li&gt;&lt;li&gt;Adjacent horizontal pairs have ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= √2 + 1.&lt;/span&gt;&lt;/li&gt;&lt;li&gt;Adjacent vertical pairs have ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;√2&lt;/span&gt;.&lt;/li&gt;&lt;li&gt;Extending the table to a third row yields consisting of the intervals in the first row multiplied by 2, and so on.&lt;/li&gt;&lt;/ul&gt;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 have comparable magnitude – highly desirable properties for the formation of musical scales.&lt;br /&gt;
 In this fractal temperament, multiplying or dividing any interval by the factor &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= √2 + 1&lt;/span&gt; produces another interval in the temperament. Any tempered interval &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J’&lt;/span&gt;&lt;/em&gt; can be split into three parts, two of equal size &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J’&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;/&lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s&lt;/span&gt; and the other of size &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J’&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;/&lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;2&lt;/span&gt;&lt;/em&gt;.&lt;br /&gt;
 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.&lt;br /&gt;
@@ Line 1,046: / Line 1,044: @@
 &lt;br /&gt;
 Figure 2 is a &lt;em&gt;continued fraction jigsaw&lt;/em&gt; showing the sizes of the octave (o), fourth (f), tone (T), limma (s&lt;span style="vertical-align: super;"&gt;p&lt;/span&gt;), Pythagorean comma (p) and 29-tone comma (p&lt;span style="font-size: 60%;"&gt;29&lt;/span&gt;) as tempered by 41edo - an approximation to silver temperament. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.&lt;br /&gt;
+&lt;br /&gt;
 &lt;!-- ws:start:WikiTextLocalImageRule:482:&amp;lt;img src=&amp;quot;/file/view/Continued%20fraction%20jigsaw%2041edo.png/541636098/800x396/Continued%20fraction%20jigsaw%2041edo.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 396px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Continued%20fraction%20jigsaw%2041edo.png/541636098/800x396/Continued%20fraction%20jigsaw%2041edo.png" alt="Continued fraction jigsaw 41edo.png" title="Continued fraction jigsaw 41edo.png" style="height: 396px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:482 --&gt;&lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,073: / Line 1,072: @@
          &lt;td&gt;Sequence 1:&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;&lt;em&gt;t/2 - 3c&lt;/em&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt; &lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;&lt;em&gt;t&lt;/em&gt;/2 - 3&lt;em&gt;c&lt;/em&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt; &lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;2&lt;/em&gt;&lt;em&gt;c&lt;/em&gt;&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;2&lt;em&gt;c&lt;/em&gt;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;t/2 - c&lt;/em&gt;&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;t&lt;/em&gt;/2 &lt;em&gt;- c&lt;/em&gt;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;T - t/2&lt;/em&gt; &lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;T - t&lt;/em&gt;/2 &lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;t&lt;/em&gt;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;T + t/2&lt;/em&gt;&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;T + t&lt;/em&gt;/2&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;M + t/2&lt;/em&gt; &lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;M + t&lt;/em&gt;/2 &lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;2M&lt;/em&gt; &lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;2&lt;em&gt;M&lt;/em&gt; &lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,093: / Line 1,092: @@
          &lt;td&gt;Sequence 2:&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;magic&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;magic&lt;/em&gt;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;diesis&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;diesis&lt;/em&gt;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;chroma&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;chroma&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;/em&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;semitone&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;semitone&lt;/em&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;t&lt;/em&gt;&lt;/span&gt;&lt;br /&gt;
@@ Line 1,113: / Line 1,112: @@
          &lt;td&gt;Difference:&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;-3σ/2&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;-3&lt;em&gt;σ&lt;/em&gt;/2&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;σ&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;σ&lt;/em&gt;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;-σ/2&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;-&lt;em&gt;σ&lt;/em&gt;/2&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: "Palatino Linotype","Book Antiqua",Palatino,serif;"&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;σ/2&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;&lt;em&gt;σ&lt;/em&gt;/2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;span style="font-family: "Lucida Sans Unicode","Lucida Grande",sans-serif; font-size: 80%;"&gt;0 &lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;0&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;σ/2&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;σ&lt;/em&gt;/2&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;σ/2&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;σ&lt;/em&gt;/2&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;σ &lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;σ&lt;/em&gt; &lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,172: / Line 1,171: @@
 &lt;/table&gt;
-where &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f = 4/3, T = 9/8, t = 10/9, M = 5/4, magic = 3125/3072, diesis = 128/125, semitone = 16/15, mp = 32/27, c = syntonic comma = 81/80, m6p = 128/81, σ = schisma = 32805/327&lt;u&gt;68.&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
+where &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;f&lt;/em&gt; = &lt;u&gt;4/3&lt;/u&gt;, &lt;em&gt;T&lt;/em&gt; = &lt;u&gt;9/8&lt;/u&gt;, &lt;em&gt;t&lt;/em&gt; = &lt;u&gt;10/9&lt;/u&gt;, &lt;em&gt;M&lt;/em&gt; = &lt;u&gt;5/4&lt;/u&gt;, &lt;em&gt;magic&lt;/em&gt; = &lt;u&gt;3125/3072&lt;/u&gt;, &lt;em&gt;diesis&lt;/em&gt; = &lt;u&gt;128/125&lt;/u&gt;, &lt;em&gt;chroma&lt;/em&gt; = &lt;u&gt;25/24&lt;/u&gt;, &lt;em&gt;semitone&lt;/em&gt; = &lt;u&gt;16/15&lt;/u&gt;, &lt;em&gt;mp&lt;/em&gt; = &lt;u&gt;32/27&lt;/u&gt;, &lt;em&gt;c&lt;/em&gt; = &lt;em&gt;syntonic comma&lt;/em&gt; = &lt;u&gt;81/80&lt;/u&gt;, &lt;em&gt;m6p&lt;/em&gt; = &lt;u&gt;128/81&lt;/u&gt;, &lt;em&gt;σ&lt;/em&gt; = &lt;em&gt;schisma&lt;/em&gt; = &lt;u&gt;32805/32768.&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
 The ratios between successive intervals in Sequence 1 are shown in the row labelled ‘Seq 1 ratios’, and are indeed close to &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;&lt;/em&gt;.&lt;br /&gt;
 Sequence 2 is another Fibonacci sequence of intervals which differ from those in Sequence 1 by small amounts of the order of one schisma (&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;σ&lt;/span&gt;&lt;/em&gt;), as indicated by the row marked ‘Difference’ (which is itself a Fibonacci sequence).&lt;br /&gt;
@@ Line 1,178: / Line 1,177: @@
 A suitable name for 5-limit tunings in which the intervals in either Sequence 1 or Sequence 2, or both, are tempered to exactly &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;&lt;/em&gt; would be ‘golden temperaments’.&lt;br /&gt;
 Tempering the Sequence 2 ratios to &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;&lt;/em&gt; while tuning the octave pure and tempering out the syntonic comma yields golden meantone temperament.&lt;br /&gt;
-Tempering the Sequence 1 ratios to &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;&lt;/em&gt; 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 error in the intervals &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;s, t, m&lt;/span&gt;&lt;/em&gt; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;M&lt;/span&gt;&lt;/em&gt; are all ±0.02106 cents.&lt;br /&gt;
+Tempering the Sequence 1 ratios to &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;&lt;/em&gt; 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 &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;s, t&lt;/span&gt;&lt;/em&gt;, &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;M&lt;/span&gt;&lt;/em&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;m&lt;/em&gt;=&lt;u&gt;6/5&lt;/u&gt;&lt;/span&gt; are all ±0.02106 cents.&lt;br /&gt;
 Tempering out the schisma tunes Sequences 1 and 2 identically so that the ratios between consecutive intervals can be fixed at &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;&lt;/em&gt; 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).&lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,195: / Line 1,194: @@
          &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; line-height: 0px; overflow: hidden;"&gt;#&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;1&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;2&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;2&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;3&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;3&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;em&gt;q&lt;/em&gt;1&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;em&gt;q&lt;/em&gt;1&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;em&gt;q&lt;/em&gt;2&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;em&gt;q&lt;/em&gt;2&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;em&gt;q&lt;/em&gt;3 &lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;em&gt;q&lt;/em&gt;3 &lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;1' &lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;1' &lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;2' &lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;2' &lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;3'&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;3'&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;6/5&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;u&gt;6/5&lt;/u&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;5/4&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;u&gt;5/4&lt;/u&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;4/3&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;u&gt;4/3&lt;/u&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;1/2√30&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;1/2√30&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;1/4√5&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;1/4√5&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;1/4√3&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;1/4√3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;3&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;4&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;4&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;5&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;5&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;4/3&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;u&gt;4/3&lt;/u&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;12/5&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;u&gt;12/5&lt;/u&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;5/2&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;u&gt;5/2&lt;/u&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;1/4√3&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;1/4√3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;7/4√15&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;7/4√15&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;3/2√10&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;3/2√10&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 1,253: / Line 1,252: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;8/5&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;u&gt;8/5&lt;/u&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;12/5&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;u&gt;12/5&lt;/u&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;8/3&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;&lt;u&gt;8/3&lt;/u&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;3/4√10&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;3/4√10&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;7/4√15&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;7/4√15&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;5/4√6&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;5/4√6&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;8&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;15&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;15&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;17&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia;"&gt;#&lt;/span&gt;17&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,279: / Line 1,278: @@
 &lt;span style="color: #333333;"&gt;The minute size of this comma can be explained using qu&lt;/span&gt;adratic approximants.&lt;br /&gt;
 It can be shown, using a suitable &lt;a class="wiki_link" href="/Comma-based%20lattices"&gt;comma-based lattice&lt;/a&gt;, that every comma tempered out by 34edo can be expressed as an integer linear combination of the &lt;em&gt;gammic&lt;/em&gt; comma |-29 -11 20&amp;gt; (4.769 cents) and the &lt;em&gt;semisuper&lt;/em&gt; comma |23 6 -14&amp;gt; (3.338 cents). In particular,&lt;br /&gt;
-&lt;span style="color: #333333;"&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;em&gt;selenia&lt;/em&gt; = 7 &lt;em&gt;gammic&lt;/em&gt; – 10 &lt;em&gt;semisuper&lt;/em&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;span style="color: #333333;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;&lt;em&gt;selenia&lt;/em&gt; = 7 &lt;em&gt;gammic&lt;/em&gt; – 10 &lt;em&gt;semisuper&lt;/em&gt;&lt;/span&gt;&lt;br /&gt;
 &lt;span style="color: #333333;"&gt;So to prove that &lt;em&gt;selenia&lt;/em&gt; is small we must show that &lt;em&gt;gammic&lt;/em&gt;&lt;em&gt;/semisuper&lt;/em&gt; ≈ 10/7.&lt;/span&gt;&lt;br /&gt;
-&lt;span style="color: #333333;"&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;em&gt;Gammic&lt;/em&gt; and &lt;em&gt;semisuper&lt;/em&gt; are both &lt;u&gt;bimodular commas&lt;/u&gt;:&lt;/span&gt;&lt;br /&gt;
+&lt;span style="color: #333333;"&gt;&lt;em&gt;Gammic&lt;/em&gt; and &lt;em&gt;semisuper&lt;/em&gt; are both &lt;u&gt;bimodular commas&lt;/u&gt;:&lt;/span&gt;&lt;br /&gt;
-&lt;span style="color: #333333;"&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;em&gt;gammic&lt;/em&gt; = &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;b&lt;/em&gt;(&lt;u&gt;6/5&lt;/u&gt;,&lt;u&gt;5/4&lt;/u&gt;)&lt;/span&gt;&lt;br /&gt;
+&lt;span style="color: #333333;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;&lt;em&gt;gammic&lt;/em&gt; = &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;b&lt;/em&gt;(&lt;u&gt;6/5&lt;/u&gt;,&lt;u&gt;5/4&lt;/u&gt;)&lt;/span&gt;&lt;br /&gt;
-&lt;span style="color: #333333;"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;em&gt;semisuper&lt;/em&gt; = &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;b&lt;/em&gt;(&lt;u&gt;25/24&lt;/u&gt;,&lt;u&gt;4/3&lt;/u&gt;)&lt;/span&gt;&lt;br /&gt;
+&lt;span style="color: #333333;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;&lt;em&gt;semisuper&lt;/em&gt; = &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;b&lt;/em&gt;(&lt;u&gt;25/24&lt;/u&gt;,&lt;u&gt;4/3&lt;/u&gt;)&lt;/span&gt;&lt;br /&gt;
 &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Using a result given in the section on bimodular commas, the size of the bimodular comma&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; &lt;em&gt;b&lt;/em&gt;(&lt;em&gt;J&lt;/em&gt;1,&lt;em&gt;J&lt;/em&gt;2)&lt;/span&gt;&lt;span style="color: #333333;"&gt; can be estimated using&lt;/span&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:43:
@@ Line 1,289: / Line 1,288: @@
 \qquad b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 – J_1^2) b_m&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\qquad b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 – J_1^2) b_m&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:43 --&gt;&lt;br /&gt;
-&lt;br /&gt;
 &lt;span style="color: #333333;"&gt;Estimating &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;2&lt;/span&gt;&lt;span style="color: #333333;"&gt; and &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;1&lt;/span&gt;&lt;span style="color: #333333;"&gt; with their quadratic approximants we then have&lt;/span&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:44:
@@ Line 1,295: / Line 1,293: @@
 \qquad b(J_1,J_2) ≈ \frac{1}{3} (q_2^2 – q_1^2) b_m&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\qquad b(J_1,J_2) ≈ \frac{1}{3} (q_2^2 – q_1^2) b_m&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:44 --&gt;&lt;br /&gt;
-&lt;br /&gt;
 &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;For &lt;em&gt;gammic&lt;/em&gt;:&lt;/span&gt;&lt;br /&gt;
-&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;₁= 6/5, &lt;em&gt;J&lt;/em&gt;₂= 5/4&lt;/span&gt;&lt;br /&gt;
+&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;₁= 6/5, &lt;em&gt;J&lt;/em&gt;₂= 5/4&lt;/span&gt;&lt;br /&gt;
-&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;em&gt;v&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₁ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/11, &lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₂ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/9, &lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 60%;"&gt;m&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; = 1&lt;/span&gt;&lt;br /&gt;
+&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;&lt;em&gt;v&lt;/em&gt;&lt;/span&gt;₁ &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/11, &lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;&lt;/em&gt;₂ &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/9, &lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 60%;"&gt;m&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; = 1&lt;/span&gt;&lt;br /&gt;
-&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;em&gt;q&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;₁² = &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(1/4)(1/30),&lt;/span&gt; &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;₂&lt;em&gt;² =&lt;/em&gt; &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(1/4)(1/20)&lt;/span&gt;&lt;br /&gt;
+&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;&lt;em&gt;q&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;₁² = &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(1/4)(1/30),&lt;/span&gt; &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;₂&lt;em&gt;² =&lt;/em&gt; &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(1/4)(1/20)&lt;/span&gt;&lt;br /&gt;
-&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;em&gt;gammic&lt;/em&gt; = &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;b&lt;/em&gt;(&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₁&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;,&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₂&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;) ≈ (1/12) (1/30 – 1/20) = (1/12) (1/60)&lt;/span&gt;&lt;br /&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;&lt;em&gt;gammic&lt;/em&gt; = &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;b&lt;/em&gt;(&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;₁&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;,&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;₂&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;) ≈ (1/12) (1/30 – 1/20) = (1/12) (1/60)&lt;/span&gt;&lt;br /&gt;
-&lt;br /&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;For &lt;em&gt;semisuper:&lt;/em&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;and for &lt;em&gt;semisuper:&lt;/em&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;₁= 25/24, &lt;em&gt;J&lt;/em&gt;₂= 4/3&lt;/span&gt;&lt;br /&gt;
-&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;₁= 25/24, &lt;em&gt;J&lt;/em&gt;₂= 4/3&lt;/span&gt;&lt;br /&gt;
+&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;em&gt;v&lt;/em&gt;&lt;/span&gt;₁ &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/49, &lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif;"&gt;v&lt;/span&gt;&lt;/em&gt;₂ &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/7, &lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif;"&gt;b&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 60%;"&gt;m&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; = 1/7&lt;/span&gt;&lt;br /&gt;
-&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;em&gt;v&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₁ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/49, &lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif;"&gt;v&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₂ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/7, &lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif;"&gt;b&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 60%;"&gt;m&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; = 1/7&lt;/span&gt;&lt;br /&gt;
 &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;em&gt;q&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;₁² = &lt;/span&gt;&lt;span style="font-family: Georgia,serif;"&gt;(1/4)(1/600),&lt;/span&gt; &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;₂&lt;em&gt;² =&lt;/em&gt; &lt;/span&gt;&lt;span style="font-family: Georgia,serif;"&gt;(1/4)(1/12)&lt;/span&gt;&lt;br /&gt;
-&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;em&gt;semisuper&lt;/em&gt; = &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;b&lt;/em&gt;(&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₁&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;,&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₂&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;) ≈ (1/12) (1/12 – 1/600)(1/7) = (1/12) (7/600)&lt;/span&gt;&lt;br /&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;em&gt;semisuper&lt;/em&gt; = &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;b&lt;/em&gt;(&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;₁&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;,&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;₂&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;) ≈ (1/12) (1/12 – 1/600)(1/7) = (1/12) (7/600)&lt;/span&gt;&lt;br /&gt;
 &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Therefore&lt;/span&gt;&lt;br /&gt;
 &lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;&lt;em&gt;gammic/semisuper&lt;/em&gt; ≈ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;10/7&lt;/span&gt;&lt;br /&gt;
 &lt;span style="color: #333333;"&gt;as required.&lt;/span&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;span style="color: #333333;"&gt;To estimate the size of &lt;em&gt;selenia&lt;/em&gt; we must quantify for the error in this ratio. A more accurate analysis gives&lt;/span&gt;&lt;br /&gt;
+&lt;span style="color: #333333;"&gt;To estimate the size of &lt;em&gt;selenia&lt;/em&gt; we must quantify the error in this ratio. A more accurate analysis gives&lt;/span&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:45:
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@@ Line 1,321: / Line 1,317: @@
 &lt;!-- ws:start:WikiTextMathRule:46:
 [[math]]&amp;lt;br/&amp;gt;
-\qquad f = 1 – \tfrac{2}{15} (q_1^2 + q_2^2)&amp;lt;br/&amp;gt;[[math]]
+\qquad f = 1 – \tfrac{2}{5} (q_1^2 + q_2^2)&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;\qquad f = 1 – \tfrac{2}{15} (q_1^2 + q_2^2)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:46 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;\qquad f = 1 – \tfrac{2}{5} (q_1^2 + q_2^2)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:46 --&gt;&lt;br /&gt;
 &lt;span style="color: #333333;"&gt;Thus a better estimate for &lt;em&gt;gammic/semisuper&lt;/em&gt; is&lt;/span&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:47:
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 &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;from which it follows that&lt;/span&gt;&lt;br /&gt;
 &lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;selenia&lt;/em&gt; = 7 &lt;em&gt;gammic&lt;/em&gt; - 10 &lt;em&gt;semisuper&lt;/em&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;####### ##&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;≈ 7 &lt;em&gt;gammic&lt;/em&gt; (1 - &lt;em&gt;f&lt;/em&gt;&lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;semisuper&lt;/span&gt;&lt;/em&gt; &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;/ &lt;em&gt;f&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;&lt;em&gt;gammic&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;)&lt;/span&gt;&lt;br /&gt;
+&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;######## &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;≈ 7 &lt;em&gt;gammic&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt; (&lt;em&gt;f&lt;/em&gt;&lt;/span&gt;&lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;gammic&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; - &lt;em&gt;f&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;&lt;em&gt;semisuper&lt;/em&gt;)&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;/&lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;gammic&lt;/span&gt;&lt;/em&gt;&lt;br /&gt;
-&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Putting in the numbers we find&lt;/span&gt;&lt;br /&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Putting in the numbers:&lt;/span&gt;&lt;br /&gt;
 &lt;em&gt;&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;&lt;em&gt;gammic&lt;/em&gt; = 1 – (2/5) (1/4) (1/30 + 1/20) = 1 – 1/120&lt;/span&gt;&lt;br /&gt;
 &lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;&lt;em&gt;semisuper&lt;/em&gt; = 1 – (2/5)(1/4) (1/600 + 1/12) = 1 – (1/120) (51/50)&lt;/span&gt;&lt;br /&gt;
 &lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;gammic &lt;/span&gt;&lt;span style="font-family: Georgia,serif;"&gt;- &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;semisuper &lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/6000&lt;/span&gt;&lt;br /&gt;
 &lt;span style="color: #333333;"&gt;Therefore&lt;/span&gt;&lt;br /&gt;
-&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;selenia&lt;/em&gt; ≈ 7 (1/6000) (120/119) &lt;em&gt;gammic&lt;/em&gt; = &lt;em&gt;gammic&lt;/em&gt;/850 = 0.00561&lt;/span&gt;&lt;span style="color: #333333;"&gt; cents&lt;/span&gt;&lt;br /&gt;
+&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;selenia&lt;/em&gt; ≈ 7 &lt;em&gt;gammic&lt;/em&gt; (1/6000) (120/119) = &lt;em&gt;gammic&lt;/em&gt;/850 = 0.00561&lt;/span&gt;&lt;span style="color: #333333;"&gt; cents&lt;/span&gt;&lt;br /&gt;
 &lt;span style="color: #333333;"&gt;which &lt;/span&gt;is within 20% of the accurate value, 0.00476 cents. (The discrepancy is due to the influence of terms in &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 80%; vertical-align: super;"&gt;6&lt;/span&gt;&lt;em&gt;,&lt;/em&gt; which become significant when the &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;/em&gt; values are very similar.)&lt;br /&gt;
 In summary, the reason &lt;em&gt;selenia&lt;/em&gt; is small (compared to &lt;em&gt;gammic&lt;/em&gt; and &lt;em&gt;semisuper&lt;/em&gt;) is because the quadratic approximants of &lt;em&gt;gammic&lt;/em&gt; and &lt;em&gt;semisuper&lt;/em&gt; are in the ratio 10/7. The reason it is &lt;em&gt;very&lt;/em&gt; small (of order &lt;em&gt;gammic&lt;/em&gt;/1000 rather than &lt;em&gt;gammic&lt;/em&gt;/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:&lt;br /&gt;
@@ Line 1,346: / Line 1,342: @@
 and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(&lt;em&gt;q&lt;/em&gt;(25/24))&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;2&lt;/span&gt; , being small in comparison to the other terms, compromises this equality only slightly.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:93:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc22"&gt;&lt;a name="Source"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:93 --&gt;Source&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:93:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc22"&gt;&lt;a name="Source"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:93 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Source&lt;/span&gt;&lt;/h1&gt;
   This article is based on original research by Martin Gough. See &lt;a href="/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf" onclick="ws.common.trackFileLink('/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf');"&gt;this paper&lt;/a&gt; for a fuller account of bimodular approximants.&lt;/body&gt;&lt;/html&gt;</pre></div>