Logarithmic approximants: Difference between revisions
Wikispaces>MartinGough **Imported revision 541661880 - Original comment: ** |
Wikispaces>MartinGough **Imported revision 541704146 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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- | : This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-02-22 11:41:16 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>541704146</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ... | \qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ... | ||
[[math]] | [[math]] | ||
The function <span style="font-family: Georgia,serif; font-size: 110%;">//v(r)//</span> is the order (1,1) Padé approximant of the function <span style="font-family: Georgia,serif; font-size: 110%;"> //J(r) =//½ ln //r// </span> in the region of <span style="font-family: Georgia,serif; font-size: 110%;">//r// = 1</span>, which has the property of matching the function value and its first and second derivatives at this value of //<span style="font-family: Georgia,serif; font-size: 110%;">r</span>//. The bimodular approximant function is thus accurate to second order in <span style="font-family: Georgia,serif; font-size: 110%;">//r// – 1</span>. | The function <span style="font-family: Georgia,serif; font-size: 110%;">//v(r)//</span> is the order (1,1) [[http://en.wikipedia.org/wiki/Pad%C3%A9_approximant|Padé approximant]] of the function <span style="font-family: Georgia,serif; font-size: 110%;"> //J(r) =//½ ln //r// </span> in the region of <span style="font-family: Georgia,serif; font-size: 110%;">//r// = 1</span>, which has the property of matching the function value and its first and second derivatives at this value of //<span style="font-family: Georgia,serif; font-size: 110%;">r</span>//. The bimodular approximant function is thus accurate to second order in <span style="font-family: Georgia,serif; font-size: 110%;">//r// – 1</span>. | ||
As an example, the size of the perfect fifth (in dNp units) is | As an example, the size of the perfect fifth (in dNp units) is | ||
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Tuning the perfect fourth and perfect fifth in the ratio of their approximants (1/7 : 1/5 = 5 : 7) and adjusting their sum to a pure octave yields 12edo (considered as a 3-limit temperament). This is an example of the high accuracy typically obtainable from a tempering policy which takes two intervals which are similar in size and not too large, tunes them in their approximant ratio, and normalises their sum to a pure interval. | Tuning the perfect fourth and perfect fifth in the ratio of their approximants (1/7 : 1/5 = 5 : 7) and adjusting their sum to a pure octave yields 12edo (considered as a 3-limit temperament). This is an example of the high accuracy typically obtainable from a tempering policy which takes two intervals which are similar in size and not too large, tunes them in their approximant ratio, and normalises their sum to a pure interval. | ||
Aspects of 12edo considered as a 5-limit temperament can be explained by noting that it tunes the major third, major sixth and octave in the ratio of their approximants (1/9 : 1/4 : 1/3 = 4 : 9 : 12). The accuracy here is lower because the octave is of a size where the approximant has a significant error, and tuning the octave pure assigns the entire error to the smaller intervals. | Aspects of 12edo considered as a 5-limit temperament can be explained by noting that it tunes the major third, major sixth and octave in the ratio of their approximants (1/9 : 1/4 : 1/3 = 4 : 9 : 12). The accuracy here is lower because the octave is of a size where the approximant has a significant error, and tuning the octave pure assigns the entire error to the smaller intervals. | ||
Tuning the major third and perfect fifth in the in the ratio of their approximants (1/9 : 1/5) and tuning the fifth pure yields Carlos | Tuning the major third and perfect fifth in the in the ratio of their approximants (1/9 : 1/5) and tuning the fifth pure yields [[Carlos Alpha]]. | ||
Tuning the minor third and perfect fifth in the in the ratio of their approximants (1/11 : 1/5) and tuning the fifth pure yields Carlos | Tuning the minor third and perfect fifth in the in the ratio of their approximants (1/11 : 1/5) and tuning the fifth pure yields [[Carlos Beta]]. | ||
Tuning the minor third and major third in the ratio of their approximants (1/11 : 1/9) and adjusting their sum to a perfect fifth yields Carlos | Tuning the minor third and major third in the ratio of their approximants (1/11 : 1/9) and adjusting their sum to a perfect fifth yields [[Carlos Gamma]] . This temperament has high accuracy because it conforms to the policy noted above. | ||
Tuning the octave pure while preserving the ratios specified above yields, respectively, 31edo, 19edo and 34edo. | Tuning the octave pure while preserving the ratios specified above yields, respectively, 31edo, 19edo and 34edo. | ||
Tuning the intervals __9/7__, __7/5__ and __5/3__ in the ratio of their approximants (1/8 : 1/6 : 1/4 = 3 : 4 : 6) and adjusting their sum to a perfect twelfth yields the equally tempered Bohlen-Pierce scale. | Tuning the intervals __9/7__, __7/5__ and __5/3__ in the ratio of their approximants (1/8 : 1/6 : 1/4 = 3 : 4 : 6) and adjusting their sum to a perfect twelfth yields the [[Bohlen-Pierce|equally tempered Bohlen-Pierce scale]]. | ||
Tuning the intervals __11/9__, __9/7__, __3/2__ and __5/3__ in the ratio of their approximants (1/10 : 1/8 : 1/5 : 1/4 = 4 : 5 : 8 : 10) and adjusting their sum to a major tenth yields 88 cent equal temperament. | Tuning the intervals __11/9__, __9/7__, __3/2__ and __5/3__ in the ratio of their approximants (1/10 : 1/8 : 1/5 : 1/4 = 4 : 5 : 8 : 10) and adjusting their sum to a major tenth yields [[88cET|88 cent equal temperament]]. | ||
Relationships of this sort can be identified in all equal temperaments. | Relationships of this sort can be identified in all equal temperaments. | ||
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and therefore | and therefore | ||
[[math]] | [[math]] | ||
\qquad | \qquad b(J_1,J_2) ≈ \tfrac{1}{3} (J_1+J_2)(J_2-J_1) b_m | ||
[[math]] | [[math]] | ||
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===<span style="font-family: Arial,Helvetica,sans-serif;">Remarks</span>=== | ===<span style="font-family: Arial,Helvetica,sans-serif;">Remarks</span>=== | ||
If the harmonics have indices //<span style="font-family: Georgia,serif; font-size: 110%;">n – m, n</span>// and //<span style="font-family: Georgia,serif; font-size: 110%;">n + m</span>//, the two intervals have reduced frequency ratios //<span style="font-family: Georgia,serif; font-size: 110%;">n/(n – m)</span>// and //<span style="font-family: Georgia,serif; font-size: 110%;">(n + m)/n</span>//. It can be assumed that //<span style="font-family: Georgia,serif; font-size: 110%;">n</span>// and //<span style="font-family: Georgia,serif; font-size: 110%;">m</span>// have no common factor. | If the harmonics have indices //<span style="font-family: Georgia,serif; font-size: 110%;">n – m, n</span>// and //<span style="font-family: Georgia,serif; font-size: 110%;">n + m</span>//, the two intervals have reduced frequency ratios //<span style="font-family: Georgia,serif; font-size: 110%;">n/(n – m)</span>// and //<span style="font-family: Georgia,serif; font-size: 110%;">(n + m)/n</span>//. It can be assumed that //<span style="font-family: Georgia,serif; font-size: 110%;">n</span>// and //<span style="font-family: Georgia,serif; font-size: 110%;">m</span>// have no common factor. | ||
//<span style="font-family: Georgia,serif; font-size: 110%;">m</span>// is the epimoricity of the intervals. When //<span style="font-family: Georgia,serif; font-size: 110%;">m</span>// = 1 the intervals are adjacent superparticular intervals. | //<span style="font-family: Georgia,serif; font-size: 110%;">m</span>// is the [[Superpartient|degree of epimoricity]] of the intervals. When //<span style="font-family: Georgia,serif; font-size: 110%;">m</span>// = 1 the intervals are adjacent superparticular intervals. | ||
The geometric mean of the frequency ratios is the frequency ratio corresponding to the arithmetic mean of the intervals. | The geometric mean of the frequency ratios is the frequency ratio corresponding to the arithmetic mean of the intervals. | ||
===<span style="font-family: Arial,Helvetica,sans-serif;">Proof</span>=== | ===<span style="font-family: Arial,Helvetica,sans-serif;">Proof</span>=== | ||
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<span style="font-family: Georgia,serif; font-size: 110%;">√5/2 = 1.11803.</span> | <span style="font-family: Georgia,serif; font-size: 110%;">√5/2 = 1.11803.</span> | ||
==<span style="font-family: Arial,Helvetica,sans-serif;"> | ==<span style="font-family: Arial,Helvetica,sans-serif;">Argent temperament</span>== | ||
As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, 1.4000. | As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, 1.4000. | ||
It can be shown that the error in a pair of intervals tuned in the ratio of their approximants is minimised if the sum of the intervals is normalised – in this case to a pure octave. If this is done while maintaining the √2 ratio the perfect fifth and fourth are tempered to | It can be shown that the error in a pair of intervals tuned in the ratio of their approximants is minimised if the sum of the intervals is normalised – in this case to a pure octave. If this is done while maintaining the √2 ratio the perfect fifth and fourth are tempered to | ||
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<span style="color: #ffffff;">###</span>Perfect fourth = __4/3__ = 497.056 cents | <span style="color: #ffffff;">###</span>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). | 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 | A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the [[http://en.wikipedia.org/wiki/Silver_ratio|silver ratio]] (sometimes called the silver mean), //<span style="font-family: Georgia,serif; font-size: 110%;">δ</span>//<span style="vertical-align: sub;">s </span>= <span style="font-family: Georgia,serif; font-size: 110%;">√2 + 1 = 2.4142</span>. On this basis, and by analogy with [[Golden Meantone|golden meantone]] temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘argent temperament' is proposed instead. | ||
Argent temperament has interesting fractal properties which help to explain why 3-limit tuning forms aesthetically pleasing scales. | |||
The continued fraction expansion of the silver ratio has a particularly simple form: | The continued fraction expansion of the silver ratio has a particularly simple form: | ||
[[math]] | [[math]] | ||
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\qquad L – 2s = (\delta_s – 2)s = s/\delta_s | \qquad L – 2s = (\delta_s – 2)s = s/\delta_s | ||
[[math]] | [[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: | (since 1///<span style="font-family: Georgia,serif; font-size: 110%;">δ</span>//<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s </span><span style="font-family: Georgia,serif; font-size: 110%;">= √2 - 1 = //δ//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span><span style="font-family: Georgia,serif; font-size: 110%;"> - 2</span>) and consequently this process can be continued indefinitely to generate sequences of decreasing intervals as follows. The names are assigned according to Pythagorean conventions (the limma being the Pythagorean semitone __<span style="font-family: Arial,Helvetica,sans-serif;">256/243</span>__) followed by tempered and just sizes in cents: | ||
|| Octave | || Octave | ||
1200.00 | 1200.00 | ||
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In this fractal temperament, multiplying or dividing any interval by the factor //<span style="font-family: Georgia,serif; font-size: 110%;">δ</span>//<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">//s// </span><span style="font-family: Georgia,serif; font-size: 110%;">= √2 + 1</span> produces another interval in the temperament. Any tempered interval //<span style="font-family: Georgia,serif; font-size: 110%;">J’</span>// can be split into three parts, two of equal size //<span style="font-family: Georgia,serif; font-size: 110%;">J’</span>//<span style="font-family: Georgia,serif; font-size: 110%;">/</span>//<span style="font-family: Georgia,serif; font-size: 110%;">δ</span>//<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span> and the other of size //<span style="font-family: Georgia,serif; font-size: 110%;">J’</span>//<span style="font-family: Georgia,serif; font-size: 110%;">/</span>//<span style="font-family: Georgia,serif; font-size: 110%;">δ</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span>//. | In this fractal temperament, multiplying or dividing any interval by the factor //<span style="font-family: Georgia,serif; font-size: 110%;">δ</span>//<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">//s// </span><span style="font-family: Georgia,serif; font-size: 110%;">= √2 + 1</span> produces another interval in the temperament. Any tempered interval //<span style="font-family: Georgia,serif; font-size: 110%;">J’</span>// can be split into three parts, two of equal size //<span style="font-family: Georgia,serif; font-size: 110%;">J’</span>//<span style="font-family: Georgia,serif; font-size: 110%;">/</span>//<span style="font-family: Georgia,serif; font-size: 110%;">δ</span>//<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span> and the other of size //<span style="font-family: Georgia,serif; font-size: 110%;">J’</span>//<span style="font-family: Georgia,serif; font-size: 110%;">/</span>//<span style="font-family: Georgia,serif; font-size: 110%;">δ</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span>//. | ||
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. | A similar principle applies to multiplication and division by the factor √2, except that intervals in the top row of the table cannot be divided by √2 to yield another interval in the temperament. These properties means that the temperament would support compositional techniques based on novel types of intervallic augmentation and diminution. | ||
Successive convergents of the silver | Successive convergents of the silver ratio produce ratios involving [[http://en.wikipedia.org/wiki/Pell_number|Pell numbers]]. | ||
<span style="color: #ffffff;">###</span>√2 + 1 ≈ 2, 5/2, 12/5, 29/12, 70/29…, | <span style="color: #ffffff;">###</span>√2 + 1 ≈ 2, 5/2, 12/5, 29/12, 70/29…, | ||
Other approximations to the silver | Other approximations to the silver ratio are provided by ratios of consecutive half Pell-Lucas numbers, which are formed by adding consecutive Pell numbers | ||
<span style="color: #ffffff;">###</span>√2 + 1 ≈ 3, 7/3, 17/7, 41/17, 99/41…, | <span style="color: #ffffff;">###</span>√2 + 1 ≈ 3, 7/3, 17/7, 41/17, 99/41…, | ||
This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, 41edo, 70edo etc.). | This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, 41edo, 70edo etc.). | ||
The accuracy of the | The accuracy of the argent fifth means that the scheme produces workable approximations to the true sizes of the 3-limit intervals featured in the table. However, if the table is extended one further step to the right, errors of sign begin to occur (the next column containing the 29-tone comma and //minus// the 41-tone comma). | ||
Figure 2 is a //continued fraction jigsaw// showing the sizes of the octave (o), fourth (f), tone (T), limma (s<span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;">p</span>), Pythagorean comma (p) and 29-tone comma (p<span style="font-size: 60%;">29</span>) as tempered by 41edo - an approximation to | Figure 2 is a //continued fraction jigsaw// showing the sizes of the octave (o), fourth (f), tone (T), limma (s<span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;">p</span>), Pythagorean comma (p) and 29-tone comma (p<span style="font-size: 60%;">29</span>) as tempered by 41edo - an approximation to argent temperament. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc. | ||
[[image:Continued fraction jigsaw 41edo.png width="800" height="396"]] | [[image:Continued fraction jigsaw 41edo.png width="800" height="396"]] | ||
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<span style="color: #ffffff;">######</span>Figure 2. Continued fraction jigsaw for 41edo | <span style="color: #ffffff;">######</span>Figure 2. Continued fraction jigsaw for 41edo | ||
Figure 3 is a geometrical representation of | Figure 3 is a geometrical representation of argent temperament in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = large tone, m<span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;">p</span><span style="color: #ffffff;">#</span>= Pythagorean minor third, s<span style="font-size: 80%; vertical-align: super;">p</span><span style="color: #ffffff;">#</span>= Pythagorean limma, X<span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;">p</span><span style="color: #ffffff;">#</span>= Pythagorean apotome, p = Pythagorean comma. | ||
[[image:Silver temperament graphic.png width="800" height="587"]] | [[image:Silver temperament graphic.png width="800" height="587"]] | ||
<span style="color: #ffffff;">######</span>Figure 3. Geometrical representation of | <span style="color: #ffffff;">######</span>Figure 3. Geometrical representation of argent temperament | ||
By the [[http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem|Gelfond-Schneider theorem ]] the frequency ratios of all argent intervals (<span style="font-family: Georgia,serif; font-size: 110%;">//r// = 2</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">√2//a//+//b//</span>, where//<span style="font-family: Georgia,serif; font-size: 110%;"> a</span>// and //<span style="font-family: Georgia,serif; font-size: 110%;">b</span>// are integers) are transcendental, with the exception of octave multiples (<span style="font-family: Georgia,serif; font-size: 110%;">//a// = 0</span>). | |||
==Golden temperaments== | ==Golden temperaments== | ||
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|| Sequence 1:<span style="color: #ffffff;">#</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff;">#</span>//t///2 - 3//c//<span style="color: #ffffff; font-family: Georgia,serif;">#</span> </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>2//c//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//t///2 //- c//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//T - t///2 </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//t//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//T + t///2<span style="color: #ffffff; font-family: Georgia,serif;">#</span></span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//M + t///2<span style="color: #ffffff; font-family: Georgia,serif;">#</span> </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>2//M// </span> || | || Sequence 1:<span style="color: #ffffff;">#</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff;">#</span>//t///2 - 3//c//<span style="color: #ffffff; font-family: Georgia,serif;">#</span> </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>2//c//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//t///2 //- c//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//T - t///2 </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//t//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//T + t///2<span style="color: #ffffff; font-family: Georgia,serif;">#</span></span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//M + t///2<span style="color: #ffffff; font-family: Georgia,serif;">#</span> </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>2//M// </span> || | ||
|| Sequence 2:<span style="color: #ffffff;">#</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//magic//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//diesis//</span> || //<span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>chroma<span style="color: #ffffff; font-family: Georgia,serif;">#</span></span>// || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//semitone//<span style="color: #ffffff; font-family: Georgia,serif;">#</span></span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//t//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//mp//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//f - c//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//m6p - c//</span><span style="color: #ffffff;">#</span> || | || Sequence 2:<span style="color: #ffffff;">#</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//magic//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//diesis//</span> || //<span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>chroma<span style="color: #ffffff; font-family: Georgia,serif;">#</span></span>// || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//semitone//<span style="color: #ffffff; font-family: Georgia,serif;">#</span></span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//t//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//mp//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//f - c//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//m6p - c//</span><span style="color: #ffffff;">#</span> || | ||
|| Difference: || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>-3//σ///2</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//σ//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>-//σ///2</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//σ///2</span> || <span style="color: #ffffff; font-family: Georgia,serif;">#</span | || Difference: || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>-3//σ///2</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//σ//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>-//σ///2</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//σ///2</span> || <span style="color: #ffffff; font-family: Georgia,serif;">#</span>0 || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//σ///2</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//σ///2</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//σ// </span> || | ||
|| Seq 1 ratios: || || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6120</span><span style="color: #ffffff; font-family: Georgia,serif;">##</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6204</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6171</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6184<span style="color: #ffffff; font-family: Georgia,serif;">#</span> </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6179</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6181</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6180</span> || | || Seq 1 ratios: || || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6120</span><span style="color: #ffffff; font-family: Georgia,serif;">##</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6204</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6171</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6184<span style="color: #ffffff; font-family: Georgia,serif;">#</span> </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6179</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6181</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6180</span> || | ||
|| Seq 2 ratios: || || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.3865</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.7212</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.5810</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6325</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6125</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6201 </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6172 </span> || | || Seq 2 ratios: || || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.3865</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.7212</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.5810</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6325</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6125</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6201 </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6172 </span> || | ||
| Line 409: | Line 411: | ||
\qquad q_1^2 + q_2^2 = q_3^2 | \qquad q_1^2 + q_2^2 = q_3^2 | ||
[[math]] | [[math]] | ||
they can be said to form a Pythagorean triple. | they can be said to form a [[http://en.wikipedia.org/wiki/Pythagorean_triple|Pythagorean triple]]. | ||
The following are three examples. In the first and third cases, their counterparts in 12edo, <span style="font-family: Georgia,serif; font-size: 110%;">//J//1', //J//2'</span> and <span style="font-family: Georgia,serif; font-size: 110%;">//J//3'</span> , are also Pythagorean triples: | The following are three examples. In the first and third cases, their counterparts in 12edo, <span style="font-family: Georgia,serif; font-size: 110%;">//J//1', //J//2'</span> and <span style="font-family: Georgia,serif; font-size: 110%;">//J//3'</span>, are also Pythagorean triples: | ||
|| <span style="font-family: Georgia,serif;"><span style="color: #ffffff; line-height: 0px; overflow: hidden;">#</span>//J//1</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//J//2</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//J//3</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//q//1</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//q//2</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//q//3 </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//J//1' </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//J//2' </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//J//3'<span style="color: #ffffff; font-family: Georgia;">#</span></span> || | || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; line-height: 0px; overflow: hidden;">#</span>//J//1</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//J//2</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//J//3</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//q//1</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//q//2</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//q//3 </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//J//1' </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//J//2' </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//J//3'<span style="color: #ffffff; font-family: Georgia;">#</span></span> || | ||
|| <span style="color: #ffffff; font-family: Georgia;">#</span>__6/5__<span style="color: #ffffff;">#</span> || <span style="color: #ffffff; font-family: Georgia;">#</span>__5/4__ || <span style="color: #ffffff; font-family: Georgia;">#</span>__4/3__<span style="color: #ffffff; font-family: Georgia;">#</span> || <span style="color: #ffffff; font-family: Georgia;">#</span>1/2√30<span style="color: #ffffff; font-family: Georgia;">#</span> || <span style="color: #ffffff; font-family: Georgia;">#</span>1/4√5 || <span style="color: #ffffff; font-family: Georgia;">#</span>1/4√3 || <span style="color: #ffffff; font-family: Georgia;">#</span>3 || <span style="color: #ffffff; font-family: Georgia;">#</span>4 || <span style="color: #ffffff; font-family: Georgia;">#</span>5 || | || <span style="color: #ffffff; font-family: Georgia;">#</span>__6/5__<span style="color: #ffffff;">#</span> || <span style="color: #ffffff; font-family: Georgia;">#</span>__5/4__ || <span style="color: #ffffff; font-family: Georgia;">#</span>__4/3__<span style="color: #ffffff; font-family: Georgia;">#</span> || <span style="color: #ffffff; font-family: Georgia;">#</span>1/2√30<span style="color: #ffffff; font-family: Georgia;">#</span> || <span style="color: #ffffff; font-family: Georgia;">#</span>1/4√5 || <span style="color: #ffffff; font-family: Georgia;">#</span>1/4√3 || <span style="color: #ffffff; font-family: Georgia;">#</span>3 || <span style="color: #ffffff; font-family: Georgia;">#</span>4 || <span style="color: #ffffff; font-family: Georgia;">#</span>5 || | ||
| Line 418: | Line 420: | ||
==A small 34edo comma== | ==A small 34edo comma== | ||
<span style="color: #333333;">As [[Gene Ward Smith]] has noted, the </span>5-limit comma <span style="color: #333333;">|-433 -137 280> (‘//selenia//’) is remarkably small at just 0.004764 cents. The minute size of this comma can be explained using qu</span>adratic approximants. | <span style="color: #333333;">As [[Gene Ward Smith]] has noted, the </span>5-limit comma <span style="color: #333333;">|-433 -137 280> (‘//selenia//’) is remarkably small at just 0.004764 cents. The minute size of this comma can be explained using qu</span>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> (4.769 cents) and the //semisuper// comma |23 6 -14> (3.338 cents). In particular, | It can be shown, using a suitable [[Comma-based lattices|comma-based lattice]], that every comma tempered out by 34edo can be expressed as an integer linear combination of the [[Gammic node|//gammic// comma ]]|-29 -11 20> (4.769 cents) and the //semisuper// comma (AKA //[[vishnuzma]]//) |23 6 -14> (3.338 cents). In particular, | ||
<span style="color: #333333;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span>//selenia// = 7 //gammic// – 10 //semisuper//</span> | <span style="color: #333333;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span>//selenia// = 7 //gammic// – 10 //semisuper//</span> | ||
<span style="color: #333333;">So to prove that //selenia// is small we must show that //gammic/////semisuper// ≈ 10/7.</span> | <span style="color: #333333;">So to prove that //selenia// is small we must show that //gammic/////semisuper// ≈ 10/7.</span> | ||
| Line 464: | Line 466: | ||
<span style="font-family: Arial,Helvetica,sans-serif;">Putting in the numbers:</span> | <span style="font-family: Arial,Helvetica,sans-serif;">Putting in the numbers:</span> | ||
//<span style="color: #ffffff;">###</span><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">gammic </span>//<span style="font-family: Georgia,serif; font-size: 110%;">=</span> <span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1 – (2/5) (1/4) (1/30 + 1/20) = 1 – 1/120</span> | //<span style="color: #ffffff;">###</span><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">gammic </span>//<span style="font-family: Georgia,serif; font-size: 110%;">=</span> <span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1 – (2/5) (1/4) (1/30 + 1/20) = 1 – 1/120</span> | ||
<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span>//<span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">semisuper </span>//<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">= | <span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span>//<span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">semisuper </span>//<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">= 1 – (2/5)(1/4) (1/600 + 1/12) = 1 – (1/120) (51/50)</span> | ||
<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span>//<span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">gammic</span> <span style="font-family: Georgia,serif;">- </span><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">semisuper </span>//<span style="font-family: Georgia,serif; font-size: 110%;">= | <span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span>//<span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">gammic</span> <span style="font-family: Georgia,serif;">- </span><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">semisuper </span>//<span style="font-family: Georgia,serif; font-size: 110%;">= 1/6000</span> | ||
<span style="color: #333333;">Therefore</span> | <span style="color: #333333;">Therefore</span> | ||
<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><span style="font-family: Georgia,serif; font-size: 110%;">//selenia// ≈ 7 //gammic// (1/6000) (120/119) = //gammic///850 = 0.00561</span><span style="color: #333333;"> cents</span> | <span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><span style="font-family: Georgia,serif; font-size: 110%;">//selenia// ≈ 7 //gammic// (1/6000) (120/119) = //gammic///850 = 0.00561</span><span style="color: #333333;"> cents</span> | ||
| Line 475: | Line 477: | ||
and <span style="font-family: Georgia,serif; font-size: 110%;">(//q//(25/24))</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span> , being small in comparison to the other terms, compromises this equality only slightly. | and <span style="font-family: Georgia,serif; font-size: 110%;">(//q//(25/24))</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span> , being small in comparison to the other terms, compromises this equality only slightly. | ||
= | =Sources and acknowledgements= | ||
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> | 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. | ||
The tuning referred to here as argent temperament was described by [[graham breed|Graham Breed]] and Paul Hahn in posts (#12599, #12670) to the Yahoo tuning list on 10 and 12 August 2000. | |||
Thanks to [[Gene Ward Smith]] for the Gelfond-Schneider result.</pre></div> | |||
<h4>Original HTML content:</h4> | <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><!-- ws:start:WikiTextHeadingRule:49:&lt;h1&gt; --><h1 id="toc0"><a name="x1. Introduction"></a><!-- ws:end:WikiTextHeadingRule:49 --><strong><span style="font-size: 20px;">1. Introduction</span></strong></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><!-- ws:start:WikiTextHeadingRule:49:&lt;h1&gt; --><h1 id="toc0"><a name="x1. Introduction"></a><!-- ws:end:WikiTextHeadingRule:49 --><strong><span style="font-size: 20px;">1. Introduction</span></strong></h1> | ||
| Line 553: | Line 557: | ||
\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...&lt;br/&gt;[[math]] | \qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...</script><!-- ws:end:WikiTextMathRule:8 --><br /> | --><script type="math/tex">\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...</script><!-- ws:end:WikiTextMathRule:8 --><br /> | ||
The function <span style="font-family: Georgia,serif; font-size: 110%;"><em>v(r)</em></span> is the order (1,1) Padé approximant of the function <span style="font-family: Georgia,serif; font-size: 110%;"> <em>J(r) =</em>½ ln <em>r</em> </span> in the region of <span style="font-family: Georgia,serif; font-size: 110%;"><em>r</em> = 1</span>, which has the property of matching the function value and its first and second derivatives at this value of <em><span style="font-family: Georgia,serif; font-size: 110%;">r</span></em>. The bimodular approximant function is thus accurate to second order in <span style="font-family: Georgia,serif; font-size: 110%;"><em>r</em> – 1</span>.<br /> | The function <span style="font-family: Georgia,serif; font-size: 110%;"><em>v(r)</em></span> is the order (1,1) <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pad%C3%A9_approximant" rel="nofollow">Padé approximant</a> of the function <span style="font-family: Georgia,serif; font-size: 110%;"> <em>J(r) =</em>½ ln <em>r</em> </span> in the region of <span style="font-family: Georgia,serif; font-size: 110%;"><em>r</em> = 1</span>, which has the property of matching the function value and its first and second derivatives at this value of <em><span style="font-family: Georgia,serif; font-size: 110%;">r</span></em>. The bimodular approximant function is thus accurate to second order in <span style="font-family: Georgia,serif; font-size: 110%;"><em>r</em> – 1</span>.<br /> | ||
<br /> | <br /> | ||
As an example, the size of the perfect fifth (in dNp units) is<br /> | As an example, the size of the perfect fifth (in dNp units) is<br /> | ||
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Tuning the perfect fourth and perfect fifth in the ratio of their approximants (1/7 : 1/5 = 5 : 7) and adjusting their sum to a pure octave yields 12edo (considered as a 3-limit temperament). This is an example of the high accuracy typically obtainable from a tempering policy which takes two intervals which are similar in size and not too large, tunes them in their approximant ratio, and normalises their sum to a pure interval.<br /> | Tuning the perfect fourth and perfect fifth in the ratio of their approximants (1/7 : 1/5 = 5 : 7) and adjusting their sum to a pure octave yields 12edo (considered as a 3-limit temperament). This is an example of the high accuracy typically obtainable from a tempering policy which takes two intervals which are similar in size and not too large, tunes them in their approximant ratio, and normalises their sum to a pure interval.<br /> | ||
Aspects of 12edo considered as a 5-limit temperament can be explained by noting that it tunes the major third, major sixth and octave in the ratio of their approximants (1/9 : 1/4 : 1/3 = 4 : 9 : 12). The accuracy here is lower because the octave is of a size where the approximant has a significant error, and tuning the octave pure assigns the entire error to the smaller intervals.<br /> | Aspects of 12edo considered as a 5-limit temperament can be explained by noting that it tunes the major third, major sixth and octave in the ratio of their approximants (1/9 : 1/4 : 1/3 = 4 : 9 : 12). The accuracy here is lower because the octave is of a size where the approximant has a significant error, and tuning the octave pure assigns the entire error to the smaller intervals.<br /> | ||
Tuning the major third and perfect fifth in the in the ratio of their approximants (1/9 : 1/5) and tuning the fifth pure yields Carlos | Tuning the major third and perfect fifth in the in the ratio of their approximants (1/9 : 1/5) and tuning the fifth pure yields <a class="wiki_link" href="/Carlos%20Alpha">Carlos Alpha</a>.<br /> | ||
Tuning the minor third and perfect fifth in the in the ratio of their approximants (1/11 : 1/5) and tuning the fifth pure yields Carlos | Tuning the minor third and perfect fifth in the in the ratio of their approximants (1/11 : 1/5) and tuning the fifth pure yields <a class="wiki_link" href="/Carlos%20Beta">Carlos Beta</a>.<br /> | ||
Tuning the minor third and major third in the ratio of their approximants (1/11 : 1/9) and adjusting their sum to a perfect fifth yields Carlos | Tuning the minor third and major third in the ratio of their approximants (1/11 : 1/9) and adjusting their sum to a perfect fifth yields <a class="wiki_link" href="/Carlos%20Gamma">Carlos Gamma</a> . This temperament has high accuracy because it conforms to the policy noted above.<br /> | ||
Tuning the octave pure while preserving the ratios specified above yields, respectively, 31edo, 19edo and 34edo.<br /> | Tuning the octave pure while preserving the ratios specified above yields, respectively, 31edo, 19edo and 34edo.<br /> | ||
Tuning the intervals <u>9/7</u>, <u>7/5</u> and <u>5/3</u> in the ratio of their approximants (1/8 : 1/6 : 1/4 = 3 : 4 : 6) and adjusting their sum to a perfect twelfth yields the equally tempered Bohlen-Pierce scale.<br /> | Tuning the intervals <u>9/7</u>, <u>7/5</u> and <u>5/3</u> in the ratio of their approximants (1/8 : 1/6 : 1/4 = 3 : 4 : 6) and adjusting their sum to a perfect twelfth yields the <a class="wiki_link" href="/Bohlen-Pierce">equally tempered Bohlen-Pierce scale</a>.<br /> | ||
Tuning the intervals <u>11/9</u>, <u>9/7</u>, <u>3/2</u> and <u>5/3</u> in the ratio of their approximants (1/10 : 1/8 : 1/5 : 1/4 = 4 : 5 : 8 : 10) and adjusting their sum to a major tenth yields 88 cent equal temperament.<br /> | Tuning the intervals <u>11/9</u>, <u>9/7</u>, <u>3/2</u> and <u>5/3</u> in the ratio of their approximants (1/10 : 1/8 : 1/5 : 1/4 = 4 : 5 : 8 : 10) and adjusting their sum to a major tenth yields <a class="wiki_link" href="/88cET">88 cent equal temperament</a>.<br /> | ||
Relationships of this sort can be identified in all equal temperaments.<br /> | Relationships of this sort can be identified in all equal temperaments.<br /> | ||
<br /> | <br /> | ||
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<!-- ws:start:WikiTextMathRule:19: | <!-- ws:start:WikiTextMathRule:19: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\qquad | \qquad b(J_1,J_2) ≈ \tfrac{1}{3} (J_1+J_2)(J_2-J_1) b_m&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\qquad | --><script type="math/tex">\qquad b(J_1,J_2) ≈ \tfrac{1}{3} (J_1+J_2)(J_2-J_1) b_m</script><!-- ws:end:WikiTextMathRule:19 --><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:61:&lt;h3&gt; --><h3 id="toc6"><a name="x2. Bimodular approximants-Bimodular commas-Examples"></a><!-- ws:end:WikiTextHeadingRule:61 -->Examples</h3> | <!-- ws:start:WikiTextHeadingRule:61:&lt;h3&gt; --><h3 id="toc6"><a name="x2. Bimodular approximants-Bimodular commas-Examples"></a><!-- ws:end:WikiTextHeadingRule:61 -->Examples</h3> | ||
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<!-- ws:start:WikiTextHeadingRule:79:&lt;h3&gt; --><h3 id="toc15"><a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Remarks"></a><!-- ws:end:WikiTextHeadingRule:79 --><span style="font-family: Arial,Helvetica,sans-serif;">Remarks</span></h3> | <!-- ws:start:WikiTextHeadingRule:79:&lt;h3&gt; --><h3 id="toc15"><a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Remarks"></a><!-- ws:end:WikiTextHeadingRule:79 --><span style="font-family: Arial,Helvetica,sans-serif;">Remarks</span></h3> | ||
If the harmonics have indices <em><span style="font-family: Georgia,serif; font-size: 110%;">n – m, n</span></em> and <em><span style="font-family: Georgia,serif; font-size: 110%;">n + m</span></em>, the two intervals have reduced frequency ratios <em><span style="font-family: Georgia,serif; font-size: 110%;">n/(n – m)</span></em> and <em><span style="font-family: Georgia,serif; font-size: 110%;">(n + m)/n</span></em>. It can be assumed that <em><span style="font-family: Georgia,serif; font-size: 110%;">n</span></em> and <em><span style="font-family: Georgia,serif; font-size: 110%;">m</span></em> have no common factor.<br /> | If the harmonics have indices <em><span style="font-family: Georgia,serif; font-size: 110%;">n – m, n</span></em> and <em><span style="font-family: Georgia,serif; font-size: 110%;">n + m</span></em>, the two intervals have reduced frequency ratios <em><span style="font-family: Georgia,serif; font-size: 110%;">n/(n – m)</span></em> and <em><span style="font-family: Georgia,serif; font-size: 110%;">(n + m)/n</span></em>. It can be assumed that <em><span style="font-family: Georgia,serif; font-size: 110%;">n</span></em> and <em><span style="font-family: Georgia,serif; font-size: 110%;">m</span></em> have no common factor.<br /> | ||
<em><span style="font-family: Georgia,serif; font-size: 110%;">m</span></em> is the epimoricity of the intervals. When <em><span style="font-family: Georgia,serif; font-size: 110%;">m</span></em> = 1 the intervals are adjacent superparticular intervals.<br /> | <em><span style="font-family: Georgia,serif; font-size: 110%;">m</span></em> is the <a class="wiki_link" href="/Superpartient">degree of epimoricity</a> of the intervals. When <em><span style="font-family: Georgia,serif; font-size: 110%;">m</span></em> = 1 the intervals are adjacent superparticular intervals.<br /> | ||
The geometric mean of the frequency ratios is the frequency ratio corresponding to the arithmetic mean of the intervals.<br /> | The geometric mean of the frequency ratios is the frequency ratio corresponding to the arithmetic mean of the intervals.<br /> | ||
<!-- ws:start:WikiTextHeadingRule:81:&lt;h3&gt; --><h3 id="toc16"><a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Proof"></a><!-- ws:end:WikiTextHeadingRule:81 --><span style="font-family: Arial,Helvetica,sans-serif;">Proof</span></h3> | <!-- ws:start:WikiTextHeadingRule:81:&lt;h3&gt; --><h3 id="toc16"><a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Proof"></a><!-- ws:end:WikiTextHeadingRule:81 --><span style="font-family: Arial,Helvetica,sans-serif;">Proof</span></h3> | ||
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<span style="font-family: Georgia,serif; font-size: 110%;">√5/2 = 1.11803.</span><br /> | <span style="font-family: Georgia,serif; font-size: 110%;">√5/2 = 1.11803.</span><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:85:&lt;h2&gt; --><h2 id="toc18"><a name="x4. Quadratic approximants- | <!-- ws:start:WikiTextHeadingRule:85:&lt;h2&gt; --><h2 id="toc18"><a name="x4. Quadratic approximants-Argent temperament"></a><!-- ws:end:WikiTextHeadingRule:85 --><span style="font-family: Arial,Helvetica,sans-serif;">Argent temperament</span></h2> | ||
As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, 1.4000.<br /> | As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, 1.4000.<br /> | ||
It can be shown that the error in a pair of intervals tuned in the ratio of their approximants is minimised if the sum of the intervals is normalised – in this case to a pure octave. If this is done while maintaining the √2 ratio the perfect fifth and fourth are tempered to<br /> | It can be shown that the error in a pair of intervals tuned in the ratio of their approximants is minimised if the sum of the intervals is normalised – in this case to a pure octave. If this is done while maintaining the √2 ratio the perfect fifth and fourth are tempered to<br /> | ||
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<span style="color: #ffffff;">###</span>Perfect fourth = <u>4/3</u> = 497.056 cents<br /> | <span style="color: #ffffff;">###</span>Perfect fourth = <u>4/3</u> = 497.056 cents<br /> | ||
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).<br /> | 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).<br /> | ||
A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the | A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Silver_ratio" rel="nofollow">silver ratio</a> (sometimes called the silver mean), <em><span style="font-family: Georgia,serif; font-size: 110%;">δ</span></em><span style="vertical-align: sub;">s </span>= <span style="font-family: Georgia,serif; font-size: 110%;">√2 + 1 = 2.4142</span>. On this basis, and by analogy with <a class="wiki_link" href="/Golden%20Meantone">golden meantone</a> temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘argent temperament' is proposed instead.<br /> | ||
Argent temperament has interesting fractal properties which help to explain why 3-limit tuning forms aesthetically pleasing scales.<br /> | |||
The continued fraction expansion of the silver ratio has a particularly simple form:<br /> | The continued fraction expansion of the silver ratio has a particularly simple form:<br /> | ||
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\qquad L – 2s = (\delta_s – 2)s = 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 = s/\delta_s</script><!-- ws:end:WikiTextMathRule:39 --><br /> | --><script type="math/tex">\qquad L – 2s = (\delta_s – 2)s = s/\delta_s</script><!-- ws:end:WikiTextMathRule:39 --><br /> | ||
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:<br /> | (since 1<em>/<span style="font-family: Georgia,serif; font-size: 110%;">δ</span></em><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s </span><span style="font-family: Georgia,serif; font-size: 110%;">= √2 - 1 = <em>δ</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span><span style="font-family: Georgia,serif; font-size: 110%;"> - 2</span>) and consequently this process can be continued indefinitely to generate sequences of decreasing intervals as follows. The names are assigned according to Pythagorean conventions (the limma being the Pythagorean semitone <u><span style="font-family: Arial,Helvetica,sans-serif;">256/243</span></u>) followed by tempered and just sizes in cents:<br /> | ||
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In this fractal temperament, multiplying or dividing any interval by the factor <em><span style="font-family: Georgia,serif; font-size: 110%;">δ</span></em><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"><em>s</em> </span><span style="font-family: Georgia,serif; font-size: 110%;">= √2 + 1</span> produces another interval in the temperament. Any tempered interval <em><span style="font-family: Georgia,serif; font-size: 110%;">J’</span></em> can be split into three parts, two of equal size <em><span style="font-family: Georgia,serif; font-size: 110%;">J’</span></em><span style="font-family: Georgia,serif; font-size: 110%;">/</span><em><span style="font-family: Georgia,serif; font-size: 110%;">δ</span></em><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span> and the other of size <em><span style="font-family: Georgia,serif; font-size: 110%;">J’</span></em><span style="font-family: Georgia,serif; font-size: 110%;">/</span><em><span style="font-family: Georgia,serif; font-size: 110%;">δ</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span></em>.<br /> | In this fractal temperament, multiplying or dividing any interval by the factor <em><span style="font-family: Georgia,serif; font-size: 110%;">δ</span></em><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"><em>s</em> </span><span style="font-family: Georgia,serif; font-size: 110%;">= √2 + 1</span> produces another interval in the temperament. Any tempered interval <em><span style="font-family: Georgia,serif; font-size: 110%;">J’</span></em> can be split into three parts, two of equal size <em><span style="font-family: Georgia,serif; font-size: 110%;">J’</span></em><span style="font-family: Georgia,serif; font-size: 110%;">/</span><em><span style="font-family: Georgia,serif; font-size: 110%;">δ</span></em><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span> and the other of size <em><span style="font-family: Georgia,serif; font-size: 110%;">J’</span></em><span style="font-family: Georgia,serif; font-size: 110%;">/</span><em><span style="font-family: Georgia,serif; font-size: 110%;">δ</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span></em>.<br /> | ||
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.<br /> | 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.<br /> | ||
Successive convergents of the silver | Successive convergents of the silver ratio produce ratios involving <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pell_number" rel="nofollow">Pell numbers</a>.<br /> | ||
<span style="color: #ffffff;">###</span>√2 + 1 ≈ 2, 5/2, 12/5, 29/12, 70/29…,<br /> | <span style="color: #ffffff;">###</span>√2 + 1 ≈ 2, 5/2, 12/5, 29/12, 70/29…,<br /> | ||
Other approximations to the silver | Other approximations to the silver ratio are provided by ratios of consecutive half Pell-Lucas numbers, which are formed by adding consecutive Pell numbers<br /> | ||
<span style="color: #ffffff;">###</span>√2 + 1 ≈ 3, 7/3, 17/7, 41/17, 99/41…,<br /> | <span style="color: #ffffff;">###</span>√2 + 1 ≈ 3, 7/3, 17/7, 41/17, 99/41…,<br /> | ||
This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, 41edo, 70edo etc.).<br /> | This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, 41edo, 70edo etc.).<br /> | ||
The accuracy of the | The accuracy of the argent fifth means that the scheme produces workable approximations to the true sizes of the 3-limit intervals featured in the table. However, if the table is extended one further step to the right, errors of sign begin to occur (the next column containing the 29-tone comma and <em>minus</em> the 41-tone comma).<br /> | ||
<br /> | <br /> | ||
Figure 2 is a <em>continued fraction jigsaw</em> showing the sizes of the octave (o), fourth (f), tone (T), limma (s<span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;">p</span>), Pythagorean comma (p) and 29-tone comma (p<span style="font-size: 60%;">29</span>) as tempered by 41edo - an approximation to | Figure 2 is a <em>continued fraction jigsaw</em> showing the sizes of the octave (o), fourth (f), tone (T), limma (s<span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;">p</span>), Pythagorean comma (p) and 29-tone comma (p<span style="font-size: 60%;">29</span>) as tempered by 41edo - an approximation to argent temperament. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule:482:&lt;img src=&quot;/file/view/Continued%20fraction%20jigsaw%2041edo.png/541636098/800x396/Continued%20fraction%20jigsaw%2041edo.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 396px; width: 800px;&quot; /&gt; --><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;" /><!-- ws:end:WikiTextLocalImageRule:482 --><br /> | <!-- ws:start:WikiTextLocalImageRule:482:&lt;img src=&quot;/file/view/Continued%20fraction%20jigsaw%2041edo.png/541636098/800x396/Continued%20fraction%20jigsaw%2041edo.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 396px; width: 800px;&quot; /&gt; --><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;" /><!-- ws:end:WikiTextLocalImageRule:482 --><br /> | ||
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<span style="color: #ffffff;">######</span>Figure 2. Continued fraction jigsaw for 41edo<br /> | <span style="color: #ffffff;">######</span>Figure 2. Continued fraction jigsaw for 41edo<br /> | ||
<br /> | <br /> | ||
Figure 3 is a geometrical representation of | Figure 3 is a geometrical representation of argent temperament in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = large tone, m<span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;">p</span><span style="color: #ffffff;">#</span>= Pythagorean minor third, s<span style="font-size: 80%; vertical-align: super;">p</span><span style="color: #ffffff;">#</span>= Pythagorean limma, X<span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;">p</span><span style="color: #ffffff;">#</span>= Pythagorean apotome, p = Pythagorean comma.<br /> | ||
<!-- ws:start:WikiTextLocalImageRule:483:&lt;img src=&quot;/file/view/Silver%20temperament%20graphic.png/541613984/800x587/Silver%20temperament%20graphic.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 587px; width: 800px;&quot; /&gt; --><img src="/file/view/Silver%20temperament%20graphic.png/541613984/800x587/Silver%20temperament%20graphic.png" alt="Silver temperament graphic.png" title="Silver temperament graphic.png" style="height: 587px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:483 --><br /> | <!-- ws:start:WikiTextLocalImageRule:483:&lt;img src=&quot;/file/view/Silver%20temperament%20graphic.png/541613984/800x587/Silver%20temperament%20graphic.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 587px; width: 800px;&quot; /&gt; --><img src="/file/view/Silver%20temperament%20graphic.png/541613984/800x587/Silver%20temperament%20graphic.png" alt="Silver temperament graphic.png" title="Silver temperament graphic.png" style="height: 587px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:483 --><br /> | ||
<span style="color: #ffffff;">######</span>Figure 3. Geometrical representation of | <span style="color: #ffffff;">######</span>Figure 3. Geometrical representation of argent temperament<br /> | ||
<br /> | |||
By the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem" rel="nofollow">Gelfond-Schneider theorem </a> the frequency ratios of all argent intervals (<span style="font-family: Georgia,serif; font-size: 110%;"><em>r</em> = 2</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">√2<em>a</em>+<em>b</em></span>, where<em><span style="font-family: Georgia,serif; font-size: 110%;"> a</span></em> and <em><span style="font-family: Georgia,serif; font-size: 110%;">b</span></em> are integers) are transcendental, with the exception of octave multiples (<span style="font-family: Georgia,serif; font-size: 110%;"><em>a</em> = 0</span>).<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:87:&lt;h2&gt; --><h2 id="toc19"><a name="x4. Quadratic approximants-Golden temperaments"></a><!-- ws:end:WikiTextHeadingRule:87 -->Golden temperaments</h2> | <!-- ws:start:WikiTextHeadingRule:87:&lt;h2&gt; --><h2 id="toc19"><a name="x4. Quadratic approximants-Golden temperaments"></a><!-- ws:end:WikiTextHeadingRule:87 -->Golden temperaments</h2> | ||
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<td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>σ</em>/2</span><br /> | <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>σ</em>/2</span><br /> | ||
</td> | </td> | ||
<td><span style="color: #ffffff; font-family: Georgia,serif;">#</span | <td><span style="color: #ffffff; font-family: Georgia,serif;">#</span>0<br /> | ||
</td> | </td> | ||
<td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>σ</em>/2</span><br /> | <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>σ</em>/2</span><br /> | ||
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\qquad q_1^2 + q_2^2 = q_3^2&lt;br/&gt;[[math]] | \qquad q_1^2 + q_2^2 = q_3^2&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\qquad q_1^2 + q_2^2 = q_3^2</script><!-- ws:end:WikiTextMathRule:42 --><br /> | --><script type="math/tex">\qquad q_1^2 + q_2^2 = q_3^2</script><!-- ws:end:WikiTextMathRule:42 --><br /> | ||
they can be said to form a Pythagorean triple.<br /> | they can be said to form a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pythagorean_triple" rel="nofollow">Pythagorean triple</a>.<br /> | ||
The following are three examples. In the first and third cases, their counterparts in 12edo, <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em>1', <em>J</em>2'</span> and <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em>3'</span> , are also Pythagorean triples:<br /> | The following are three examples. In the first and third cases, their counterparts in 12edo, <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em>1', <em>J</em>2'</span> and <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em>3'</span>, are also Pythagorean triples:<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:91:&lt;h2&gt; --><h2 id="toc21"><a name="x4. Quadratic approximants-A small 34edo comma"></a><!-- ws:end:WikiTextHeadingRule:91 -->A small 34edo comma</h2> | <!-- ws:start:WikiTextHeadingRule:91:&lt;h2&gt; --><h2 id="toc21"><a name="x4. Quadratic approximants-A small 34edo comma"></a><!-- ws:end:WikiTextHeadingRule:91 -->A small 34edo comma</h2> | ||
<span style="color: #333333;">As <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a> has noted, the </span>5-limit comma <span style="color: #333333;">|-433 -137 280&gt; (‘<em>selenia</em>’) is remarkably small at just 0.004764 cents. The minute size of this comma can be explained using qu</span>adratic approximants.<br /> | <span style="color: #333333;">As <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a> has noted, the </span>5-limit comma <span style="color: #333333;">|-433 -137 280&gt; (‘<em>selenia</em>’) is remarkably small at just 0.004764 cents. The minute size of this comma can be explained using qu</span>adratic approximants.<br /> | ||
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 < | 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 <a class="wiki_link" href="/Gammic%20node">//gammic// comma </a>|-29 -11 20&gt; (4.769 cents) and the <em>semisuper</em> comma (AKA <em><a class="wiki_link" href="/vishnuzma">vishnuzma</a></em>) |23 6 -14&gt; (3.338 cents). In particular,<br /> | ||
<span style="color: #333333;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span><em>selenia</em> = 7 <em>gammic</em> – 10 <em>semisuper</em></span><br /> | <span style="color: #333333;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span><em>selenia</em> = 7 <em>gammic</em> – 10 <em>semisuper</em></span><br /> | ||
<span style="color: #333333;">So to prove that <em>selenia</em> is small we must show that <em>gammic</em><em>/semisuper</em> ≈ 10/7.</span><br /> | <span style="color: #333333;">So to prove that <em>selenia</em> is small we must show that <em>gammic</em><em>/semisuper</em> ≈ 10/7.</span><br /> | ||
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<span style="font-family: Arial,Helvetica,sans-serif;">Putting in the numbers:</span><br /> | <span style="font-family: Arial,Helvetica,sans-serif;">Putting in the numbers:</span><br /> | ||
<em><span style="color: #ffffff;">###</span><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">gammic </span></em><span style="font-family: Georgia,serif; font-size: 110%;">=</span> <span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1 – (2/5) (1/4) (1/30 + 1/20) = 1 – 1/120</span><br /> | <em><span style="color: #ffffff;">###</span><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">gammic </span></em><span style="font-family: Georgia,serif; font-size: 110%;">=</span> <span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1 – (2/5) (1/4) (1/30 + 1/20) = 1 – 1/120</span><br /> | ||
<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><em><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">semisuper </span></em><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">= | <span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><em><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">semisuper </span></em><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">= 1 – (2/5)(1/4) (1/600 + 1/12) = 1 – (1/120) (51/50)</span><br /> | ||
<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><em><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">gammic</span> <span style="font-family: Georgia,serif;">- </span><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">semisuper </span></em><span style="font-family: Georgia,serif; font-size: 110%;">= | <span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><em><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">gammic</span> <span style="font-family: Georgia,serif;">- </span><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">semisuper </span></em><span style="font-family: Georgia,serif; font-size: 110%;">= 1/6000</span><br /> | ||
<span style="color: #333333;">Therefore</span><br /> | <span style="color: #333333;">Therefore</span><br /> | ||
<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><span style="font-family: Georgia,serif; font-size: 110%;"><em>selenia</em> ≈ 7 <em>gammic</em> (1/6000) (120/119) = <em>gammic</em>/850 = 0.00561</span><span style="color: #333333;"> cents</span><br /> | <span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><span style="font-family: Georgia,serif; font-size: 110%;"><em>selenia</em> ≈ 7 <em>gammic</em> (1/6000) (120/119) = <em>gammic</em>/850 = 0.00561</span><span style="color: #333333;"> cents</span><br /> | ||
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and <span style="font-family: Georgia,serif; font-size: 110%;">(<em>q</em>(25/24))</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span> , being small in comparison to the other terms, compromises this equality only slightly.<br /> | and <span style="font-family: Georgia,serif; font-size: 110%;">(<em>q</em>(25/24))</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span> , being small in comparison to the other terms, compromises this equality only slightly.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:93:&lt;h1&gt; --><h1 id="toc22"><a name=" | <!-- ws:start:WikiTextHeadingRule:93:&lt;h1&gt; --><h1 id="toc22"><a name="Sources and acknowledgements"></a><!-- ws:end:WikiTextHeadingRule:93 -->Sources and acknowledgements</h1> | ||
This article is based on original research by <a class="wiki_link" href="/Martin%20Gough">Martin Gough</a>. 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.<br /> | |||
The tuning referred to here as argent temperament was described by <a class="wiki_link" href="/graham%20breed">Graham Breed</a> and Paul Hahn in posts (#12599, #12670) to the Yahoo tuning list on 10 and 12 August 2000.<br /> | |||
Thanks to <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a> for the Gelfond-Schneider result.</body></html></pre></div> | |||