Cangwu badness - Revision history

Sintel: fix headings and some math formatting

2026-05-30T21:53:54Z

fix headings and some math formatting

← Older revision		Revision as of 21:53, 30 May 2026
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	''Cangwu badness'' is a polynomial function [[Badness\|badness]] measure; the name stems from Cangwu Green Park, Lianyungang, China, where [[Graham_Breed\|Graham Breed]] thought it up.		''Cangwu badness'' is a polynomial function [[Badness\|badness]] measure; the name stems from Cangwu Green Park, Lianyungang, China, where [[Graham_Breed\|Graham Breed]] thought it up.

	= Definitions =		== Definitions ==

	In the following definitions, suppose that <math>C(T)</math> is the [[Tenney-Euclidean_temperament_measures#TE_Complexity\|TE Complexity]], and <math>E_R(T)</math> is the [[Tenney-Euclidean_temperament_measures#TE_simple_badness\|"relativized error"]] of temperament <math>T</math>. The "relativized error" for EDOs gives the error as a proportion of the step size, and yields a similar result for higher-rank temperaments; we may also call it the "TE Simple Badness." The Cangwu badness, then, can be most simply thought of as a weighted RMS of the two. The main feature of Cangwu badness is that there is a free '''weighting''' parameter determining how much we are weighting the error vs complexity into the calculation.		In the following definitions, suppose that <math>C(T)</math> is the [[Tenney-Euclidean_temperament_measures#TE_Complexity\|TE Complexity]], and <math>E_R(T)</math> is the [[Tenney-Euclidean_temperament_measures#TE_simple_badness\|"relativized error"]] of temperament <math>T</math>. The "relativized error" for EDOs gives the error as a proportion of the step size, and yields a similar result for higher-rank temperaments; we may also call it the "TE Simple Badness." The Cangwu badness, then, can be most simply thought of as a weighted RMS of the two. The main feature of Cangwu badness is that there is a free '''weighting''' parameter determining how much we are weighting the error vs complexity into the calculation.
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	== Simplified Version of Graham's Definition ==		=== Simplified Version of Graham's Definition ===

	Let <math>\epsilon</math> be a free parameter in <math>[0,1]</math>. The Cangwu badness is then simply defined via the weighted RMS		Let <math>\epsilon</math> be a free parameter in <math>[0,1]</math>. The Cangwu badness is then simply defined via the weighted RMS

	$$		:<math>
	\sqrt{(1-\epsilon^2)E_R(T)^2 + \epsilon^2 C(T)^2}		\sqrt{(1-\epsilon^2)E_R(T)^2 + \epsilon^2 C(T)^2}
	$$		</math>

	where the squaring is just due to convention and/or for numerical stability reasons.		where the squaring is just due to convention and/or for numerical stability reasons.
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	== Simplified Version of Graham's Definition With "Intended Cents of Error" Parameter ==		=== Simplified Version of Graham's Definition With "Intended Cents of Error" Parameter ===

	Graham tends to use a change of variables so that the free parameter can be viewed as a number of cents representing the "intended target error" in temperament searches. To that extent, we define the change of variables		Graham tends to use a change of variables so that the free parameter can be viewed as a number of cents representing the "intended target error" in temperament searches. To that extent, we define the change of variables

	$$		:<math>
			\begin{aligned}
	\epsilon^2 = \frac{E_k^2}{1+E_k^2} \\		\epsilon^2 = \frac{E_k^2}{1+E_k^2} \\
	E_k^2 = \frac{\epsilon^2}{1-\epsilon^2}		E_k^2 = \frac{\epsilon^2}{1-\epsilon^2}
	$$		\end{aligned}
			</math>

	So that <math>E_k</math> can now be thought of as a real number in <math>[0,\infty)</math>. This leads to the formula		So that <math>E_k</math> can now be thought of as a real number in <math>[0,\infty)</math>. This leads to the formula

	$$		:<math>
	\sqrt{\frac{E_R(T)^2 + E_k^2 C(T)^2}{1+E_k^2}}		\sqrt{\frac{E_R(T)^2 + E_k^2 C(T)^2}{1+E_k^2}}
	$$		</math>

	This formula is also equivalent to the one used in Graham's python library in terms of the <math>E_k</math> parameter, and is again equivalent to the version in his articles except for the aforementioned scalar multiplication.		This formula is also equivalent to the one used in Graham's python library in terms of the <math>E_k</math> parameter, and is again equivalent to the version in his articles except for the aforementioned scalar multiplication.


	== Graham's Original Definition ==		=== Graham's Original Definition ===

	Graham's original definition, in terms of <math>E_k</math>, can be written as		Graham's original definition, in terms of <math>E_k</math>, can be written as

	$$		:<math>
	\sqrt{\frac{E_R(T)^2 + E_k^2 C(T)^2}{1+E_k^2}} \cdot \left(\sqrt{1+E_k^2}\right)^{\text{rank(t)}}		\sqrt{\frac{E_R(T)^2 + E_k^2 C(T)^2}{1+E_k^2}} \cdot \left(\sqrt{1+E_k^2}\right)^{\text{rank}(T)}
	$$		</math>

	This can be shown to be identical to the metric given in Graham's three articles linked above, although he originally defined it in terms of a large matrix formula. It is also equivalent to the metrics previously given, except for the presence of the added term		This can be shown to be identical to the metric given in Graham's three articles linked above, although he originally defined it in terms of a large matrix formula. It is also equivalent to the metrics previously given, except for the presence of the added term

	$$		:<math>
	\left(\sqrt{1+E_k^2}\right)^{\text{rank(t)}}		\left(\sqrt{1+E_k^2}\right)^{\text{rank}(T)}
	$$		</math>

	which, for any choice of <math>E_k</math> or <math>\text{rank}(t)</math>, multiplies all temperaments only by a scalar.		which, for any choice of <math>E_k</math> or <math>\text{rank}(T)</math>, multiplies all temperaments only by a scalar.


	== Gene Smith's Matrix Determinant Definition ==		=== Gene Smith's Matrix Determinant Definition ===

	[[Gene Ward Smith]] wrote the below presentation of Cangwu badness in terms of the characteristic polynomial of a matrix. The definition is again equivalent to the above, except it is equivalent to the square of the above Cangwu badness, and again involving equivalence only up to a scalar multiplication.		[[Gene Ward Smith]] wrote the below presentation of Cangwu badness in terms of the characteristic polynomial of a matrix. The definition is again equivalent to the above, except it is equivalent to the square of the above Cangwu badness, and again involving equivalence only up to a scalar multiplication.
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	Gene's definition of the Cangwu badness is in terms		Gene's definition of the Cangwu badness is in terms

	$$		:<math>
	C(x) = \det\left(\left[(1+x)\frac{v_i \cdot v_j}{n} - a_ia_j\right]\right)		C(x) = \det\left(\left[(1+x)\frac{v_i \cdot v_j}{n} - a_ia_j\right]\right)
	$$		</math>

	where the vᵢ are r independent weighted vals of dimension n defining a rank r regular temperament, and the aᵢ are the average of the values of vᵢ; that is, the sum of the values of vᵢ divided by n.		where the vᵢ are r independent weighted vals of dimension n defining a rank r regular temperament, and the aᵢ are the average of the values of vᵢ; that is, the sum of the values of vᵢ divided by n.
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	From this definition, it follows that C(0) is proportional to the square of [[Tenney-Euclidean_temperament_measures#TE simple badness\|simple badness]], aka relative error. The leading coefficient of the polynomial defining C(x) is proportional to the square of [[Tenney-Euclidean_temperament_measures#TE Complexity\|TE complexity]]. Hence as x grows larger, C(x) increasingly weights smaller complexity temperaments as better, and in the limit becomes a complexity measure, whereas at 0 C(0) is a badness measure which strongly favors more accurate temperaments, being in fact a measure of relative error.		From this definition, it follows that C(0) is proportional to the square of [[Tenney-Euclidean_temperament_measures#TE simple badness\|simple badness]], aka relative error. The leading coefficient of the polynomial defining C(x) is proportional to the square of [[Tenney-Euclidean_temperament_measures#TE Complexity\|TE complexity]]. Hence as x grows larger, C(x) increasingly weights smaller complexity temperaments as better, and in the limit becomes a complexity measure, whereas at 0 C(0) is a badness measure which strongly favors more accurate temperaments, being in fact a measure of relative error.

	=== Cangwu Dominance ===		== Cangwu Dominance ==
	If Ca(x) is the Cangwu badness for rank r temperament 'a', and Cb(x) for 'b', we can say a ''dominates'' b if Cb(x) - Ca(x) is a positive function for x ≥ 0, which says that the badness of 'a' is always less than the badness of 'b' for every choice of the parameter x. If a temperament is not dominated by any temperament of the same rank (and on the same subgroup, if we are considering subgroups) we may say it is ''indomitable''. An alternative procedure is to divide the cangwu badness polynomial by the coefficient of the highest degree term, getting a monic polynomial with constant term proportional to the the square of TE absolute error. When one temperament dominates another using this polynomial, it is lower in both error and complexity.		If Ca(x) is the Cangwu badness for rank r temperament 'a', and Cb(x) for 'b', we can say a ''dominates'' b if Cb(x) - Ca(x) is a positive function for x ≥ 0, which says that the badness of 'a' is always less than the badness of 'b' for every choice of the parameter x. If a temperament is not dominated by any temperament of the same rank (and on the same subgroup, if we are considering subgroups) we may say it is ''indomitable''. An alternative procedure is to divide the cangwu badness polynomial by the coefficient of the highest degree term, getting a monic polynomial with constant term proportional to the the square of TE absolute error. When one temperament dominates another using this polynomial, it is lower in both error and complexity.

Sintel: -legacy

2025-03-29T17:56:21Z

-legacy

← Older revision		Revision as of 17:56, 29 March 2025
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	~~{{Legacy}}~~
	''Cangwu badness'' is a polynomial function [[Badness\|badness]] measure; the name stems from Cangwu Green Park, Lianyungang, China, where [[Graham_Breed\|Graham Breed]] thought it up.		''Cangwu badness'' is a polynomial function [[Badness\|badness]] measure; the name stems from Cangwu Green Park, Lianyungang, China, where [[Graham_Breed\|Graham Breed]] thought it up.

Lériendil at 16:51, 17 February 2025

2025-02-17T16:51:52Z

← Older revision		Revision as of 16:51, 17 February 2025
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	{{Legacy~~\|Cangwu_badness~~}}		{{Legacy}}
	''Cangwu badness'' is a polynomial function [[Badness\|badness]] measure; the name stems from Cangwu Green Park, Lianyungang, China, where [[Graham_Breed\|Graham Breed]] thought it up.		''Cangwu badness'' is a polynomial function [[Badness\|badness]] measure; the name stems from Cangwu Green Park, Lianyungang, China, where [[Graham_Breed\|Graham Breed]] thought it up.

Lériendil: deploying new cat

2025-02-17T16:15:31Z

deploying new cat

← Older revision		Revision as of 16:15, 17 February 2025
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			{{Legacy\|Cangwu_badness}}
	''Cangwu badness'' is a polynomial function [[Badness\|badness]] measure; the name stems from Cangwu Green Park, Lianyungang, China, where [[Graham_Breed\|Graham Breed]] thought it up.		''Cangwu badness'' is a polynomial function [[Badness\|badness]] measure; the name stems from Cangwu Green Park, Lianyungang, China, where [[Graham_Breed\|Graham Breed]] thought it up.

Lériendil: Changed protection level for "Cangwu badness" ([Edit=Allow only administrators] (expires 17:02, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:02, 17 February 2025 (UTC)))

2025-02-17T16:02:31Z

Changed protection level for "Cangwu badness" ([Edit=Allow only administrators] (expires 17:02, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:02, 17 February 2025 (UTC)))

← Older revision	Revision as of 16:02, 17 February 2025
(No difference)

Mike Battaglia: Protected "Cangwu badness" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

2021-11-15T03:16:09Z

Protected "Cangwu badness" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

← Older revision	Revision as of 03:16, 15 November 2021
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Mike Battaglia: /* Gene Smith's Matrix Determinant Definition */

2019-06-29T20:14:04Z

Gene Smith's Matrix Determinant Definition

← Older revision		Revision as of 20:14, 29 June 2019
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	== Gene Smith's Matrix Determinant Definition ==		== Gene Smith's Matrix Determinant Definition ==

	[[Gene Smith]] wrote the below presentation of Cangwu badness in terms of the characteristic polynomial of a matrix. The definition is again equivalent to the above, except it is equivalent to the square of the above Cangwu badness, and again involving equivalence only up to a scalar multiplication.		[[Gene Ward Smith]] wrote the below presentation of Cangwu badness in terms of the characteristic polynomial of a matrix. The definition is again equivalent to the above, except it is equivalent to the square of the above Cangwu badness, and again involving equivalence only up to a scalar multiplication.

	Note that the notation below is not, in general, the same as the notation above; Gene's original notation has been left as written, where the free parameter is called <math>x</math>.		Note that the notation below is not, in general, the same as the notation above; Gene's original notation has been left as written, where the free parameter is called <math>x</math>.

Mike Battaglia: Cangwu update

2019-06-29T20:10:36Z

Cangwu update

@@ Line 1: / Line 1: @@
-''Cangwu badness'' is a polynomial function [[Badness|badness]] measure; the name stems from Cangwu Green Park, Lianyungang, China, where [[Graham_Breed|Graham Breed]] thought it up. It is defined in terms of a matrix determinant as
+''Cangwu badness'' is a polynomial function [[Badness|badness]] measure; the name stems from Cangwu Green Park, Lianyungang, China, where [[Graham_Breed|Graham Breed]] thought it up.
-<math>\displaystyle C(x) = det([(1+x)v_i \cdot v_j/n - a_ia_j])</math>
+= Definitions =
 where the vᵢ are r independent weighted vals of dimension n defining a rank r regular temperament, and the aᵢ are the average of the values of vᵢ; that is, the sum of the values of vᵢ divided by n.
@@ Line 7: / Line 74: @@
 From this definition, it follows that C(0) is proportional to the square of [[Tenney-Euclidean_temperament_measures#TE simple badness|simple badness]], aka relative error. The leading coefficient of the polynomial defining C(x) is proportional to the square of [[Tenney-Euclidean_temperament_measures#TE Complexity|TE complexity]]. Hence as x grows larger, C(x) increasingly weights smaller complexity temperaments as better, and in the limit becomes a complexity measure, whereas at 0 C(0) is a badness measure which strongly favors more accurate temperaments, being in fact a measure of relative error.
 If Ca(x) is the Cangwu badness for rank r temperament 'a', and Cb(x) for 'b', we can say a ''dominates'' b if Cb(x) - Ca(x) is a positive function for x ≥ 0, which says that the badness of 'a' is always less than the badness of 'b' for every choice of the parameter x. If a temperament is not dominated by any temperament of the same rank (and on the same subgroup, if we are considering subgroups) we may say it is ''indomitable''. An alternative procedure is to divide the cangwu badness polynomial by the coefficient of the highest degree term, getting a monic polynomial with constant term proportional to the the square of TE absolute error. When one temperament dominates another using this polynomial, it is lower in both error and complexity.
 Examples of 5-limit indomitable temperaments are:

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

← Older revision		Revision as of 00:00, 17 July 2018
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	~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~		''Cangwu badness'' is a polynomial function [[Badness\|badness]] measure; the name stems from Cangwu Green Park, Lianyungang, China, where [[Graham_Breed\|Graham Breed]] thought it up. It is defined in terms of a matrix determinant as
	~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~
	~~: This revision was by author [[User:clumma\|clumma]] and made on <tt>2014-12-14 22:29:39 UTC</tt>.<br>~~
	~~: The original revision id was <tt>535152242</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">//Cangwu badness// is a polynomial function [[badness]] measure; the name stems from Cangwu Green Park, Lianyungang, China, where [[Graham Breed]] thought it up. It is defined in terms of a matrix determinant as

	[[math]]		<math>\displaystyle C(x) = det([(1+x)v_i \cdot v_j/n - a_ia_j])</math>
	\displaystyle C(x) = det([(1+x)v_i \cdot v_j/n - a_ia_j])
	[[math]]

	where the vᵢ are r independent weighted vals of dimension n defining a rank r regular temperament, and the aᵢ are the average of the values of vᵢ; that is, the sum of the values of vᵢ divided by n.		where the vᵢ are r independent weighted vals of dimension n defining a rank r regular temperament, and the aᵢ are the average of the values of vᵢ; that is, the sum of the values of vᵢ divided by n.

	From this definition, it follows that C(0) is proportional to the square of [[Tenney-~~Euclidean temperament measures~~#TE simple badness\|simple badness]], aka relative error. The leading coefficient of the polynomial defining C(x) is proportional to the square of [[Tenney-~~Euclidean temperament measures~~#TE Complexity\|TE complexity]]. Hence as x grows larger, C(x) increasingly weights smaller complexity temperaments as better, and in the limit becomes a complexity measure, whereas at 0 C(0) is a badness measure which strongly favors more accurate temperaments, being in fact a measure of relative error.		From this definition, it follows that C(0) is proportional to the square of [[Tenney-Euclidean_temperament_measures#TE simple badness\|simple badness]], aka relative error. The leading coefficient of the polynomial defining C(x) is proportional to the square of [[Tenney-Euclidean_temperament_measures#TE Complexity\|TE complexity]]. Hence as x grows larger, C(x) increasingly weights smaller complexity temperaments as better, and in the limit becomes a complexity measure, whereas at 0 C(0) is a badness measure which strongly favors more accurate temperaments, being in fact a measure of relative error.

	If Ca(x) is the Cangwu badness for rank r temperament 'a', and Cb(x) for 'b', we can say a //dominates// b if Cb(x) - Ca(x) is a positive function for x ≥ 0, which says that the badness of 'a' is always less than the badness of 'b' for every choice of the parameter x. If a temperament is not dominated by any temperament of the same rank (and on the same subgroup, if we are considering subgroups) we may say it is //indomitable//. An alternative procedure is to divide the cangwu badness polynomial by the coefficient of the highest degree term, getting a monic polynomial with constant term proportional to the the square of TE absolute error. When one temperament dominates another using this polynomial, it is lower in both error and complexity.		If Ca(x) is the Cangwu badness for rank r temperament 'a', and Cb(x) for 'b', we can say a ''dominates'' b if Cb(x) - Ca(x) is a positive function for x ≥ 0, which says that the badness of 'a' is always less than the badness of 'b' for every choice of the parameter x. If a temperament is not dominated by any temperament of the same rank (and on the same subgroup, if we are considering subgroups) we may say it is ''indomitable''. An alternative procedure is to divide the cangwu badness polynomial by the coefficient of the highest degree term, getting a monic polynomial with constant term proportional to the the square of TE absolute error. When one temperament dominates another using this polynomial, it is lower in both error and complexity.

	Examples of 5-limit indomitable temperaments are:		Examples of 5-limit indomitable temperaments are:

	Father 16/15 \|4 -1 -1>		Father 16/15 \|4 -1 -1>

	Dicot 25/24 \|-3 -1 2>		Dicot 25/24 \|-3 -1 2>

	Meantone 81/80 \|-4 4 -1>		Meantone 81/80 \|-4 4 -1>

	Srutal/Diaschismic 2048/2025 \|11 -4 -2>		Srutal/Diaschismic 2048/2025 \|11 -4 -2>

	Hanson/Kleismic 15625/15552 \|-6 -5 6>		Hanson/Kleismic 15625/15552 \|-6 -5 6>

	Helmholtz/Schismic 32805/32768 \|-15 8 1>		Helmholtz/Schismic 32805/32768 \|-15 8 1>

	Hemithirds \|38 -2 -15>		Hemithirds \|38 -2 -15>

	Ennealimmal \|1 -27 18>		Ennealimmal \|1 -27 18>

	Kwazy \|-53 10 16>		Kwazy \|-53 10 16>

	Monzismic \|54 -37 2>		Monzismic \|54 -37 2>

	Senior \|-17 62 -35>		Senior \|-17 62 -35>

	Pirate \|-90 -15 49>		Pirate \|-90 -15 49>

	Atomic \|161 -84 -12>		Atomic \|161 -84 -12>

	Some 7-limit examples are beep, meantone, miracle, pontiac and ennealimmal.		Some 7-limit examples are beep, meantone, miracle, pontiac and ennealimmal.

	If two temperaments of the same rank are such that neither dominants the other, we may subtract one Cangwu badness polynomial from the other and find the positive root of the result. This gives a value of the parameter 'x' at which the two temperaments are rated equal in badness, which can be applied to rate other temperaments by badness. For example, if 5-limit father and helmholtz are made equally bad, then meantone, augmented, dicot, porcupine, srutal, diminished, magic, hanson and mavila, in that order, rate as better.~~</pre></div>~~		If two temperaments of the same rank are such that neither dominants the other, we may subtract one Cangwu badness polynomial from the other and find the positive root of the result. This gives a value of the parameter 'x' at which the two temperaments are rated equal in badness, which can be applied to rate other temperaments by badness. For example, if 5-limit father and helmholtz are made equally bad, then meantone, augmented, dicot, porcupine, srutal, diminished, magic, hanson and mavila, in that order, rate as better.
	~~<h4>Original HTML content:</h4>~~		[[Category:badness]]
	<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>Cangwu badness</title></head><body><em>Cangwu~~ badness</em> is a polynomial function <a class="wiki_link" href="/badness">badness</a> measure; the name stems from Cangwu Green Park, Lianyungang, China, where <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a> thought it up. It is defined in terms of a matrix determinant as<br />
	~~<br />~~
	~~<!-- ws:start:WikiTextMathRule:0:~~
	~~[[math~~]]~~&lt;br/&gt;~~
	~~\displaystyle C(x) = det([(1+x)v_i \cdot v_j/n - a_ia_j])&lt;br/&gt;[[math]]~~
	~~--><script type="math/tex">\displaystyle C(x) = det([(1+x)v_i \cdot v_j/n - a_ia_j])</script><!-- ws:end:WikiTextMathRule:0 --><br />~~
	~~<br />~~
	where the vᵢ are r independent weighted vals of dimension n defining a rank r regular temperament, and the aᵢ are the average of the values of vᵢ; that is, the sum of the values of vᵢ divided by n.<br />
	~~<br />~~
	From this definition, it follows that C(0) is proportional to the square of <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness">simple badness</a>, aka relative error. The leading coefficient of the polynomial defining C(x) is proportional to the square of <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE Complexity">TE complexity</a>. Hence as x grows larger, C(x) increasingly weights smaller complexity temperaments as better, and in the limit becomes a complexity measure, whereas at 0 C(0) is a badness measure which strongly favors more accurate temperaments, being in fact a measure of relative error.<br />
	~~<br />~~
	If Ca(x) is the Cangwu badness for rank r temperament 'a', and Cb(x) for 'b', we can say a <em>dominates</em> b if Cb(x) - Ca(x) is a positive function for x ≥ 0, which says that the badness of 'a' is always less than the badness of 'b' for every choice of the parameter x. If a temperament is not dominated by any temperament of the same rank (and on the same subgroup, if we are considering subgroups) we may say it is <em>indomitable</em>. An alternative procedure is to divide the cangwu badness polynomial by the coefficient of the highest degree term, getting a monic polynomial with constant term proportional to the the square of TE absolute error. When one temperament dominates another using this polynomial, it is lower in both error and complexity.<br />
	~~<br />~~
	~~Examples of 5-limit indomitable temperaments are:<br />~~
	~~<br />~~
	~~Father 16/15 \|4 -1 -1&gt;<br />~~
	~~Dicot 25/24 \|-3 -1 2&gt;<br />~~
	~~Meantone 81/80 \|-4 4 -1&gt;<br />~~
	~~Srutal/Diaschismic 2048/2025 \|11 -4 -2&gt;<br />~~
	~~Hanson/Kleismic 15625/15552 \|-6 -5 6&gt;<br />~~
	~~Helmholtz/Schismic 32805/32768 \|-15 8 1&gt;<br />~~
	~~Hemithirds \|38 -2 -15&gt;<br />~~
	~~Ennealimmal \|1 -27 18&gt;<br />~~
	~~Kwazy \|-53 10 16&gt;<br />~~
	~~Monzismic \|54 -37 2&gt;<br />~~
	~~Senior \|-17 62 -35&gt;<br />~~
	~~Pirate \|-90 -15 49&gt;<br />~~
	~~Atomic \|161 -84 -12&gt;<br />~~
	~~<br />~~
	~~Some 7-limit examples are beep, meantone, miracle, pontiac and ennealimmal.<br />~~
	~~<br />~~
	If two temperaments of the same rank are such that neither dominants the other, we may subtract one Cangwu badness polynomial from the other and find the positive root of the result. This gives a value of the parameter 'x' at which the two temperaments are rated equal in badness, which can be applied to rate other temperaments by badness. For example, if 5-limit father and helmholtz are made equally bad, then meantone, augmented, dicot, porcupine, srutal, diminished, magic, hanson and mavila, in that order, rate as better.</body></html></pre></div>

Wikispaces>clumma: Imported revision 535152242 - Original comment:

2014-12-14T22:29:39Z

**Imported revision 535152242 - Original comment: **

← Older revision		Revision as of 22:29, 14 December 2014
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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:~~genewardsmith~~\|~~genewardsmith~~]] and made on <tt>2014-05-~~24 17~~:13:52 UTC</tt>.<br>		: This revision was by author [[User:clumma\|clumma]] and made on <tt>2014-12-14 22:29:39 UTC</tt>.<br>
	: The original revision id was <tt>~~511016978~~</tt>.<br>		: The original revision id was <tt>535152242</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>
	<h4>Original Wikitext content:</h4>		<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">		<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">//Cangwu badness// is a polynomial function [[badness]] measure; the name stems from Cangwu Green Park, Lianyungang, China, where [[Graham Breed]] thought it up. It is defined in terms of a matrix determinant as
	~~[[image:mathhazard.jpg align="left"]]~~
	//Cangwu badness// is a polynomial function [[badness]] measure; the name stems from Cangwu Green Park, Lianyungang, China, where [[Graham Breed]] thought it up. It is defined in terms of a matrix determinant as

	[[math]]		[[math]]
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	If two temperaments of the same rank are such that neither dominants the other, we may subtract one Cangwu badness polynomial from the other and find the positive root of the result. This gives a value of the parameter 'x' at which the two temperaments are rated equal in badness, which can be applied to rate other temperaments by badness. For example, if 5-limit father and helmholtz are made equally bad, then meantone, augmented, dicot, porcupine, srutal, diminished, magic, hanson and mavila, in that order, rate as better.</pre></div>		If two temperaments of the same rank are such that neither dominants the other, we may subtract one Cangwu badness polynomial from the other and find the positive root of the result. This gives a value of the parameter 'x' at which the two temperaments are rated equal in badness, which can be applied to rate other temperaments by badness. For example, if 5-limit father and helmholtz are made equally bad, then meantone, augmented, dicot, porcupine, srutal, diminished, magic, hanson and mavila, in that order, rate as better.</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>Cangwu badness</title></head><body>~~<br />~~		<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>Cangwu badness</title></head><body><em>Cangwu badness</em> is a polynomial function <a class="wiki_link" href="/badness">badness</a> measure; the name stems from Cangwu Green Park, Lianyungang, China, where <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a> thought it up. It is defined in terms of a matrix determinant as<br />
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	<em>Cangwu badness</em> is a polynomial function <a class="wiki_link" href="/badness">badness</a> measure; the name stems from Cangwu Green Park, Lianyungang, China, where <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a> thought it up. It is defined in terms of a matrix determinant as<br />
	<br />		<br />
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