Diamond tradeoff - Revision history

ArrowHead294 at 16:35, 15 January 2025

2025-01-15T16:35:24Z

← Older revision		Revision as of 16:35, 15 January 2025
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	Reading the columns from <math>G</math>, the first one {{vector\|1 0 0}} confirms our period of 2/1, and the second column {{vector\|-1 1 0}} gives our generator 3/2. Which is unsurprising. In cents, that's {{nowrap\|1200 ~~×~~ log<sub>2</sub>(3/2) ~~≈~~ 701.955{{c}}}}. The next unchanged interval will give a more interesting result.		Reading the columns from <math>G</math>, the first one {{vector\|1 0 0}} confirms our period of 2/1, and the second column {{vector\|-1 1 0}} gives our generator 3/2. Which is unsurprising. In cents, that's {{nowrap\|1200 × log<sub>2</sub>(3/2) ≈ 701.955{{cent}}}}. The next unchanged interval will give a more interesting result.

	==== Second diamond extrema ====		==== Second diamond extrema ====
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	This tells us our generator is <math>\monzo{0 & 0 & \frac{1}{4}}</math>, or 5<sup>{{frac\|1\|4}}</sup>. In cents, that's {{nowrap\|1200 ~~×~~ log<sub>2</sub>(5<sup>{{frac\|1\|4}}</sup>) ~~≈~~ 696.578{{c}}}}.		This tells us our generator is <math>\monzo{0 & 0 & \frac{1}{4}}</math>, or 5<sup>{{frac\|1\|4}}</sup>. In cents, that's {{nowrap\|1200 × log<sub>2</sub>(5<sup>{{frac\|1\|4}}</sup>) ≈ 696.578{{c}}}}.

	==== Third diamond extrema ====		==== Third diamond extrema ====
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	This tells us our generator is <math>\monzo{\frac{1}{3} & -\frac{1}{3} & \frac{1}{3}}</math>, or expressed another way, ({{frac\|10\|3}})<sup>{{frac\|1\|3}}</sup>. In cents, that's {{nowrap\|1200 ~~×~~ log<sub>2</sub>(({{frac\|10\|3}})<sup>{{frac\|1\|3}}</sup>) ~~≈~~ 694.786{{c}}}}.		This tells us our generator is <math>\monzo{\frac{1}{3} & -\frac{1}{3} & \frac{1}{3}}</math>, or expressed another way, ({{frac\|10\|3}})<sup>{{frac\|1\|3}}</sup>. In cents, that's {{nowrap\|1200 × log<sub>2</sub>(({{frac\|10\|3}})<sup>{{frac\|1\|3}}</sup>) ≈ 694.786{{c}}}}.

	==== Determining the final range ====		==== Determining the final range ====

ArrowHead294: Writing \frac without brackets in LaTeX is bad practice.

2024-11-13T18:02:26Z

Writing \frac without brackets in LaTeX is bad practice.

← Older revision		Revision as of 18:02, 13 November 2024
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	(MU)^{-1} \\		(MU)^{-1} \\
	\left[ \begin{array} {rrr}		\left[ \begin{array} {rrr}
	1 & \~~frac12~~ \\		1 & \frac{1}{2} \\
	0 & \~~frac14~~ \\		0 & \frac{1}{4} \\
	\end{array} \right]		\end{array} \right]
	\end{array}		\end{array}
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	(MU)^{-1} \\		(MU)^{-1} \\
	\left[ \begin{array} {rrr}		\left[ \begin{array} {rrr}
	1 & \~~frac12~~ \\		1 & \frac{1}{2} \\
	0 & \~~frac14~~ \\		0 & \frac{1}{4} \\
	\end{array} \right]		\end{array} \right]
	\end{array}		\end{array}
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	1 & 0 \\		1 & 0 \\
	0 & 0 \\		0 & 0 \\
	0 & \~~frac14~~ \\		0 & \frac{1}{4} \\
	\end{array} \right]		\end{array} \right]
	\end{array}		\end{array}
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	This tells us our generator is {~~{vector\|~~0 0 ~~<math>~~\~~frac14~~</math>}}, or 5^(1/4). In cents, that's {{nowrap\|1200 × log<sub>2</sub>(5<sup>{{frac\|1\|4}}</sup>) ≈ 696.578{{c}}}}.		This tells us our generator is <math>\monzo{0 & 0 & \frac{1}{4}}</math>, or 5<sup>{{frac\|1\|4}}</sup>. In cents, that's {{nowrap\|1200 × log<sub>2</sub>(5<sup>{{frac\|1\|4}}</sup>) ≈ 696.578{{c}}}}.

	==== Third diamond extrema ====		==== Third diamond extrema ====
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	(MU)^{-1} \\		(MU)^{-1} \\
	\left[ \begin{array} {rrr}		\left[ \begin{array} {rrr}
	1 & \~~frac23~~ \\		1 & \frac{2}{3} \\
	0 & -\~~frac13~~ \\		0 & -\frac{1}{3} \\
	\end{array} \right]		\end{array} \right]
	\end{array}		\end{array}
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	(MU)^{-1} \\		(MU)^{-1} \\
	\left[ \begin{array} {rrr}		\left[ \begin{array} {rrr}
	1 & \~~frac23~~ \\		1 & \frac{2}{3} \\
	0 & -\~~frac13~~ \\		0 & -\frac{1}{3} \\
	\end{array} \right]		\end{array} \right]
	\end{array}		\end{array}
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	G \\		G \\
	\left[ \begin{array} {rrr}		\left[ \begin{array} {rrr}
	1 & \~~frac13~~ \\		1 & \frac{1}{3} \\
	0 & -\~~frac13~~ \\		0 & -\frac{1}{3} \\
	0 & \~~frac13~~ \\		0 & \frac{1}{3} \\
	\end{array} \right]		\end{array} \right]
	\end{array}		\end{array}

Cmloegcmluin: /* First diamond extrema */

2024-11-13T15:25:13Z

First diamond extrema

← Older revision		Revision as of 15:25, 13 November 2024
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	Next, we find the generators matrix <math>G</math> corresponding to the tuning where these two intervals are unchanged. The formula for this generators matrix is <math>G = U(MU)^{-1}</math> (see [[~~Dave Keenan & Douglas Blumeyer's guide to RTT/All-interval tuning schemes~~\|here]] for a full explanation of this formula). Here, let's work it out for our chosen <math>U</math>. First, we multiply <math>MU</math>:		Next, we find the generators matrix <math>G</math> corresponding to the tuning where these two intervals are unchanged. The formula for this generators matrix is <math>G = U(MU)^{-1}</math> (see [[Generator embedding optimization#Generator embedding\|here]] for a full explanation of this formula). Here, let's work it out for our chosen <math>U</math>. First, we multiply <math>MU</math>:


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	Reading the columns from <math>G</math>, the first one {{vector\|1 0 0}} confirms our period of 2/1, and the second column {{vector\|-1 1 0}} gives our generator 3/2. Which is unsurprising. In cents, that's {{nowrap\|1200 × log<sub>2</sub>(3/2) ≈ 701.955{{c}}}}. The next unchanged interval will give a more interesting result.		Reading the columns from <math>G</math>, the first one {{vector\|1 0 0}} confirms our period of 2/1, and the second column {{vector\|-1 1 0}} gives our generator 3/2. Which is unsurprising. In cents, that's {{nowrap\|1200 × log<sub>2</sub>(3/2) ≈ 701.955{{c}}}}. The next unchanged interval will give a more interesting result.

	==== Second diamond extrema ====		==== Second diamond extrema ====

ArrowHead294 at 14:04, 13 November 2024

2024-11-13T14:04:01Z

← Older revision		Revision as of 14:04, 13 November 2024
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	A tuning for a [[regular temperament]] is '''diamond tradeoff''', or '''diamond strict''', if it is in the range where the concerning approximations to simple [[frequency ratio]]s can have their qualities "traded" against each other, that is, if some ratios are made more accurate, the others will be less so. For example, if you make the 3/2 more accurate, the 5/4 will suffer. However, outside this range, the tunings of ''all'' such intervals will be improved by moving back inside. This range therefore makes sense when one is concerned with approximating JI as closely as possible (without asserting a priori which specific consonances are the most important) because, under that criterion, it makes no logical sense to choose a tuning outside that range.		{{texmap}}
			A tuning for a [[regular temperament]] is '''diamond tradeoff''', or '''diamond strict''', if it is in the range where the concerning approximations to simple [[frequency ratio]]s can have their qualities "traded" against each other, that is, if some ratios are made more accurate, the others will be less so. For example, if you make the 3/2 more accurate, the 5/4 will suffer. However, outside this range, the tunings of ''all'' such intervals will be improved by moving back inside. This range therefore makes sense when one is concerned with approximating JI as closely as possible (without asserting ''a priori'' which specific consonances are the most important) because, under that criterion, it makes no logical sense to choose a tuning outside that range.

	However, it is quite clear that tunings outside of this diamond tradeoff range can function perfectly well as less accurate (and arguably more characterful) representations of the JI intervals specified by the temperament. That is, they are likely to be correctly recognized (whatever that actually means). For example, a 17-TET rendition of a standard piece of meantone music still makes complete musical sense, and major and minor chords still sound like major and minor chords, even though this tuning is outside the diamond tradeoff tuning range.		However, it is quite clear that tunings outside of this diamond tradeoff range can function perfectly well as less accurate (and arguably more characterful) representations of the JI intervals specified by the temperament. That is, they are likely to be correctly recognized (whatever that actually means). For example, a 17-TET rendition of a standard piece of meantone music still makes complete musical sense, and major and minor chords still sound like major and minor chords, even though this tuning is outside the diamond tradeoff tuning range.

	== Mathematical definition ==		== Mathematical definition ==
	[[Gene Ward Smith]] gives the following definition. The [[Odd limit\|''q''-odd-limit]] '''diamond tradeoff''' range of a [[rank]]-''r'' [[Prime limit\|''p''-limit]] temperament is the {{w\|convex hull}} in [[Vals and tuning space\|tuning space]] of the set of all tunings with ''r'' [[eigenmonzo\|eigenmonzos (unchanged-intervals)]] chosen as follows: one eigenmonzo 2 ~~(pure~~ octaves ~~tunings~~) and the rest of the eigenmonzos any set of ''r'' - 1 members of the ''q''-odd limit [[tonality diamond]], whenever such a tuning is defined.		[[Gene Ward Smith]] gives the following definition. The [[Odd limit\|''q''-odd-limit]] '''diamond tradeoff''' range of a [[rank]]-''r'' [[Prime limit\|''p''-limit]] temperament is the {{w\|convex hull}} in [[Vals and tuning space\|tuning space]] of the set of all tunings with ''r'' [[eigenmonzo\|eigenmonzos (unchanged-intervals)]] chosen as follows: one eigenmonzo (2/1, octaves) and the rest of the eigenmonzos any set of {{nowrap\|''r'' − 1}} members of the ''q''-odd limit [[tonality diamond]], whenever such a tuning is defined.

	== Original name ==		== Original name ==

	In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]], which defined this tuning range, it was known as the "purer" tuning range.<ref>Milne, A. J., Sethares, W. A., and Plamondon, J. (2006). X System. Technical report, Thumtronics, Inc. https://oro.open.ac.uk/21510/1/X_System.pdf</ref> On the wiki and in the regular temperament community, for many years this tuning range was referred to simply as the "nice" tuning range.		In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]], which defined this tuning range, it was known as the "purer" tuning range.<ref>Milne, A. J., Sethares, W. A., and Plamondon, J. (2006). X System. Technical report, Thumtronics, Inc. https://oro.open.ac.uk/21510/1/X_System.pdf</ref> On the wiki and in the regular temperament community, for many years this tuning range was referred to simply as the "nice" tuning range.

	== Vs. diamond monotone ==		== Vs. diamond monotone ==

	Diamond tradeoff tunings are always guaranteed to occur, but [[diamond monotone]] tunings are not.		Diamond tradeoff tunings are always guaranteed to occur, but [[diamond monotone]] tunings are not.

	== Examples ==		== Examples ==
	=== Explanation adapted from Keenan Pepper ===		=== Explanation adapted from Keenan Pepper ===

	For meantone temperament, there are three specific tunings that are special: one that tunes 4/3 and 3/2 pure, another that tunes 5/4 and 8/5 pure, and the third that tunes 6/5 and 5/3 pure. The tradeoff tuning range consists of these three points in tuning space ''and everything in between''. In this case, the three points fall along a line, where the pure-5/4 tuning is in between the pure-4/3 tuning and the pure-6/5 tuning.		For meantone temperament, there are three specific tunings that are special: one that tunes 4/3 and 3/2 pure, another that tunes 5/4 and 8/5 pure, and the third that tunes 6/5 and 5/3 pure. The tradeoff tuning range consists of these three points in tuning space ''and everything in between''. In this case, the three points fall along a line, where the pure-5/4 tuning is in between the pure-4/3 tuning and the pure-6/5 tuning.

	The next thing to understand is that tunings in the middle between the pure-4/3 tuning and the pure-6/5 tuning (including the pure-5/4 tuning, because that's in the middle) are in a sense "reasonable compromises" because whenever you're making one interval of the diamond worse, you're always making another one better. You're in the realm of tradeoffs.		The next thing to understand is that tunings in the middle between the pure-4/3 tuning and the pure-6/5 tuning (including the pure-5/4 tuning, because that's in the middle) are in a sense "reasonable compromises" because whenever you're making one interval of the diamond worse, you're always making another one better. You're in the realm of tradeoffs.

	On the other hand, if you go outside these boundaries - for example, if you make 4/3 even flatter than pure - then you're making some intervals in the 5-limit diamond worse without making ''any'' of them better. You're past the realm of compromises and now you're just damaging things for no reason.		On the other hand, if you go outside these boundaries—for example, if you make 4/3 even flatter than pure—then you're making some intervals in the 5-limit diamond worse without making ''any'' of them better. You're past the realm of compromises and now you're just damaging things for no reason.

	=== Computation ===		=== Computation ===
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	==== First diamond extrema ====		==== First diamond extrema ====

	Let's do it for 4/3 first, {{vector\|2 -1 0}}. So, we prepare a matrix out of these two unchanged intervals, 2/1 and 4/3, and call it <math>U</math>:		Let's do it for 4/3 first, {{vector\|2 -1 0}}. So, we prepare a matrix out of these two unchanged intervals, 2/1 and 4/3, and call it <math>U</math>:

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	Next, we find the generators matrix <math>G</math> corresponding to the tuning where these two intervals are unchanged. The formula for this generators matrix is <math>G = U(MU)^{-1}</math> (~~for a full explanation of this formula,~~ see [[Dave Keenan ~~and~~ Douglas Blumeyer's guide to RTT ~~tuning#Unchanged~~ interval tuning ~~strategy~~]]). Here, let's work it out for our chosen <math>U</math>. First, we multiply <math>MU</math>:		Next, we find the generators matrix <math>G</math> corresponding to the tuning where these two intervals are unchanged. The formula for this generators matrix is <math>G = U(MU)^{-1}</math> (see [[Dave Keenan & Douglas Blumeyer's guide to RTT/All-interval tuning schemes\|here]] for a full explanation of this formula). Here, let's work it out for our chosen <math>U</math>. First, we multiply <math>MU</math>:


Line 152:		Line 149:


	Reading the columns from <math>G</math>, the first one {{vector\|1 0 0}} confirms our period of 2/1, and the second column {{vector\|-1 1 0}} gives our generator 3/2. Which is unsurprising. In cents, that's ~~1200¢ × log₂~~(3/2) ≈ 701.~~955¢~~. The next unchanged interval will give a more interesting result.		Reading the columns from <math>G</math>, the first one {{vector\|1 0 0}} confirms our period of 2/1, and the second column {{vector\|-1 1 0}} gives our generator 3/2. Which is unsurprising. In cents, that's {{nowrap\|1200 × log<sub>2</sub>(3/2) ≈ 701.955{{c}}}}. The next unchanged interval will give a more interesting result.

	==== Second diamond extrema ====		==== Second diamond extrema ====

	So let's do 5/4 now, {{vector\|-2 0 1}}. We prepare a matrix out of these two unchanged intervals, 2/1 and 5/4, and call it <math>U</math>:		So let's do 5/4 now, {{vector\|-2 0 1}}. We prepare a matrix out of these two unchanged intervals, 2/1 and 5/4, and call it <math>U</math>:

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	This tells us our generator is {{vector\|0 0 <math>\frac14</math>}}, or 5^(1/4). In cents, that's ~~1200¢ × log₂~~(~~5¹⸍⁴~~) ≈ 696.~~578¢~~.		This tells us our generator is {{vector\|0 0 <math>\frac14</math>}}, or 5^(1/4). In cents, that's {{nowrap\|1200 × log<sub>2</sub>(5<sup>{{frac\|1\|4}}</sup>) ≈ 696.578{{c}}}}.

	==== Third diamond extrema ====		==== Third diamond extrema ====

	Okay, one more unchanged interval to check: 6/5, which is {{vector\|1 1 -1}}. We prepare a matrix out of these two unchanged intervals, 2/1 and 6/5, and call it <math>U</math>:		Okay, one more unchanged interval to check: 6/5, which is {{vector\|1 1 -1}}. We prepare a matrix out of these two unchanged intervals, 2/1 and 6/5, and call it <math>U</math>:

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	This tells us our generator is ~~{{vector\|~~<math>\~~frac13</math> <math>~~-\~~frac13</math> <math>~~\~~frac13~~</math>}}, or expressed another way, (10/3)^(1/3). In cents, that's ~~1200¢ × log₂~~((10/3)~~¹⸍³~~) ≈ 694.~~786¢~~.		This tells us our generator is <math>\monzo{\frac{1}{3} & -\frac{1}{3} & \frac{1}{3}}</math>, or expressed another way, ({{frac\|10\|3}})<sup>{{frac\|1\|3}}</sup>. In cents, that's {{nowrap\|1200 × log<sub>2</sub>(({{frac\|10\|3}})<sup>{{frac\|1\|3}}</sup>) ≈ 694.786{{c}}}}.

	==== Determining the final range ====		==== Determining the final range ====

	We now have our generator sizes that give us pure consonances in the tonality diamond: 701.955¢, 696.578¢, and 694.786¢. The minimum of those is 694.786¢ and the maximum is 701.955¢, so that's our diamond tradeoff range. Anywhere inside that range, we are making at least one of our diamond consonances purer; outside it, we're making them all less pure.		We now have our generator sizes that give us pure consonances in the tonality diamond: 701.955¢, 696.578¢, and 694.786¢. The minimum of those is 694.786¢ and the maximum is 701.955¢, so that's our diamond tradeoff range. Anywhere inside that range, we are making at least one of our diamond consonances purer; outside it, we're making them all less pure.

	== See also ==		== See also ==
			For examples and other information, see [[tuning ranges of regular temperaments]].

	~~For examples and other information, see the topic page [[Tuning_ranges_of_regular_temperaments\|Tuning ranges of regular temperaments]].~~		== References ==
			<references />

	~~== References ==~~
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Terms]]		[[Category:Terms]]

AndyMilne: References heading

2024-06-10T02:18:06Z

References heading

← Older revision		Revision as of 02:18, 10 June 2024
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	For examples and other information, see the topic page [[Tuning_ranges_of_regular_temperaments\|Tuning ranges of regular temperaments]].		For examples and other information, see the topic page [[Tuning_ranges_of_regular_temperaments\|Tuning ranges of regular temperaments]].

			== References ==
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Terms]]		[[Category:Terms]]

AndyMilne: Reference added

2024-06-10T02:12:25Z

Reference added

← Older revision		Revision as of 02:12, 10 June 2024
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	== Original name ==		== Original name ==

	In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]] this tuning range was known as the "purer" range. On the wiki and in the regular temperament community, for many years this tuning range was referred to simply as the "nice" tuning range.		In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]], which defined this tuning range, it was known as the "purer" tuning range.<ref>Milne, A. J., Sethares, W. A., and Plamondon, J. (2006). X System. Technical report, Thumtronics, Inc. https://oro.open.ac.uk/21510/1/X_System.pdf</ref> On the wiki and in the regular temperament community, for many years this tuning range was referred to simply as the "nice" tuning range.

	== Vs. diamond monotone ==		== Vs. diamond monotone ==

FloraC: Fix a link

2023-09-25T11:23:42Z

Fix a link

← Older revision		Revision as of 11:23, 25 September 2023
Line 4:		Line 4:

	== Mathematical definition ==		== Mathematical definition ==
	[[Gene Ward Smith]] gives the following definition. The [[Odd limit\|''q''-odd-limit]] '''diamond tradeoff''' range of a [[rank]]-''r'' [[Prime limit\|''p''-limit]] temperament is the ~~[[Wikipedia: Convex hull~~\|convex hull]] in [[Vals and tuning ~~Space~~\|tuning space]] of the set of all tunings with ''r'' [[eigenmonzo\|eigenmonzos (unchanged-intervals)]] chosen as follows: one eigenmonzo 2 (pure octaves tunings) and the rest of the eigenmonzos any set of ''r'' - 1 members of the ''q''-odd limit [[tonality diamond]], whenever such a tuning is defined.		[[Gene Ward Smith]] gives the following definition. The [[Odd limit\|''q''-odd-limit]] '''diamond tradeoff''' range of a [[rank]]-''r'' [[Prime limit\|''p''-limit]] temperament is the {{w\|convex hull}} in [[Vals and tuning space\|tuning space]] of the set of all tunings with ''r'' [[eigenmonzo\|eigenmonzos (unchanged-intervals)]] chosen as follows: one eigenmonzo 2 (pure octaves tunings) and the rest of the eigenmonzos any set of ''r'' - 1 members of the ''q''-odd limit [[tonality diamond]], whenever such a tuning is defined.

	== Original name ==		== Original name ==

FloraC: Readability improvement: explanation first, mathematical definition later

2023-09-25T10:38:25Z

Readability improvement: explanation first, mathematical definition later

← Older revision		Revision as of 10:38, 25 September 2023
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	A tuning for a [[~~rank~~]]-''r'' [[~~Prime limit\|~~''p''-~~limit~~]] [[~~regular temperament~~]] is '''diamond tradeoff'''~~, or~~ '''~~diamond strict~~'''~~, if it fits the following definition: we may define the diamond tradeoff tuning range by finding~~ the [[Wikipedia: Convex hull\|convex hull]] in [[Vals and ~~Tuning~~ Space\|tuning space]] of the set of all tunings with ''r'' [[eigenmonzo ~~subgroup~~\|~~eigenmonzo]]s~~ (unchanged-intervals) chosen as follows: one eigenmonzo 2 (pure octaves tunings) and the rest of the eigenmonzos any set of ''r'' - 1 members of the ''p''-odd limit [[tonality diamond]], whenever such a tuning is defined ~~(this definition is based on Gene Ward Smith's one)~~.		A tuning for a [[regular temperament]] is '''diamond tradeoff''', or '''diamond strict''', if it is in the range where the concerning approximations to simple [[frequency ratio]]s can have their qualities "traded" against each other, that is, if some ratios are made more accurate, the others will be less so. For example, if you make the 3/2 more accurate, the 5/4 will suffer. However, outside this range, the tunings of ''all'' such intervals will be improved by moving back inside. This range therefore makes sense when one is concerned with approximating JI as closely as possible (without asserting a priori which specific consonances are the most important) because, under that criterion, it makes no logical sense to choose a tuning outside that range.

			However, it is quite clear that tunings outside of this diamond tradeoff range can function perfectly well as less accurate (and arguably more characterful) representations of the JI intervals specified by the temperament. That is, they are likely to be correctly recognized (whatever that actually means). For example, a 17-TET rendition of a standard piece of meantone music still makes complete musical sense, and major and minor chords still sound like major and minor chords, even though this tuning is outside the diamond tradeoff tuning range.

			== Mathematical definition ==
			[[Gene Ward Smith]] gives the following definition. The [[Odd limit\|''q''-odd-limit]] '''diamond tradeoff''' range of a [[rank]]-''r'' [[Prime limit\|''p''-limit]] temperament is the [[Wikipedia: Convex hull\|convex hull]] in [[Vals and tuning Space\|tuning space]] of the set of all tunings with ''r'' [[eigenmonzo\|eigenmonzos (unchanged-intervals)]] chosen as follows: one eigenmonzo 2 (pure octaves tunings) and the rest of the eigenmonzos any set of ''r'' - 1 members of the ''q''-odd limit [[tonality diamond]], whenever such a tuning is defined.

	== Original name ==		== Original name ==
Line 9:		Line 14:
	Diamond tradeoff tunings are always guaranteed to occur, but [[diamond monotone]] tunings are not.		Diamond tradeoff tunings are always guaranteed to occur, but [[diamond monotone]] tunings are not.

	== ~~Additional notes from Andrew Milne~~ ==		== Examples ==
			=== Explanation adapted from Keenan Pepper ===
	The diamond tradeoff tuning range marks tuning boundaries inside of which the temperament's approximations to simple low-ratio frequency ratios can be "traded" against each other; that is, if I make the 3/2 more accurate, the 5/4 will suffer. However, outside this range, you will improve the tunings of ''all'' such intervals by moving back inside. This range therefore makes sense when one is concerned with approximating JI as closely as possible (without asserting a priori which specific consonances are the most important) because, under that criterion, it makes no logical sense to choose a tuning outside that range.

	However, it is quite clear that tunings outside of this diamond tradeoff range can function perfectly well as less accurate (and arguably more characterful) representations of the JI intervals specified by the temperament. That is, they are likely to be correctly recognized (whatever that actually means). For example, a 17-TET rendition of a standard piece of meantone music still makes complete musical sense, and major and minor chords still sound like major and minor chords, even though this tuning is outside the diamond tradeoff tuning range.

	== Explanation adapted from Keenan Pepper ==

	For meantone temperament, there are three specific tunings that are special: one that tunes 4/3 and 3/2 pure, another that tunes 5/4 and 8/5 pure, and the third that tunes 6/5 and 5/3 pure. The tradeoff tuning range consists of these three points in tuning space ''and everything in between''. In this case, the three points fall along a line, where the pure-5/4 tuning is in between the pure-4/3 tuning and the pure-6/5 tuning.		For meantone temperament, there are three specific tunings that are special: one that tunes 4/3 and 3/2 pure, another that tunes 5/4 and 8/5 pure, and the third that tunes 6/5 and 5/3 pure. The tradeoff tuning range consists of these three points in tuning space ''and everything in between''. In this case, the three points fall along a line, where the pure-5/4 tuning is in between the pure-4/3 tuning and the pure-6/5 tuning.
Line 23:		Line 23:
	On the other hand, if you go outside these boundaries - for example, if you make 4/3 even flatter than pure - then you're making some intervals in the 5-limit diamond worse without making ''any'' of them better. You're past the realm of compromises and now you're just damaging things for no reason.		On the other hand, if you go outside these boundaries - for example, if you make 4/3 even flatter than pure - then you're making some intervals in the 5-limit diamond worse without making ''any'' of them better. You're past the realm of compromises and now you're just damaging things for no reason.

	== ~~Example~~ ==		=== Computation ===

	Here we will demonstrate the calculation of the diamond tradeoff tuning range for meantone.		Here we will demonstrate the calculation of the diamond tradeoff tuning range for meantone.

Cmloegcmluin: add unchanged-intervals along with eigenmonzo, per prior agreement on Xenwiki Work Group

2023-05-05T23:42:26Z

add unchanged-intervals along with eigenmonzo, per prior agreement on Xenwiki Work Group

← Older revision		Revision as of 23:42, 5 May 2023
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	A tuning for a [[rank]]-''r'' [[Prime limit\|''p''-limit]] [[regular temperament]] is '''diamond tradeoff''', or '''diamond strict''', if it fits the following definition: we may define the diamond tradeoff tuning range by finding the [[Wikipedia: Convex hull\|convex hull]] in [[Vals and Tuning Space\|tuning space]] of the set of all tunings with ''r'' [[eigenmonzo subgroup\|eigenmonzo]]s chosen as follows: one eigenmonzo 2 (pure octaves tunings) and the rest of the eigenmonzos any set of ''r'' - 1 members of the ''p''-odd limit [[tonality diamond]], whenever such a tuning is defined (this definition is based on Gene Ward Smith's one).		A tuning for a [[rank]]-''r'' [[Prime limit\|''p''-limit]] [[regular temperament]] is '''diamond tradeoff''', or '''diamond strict''', if it fits the following definition: we may define the diamond tradeoff tuning range by finding the [[Wikipedia: Convex hull\|convex hull]] in [[Vals and Tuning Space\|tuning space]] of the set of all tunings with ''r'' [[eigenmonzo subgroup\|eigenmonzo]]s (unchanged-intervals) chosen as follows: one eigenmonzo 2 (pure octaves tunings) and the rest of the eigenmonzos any set of ''r'' - 1 members of the ''p''-odd limit [[tonality diamond]], whenever such a tuning is defined (this definition is based on Gene Ward Smith's one).

	== Original name ==		== Original name ==
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	This is a 5-limit temperament, so we consider the 5-limit tonality diamond: {1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}. Of these seven pitches, there are only three we care about. We don't care about the unison, and half of the remaining pitches are octave-complements of the others are thus irrelevant. So, we'll only look at {4/3, 5/4, 6/5}.		This is a 5-limit temperament, so we consider the 5-limit tonality diamond: {1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}. Of these seven pitches, there are only three we care about. We don't care about the unison, and half of the remaining pitches are octave-complements of the others are thus irrelevant. So, we'll only look at {4/3, 5/4, 6/5}.

	For each of these three diamond consonances, we want to find what generator is required in order that this pitch remains pure after tempering, or in other words, that it is an unchanged interval (sometimes called an [[eigenmonzo]]). And we want to know this for the situation where octaves are pure.		For each of these three diamond consonances, we want to find what generator is required in order that this pitch remains pure after tempering, or in other words, that it is an unchanged interval (sometimes called an [[eigenmonzo]], or unchanged-interval). And we want to know this for the situation where octaves are pure.

	==== First diamond extrema ====		==== First diamond extrema ====

Cmloegcmluin: focus and clean up formatting of example

2022-05-27T00:28:52Z

focus and clean up formatting of example

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