Generator embedding optimization - Revision history

FloraC: Linking

2025-09-11T06:25:34Z

Linking

← Older revision		Revision as of 06:25, 11 September 2025
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	=== Advanced tie-breaking example: Setup ===		=== Advanced tie-breaking example: Setup ===
	To help illustrate advanced tie-breaking, we're going to look at a minimax-C tuning of [[augmented]] temperament, with mapping {{rket\|{{map\|3 0 7}} {{map\|0 1 0}}}}. In particular, we're going to use the somewhat arbitrary target-interval set <math>\{ \frac32, \frac52, \frac53, \frac83, \frac95, \frac{16}{5}, \frac{15}{8}, \frac{18}{5} \}</math>. As a rank-2 temperament, we're going to be searching 3D tuning damage space. This temperament divides the octave into three parts, so our ballpark <math>g_1</math> is 400 ¢, and our second generator <math>g_2</math> is a free generator for prime 3, so it's going to be ballpark its pure tuning of 1901.955 ¢.		To help illustrate advanced tie-breaking, we're going to look at a minimax-C tuning of [[augmented (temperament)\|augmented]] temperament, with mapping {{rket\|{{map\|3 0 7}} {{map\|0 1 0}}}}. In particular, we're going to use the somewhat arbitrary target-interval set <math>\{ \frac32, \frac52, \frac53, \frac83, \frac95, \frac{16}{5}, \frac{15}{8}, \frac{18}{5} \}</math>. As a rank-2 temperament, we're going to be searching 3D tuning damage space. This temperament divides the octave into three parts, so our ballpark <math>g_1</math> is 400 ¢, and our second generator <math>g_2</math> is a free generator for prime 3, so it's going to be ballpark its pure tuning of 1901.955 ¢.

	Here's an aerial view on the tuning damage space, where we've "clipped" every damage graph hyper-V where it has gone out-of-bounds, above 150 ¢(C) damage; that is, we've colored it in grey and flattened it across the top of the box of our visualization. This lets us focus in clearly on the region of real interest, which is where all target-intervals' damages are less than this cap at the same time. This is the multicolored crater in the middle here:		Here's an aerial view on the tuning damage space, where we've "clipped" every damage graph hyper-V where it has gone out-of-bounds, above 150 ¢(C) damage; that is, we've colored it in grey and flattened it across the top of the box of our visualization. This lets us focus in clearly on the region of real interest, which is where all target-intervals' damages are less than this cap at the same time. This is the multicolored crater in the middle here:

Sintel: /* With held-intervals */ Ref should be note

2025-04-19T21:55:39Z

With held-intervals: Ref should be note

← Older revision		Revision as of 21:55, 19 April 2025
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	== With held-intervals ==		== With held-intervals ==
	The zero-damage method is easily modified to handle held-intervals along with target-intervals.<ref>In fact, the Target tunings page of the wiki uses this more complicated approach in order to realize pure octaves, and so the authors of this page had to reverse engineer from it how to make it work ''without'' any held-intervals.</ref> In short, rather than assembling our set of unchanged-interval bases <math>\mathrm{U}_1</math> through <math>\mathrm{U}_n</math> (where <math>n = {{k}\choose{r}}</math>) corresponding to the zero-damage points by finding every combination of <math>r</math> different ones of our <math>k</math> target-intervals (one for each generator to be responsible for tuning exactly), instead we must first reserve <math>h</math> (''held''-unchanged-interval count) columns of each <math>\mathrm{U}_n</math> for the held-intervals, leaving only the remaining <math>r - h</math> columns to be assembled from the target-intervals as normal. So, we'll only have <math>{{k}\choose{r - h}}</math> candidate tunings / zero-damage points / unchanged-interval bases in this case.		The zero-damage method is easily modified to handle held-intervals along with target-intervals.<ref group="note">In fact, the Target tunings page of the wiki uses this more complicated approach in order to realize pure octaves, and so the authors of this page had to reverse engineer from it how to make it work ''without'' any held-intervals.</ref> In short, rather than assembling our set of unchanged-interval bases <math>\mathrm{U}_1</math> through <math>\mathrm{U}_n</math> (where <math>n = {{k}\choose{r}}</math>) corresponding to the zero-damage points by finding every combination of <math>r</math> different ones of our <math>k</math> target-intervals (one for each generator to be responsible for tuning exactly), instead we must first reserve <math>h</math> (''held''-unchanged-interval count) columns of each <math>\mathrm{U}_n</math> for the held-intervals, leaving only the remaining <math>r - h</math> columns to be assembled from the target-intervals as normal. So, we'll only have <math>{{k}\choose{r - h}}</math> candidate tunings / zero-damage points / unchanged-interval bases in this case.

	In other words, if <math>\mathrm{U}_n</math> is one of the unchanged-interval bases characterizing a candidate miniaverage tuning, then it must contain <math>\mathrm{H}</math> itself, the held-interval basis, which does not yet fully characterize our tuning, leaving some wiggle room (otherwise we'd just use the "[[#Only held-intervals method\|only held-intervals]]" approach, discussed later).		In other words, if <math>\mathrm{U}_n</math> is one of the unchanged-interval bases characterizing a candidate miniaverage tuning, then it must contain <math>\mathrm{H}</math> itself, the held-interval basis, which does not yet fully characterize our tuning, leaving some wiggle room (otherwise we'd just use the "[[#Only held-intervals method\|only held-intervals]]" approach, discussed later).
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	And convert those to generator tuning maps: {{map\|1200 701.955}}, {{map\|1200 696.578}}, and {{map\|1200 694.786}}. Note that every one of these has a pure-octave period. Then check the damage sums: 353.942 ¢(U), 89.083 ¢(U), and 110.390 ¢(U), respectively. So that tells us that we want the middle result of these three, {{map\|1200 696.578}}, as the minimization of the <math>1</math>-mean of unity-weight damage to the 6-TILT, when we're constrained to the octave being unchanged.		And convert those to generator tuning maps: {{map\|1200 701.955}}, {{map\|1200 696.578}}, and {{map\|1200 694.786}}. Note that every one of these has a pure-octave period. Then check the damage sums: 353.942 ¢(U), 89.083 ¢(U), and 110.390 ¢(U), respectively. So that tells us that we want the middle result of these three, {{map\|1200 696.578}}, as the minimization of the <math>1</math>-mean of unity-weight damage to the 6-TILT, when we're constrained to the octave being unchanged.

	For a rank-3 temperament, with 2 held-intervals, we'd again have 8 choose 1 = 8 tunings to check. With 1 held-interval, we'd have 8 choose 2 = 28 tunings to check.		For a rank-3 temperament, with 2 held-intervals, we'd again have 8 choose 1 = 8 tunings to check. With 1 held-interval, we'd have 8 choose 2 = 28 tunings to check.

	== For all-interval tuning schemes ==		== For all-interval tuning schemes ==

Cmloegcmluin: /* Generator embedding */

2024-11-13T15:24:41Z

Generator embedding

← Older revision		Revision as of 15:24, 13 November 2024
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	== Generator embedding ==		== Generator embedding ==
	So, substituting <math>\mathrm{U}</math> in for <math>\mathrm{H}</math> in the formula we learned from the D&D's guide article:		So, substituting <math>\mathrm{U}</math> in for <math>\mathrm{H}</math> in the formula we learned [[Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning computation#Only held-intervals method\|from the D&D's guide article]]:


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	G = \mathrm{U}(M\mathrm{U})^{-1}		G = \mathrm{U}(M\mathrm{U})^{-1}
	</math>		</math>


	= Pseudoinverse method =		= Pseudoinverse method =

ArrowHead294: Update links

2024-11-02T18:38:18Z

Update links

← Older revision		Revision as of 18:38, 2 November 2024
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	As discussed in [[Dave Keenan & Douglas Blumeyer's guide to RTT~~: tuning~~ fundamentals#Absolute errors]], these vertical bars mean to take the absolute value of each entry of this vector, not to take its magnitude.		As discussed in [[Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning fundamentals#Absolute errors]], these vertical bars mean to take the absolute value of each entry of this vector, not to take its magnitude.

	As discussed elsewhere, we can simplify this to:		As discussed elsewhere, we can simplify this to:
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	But sometimes we have a tie between tunings for least average damage, though. For example, if we had we done a unity-weight tuning, in which case <math>W = I</math>, and included the interval <math>\frac85</math> in our set, we would have found that quarter-comma meantone tied with another tuning, one with generators of <math>\sqrt[5]{\frac{2560}{81}}</math> and <math>\sqrt[5]{\frac{200}{27}}</math>, which are approximately 1195.7 ¢ and 693.352 ¢.		But sometimes we have a tie between tunings for least average damage, though. For example, if we had we done a unity-weight tuning, in which case <math>W = I</math>, and included the interval <math>\frac85</math> in our set, we would have found that quarter-comma meantone tied with another tuning, one with generators of <math>\sqrt[5]{\frac{2560}{81}}</math> and <math>\sqrt[5]{\frac{200}{27}}</math>, which are approximately 1195.7 ¢ and 693.352 ¢.

	In this case, we fall back to our general method, which is equipped to find the true optimum somewhere in between these two extreme ends of goodness, albeit as an approximate solution.<ref group="note">The article for the minimax tuning scheme, [[Target tunings]], suggests that you fall back to the miniRMS method to tie-break between these, but that sort of misses the point of the problem. The two tied points are on extreme opposite ends of the slice of good solutions, and the optimum solution lies somewhere in between them. We don't want the tie-break to choose one or the other extreme; we want to find a better solution somewhere in between them.</ref> This method is discussed here: [[Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning computation#Tie_breaking:_power_limit_method\|power limit method]]. Or, if you'd like a refresher on how to think about non-unique tunings, please see [[Dave Keenan & Douglas Blumeyer's guide to RTT~~: tuning~~ fundamentals#Non-unique tunings]].		In this case, we fall back to our general method, which is equipped to find the true optimum somewhere in between these two extreme ends of goodness, albeit as an approximate solution.<ref group="note">The article for the minimax tuning scheme, [[Target tunings]], suggests that you fall back to the miniRMS method to tie-break between these, but that sort of misses the point of the problem. The two tied points are on extreme opposite ends of the slice of good solutions, and the optimum solution lies somewhere in between them. We don't want the tie-break to choose one or the other extreme; we want to find a better solution somewhere in between them.</ref> This method is discussed here: [[Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning computation#Tie_breaking:_power_limit_method\|power limit method]]. Or, if you'd like a refresher on how to think about non-unique tunings, please see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning fundamentals#Non-unique tunings]].

	We note that there may be a way to find an exact solution to a nested miniaverage, in a similar fashion to the nested minimax discussed in the coinciding-damage method section below, but it raises some conceptual issues about what a nested miniaverage even means.<ref group="note">There does not seem to be any consensus about how to identify a true optimum in the case of multiple solutions when <math>p=1</math>. See https://en.wikipedia.org/wiki/Least_absolute_deviations#Properties, https://www.researchgate.net/publication/223752233_Dealing_with_the_multiplicity_of_solutions_of_the_l1_and_l_regression_models, and https://stats.stackexchange.com/questions/275931/is-it-possible-to-force-least-absolute-deviations-lad-regression-to-return-the.</ref> We have done some pondering of this problem but it remains open; we didn't prioritize solving it, on account of the fact that nobody uses miniaverage tunings anyway.		We note that there may be a way to find an exact solution to a nested miniaverage, in a similar fashion to the nested minimax discussed in the coinciding-damage method section below, but it raises some conceptual issues about what a nested miniaverage even means.<ref group="note">There does not seem to be any consensus about how to identify a true optimum in the case of multiple solutions when <math>p=1</math>. See https://en.wikipedia.org/wiki/Least_absolute_deviations#Properties, https://www.researchgate.net/publication/223752233_Dealing_with_the_multiplicity_of_solutions_of_the_l1_and_l_regression_models, and https://stats.stackexchange.com/questions/275931/is-it-possible-to-force-least-absolute-deviations-lad-regression-to-return-the.</ref> We have done some pondering of this problem but it remains open; we didn't prioritize solving it, on account of the fact that nobody uses miniaverage tunings anyway.

ArrowHead294 at 18:34, 2 November 2024

2024-11-02T18:34:12Z

← Older revision		Revision as of 18:34, 2 November 2024
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	Two of these four specialized methods were briefly discussed in D&D's guide, along with the general method, because these specialized methods are actually even quicker and easier than the general method. These two are the only held-intervals method, and the pseudoinverse method. But there's still plenty more insight to be had into how and why exactly these methods work, in particular for the pseudoinverse method, so we'll be doing a much deeper dive into it in this article here than was done in D&D's guide.		Two of these four specialized methods were briefly discussed in D&D's guide, along with the general method, because these specialized methods are actually even quicker and easier than the general method. These two are the only held-intervals method, and the pseudoinverse method. But there's still plenty more insight to be had into how and why exactly these methods work, in particular for the pseudoinverse method, so we'll be doing a much deeper dive into it in this article here than was done in D&D's guide.

	The other two of these four specialized ~~methods — the~~ zero-damage method, and the coinciding-damage method—are significantly more challenging to understand than the general method. Most students of RTT would not gain enough musical insight by familiarizing themselves with them to have justified the investment. This is why these two methods were not discussed in D&D's guide. However, if you feel compelled to understand the nuts and bolts of these methods anyway, then those sections of the article may well appeal to you.		The other two of these four specialized methods—the zero-damage method, and the coinciding-damage method—are significantly more challenging to understand than the general method. Most students of RTT would not gain enough musical insight by familiarizing themselves with them to have justified the investment. This is why these two methods were not discussed in D&D's guide. However, if you feel compelled to understand the nuts and bolts of these methods anyway, then those sections of the article may well appeal to you.

	This article is titled "Generator embedding optimization" because of a key feature these four specialized methods share: they can all give their solutions as [[generator embedding matrix\|generator embedding]]s, i.e. lists of [[prime-count vector]]s, one for each generator, where typically these prime-count vectors have non-integer entries (and are thus not [[JI]]). This is different from the general method, which can only give [[generator tuning map]]s, i.e. sizes in cents for each generator. As we'll see, a tuning optimization method's ability to give solutions as generator embeddings is equivalent to its ability to give solutions that are exact.		This article is titled "Generator embedding optimization" because of a key feature these four specialized methods share: they can all give their solutions as [[generator embedding matrix\|generator embedding]]s, i.e. lists of [[prime-count vector]]s, one for each generator, where typically these prime-count vectors have non-integer entries (and are thus not [[JI]]). This is different from the general method, which can only give [[generator tuning map]]s, i.e. sizes in cents for each generator. As we'll see, a tuning optimization method's ability to give solutions as generator embeddings is equivalent to its ability to give solutions that are exact.

ArrowHead294: Reformat markup, update links

2024-11-02T18:33:47Z

Reformat markup, update links

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Cmloegcmluin: target set → target-interval set

2023-02-18T01:37:24Z

target set → target-interval set

← Older revision		Revision as of 01:37, 18 February 2023
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	When computing an all-interval tuning where the dual norm power is <math>∞</math>, we use a variation on the method we used for ordinary tunings when the optimization power was <math>∞</math>.		When computing an all-interval tuning where the dual norm power is <math>∞</math>, we use a variation on the method we used for ordinary tunings when the optimization power was <math>∞</math>.

	In this case, our optimization power is also still <math>∞</math>. That is to say, that in this case we're doing the same computation we would have been doing if we had a finite target set, but now we're doing it as if the primes alone were our target set.		In this case, our optimization power is also still <math>∞</math>. That is to say, that in this case we're doing the same computation we would have been doing if we had a finite target-interval set, but now we're doing it as if the primes alone were our target-interval set.

	Let's get the minimax-S tuning of meantone. With three proxy target-intervals and two generators, we end up with four constraint matrices:		Let's get the minimax-S tuning of meantone. With three proxy target-intervals and two generators, we end up with four constraint matrices:

Dave Keenan: /* Convert to generators */ Changed "generator preimage transversal" to "generator detempering".

2023-02-15T00:37:28Z

Convert to generators: Changed "generator preimage transversal" to "generator detempering".

← Older revision		Revision as of 00:37, 15 February 2023
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	===Convert to generators===		===Convert to generators===

	Now we just need to convert each of these unchanged-interval bases <math>\mathrm{U}_{(i,j)}</math> to a corresponding generator embedding <math>G</math>. To do this, we use the formula <math>G = \mathrm{U}(M\mathrm{U})^{-1}</math>, where <math>M</math> is the temperament mapping (the derivation of this formula, and examples of working through this calculation, are both described later in this article here: [[#Only unchanged-intervals method]]).<ref>Note that this technique for converting zero-damage points to generator tunings is much simpler than the technique described on the [[Target tunings]] page. The Target tunings page uses eigendecomposition, which unnecessarily requires you to find the commas for the temperament, compute a full [[projection matrix]] <math>P</math>, and then when you need to spit a generator tuning map <math>𝒈</math> out at the end, requires the computation of a [[generator ~~preimage transversal~~]] to do so (moreover, it doesn't explain or even mention eigendecomposition; it assumes the reader knows how and when to do them, cutting off at the point of listing the eigenvectors — a big thanks to Sintel for unpacking the thought process in that article for us). The technique described here skips the commas, computing the generator embedding <math>G</math> directly rather than via <math>P = GM</math>, and then when you need to spit a generator tuning map out at the end, it's just <math>𝒈 = 𝒋G</math>, which is much simpler than the generator ~~preimage transversal~~ computation.		Now we just need to convert each of these unchanged-interval bases <math>\mathrm{U}_{(i,j)}</math> to a corresponding generator embedding <math>G</math>. To do this, we use the formula <math>G = \mathrm{U}(M\mathrm{U})^{-1}</math>, where <math>M</math> is the temperament mapping (the derivation of this formula, and examples of working through this calculation, are both described later in this article here: [[#Only unchanged-intervals method]]).<ref>Note that this technique for converting zero-damage points to generator tunings is much simpler than the technique described on the [[Target tunings]] page. The Target tunings page uses eigendecomposition, which unnecessarily requires you to find the commas for the temperament, compute a full [[projection matrix]] <math>P</math>, and then when you need to spit a generator tuning map <math>𝒈</math> out at the end, requires the computation of a [[generator detempering]] to do so (moreover, it doesn't explain or even mention eigendecomposition; it assumes the reader knows how and when to do them, cutting off at the point of listing the eigenvectors — a big thanks to Sintel for unpacking the thought process in that article for us). The technique described here skips the commas, computing the generator embedding <math>G</math> directly rather than via <math>P = GM</math>, and then when you need to spit a generator tuning map out at the end, it's just <math>𝒈 = 𝒋G</math>, which is much simpler than the generator detempering computation.
	<br><br>		<br><br>
	The Target tunings approach and this approach are quite similar conceptually. Here's the Target tunings approach:		The Target tunings approach and this approach are quite similar conceptually. Here's the Target tunings approach:

Cmloegcmluin: /* Tie-breaking */ misc issues

2023-01-31T23:08:44Z

Tie-breaking: misc issues

Show changes

Cmloegcmluin: Exact solutions with advanced tie-breaking

2023-01-31T22:48:14Z

Exact solutions with advanced tie-breaking

← Older revision		Revision as of 22:48, 31 January 2023
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	The ADSLOD here, by the way, is [92.557 92.557 81.117 81.117 57.928 ... ] So it's a tie of 81.117 ¢(C) for the second-most minimax damage to <math>\frac95</math> and <math>\frac{16}{5}</math>. No other tuning can beat this 81.117 number, even just three entries down the list. And so we're done.		The ADSLOD here, by the way, is [92.557 92.557 81.117 81.117 57.928 ... ] So it's a tie of 81.117 ¢(C) for the second-most minimax damage to <math>\frac95</math> and <math>\frac{16}{5}</math>. No other tuning can beat this 81.117 number, even just three entries down the list. And so we're done.

			===Exact solutions with advanced tie-breaking===

			As for recovering <math>G</math>, though. You know, the whole point of this article — finding exact tunings — we wouldn't want to give up on that just because we had to use advanced tie-breaking, would we?

			So we've been looking for <math>𝒈</math> which are blends of other <math>𝒈</math>'s. But we need to look for <math>G</math>'s that are blends of other <math>G</math>'s! Doing that directly would explode the dimensionality of the space we're searching, by the rank <math>r</math> times the length of the blend vector <math>𝒃</math>, that is, <math>𝑟×(τ - 1)</math>. And would it even be meaningful to independently search the powers of the primes that comprise each entry of a <math>𝒈</math>? Probably not. The compressed information in <math>𝒈</math> is all that really matters for defining the constrained search region. So what if instead we still search by <math>𝒈</math>, but what if the blend we find for each <math>K</math> can be applied to <math>G</math>'s instead of <math>𝒈</math>'s?

			Let's test on an example.


			<math>

			G_0 =
			\left[ \begin{array} {r}
			1 & 0 \\
			0 & 0 \\
			0 & \frac14 \\
			\end{array} \right]

			\quad

			𝒈_0 =
			\left[ \begin{array} {r}
			1200.000 & 696.578 \\
			\end{array} \right]

			\\[20pt]

			G_1 =
			\left[ \begin{array} {r}
			\frac73 & \frac13 \\
			-\frac43 & -\frac13 \\
			\frac13 & \frac13 \\
			\end{array} \right]

			\quad

			𝒈_1 =
			\left[ \begin{array} {r}
			1192.831 & 694.786 \\
			\end{array} \right]

			</math>


			So <math>𝜹_1 = 𝒈_1 - 𝒈_0</math> = {{rbra\|-7.169 -1.792}}. Suppose we get <math>𝒃</math> = [0.5]. We know that <math>𝒃D</math> = {{rbra\|-3.584 -0.896}}. So <math>𝒈</math> should be <math>𝒈_0 + 𝒃D</math> = {{rbra\|1200 696.578}} + {{rbra\|-3.584 -0.896}} = {{rbra\|1196.416 695.682}}.

			But do we find the same tuning with <math>G = G_0 + b_1(G_1 - G_0) + b_2(G_2 - G_0) + \ldots + b_{τ-1}(G_{τ-1} - G_0)</math>? That's the key question. (In this case, we have to bust the matrix multiplication up. That is, there's no way to replace the rows of D with entire matrices. Cumbersome, but reality.)

			In this case we only have the one delta, <math>G_1 - G_0 =</math>


			<math>

			\left[ \begin{array} {r}
			\frac73 & \frac13 \\
			-\frac43 & -\frac13 \\
			\frac13 & \frac13 \\
			\end{array} \right]

			-

			\left[ \begin{array} {r}
			1 & 0 \\
			0 & 0 \\
			0 & \frac14 \\
			\end{array} \right]

			=

			\left[ \begin{array} {r}
			\frac43 & \frac13 \\
			-\frac43 & -\frac13 \\
			\frac13 & \frac{1}{12} \\
			\end{array} \right]
			</math>


			And so <math>b_1(G_1 - G_0)</math>, or half of that, is:


			<math>
			\left[ \begin{array} {r}
			\frac23 & \frac16 \\
			-\frac23 & -\frac16 \\
			\frac16 & \frac{1}{24} \\
			\end{array} \right]
			</math>


			And add that to <math>G_0</math> then:


			<math>
			\left[ \begin{array} {r}
			1 & 0 \\
			0 & 0 \\
			0 & \frac14 \\
			\end{array} \right]

			+

			\left[ \begin{array} {r}
			\frac23 & \frac16 \\
			-\frac23 & -\frac16 \\
			\frac16 & \frac{1}{24} \\
			\end{array} \right]

			=

			\left[ \begin{array} {r}
			\frac53 & \frac16 \\
			-\frac23 & -\frac16 \\
			\frac16 & \frac{9}{24} \\
			\end{array} \right]
			</math>


			So <math>\textbf{g}_1</math> here, the first column, {{vector\|<math>\frac53</math> <math>-\frac23</math> <math>\frac16</math>}}, is <math>2^{\frac53}3^{-\frac23}5^{\frac16} \approx 1.996</math>. So <math>g_1</math> = 1196.416 ¢.

			And <math>\textbf{g}_2</math> here, the second column, {{vector\|<math>\frac16</math> <math>-\frac16</math> <math>\frac{9}{24}</math>⟩, is <math>2^{\frac16}3^{-\frac16}5^{\frac{9}{24}} \approx 1.495</math>. So <math>g_2</math> = 695.682 ¢.

			Perfect! We wanted {{rbra\|1196.416 695.682}} and we got it.

			Now maybe this doesn't fully test the system, since we only convexly combined two tunings, but this is probably sound for general use. At least, the test suite of the RTT Library in Wolfram Language included several examples that should have failed upon switching to this way of computing true optimum tunings if this were a problem, but they did not.

	==For all-interval tuning schemes==		==For all-interval tuning schemes==