Talk:Optimal ET sequence: Difference between revisions

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::::: I recall reading that there are infinitely many ET for each EDO (or equal tuning) out there, alluding to the fact that an ET is defined by a temperament map (val), and you could technically use any kind of map, even wonky ones, with any tuning. The use of wart notation in this sequence confirms that we're really enumerating equal ''temperaments'', not equal ''tunings'' (both of which can abbreviate to ET, that's annoying, but I'm using ET for equal temperament here). After all, putting something like 17c in a sequence is exactly the same as spelling out the corresponding map in full, it's just that wart notation makes the sequence look like a list of equal ''tunings'' when it actually isn't. Now, uniform maps (GPVs) are a special kind of ET that excludes the "wonky ones", and in a given prime limit (or domain), there are finitely many uniform maps that map a given prime to the same number of steps; [[:File:Generalized Patent Vals.png]] offers a nice visualization of this. I believe it's reasonable to only include uniform maps in these lists, because most people interested in temperament data want maps that actually try to approximate JI logically, and it would only be less clear what exactly is being listed if we switched from GPV to ET. If people understand it as "supporting ET", while it's not the full picture, it's not completely off track either, unless I am myself off track of course. My current concern would be "GPV" vs. "uniform map" (UM?). "Optimal uniform map sequence" is a mouthful, and "Optimal UM sequence" looks weird because it's new although I bet I could get used to it quickly. On the other hand, since it's an "optimal sequence", one might expect that the ETs listed would also be uniform maps. From that perspective, I agree that "optimal ET sequence" is the most straightforward choice. However, I wonder if non-uniform maps can make it in the sequences even with the restriction of decreasing error, so maybe someone could help me clear that up? --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 03:10, 4 May 2023 (UTC)

:::: I've never heard of ET standing for "equal tuning", only "equal temperament". I do not say that ETs ''are'' maps (vals), but rather that ETs ''have'' maps (vals), because an ET can just as well be defined by its comma basis (monzo list). So, to me, those sequences are not sequences of any kind of map or val. That would look like: ⟨12 19 28], ⟨17 27 40], ... They are sequences of ETs. I agree it would be interesting to know if an ET with a non-uniform map could ever make it into such list, given only the requirement of decreasing error with increasing ET number. I'm sorry I can't answer that. I expect that Flora could easily perform some experiments in that regard. But we don't need to know that to resolve this issue, since we agree we can simply add the uniformity (GPV) requirement as part of what it means to be "optimal" here. [[User:Dave Keenan|Dave Keenan]] ([[User talk:Dave Keenan|talk]]) 04:06, 4 May 2023 (UTC)

:::: I've never heard of ET standing for "equal tuning", only "equal temperament". I do not say that ETs ''are'' maps (vals), but rather that ETs ''have'' maps (vals), because an ET can just as well be defined by its comma basis (monzo list). So, to me, those sequences are not sequences of any kind of map or val. That would look like: ⟨12 19 28], ⟨17 27 40], ... They are sequences of ETs. I agree it would be interesting to know if an ET with a non-uniform map could ever make it into such list, given only the requirement of decreasing error with increasing ET number. I'm sorry I can't answer that. I expect that Flora could easily perform some experiments in that regard. But we don't need to know that to resolve this issue, since we agree we can (and would want to) simply add the uniformity (GPV) requirement as part of what it means to be "optimal" here. [[User:Dave Keenan|Dave Keenan]] ([[User talk:Dave Keenan|talk]]) 04:06, 4 May 2023 (UTC)

Revision as of 04:07, 4 May 2023 view source Dave Keenan (talk \| contribs) autopatrolled, emailconfirmed 1,671 edits Corrected "[12 19 28⟩, [17 27 40⟩" to "⟨12 19 28], ⟨17 27 40]". ← Older edit		Revision as of 04:11, 4 May 2023 view source Dave Keenan (talk \| contribs) autopatrolled, emailconfirmed 1,671 edits Changed "we agree we can" to "we agree we can (and would want to)". Newer edit →
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	::::: I recall reading that there are infinitely many ET for each EDO (or equal tuning) out there, alluding to the fact that an ET is defined by a temperament map (val), and you could technically use any kind of map, even wonky ones, with any tuning. The use of wart notation in this sequence confirms that we're really enumerating equal ''temperaments'', not equal ''tunings'' (both of which can abbreviate to ET, that's annoying, but I'm using ET for equal temperament here). After all, putting something like 17c in a sequence is exactly the same as spelling out the corresponding map in full, it's just that wart notation makes the sequence look like a list of equal ''tunings'' when it actually isn't. Now, uniform maps (GPVs) are a special kind of ET that excludes the "wonky ones", and in a given prime limit (or domain), there are finitely many uniform maps that map a given prime to the same number of steps; [[:File:Generalized Patent Vals.png]] offers a nice visualization of this. I believe it's reasonable to only include uniform maps in these lists, because most people interested in temperament data want maps that actually try to approximate JI logically, and it would only be less clear what exactly is being listed if we switched from GPV to ET. If people understand it as "supporting ET", while it's not the full picture, it's not completely off track either, unless I am myself off track of course. My current concern would be "GPV" vs. "uniform map" (UM?). "Optimal uniform map sequence" is a mouthful, and "Optimal UM sequence" looks weird because it's new although I bet I could get used to it quickly. On the other hand, since it's an "optimal sequence", one might expect that the ETs listed would also be uniform maps. From that perspective, I agree that "optimal ET sequence" is the most straightforward choice. However, I wonder if non-uniform maps can make it in the sequences even with the restriction of decreasing error, so maybe someone could help me clear that up? --[[User:Fredg999\|Fredg999]] ([[User talk:Fredg999\|talk]]) 03:10, 4 May 2023 (UTC)		::::: I recall reading that there are infinitely many ET for each EDO (or equal tuning) out there, alluding to the fact that an ET is defined by a temperament map (val), and you could technically use any kind of map, even wonky ones, with any tuning. The use of wart notation in this sequence confirms that we're really enumerating equal ''temperaments'', not equal ''tunings'' (both of which can abbreviate to ET, that's annoying, but I'm using ET for equal temperament here). After all, putting something like 17c in a sequence is exactly the same as spelling out the corresponding map in full, it's just that wart notation makes the sequence look like a list of equal ''tunings'' when it actually isn't. Now, uniform maps (GPVs) are a special kind of ET that excludes the "wonky ones", and in a given prime limit (or domain), there are finitely many uniform maps that map a given prime to the same number of steps; [[:File:Generalized Patent Vals.png]] offers a nice visualization of this. I believe it's reasonable to only include uniform maps in these lists, because most people interested in temperament data want maps that actually try to approximate JI logically, and it would only be less clear what exactly is being listed if we switched from GPV to ET. If people understand it as "supporting ET", while it's not the full picture, it's not completely off track either, unless I am myself off track of course. My current concern would be "GPV" vs. "uniform map" (UM?). "Optimal uniform map sequence" is a mouthful, and "Optimal UM sequence" looks weird because it's new although I bet I could get used to it quickly. On the other hand, since it's an "optimal sequence", one might expect that the ETs listed would also be uniform maps. From that perspective, I agree that "optimal ET sequence" is the most straightforward choice. However, I wonder if non-uniform maps can make it in the sequences even with the restriction of decreasing error, so maybe someone could help me clear that up? --[[User:Fredg999\|Fredg999]] ([[User talk:Fredg999\|talk]]) 03:10, 4 May 2023 (UTC)

	:::: I've never heard of ET standing for "equal tuning", only "equal temperament". I do not say that ETs ''are'' maps (vals), but rather that ETs ''have'' maps (vals), because an ET can just as well be defined by its comma basis (monzo list). So, to me, those sequences are not sequences of any kind of map or val. That would look like: ⟨12 19 28], ⟨17 27 40], ... They are sequences of ETs. I agree it would be interesting to know if an ET with a non-uniform map could ever make it into such list, given only the requirement of decreasing error with increasing ET number. I'm sorry I can't answer that. I expect that Flora could easily perform some experiments in that regard. But we don't need to know that to resolve this issue, since we agree we can simply add the uniformity (GPV) requirement as part of what it means to be "optimal" here. [[User:Dave Keenan\|Dave Keenan]] ([[User talk:Dave Keenan\|talk]]) 04:06, 4 May 2023 (UTC)		:::: I've never heard of ET standing for "equal tuning", only "equal temperament". I do not say that ETs ''are'' maps (vals), but rather that ETs ''have'' maps (vals), because an ET can just as well be defined by its comma basis (monzo list). So, to me, those sequences are not sequences of any kind of map or val. That would look like: ⟨12 19 28], ⟨17 27 40], ... They are sequences of ETs. I agree it would be interesting to know if an ET with a non-uniform map could ever make it into such list, given only the requirement of decreasing error with increasing ET number. I'm sorry I can't answer that. I expect that Flora could easily perform some experiments in that regard. But we don't need to know that to resolve this issue, since we agree we can (and would want to) simply add the uniformity (GPV) requirement as part of what it means to be "optimal" here. [[User:Dave Keenan\|Dave Keenan]] ([[User talk:Dave Keenan\|talk]]) 04:06, 4 May 2023 (UTC)