Fifth-chroma temperaments: Difference between revisions

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* "superneutral", which principally represents ~27/22~16/13

* "submajor", a general bucket/category for aggregating ratios like 21/17, 26/21, 31/25, 36/29, 41/33, 46/37

:: <nowiki>*</nowiki> of the edos discussed, only 94edo does not temper out (50/41)/(39/32) = [[1600/1599]] = S40, corresponding to its inconsistently-flat mapping of 25/16; this is noteworthy only because the subneutral and superneutral categories generally are supposed to be unambiguous in their interpretation; an asterisk with 84edo's mapping of prime 11 is discussed later

: <nowiki>*</nowiki> of the edos discussed, only 94edo does not temper out (50/41)/(39/32) = [[1600/1599]] = S40, corresponding to its inconsistently-flat mapping of 25/16; this is noteworthy only because the subneutral and superneutral categories generally are supposed to be unambiguous in their interpretation; an asterisk with 84edo's mapping of prime 11 is discussed later

The lists of ratios for supraminor and submajor are not complete and a given edo tuning will not necessarily find contexts where all of these ratios make sense as interpretations, but the principle is that between 6/5 and 11/9 are many ratios that are various mediants of 6/5 and 11/9, and between 16/13 and 5/4 are many ratios that are various mediants of 16/13 and 5/4, so that both general areas represent places where the source of concordance (if any) is not necessarily obvious, so that any mediant therein can potentially be suggested with sufficiently forcing amounts of harmonic context (notes in a chord with approximate frequency ratios suggesting a certain otonal/harmonic series interpretation). Therefore, it represents a flexible melodic category able to represent a wide variety of tempered harmonies contextually.

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That is, it represents the shared harmonic and spacing logic between at least* [[77edo]], [[80edo]], [[84edo]]**, [[87edo]] and [[94edo]], so that its join can be formed by any four of these five edos ''except'' for the one formed by omitting 80edo, because 77 & 84e & 87 & 94 reduces to something simpler (77 & 87 & 94), corresponding to 80edo being the only one that doesn't temper out [[385/384]].

:: <nowiki>*</nowiki> There is a more idiosyncratic edo that fits into this scheme as well: [[70edo]] also obeys the intended 13-limit mapping using the patent val, but it's more dubious as a self-contained system due to being better conceptualized as a subset of [[140edo]], so that it's more of a dual-5's dual-7's 17-limit system with high-limit capabilities inherited from 140edo, which also performs well in a wide range of large odd-limits according to the same metric discussed in the section [[#Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness]].

: <nowiki>*</nowiki> There is a more idiosyncratic edo that fits into this scheme as well: [[70edo]] also obeys the intended 13-limit mapping using the patent val, but it's more dubious as a self-contained system due to being better conceptualized as a subset of [[140edo]], so that it's more of a dual-5's dual-7's 17-limit system with high-limit capabilities inherited from 140edo, which also performs well in a wide range of large odd-limits according to the same metric discussed in the section [[#Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness]].

:: <nowiki>**</nowiki> [[84edo]] is peculiar because though using the flat 11 makes some sense in lower limits, in higher limits using the patent val tends to be more performant, so that though one is likely mapping 11/9 as the subneutral third, this mapping may sometimes be inconsistent with the mapping of 11 used when building chords, corresponding to multiple possible tunings of ~8:9:11; the sharp tuning uses 28edo's sharp ~9/8 and ~11/8 while the flat tuning uses the flat ~11/8 and 12edo's major second; interestingly, both of these map 11/9 to the subneutral third, with the trick being that the 28edo rendition uses a sharp ~9/8 that can't be achieved from the 12edo circle of fifths.

: <nowiki>**</nowiki> [[84edo]] is peculiar because though using the flat 11 makes some sense in lower limits, in higher limits using the patent val tends to be more performant, so that though one is likely mapping 11/9 as the subneutral third, this mapping may sometimes be inconsistent with the mapping of 11 used when building chords, corresponding to multiple possible tunings of ~8:9:11; the sharp tuning uses 28edo's sharp ~9/8 and ~11/8 while the flat tuning uses the flat ~11/8 and 12edo's major second; interestingly, both of these map 11/9 to the subneutral third, with the trick being that the 28edo rendition uses a sharp ~9/8 that can't be achieved from the 12edo circle of fifths.

If we also consider the aforediscussed idiosyncratic tuning 70edo, then many more edo joins are possible for describing this rank 4 (= 4-dimensional) 13-limit temperament:

Line 40:

As a reminder then, the remarkable feature is that ''all of these edo joins are equivalent''* and refer to the same 13-limit tempered structure: the one which splits 25/24 into five equal parts of ~144/143 so that 11/9 and 16/13 are made fifth-complements. This structure maps various high-limit things onto the 2.3.7.11 subgroup, and is home to a family of lower-rank temperaments that introduce various extra equivalences, discussed in [[#Fifth-chroma temperaments]].

:: <nowiki>*</nowiki> (other than the ones indicated as specifying something different/more specific)

: <nowiki>*</nowiki> (other than the ones indicated as specifying something different/more specific)

=== Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness ===

Line 48:

The default badness (out-of-tune-ness) measure is a type of Mean-Square-Error applied to the set of intervals given, where the "error" in question is cent error weighted proportional to the square-root of the odd-limit of the interval, so that it can also be thought of as minimizing the square of the cent error with each squared error contribution of a given interval weighted proportional to its odd-limit. The mean-square-error is standard, but the odd-limit weighting needs explanation: a more in-depth discussion of its principles can be found in [[Talk:Marvel#Challenge on optimality of 53edo for FloraC|the discussion page of marvel]] as the discussion also covers various tuning principles User:Godtone believes in, but the basic motivation is that the more complex an interval is, the more precise the tuning needs to be to approximate it; initially this seems extremely harsh, because one would expect that the precision required goes up with the square of the odd-limit, and though this is true for dyads, it needn't be true in practice when constructing harmonies, because the other notes that a tempered interval is placed with quickly eliminate possibilities for what the tempered interval might be approximating in favor of a comparatively small number of possible harmonic series chords, so that even weighting the unsquared error proportional to the odd-limit can give strangely biased results, but clearly more complex intervals need more tuning fidelity* even if we have a relatively forcing harmonic context, so the next best thing was the square-root which also can be interpreted as weighting the squared error proportional the odd-limit, which is how it was originally conceived.

:: <nowiki>*</nowiki> This is not to touch on the debate of what intervals it is aesthetically more ''important'' to get right, where one might consider 3/2 as requiring more precise tuning simply because it's ''more obvious'' when mistuned due to its simplicity; the perspective of the default used by <code>optimal_edo_sequence</code> is that different people often prefer different temperings of simple intervals - and not necessarily the most pure ones - so that its goal is instead trying to maximize psychoacoustic plausibility of the approximations rather than making a judgement on aesthetic, which is something that only the musician (or tuning theorist) can make a judgement on.

: <nowiki>*</nowiki> This is not to touch on the debate of what intervals it is aesthetically more ''important'' to get right, where one might consider 3/2 as requiring more precise tuning simply because it's ''more obvious'' when mistuned due to its simplicity; the perspective of the default used by <code>optimal_edo_sequence</code> is that different people often prefer different temperings of simple intervals - and not necessarily the most pure ones - so that its goal is instead trying to maximize psychoacoustic plausibility of the approximations rather than making a judgement on aesthetic, which is something that only the musician (or tuning theorist) can make a judgement on.

[[Category:Temperament collections]]

@@ Line 6: / Line 6: @@
 * "superneutral", which principally represents ~27/22~16/13
 * "submajor", a general bucket/category for aggregating ratios like 21/17, 26/21, 31/25, 36/29, 41/33, 46/37
-:: <nowiki>*</nowiki> of the edos discussed, only 94edo does not temper out (50/41)/(39/32) = [[1600/1599]] = S40, corresponding to its inconsistently-flat mapping of 25/16; this is noteworthy only because the subneutral and superneutral categories generally are supposed to be unambiguous in their interpretation; an asterisk with 84edo's mapping of prime 11 is discussed later
+: <nowiki>*</nowiki> of the edos discussed, only 94edo does not temper out (50/41)/(39/32) = [[1600/1599]] = S40, corresponding to its inconsistently-flat mapping of 25/16; this is noteworthy only because the subneutral and superneutral categories generally are supposed to be unambiguous in their interpretation; an asterisk with 84edo's mapping of prime 11 is discussed later
 The lists of ratios for supraminor and submajor are not complete and a given edo tuning will not necessarily find contexts where all of these ratios make sense as interpretations, but the principle is that between 6/5 and 11/9 are many ratios that are various mediants of 6/5 and 11/9, and between 16/13 and 5/4 are many ratios that are various mediants of 16/13 and 5/4, so that both general areas represent places where the source of concordance (if any) is not necessarily obvious, so that any mediant therein can potentially be suggested with sufficiently forcing amounts of harmonic context (notes in a chord with approximate frequency ratios suggesting a certain otonal/harmonic series interpretation). Therefore, it represents a flexible melodic category able to represent a wide variety of tempered harmonies contextually.
@@ Line 21: / Line 21: @@
 That is, it represents the shared harmonic and spacing logic between at least* [[77edo]], [[80edo]], [[84edo]]**, [[87edo]] and [[94edo]], so that its join can be formed by any four of these five edos ''except'' for the one formed by omitting 80edo, because 77 & 84e & 87 & 94 reduces to something simpler (77 & 87 & 94), corresponding to 80edo being the only one that doesn't temper out [[385/384]].
-:: <nowiki>*</nowiki> There is a more idiosyncratic edo that fits into this scheme as well: [[70edo]] also obeys the intended 13-limit mapping using the patent val, but it's more dubious as a self-contained system due to being better conceptualized as a subset of [[140edo]], so that it's more of a dual-5's dual-7's 17-limit system with high-limit capabilities inherited from 140edo, which also performs well in a wide range of large odd-limits according to the same metric discussed in the section [[#Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness]].
+: <nowiki>*</nowiki> There is a more idiosyncratic edo that fits into this scheme as well: [[70edo]] also obeys the intended 13-limit mapping using the patent val, but it's more dubious as a self-contained system due to being better conceptualized as a subset of [[140edo]], so that it's more of a dual-5's dual-7's 17-limit system with high-limit capabilities inherited from 140edo, which also performs well in a wide range of large odd-limits according to the same metric discussed in the section [[#Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness]].
-:: <nowiki>**</nowiki> [[84edo]] is peculiar because though using the flat 11 makes some sense in lower limits, in higher limits using the patent val tends to be more performant, so that though one is likely mapping 11/9 as the subneutral third, this mapping may sometimes be inconsistent with the mapping of 11 used when building chords, corresponding to multiple possible tunings of ~8:9:11; the sharp tuning uses 28edo's sharp ~9/8 and ~11/8 while the flat tuning uses the flat ~11/8 and 12edo's major second; interestingly, both of these map 11/9 to the subneutral third, with the trick being that the 28edo rendition uses a sharp ~9/8 that can't be achieved from the 12edo circle of fifths.
+: <nowiki>**</nowiki> [[84edo]] is peculiar because though using the flat 11 makes some sense in lower limits, in higher limits using the patent val tends to be more performant, so that though one is likely mapping 11/9 as the subneutral third, this mapping may sometimes be inconsistent with the mapping of 11 used when building chords, corresponding to multiple possible tunings of ~8:9:11; the sharp tuning uses 28edo's sharp ~9/8 and ~11/8 while the flat tuning uses the flat ~11/8 and 12edo's major second; interestingly, both of these map 11/9 to the subneutral third, with the trick being that the 28edo rendition uses a sharp ~9/8 that can't be achieved from the 12edo circle of fifths.
 If we also consider the aforediscussed idiosyncratic tuning 70edo, then many more edo joins are possible for describing this rank 4 (= 4-dimensional) 13-limit temperament:
@@ Line 40: / Line 40: @@
 As a reminder then, the remarkable feature is that ''all of these edo joins are equivalent''* and refer to the same 13-limit tempered structure: the one which splits 25/24 into five equal parts of ~144/143 so that 11/9 and 16/13 are made fifth-complements. This structure maps various high-limit things onto the 2.3.7.11 subgroup, and is home to a family of lower-rank temperaments that introduce various extra equivalences, discussed in [[#Fifth-chroma temperaments]].
-:: <nowiki>*</nowiki> (other than the ones indicated as specifying something different/more specific)
+: <nowiki>*</nowiki> (other than the ones indicated as specifying something different/more specific)
 === Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness ===
@@ Line 48: / Line 48: @@
 The default badness (out-of-tune-ness) measure is a type of Mean-Square-Error applied to the set of intervals given, where the "error" in question is cent error weighted proportional to the square-root of the odd-limit of the interval, so that it can also be thought of as minimizing the square of the cent error with each squared error contribution of a given interval weighted proportional to its odd-limit. The mean-square-error is standard, but the odd-limit weighting needs explanation: a more in-depth discussion of its principles can be found in [[Talk:Marvel#Challenge on optimality of 53edo for FloraC|the discussion page of marvel]] as the discussion also covers various tuning principles User:Godtone believes in, but the basic motivation is that the more complex an interval is, the more precise the tuning needs to be to approximate it; initially this seems extremely harsh, because one would expect that the precision required goes up with the square of the odd-limit, and though this is true for dyads, it needn't be true in practice when constructing harmonies, because the other notes that a tempered interval is placed with quickly eliminate possibilities for what the tempered interval might be approximating in favor of a comparatively small number of possible harmonic series chords, so that even weighting the unsquared error proportional to the odd-limit can give strangely biased results, but clearly more complex intervals need more tuning fidelity* even if we have a relatively forcing harmonic context, so the next best thing was the square-root which also can be interpreted as weighting the squared error proportional the odd-limit, which is how it was originally conceived.
-:: <nowiki>*</nowiki> This is not to touch on the debate of what intervals it is aesthetically more ''important'' to get right, where one might consider 3/2 as requiring more precise tuning simply because it's ''more obvious'' when mistuned due to its simplicity; the perspective of the default used by <code>optimal_edo_sequence</code> is that different people often prefer different temperings of simple intervals - and not necessarily the most pure ones - so that its goal is instead trying to maximize psychoacoustic plausibility of the approximations rather than making a judgement on aesthetic, which is something that only the musician (or tuning theorist) can make a judgement on.
+: <nowiki>*</nowiki> This is not to touch on the debate of what intervals it is aesthetically more ''important'' to get right, where one might consider 3/2 as requiring more precise tuning simply because it's ''more obvious'' when mistuned due to its simplicity; the perspective of the default used by <code>optimal_edo_sequence</code> is that different people often prefer different temperings of simple intervals - and not necessarily the most pure ones - so that its goal is instead trying to maximize psychoacoustic plausibility of the approximations rather than making a judgement on aesthetic, which is something that only the musician (or tuning theorist) can make a judgement on.
 [[Category:Temperament collections]]