Würschmidt family: Difference between revisions

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2-würschmidt, the temperament with all the same commas as würschmidt but a generator of twice the size, is equivalent to [[skwares]] as a 2.3.7.11 subgroup temperament.

The S-expression-based comma list of the 11-limit würschmidt discussed here is {[[176/175|S8/S10]], [[243/242|S9/S11]], [[225/224|S15]]}. Tempering out [[81/80|S9]] or [[121/120|S11]] results in [[31edo]], and in complementary fashion, tempering out [[64/63|S8]] or [[100/99|S10]] results in [[34edo]], but specifically, the 34d [[val]] where we accept 17edo's mapping of ~7. Their val sum, 31 + 34d = 65d, thus observes all of these [[square superparticular]]s by equating them as S8 = S9 = S10 = S11, hence its S-expression-based comma list is {{nowrap| {[[5120/5103|S8/S9]], [[8019/8000|S9/S10]], [[4000/3993|S10/S11]]} }}, which may be expressed in shortened form as {{nowrap| {S8/9/10/11} }}. (The advantage of this form is we can easily see that all of the [[semiparticular]] commas expected are implied as well as any other commas expressible as the difference between two square superparticular commas by reading them off as ratios like 8/10 (S8/S10) and 9/11 (S9/S11).). As a result, 65edo is especially structurally natural for this temperament, though high damage on the 7 no matter what mapping you use (with the sharp 7 being used for this temperament); even so, it's fairly close to the optimal tuning already if you are fine with a significantly flat ~9/7, which has the advantage of ~14/11 more in tune.

The S-expression-based comma list of the 11-limit würschmidt discussed here is {[[176/175|S8/S10]], [[243/242|S9/S11]], [[225/224|S15]]}. Tempering out [[81/80|S9]] or [[121/120|S11]] results in [[31edo]], and in complementary fashion, tempering out [[64/63|S8]] or [[100/99|S10]] results in [[34edo]], but specifically, the 34d [[val]] where we accept 17edo's mapping of ~7. Their val sum, 31 + 34d = 65d, thus observes all of these [[square superparticular]]s by equating them as S8 = S9 = S10 = S11, hence its S-expression-based comma list is {{nowrap| {[[5120/5103|S8/S9]], [[8019/8000|S9/S10]], [[4000/3993|S10/S11]]} }}, which may be expressed in shortened form as {{nowrap| {S8/9/10/11} }}*. As a result, [[65edo]] is especially structurally natural for this temperament, though high damage on the 7 no matter what mapping you use (with the sharp 7 being used for this temperament); even so, it's fairly close to the optimal tuning already if you are fine with a significantly flat ~9/7, which has the advantage of ~14/11 more in tune. However, as 31edo is relatively in-tune already, 65d + 31 = [[96edo]] is also a reasonable choice, as it has the advantage of being [[patent val]] in the 11-limit, though it uses a different (more accurate) mapping for 13.

(<nowiki>*</nowiki> The advantage of this form is we can easily see that all of the [[semiparticular]] commas expected are implied as well as any other commas expressible as the difference between two square superparticular commas by reading them off as ratios like 8/10 (S8/S10) and 9/11 (S9/S11).)

[[Subgroup]]: 2.3.5.7

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 -würschmidt, the temperament with all the same commas as würschmidt but a generator of twice the size, is equivalent to [[skwares]] as a 2.3.7.11 subgroup temperament.
-The S-expression-based comma list of the 11-limit würschmidt discussed here is {[[176/175|S8/S10]], [[243/242|S9/S11]], [[225/224|S15]]}. Tempering out [[81/80|S9]] or [[121/120|S11]] results in [[31edo]], and in complementary fashion, tempering out [[64/63|S8]] or [[100/99|S10]] results in [[34edo]], but specifically, the 34d [[val]] where we accept 17edo's mapping of ~7. Their val sum, 31 + 34d = 65d, thus observes all of these [[square superparticular]]s by equating them as S8 = S9 = S10 = S11, hence its S-expression-based comma list is {{nowrap| {[[5120/5103|S8/S9]], [[8019/8000|S9/S10]], [[4000/3993|S10/S11]]} }}, which may be expressed in shortened form as {{nowrap| {S8/9/10/11} }}. (The advantage of this form is we can easily see that all of the [[semiparticular]] commas expected are implied as well as any other commas expressible as the difference between two square superparticular commas by reading them off as ratios like 8/10 (S8/S10) and 9/11 (S9/S11).). As a result, 65edo is especially structurally natural for this temperament, though high damage on the 7 no matter what mapping you use (with the sharp 7 being used for this temperament); even so, it's fairly close to the optimal tuning already if you are fine with a significantly flat ~9/7, which has the advantage of ~14/11 more in tune.
+The S-expression-based comma list of the 11-limit würschmidt discussed here is {[[176/175|S8/S10]], [[243/242|S9/S11]], [[225/224|S15]]}. Tempering out [[81/80|S9]] or [[121/120|S11]] results in [[31edo]], and in complementary fashion, tempering out [[64/63|S8]] or [[100/99|S10]] results in [[34edo]], but specifically, the 34d [[val]] where we accept 17edo's mapping of ~7. Their val sum, 31 + 34d = 65d, thus observes all of these [[square superparticular]]s by equating them as S8 = S9 = S10 = S11, hence its S-expression-based comma list is {{nowrap| {[[5120/5103|S8/S9]], [[8019/8000|S9/S10]], [[4000/3993|S10/S11]]} }}, which may be expressed in shortened form as {{nowrap| {S8/9/10/11} }}*. As a result, [[65edo]] is especially structurally natural for this temperament, though high damage on the 7 no matter what mapping you use (with the sharp 7 being used for this temperament); even so, it's fairly close to the optimal tuning already if you are fine with a significantly flat ~9/7, which has the advantage of ~14/11 more in tune. However, as 31edo is relatively in-tune already, 65d + 31 = [[96edo]] is also a reasonable choice, as it has the advantage of being [[patent val]] in the 11-limit, though it uses a different (more accurate) mapping for 13.
+(<nowiki>*</nowiki> The advantage of this form is we can easily see that all of the [[semiparticular]] commas expected are implied as well as any other commas expressible as the difference between two square superparticular commas by reading them off as ratios like 8/10 (S8/S10) and 9/11 (S9/S11).)
 [[Subgroup]]: 2.3.5.7