145edo: Difference between revisions

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== Theory ==

{{Nowrap| 145 {{=}} 5 × 29 }}, and 145edo shares the same perfect fifth with [[29edo]]. It is generally a sharp-tending system, with [[prime harmonic]]s 3 to 23 all tuned sharp except for [[7/1|7]], which is slightly flat. It is [[consistent]] to the [[11-odd-limit]], or the no-13 no-15 [[23-odd-limit]], with [[13/7]], [[15/8]] and their [[octave complement]]s being the only intervals going over the line.

{{Nowrap| 145 {{=}} 5 × 29 }}, and 145edo shares the same perfect fifth with [[29edo]]. It is generally a sharp-tending system, with [[prime harmonic]]s 3 to 23 all tuned sharp except for [[7/1|7]], which is slightly flat. It is [[consistent]] to the [[11-odd-limit]], or the no-13 no-15 [[23-odd-limit]], with [[13/7]], [[15/14]], [[15/8]] and their [[octave complement]]s being the only intervals going over the line.

As an equal temperament, 145et [[tempering out|tempers out]] [[1600000/1594323]] in the [[5-limit]]; [[4375/4374]] and [[5120/5103]] in the [[7-limit]]; [[441/440]] and [[896/891]] in the [[11-limit]]; [[196/195]], [[352/351]], [[364/363]], [[676/675]], [[847/845]], and [[1001/1000]] in the [[13-limit]]; [[595/594]] in the [[17-limit]]; [[343/342]] and [[476/475]] in the [[19-limit]].

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=== Octave stretch ===

145edo's approximated harmonics 3, 5, 11, 13, 17, 19, and 23 can be improved at the cost of a little worse 7, and moreover the approximated harmonic 13 can be brought to consistency, if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable. [[375ed6]] is about at the sweet spot for this.

=== Subsets and supersets ===

Line 33:

| 2.3.5

| 1600000/1594323, {{monzo| 28 -3 -10 }}

| {{~~mapping~~| 145 230 337 }}

| {{Mapping| 145 230 337 }}

| -0.695

| 0.498

Line 40:

| 2.3.5.7

| 4375/4374, 5120/5103, 50421/50000

| {{~~mapping~~| 145 230 337 407 }}

| {{Mapping| 145 230 337 407 }}

| -0.472

| 0.578

Line 47:

| 2.3.5.7.11

| 441/440, 896/891, 3388/3375, 4375/4374

| {{~~mapping~~| 145 230 337 407 502 }}

| {{Mapping| 145 230 337 407 502 }}

| -0.561

| 0.547

Line 54:

| 2.3.5.7.11.13

| 196/195, 352/351, 364/363, 676/675, 4375/4374

| {{~~mapping~~| 145 230 337 407 502 537 }}

| {{Mapping| 145 230 337 407 502 537 }}

| -0.630

| 0.522

Line 61:

| 2.3.5.7.11.13.17

| 196/195, 256/255, 352/351, 364/363, 676/675, 1156/1155

| {{~~mapping~~| 145 230 337 407 502 537 593 }}

| {{Mapping| 145 230 337 407 502 537 593 }}

| -0.632

| 0.484

| 5.85

|-

| 2.3.5.7.11.13.17.19

| 196/195, 256/255, 343/342, 352/351, 361/360, 364/363, 476/475

| {{Mapping| 145 230 337 407 502 537 593 616 }}

| -0.565

| 0.486

| 5.87

|-

| 2.3.5.7.11.13.17.19.23

| 196/195, 256/255, 276/275, 352/351, 361/360, 364/363, 460/459, 476/475

| {{Mapping| 145 230 337 407 502 537 593 616 656 }}

| -0.519

| 0.476

| 5.75

|}

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| 339.31

| 128/105

| [[Amity]] / ~~catamite~~

| [[Amity]] / stalagmite

|-

| 5

@@ Line 3: / Line 3: @@
 == Theory ==
-{{Nowrap| 145 {{=}} 5 × 29 }}, and 145edo shares the same perfect fifth with [[29edo]]. It is generally a sharp-tending system, with [[prime harmonic]]s 3 to 23 all tuned sharp except for [[7/1|7]], which is slightly flat. It is [[consistent]] to the [[11-odd-limit]], or the no-13 no-15 [[23-odd-limit]], with [[13/7]], [[15/8]] and their [[octave complement]]s being the only intervals going over the line.
+{{Nowrap| 145 {{=}} 5 × 29 }}, and 145edo shares the same perfect fifth with [[29edo]]. It is generally a sharp-tending system, with [[prime harmonic]]s 3 to 23 all tuned sharp except for [[7/1|7]], which is slightly flat. It is [[consistent]] to the [[11-odd-limit]], or the no-13 no-15 [[23-odd-limit]], with [[13/7]], [[15/14]], [[15/8]] and their [[octave complement]]s being the only intervals going over the line.
 As an equal temperament, 145et [[tempering out|tempers out]] [[1600000/1594323]] in the [[5-limit]]; [[4375/4374]] and [[5120/5103]] in the [[7-limit]]; [[441/440]] and [[896/891]] in the [[11-limit]]; [[196/195]], [[352/351]], [[364/363]], [[676/675]], [[847/845]], and [[1001/1000]] in the [[13-limit]]; [[595/594]] in the [[17-limit]]; [[343/342]] and [[476/475]] in the [[19-limit]].
@@ Line 15: / Line 15: @@
 === Octave stretch ===
 edo's approximated harmonics 3, 5, 11, 13, 17, 19, and 23 can be improved at the cost of a little worse 7, and moreover the approximated harmonic 13 can be brought to consistency, if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable. [[375ed6]] is about at the sweet spot for this.
 === Subsets and supersets ===
@@ Line 33: / Line 33: @@
 | 2.3.5
 | 1600000/1594323, {{monzo| 28 -3 -10 }}
-| {{mapping| 145 230 337 }}
+| {{Mapping| 145 230 337 }}
 | -0.695
 | 0.498
@@ Line 40: / Line 40: @@
 | 2.3.5.7
 | 4375/4374, 5120/5103, 50421/50000
-| {{mapping| 145 230 337 407 }}
+| {{Mapping| 145 230 337 407 }}
 | -0.472
 | 0.578
@@ Line 47: / Line 47: @@
 | 2.3.5.7.11
 | 441/440, 896/891, 3388/3375, 4375/4374
-| {{mapping| 145 230 337 407 502 }}
+| {{Mapping| 145 230 337 407 502 }}
 | -0.561
 | 0.547
@@ Line 54: / Line 54: @@
 | 2.3.5.7.11.13
 | 196/195, 352/351, 364/363, 676/675, 4375/4374
-| {{mapping| 145 230 337 407 502 537 }}
+| {{Mapping| 145 230 337 407 502 537 }}
 | -0.630
 | 0.522
@@ Line 61: / Line 61: @@
 | 2.3.5.7.11.13.17
 | 196/195, 256/255, 352/351, 364/363, 676/675, 1156/1155
-| {{mapping| 145 230 337 407 502 537 593 }}
+| {{Mapping| 145 230 337 407 502 537 593 }}
 | -0.632
 | 0.484
 | 5.85
+|-
+| 2.3.5.7.11.13.17.19
+| 196/195, 256/255, 343/342, 352/351, 361/360, 364/363, 476/475
+| {{Mapping| 145 230 337 407 502 537 593 616 }}
+| -0.565
+| 0.486
+| 5.87
+|-
+| 2.3.5.7.11.13.17.19.23
+| 196/195, 256/255, 276/275, 352/351, 361/360, 364/363, 460/459, 476/475
+| {{Mapping| 145 230 337 407 502 537 593 616 656 }}
+| -0.519
+| 0.476
+| 5.75
 |}
@@ Line 104: / Line 118: @@
 | 339.31
 | 128/105
-| [[Amity]] / catamite
+| [[Amity]] / stalagmite
 |-
 | 5