121edo: Difference between revisions

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== Theory ==

121edo has a distinctly sharp tendency, in that the odd ~~primes~~ from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and 1375/1372 in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. The equal temperament [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

Line 23:

| 2.3

| {{monzo| 192 -121 }}

| [{{~~val~~| 121 192 }}]

| {{mapping| 121 192 }}

| -0.687

| 0.687

Line 30:

| 2.3.5

| 15625/15552, {{monzo| 31 -21 1 }}

| [{{~~val~~| 121 192 281 }}]

| {{mapping| 121 192 281 }}

| -0.524

| 0.606

Line 37:

| 2.3.5.7

| 4000/3969, 6144/6125, 10976/10935

| [{{~~val~~| 121 192 281 340 }}]

| {{mapping| 121 192 281 340 }}

| -0.667

| 0.580

Line 44:

| 2.3.5.7.11

| 540/539, 896/891, 1375/1372, 4375/4356

| [{{~~val~~| 121 192 281 340 419 }}]

| {{mapping| 121 192 281 340 419 }}

| -0.768

| 0.556

Line 51:

| 2.3.5.7.11.13

| 325/324, 352/351, 364/363, 540/539, 625/624

| [{{~~val~~| 121 192 281 340 419 448 }}]

| {{mapping| 121 192 281 340 419 448 }}

| -0.750

| 0.510

Line 58:

| 2.3.5.7.11.13.17

| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441

| [{{~~val~~| 121 192 281 340 419 448 495 }}]

| {{mapping| 121 192 281 340 419 448 495 }}

| -0.787

| 0.480

Line 65:

| 2.3.5.7.11.13.17.19

| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374

| [{{~~val~~| 121 192 281 340 419 448 495 514 }}]

| {{mapping| 121 192 281 340 419 448 495 514 }}

| -0.689

| 0.519

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

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| [[Hendecatonic]]

|}

<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== 13-limit detempering of 121et ==

@@ Line 3: / Line 3: @@
 == Theory ==
-edo has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and 1375/1372 in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].
+edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. The equal temperament [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].
 Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.
@@ Line 23: / Line 23: @@
 | 2.3
 | {{monzo| 192 -121 }}
-| [{{val| 121 192 }}]
+| {{mapping| 121 192 }}
 | -0.687
 | 0.687
@@ Line 30: / Line 30: @@
 | 2.3.5
 | 15625/15552, {{monzo| 31 -21 1 }}
-| [{{val| 121 192 281 }}]
+| {{mapping| 121 192 281 }}
 | -0.524
 | 0.606
@@ Line 37: / Line 37: @@
 | 2.3.5.7
 | 4000/3969, 6144/6125, 10976/10935
-| [{{val| 121 192 281 340 }}]
+| {{mapping| 121 192 281 340 }}
 | -0.667
 | 0.580
@@ Line 44: / Line 44: @@
 | 2.3.5.7.11
 | 540/539, 896/891, 1375/1372, 4375/4356
-| [{{val| 121 192 281 340 419 }}]
+| {{mapping| 121 192 281 340 419 }}
 | -0.768
 | 0.556
@@ Line 51: / Line 51: @@
 | 2.3.5.7.11.13
 | 325/324, 352/351, 364/363, 540/539, 625/624
-| [{{val| 121 192 281 340 419 448 }}]
+| {{mapping| 121 192 281 340 419 448 }}
 | -0.750
 | 0.510
@@ Line 58: / Line 58: @@
 | 2.3.5.7.11.13.17
 | 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
-| [{{val| 121 192 281 340 419 448 495 }}]
+| {{mapping| 121 192 281 340 419 448 495 }}
 | -0.787
 | 0.480
@@ Line 65: / Line 65: @@
 | 2.3.5.7.11.13.17.19
 | 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
-| [{{val| 121 192 281 340 419 448 495 514 }}]
+| {{mapping| 121 192 281 340 419 448 495 514 }}
 | -0.689
 | 0.519
@@ Line 75: / Line 75: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 188: / Line 188: @@
 | [[Hendecatonic]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 == 13-limit detempering of 121et ==