Xenharmonic Wiki

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== Introduction ==
Yer is an octave-reduced EFG ([[wikipedia:Euler–Fokker_genus|Euler-Fokker genus]]) of 11, 13, 17, and 19. As such it is a 2.11.13.17.19 limit just intonation tuning (ignoring the 3rd, 5th, and 7th harmonics) which is octave-repeating.


Yer is an octave-reduced [[wikipedia:Euler–Fokker_genus|Euler-Fokker genus]] of 11, 13, 17, and 19. As such it is a 2.11.13.17.19 limit just intonation tuning (ignoring the 3rd, 5th, and 7th harmonics) which is octave-repeating.
Yer has 16 pitches from the EFG, and 2 additional pitches are optional to include; these pitches are very close to other pitches already in the system (off by comma-sized amounts) and are useful as inflections or points to comma-shift on.  


{| class="wikitable"
Yer was developed by [[Douglas Blumeyer|Douglas Blumeyer]] as a system to explore the [[unnoticeable comma]] 2432/2431, the Blumeyer comma, which appears twice as a step in the scale.
|+
 
!name
== The scale ==
{| class="wikitable center-all"  
|+ style="font-size: 105%" | Steps of Yer
!Name
!11
!13
!17
!19
!EFG composition
!EFG composition
!frequency multiplier (definition)
!Frequency multiplier (definition)
!frequency multiplier (decimal)
!Frequency multiplier (decimal)
!pitch (¢)
!Pitch (¢)
!HEWM (on C)
!EHEJIPN (on C)
|-
|-
|a
|a
|
|
|
|
|1
|1
|1/1
|1/1
Line 20: Line 31:
|- style="color: #bbbbbb;"
|- style="color: #bbbbbb;"
|(q)
|(q)
| style="background-color: #bbbbbb;" |
| style="background-color: #bbbbbb;" |
| style="background-color: #bbbbbb;" |
| style="background-color: #bbbbbb;" |
|11⋅11⋅17
|11⋅11⋅17
|2057/2048
|2057/2048
Line 27: Line 42:
|-
|-
|o
|o
|
| style="background-color: #FF8000;" |
| style="background-color: #00C0C0;" |
| style="background-color: #FF0080;" |
|13⋅17⋅19
|13⋅17⋅19
|4199/4096
|4199/4096
Line 34: Line 53:
|-
|-
|d
|d
|
|
| style="background-color: #00C0C0;" |
|
|17
|17
|17/16
|17/16
Line 41: Line 64:
|-
|-
|f
|f
| style="background-color: #60C060;" |
| style="background-color: #FF8000;" |
|
|
|11⋅13
|11⋅13
|143/128
|143/128
Line 48: Line 75:
|-
|-
|l
|l
| style="background-color: #60C060;" |
| style="background-color: #FF8000;" |
| style="background-color: #00C0C0;" |
|
|11⋅13⋅17
|11⋅13⋅17
|2431/2048
|2431/2048
Line 55: Line 86:
|-
|-
|e
|e
|
|
|
| style="background-color: #FF0080;" |
|19
|19
|19/16
|19/16
Line 62: Line 97:
|-
|-
|k
|k
|
|
| style="background-color: #00C0C0;" |
| style="background-color: #FF0080;" |
|17⋅19
|17⋅19
|323/256
|323/256
Line 69: Line 108:
|-
|-
|m
|m
| style="background-color: #60C060;" |
| style="background-color: #FF8000;" |
|
| style="background-color: #FF0080;" |
|11⋅13⋅19
|11⋅13⋅19
|2717/2048
|2717/2048
Line 76: Line 119:
|-
|-
|b
|b
| style="background-color: #60C060;" |
|
|
|
|11
|11
|11/8
|11/8
Line 83: Line 130:
|- style="color: #bbbbbb;"
|- style="color: #bbbbbb;"
|(r)
|(r)
| style="background-color: #bbbbbb;" |
| style="background-color: #bbbbbb;" |
| style="background-color: #bbbbbb;" |
| style="background-color: #bbbbbb;" |
|(13⋅19)/11
|(13⋅19)/11
|247/176
|247/176
Line 90: Line 141:
|-
|-
|p
|p
| style="background-color: #60C060;" |
| style="background-color: #FF8000;" |
| style="background-color: #00C0C0;" |
| style="background-color: #FF0080;" |
|11⋅13⋅17⋅19
|11⋅13⋅17⋅19
|46189/32768
|46189/32768
Line 97: Line 152:
|-
|-
|g
|g
| style="background-color: #60C060;" |
|
| style="background-color: #00C0C0;" |
|
|11⋅17
|11⋅17
|187/128
|187/128
Line 104: Line 163:
|-
|-
|c
|c
|
| style="background-color: #FF8000;" |
|
|
|13
|13
|13/8
|13/8
Line 111: Line 174:
|-
|-
|h
|h
| style="background-color: #60C060;" |
|
|
| style="background-color: #FF0080;" |
|11⋅19
|11⋅19
|209/128
|209/128
Line 118: Line 185:
|-
|-
|i
|i
|
| style="background-color: #FF8000;" |
| style="background-color: #00C0C0;" |
|
|13⋅17
|13⋅17
|221/128
|221/128
Line 125: Line 196:
|-
|-
|n
|n
| style="background-color: #60C060;" |
|
| style="background-color: #00C0C0;" |
| style="background-color: #FF0080;" |
|11⋅17⋅19
|11⋅17⋅19
|3553/2048
|3553/2048
Line 132: Line 207:
|-
|-
|j
|j
|
| style="background-color: #FF8000;" |
|
| style="background-color: #FF0080;" |
|13⋅19
|13⋅19
|247/128
|247/128
Line 139: Line 218:
|}
|}


Throughout this post, the same four colors will be consistently associated with the four harmonics: green is 11, orange is 13, cyan is 17, and magenta is 19.
The diagrams throughout this wiki entry were designed by Douglas Blumeyer. The same four colors are consistently associated with the four harmonics: green is 11, orange is 13, cyan is 17, and magenta is 19.


[[File:Yer - cycle view.jpg|thumb|
[[File:Yer - cycle view.jpg|thumb|
Line 148: Line 227:
|none]]
|none]]


It has a minor third of 19:16 and a minor second 17:16.
Yer features a minor third of 19:16 and a minor second 17:16.


== EFG as combination of CPS's ==
== EFG as combination of CPS's ==
Line 160: Line 239:


== Blumeyer comma ==
== Blumeyer comma ==
Those pitches right on top of each other are a feature, not a bug. The pitch 11 * 13 * 17 is 2431, only one off from the pitch 19, which when octave adjusted is 2432. So we end up with this superparticular ratio, 2432:2431, called the Blumeyer comma.
Those pitches right on top of each other are a feature, not a bug. The pitch 11 * 13 * 17 is 2431, only one off from the pitch 19, which when octave adjusted is 2432. So we end up with this superparticular ratio, 2432:2431, called the Blumeyer comma.


{| class="wikitable"
{| class="wikitable"
|+
|+
!name
!Name
!value
!Value
!cents
!Cents
!monzo
!Prime exponent vector
|-
|-
|Blumeyer comma
|Blumeyer comma
|2432/2431
|[[2432/2431]]
|0.71200249782
|0.71200249782
|<nowiki>| 7 0 0 0 -1 -1 -1 1 ></nowiki>
|<nowiki>[ 7 0 0 0 -1 -1 -1 1 ></nowiki>
|}
|}
[[File:Blumeyer comma.png|none|thumb|
[[File:Blumeyer comma.png|none|thumb|
Blumeyer comma visualized
Blumeyer comma visualized
]]
]]Perhaps, given its tiny size, it would be better called the Blumeyer schismina.


== Lattice Play ==
== Lattice Play ==
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]]
]]


== Blume comma ==
== Yama comma ==
The reason why 13 and 11 * 19 are so close is that it turns out there is a *another* comma existing in this world of 11’s, 13’s, 17’s, and 19’s — not nearly as exciting or colorful of one, but it’s there. If we move by an eleventh, a seventeenth, and another eleventh, we end up right back where you started, off by 7.591 cents. 13 and 11 * 19 are off from each other by an amount of one Blumeyer comma and one Blume comma. These two commas are just both so small that it doesn’t much matter. 7.591 + 0.712 is still just 8.303 cents.
The reason why 13 and 11 * 19 are so close is that it turns out there is *another* comma existing as a step in this tuning of 11’s, 13’s, 17’s, and 19’s. If we move by an eleventh, then a nineteenth, and then down a thirteenth, we end up right back where we started, off by 8.303 cents. That's the yama comma.
{| class="wikitable"
{| class="wikitable"
|+
|+
!name
!Name
!value
!Value
!cents
!Cents
!monzo
!Prime exponent vector
|-
|Yama comma
|[[209/208]]
|8.303296728
|<nowiki>[ -4 0 0 0 1 -1 0 1 ></nowiki>
|}
Another consequence of the yama comma is that a couple “extra” lattice connections appear. These are drawn in dotted green on the lattices in the introduction.
If we move by an eleventh, a seventeenth, and another eleventh, we end up right back where you started, off by 7.591 cents, or in other words, one yama comma minus one Blumeyer comma. We call this the Blume comma.
{| class="wikitable"
|+
!Name
!Value
!Cents
!Prime exponent vector
|-
|-
|Blume comma
|Blume comma
|2057/2048
|[[2057/2048]]
|7.59129422992
|7.59129422992
|<nowiki>| -11 0 0 0 2 0 1 ></nowiki>
|<nowiki>[ -11 0 0 0 2 0 1 ></nowiki>
|}
|}
[[File:Blume comma.png|none|thumb|
[[File:Blume comma.png|none|thumb|
Blume comma visualized
Blume comma visualized
]]
]]
Another consequence of the Blume comma is that a couple “extra” lattice connections appear. These are drawn in dotted green on the lattices in the introduction.


The idea is that if you were to try to go from 11 * 17 by an eleventh to 11 * 11 * 17, well, you’re not really allowed to do that because the EFG does not duplicate factors (you can’t have two 11’s), but since 11 * 11 * 17 is essentially 1, we’ll permit it.  
The idea is that if you were to try to go from 11 * 17 by an eleventh to 11 * 11 * 17, well, you’re not really allowed to do that because the EFG does not duplicate factors (you can’t have two 11’s), but since 11 * 11 * 17 is essentially 1, we’ll permit it.  
Line 223: Line 313:
And as long as we're changing the angle we look on the cube to bring the right pairs of pitches together, we have also taken care to balance these dotted lines with the real 11 lines, so that it’s the zig to the zag of the real 11, more strongly suggesting the dimension of the 11th harmonic’s relationship to that of the 17th (i.e that two 11's make a 17).
And as long as we're changing the angle we look on the cube to bring the right pairs of pitches together, we have also taken care to balance these dotted lines with the real 11 lines, so that it’s the zig to the zag of the real 11, more strongly suggesting the dimension of the 11th harmonic’s relationship to that of the 17th (i.e that two 11's make a 17).


Now we could have drawn dotted lines connecting 11 * 13 * 17 to 13, but declined,  
Now we could have drawn dotted lines connecting 11 * 13 * 17 to 13, but declined, considering that the ability to move between these two pitches is already achievable by modulating from from 11 * 13 * 17 to 19, then moving by that 11 to 11 * 19, then modulating to 13. The same goes for the connection between 11 * 17 * 19 and 19. This would be pretty obvious on the cycle view, because we’d just be drawing a dotted line right alongside an existing solid one.
considering that the ability to move between these two pitches is already achievable  
by modulating from from 11 * 13 * 17 to 19, then moving by that 11 to 11 * 19, then modulating to 13. The same goes for the connection between 11 * 17 * 19 and 19. This would be pretty obvious on the cycle view, because we’d just be drawing a dotted line right alongside an existing solid one.


Here’s something else interesting: we can move by four 11’s in a row. We can move from 17 * 19 to 11 * 17 * 19, which can be shifted to 13 * 17, then move to 11 * 13 * 17, which can be shifted to 19, then move to 11 * 19, which can be shifted to 13, then move to 11 * 13.
Here’s something else interesting: we can move by four 11’s in a row. We can move from 17 * 19 to 11 * 17 * 19, which can be shifted to 13 * 17, then move to 11 * 13 * 17, which can be shifted to 19, then move to 11 * 19, which can be shifted to 13, then move to 11 * 13.
Line 232: Line 320:


== Comma pumps ==  
== Comma pumps ==  
Yer is pure JI, but due to the five places where it boasts two pitches very close together but with very different harmonic compositions, it can achieve [[Zero comma pump|zero comma pumps]] by “comma shifting” at those key points, returning to exactly their original pitch.
Yer is pure JI, but due to the five places where it boasts two pitches very close together but with very different harmonic compositions, it can achieve [[Zero comma pump|zero comma pumps]] by “comma shifting” at those key points, returning to exactly their original pitch.


Line 310: Line 397:


...you can actually begin to see stuff. In the lattice view on the left, notice how the tetranies take form as tetrahedrons and the hexany as an octahedron. In the cyclical view on the right, you can see how the choose-1 and choose-3 tetranies, the blue and green shapes, are reflections of each other across a vertical line drawn down this circle, as well as how the choose 2 hexany is symmetrical across the same line, and the remaining two notes -  the unison and the aota - are located symmetrically to each other. You could write music with just these CPSs.  
...you can actually begin to see stuff. In the lattice view on the left, notice how the tetranies take form as tetrahedrons and the hexany as an octahedron. In the cyclical view on the right, you can see how the choose-1 and choose-3 tetranies, the blue and green shapes, are reflections of each other across a vertical line drawn down this circle, as well as how the choose 2 hexany is symmetrical across the same line, and the remaining two notes -  the unison and the aota - are located symmetrically to each other. You could write music with just these CPSs.  
== Intervals ==
{| class="wikitable"
|+Yer interval class matrix
!
|'''1'''
|'''13⋅17⋅19'''
|'''17'''
|'''11⋅13'''
|'''11⋅13⋅17'''
|'''19'''
|'''17⋅19'''
|'''11⋅13⋅19'''
|'''11'''
|'''11⋅13⋅17⋅19'''
|'''11⋅17'''
|'''13'''
|'''11⋅19'''
|'''13⋅17'''
|'''11⋅17⋅19'''
|'''13⋅19'''
|-
|'''1'''
|0
|42.9960874
|104.9554095
|191.8456041
|296.8010136
|297.5130161
|402.4684256
|489.3586203
|551.3179424
|594.3140298
|543.7266481
|359.4723382
|351.1690415
|254.5169287
|246.213632
|61.9593221
|-
|'''13⋅17⋅19'''
|
|0
|61.9593221
|148.8495167
|253.8049262
|254.5169287
|359.4723382
|446.3625329
|508.321855
|551.3179424
|586.7227355
|402.4684256
|394.1651289
|297.5130161
|289.2097194
|104.9554095
|-
|'''17'''
|
|
|0
|86.89019463
|191.8456041
|192.5576066
|297.5130161
|384.4032108
|446.3625329
|489.3586203
|551.3179424
|464.4277477
|456.124451
|359.4723382
|351.1690415
|166.9147316
|-
|'''11⋅13'''
|
|
|
|0
|104.9554095
|105.667412
|210.6228215
|297.5130161
|359.4723382
|402.4684256
|464.4277477
|551.3179424
|543.0146456
|446.3625329
|438.0592361
|253.8049262
|-
|'''11⋅13⋅17'''
|
|
|
|
|0
|0.7120024978
|105.667412
|192.5576066
|254.5169287
|297.5130161
|359.4723382
|543.7266481
|552.0299449
|551.3179424
|543.0146456
|358.7603357
|-
|'''19'''
|
|
|
|
|
|0
|104.9554095
|191.8456041
|253.8049262
|296.8010136
|358.7603357
|543.0146456
|551.3179424
|552.0299449
|543.7266481
|359.4723382
|-
|'''17⋅19'''
|
|
|
|
|
|
|0
|86.89019463
|148.8495167
|191.8456041
|253.8049262
|438.0592361
|446.3625329
|543.0146456
|551.3179424
|464.4277477
|-
|'''11⋅13⋅19'''
|
|
|
|
|
|
|
|0
|61.9593221
|104.9554095
|166.9147316
|351.1690415
|359.4723382
|456.124451
|464.4277477
|551.3179424
|-
|'''11'''
|
|
|
|
|
|
|
|
|0
|42.9960874
|104.9554095
|289.2097194
|297.5130161
|394.1651289
|402.4684256
|586.7227355
|-
|'''11⋅13⋅17⋅19'''
|
|
|
|
|
|
|
|
|
|0
|61.9593221
|246.213632
|254.5169287
|351.1690415
|359.4723382
|543.7266481
|-
|'''11⋅17'''
|
|
|
|
|
|
|
|
|
|
|0
|184.2543099
|192.5576066
|289.2097194
|297.5130161
|481.767326
|-
|'''13'''
|
|
|
|
|
|
|
|
|
|
|
|0
|8.303296728
|104.9554095
|113.2587062
|297.5130161
|-
|'''11⋅19'''
|
|
|
|
|
|
|
|
|
|
|
|
|0
|96.65211277
|104.9554095
|289.2097194
|-
|'''13⋅17'''
|
|
|
|
|
|
|
|
|
|
|
|
|
|0
|8.303296728
|192.5576066
|-
|'''11⋅17⋅19'''
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|0
|184.2543099
|-
|'''13⋅19'''
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|0
|}
{| class="wikitable"
|+Yer interval classes
!Ratio
!Type
!Count
!Prime exponent vector
!Cents
!Gap between available interval classes
!Lower limit interval to nearest in Yer
!Cents difference up
!Cents difference down (to prove how they're right in between)
!Inverse ratio
!Inverse cents
|-
|2431:2432
|3v1
|1
|<nowiki>[ 7 0 0 0 -1 -1 -1 1 ></nowiki>
|0.7120024978
|0.7120024978
|
|
|
|
|1199.287998
|-
|208:209
|2v1
|2
|13 -> 11 19
|8.303296728
|7.59129423
|
|
|
|
|1191.696703
|-
|4096:4199
|3v0
|2
|1 -> 13 17 19
|42.9960874
|34.69279067
|48:49
|35.69681207 -
|35.69681207 -
|
|1157.003913
|-
|247:256
|2v0
|4
|13 19
|61.9593221
|18.9632347
|24:25
|70.67242686 -
|70.67242686 -
|
|1138.040678
|-
|136:143
|2v1
|2
|17 : 11 13
|86.89019463
|24.93087253
|20:21
|84.46719347 -
|84.46719347 -
|
|1113.109805
|-
|209:221
|2v2
|1
|11 19 : 13 17
|96.65211277
|9.76191814
|
|
|
|
|1103.347887
|-
|16:17
|1v0
|8
|16:17
|104.9554095
|8.30329673
|
|
|
|
|1095.044591
|-
|143:152
|2v1
|2
|11 13 : 19
|105.667412
|0.7120025
|
|
|
|
|1094.332588
|-
|3328:3553
|3v1
|1
|13 : 11 17 19
|113.2587062
|7.5912942
|15:16, 15:14
|111.7312853 -, 119.4428083 -
|111.7312853 -, 119.4428083 -
|
|1086.741294
|-
|323:352
|2v1
|2
|11 : 17 19
|148.8495167
|35.5908105
|32:35
|155.1396203 -
|155.1396203 -
|
|1051.150483
|-
|247:272
|2v1
|2
|17 : 13 19
|166.9147316
|18.0652149
|
|
|
|
|1033.085268
|-
|187:208
|2v1
|2
|13 : 11 17
|184.2543099
|17.3395783
|9:10
|182.4037121 -
|182.4037121 -
|
|1015.74569
|-
|128:143
|2v0
|4
|1 : 11 13
|191.8456041
|7.5912942
|
|
|
|
|1008.154396
|-
|17:19
|1v1
|4
|17:19
|192.5576066
|0.7120025
|25:28
|196.1984787 -
|196.1984787 -
|
|1007.442393
|-
|286:323
|2v2
|1
|11 13 : 17 19
|210.6228215
|18.0652149
|8:9
|203.9100017 -
|203.9100017 -
|
|989.3771785
|-
|3553:4096
|3v0
|2
|1 : 11 17 19
|246.213632
|35.5908105
|7:8
|231.174094 -
|231.174094 -
|
|953.786368
|-
|19:22
|1v1
|4
|19 : 11
|253.8049262
|7.5912942
|
|
|
|
|946.1950738
|-
|221:256
|2v0
|4
|13 17
|254.5169287
|0.7120025
|7:6
|266.870906 -
|266.870906 -
|
|945.4830713
|-
|11:13
|1v1
|4
|11:13
|289.2097194
|34.6927907
|
|
|
|
|910.7902806
|-
|2048:2431
|3v0
|2
|11 13 17
|296.8010136
|7.5912942
|
|
|
|
|903.1989864
|-
|16:19
|1v0
|8
|
|297.5130161
|0.7120025
|5:6
|315.641287 -
|315.641287 -
|
|902.4869839
|-
|209:256
|2v0
|4
|11 19
|351.1690415
|53.6560254
|40:49
|351.3380991 -
|351.3380991 -
|
|848.8309585
|-
|152:187
|2v1
|2
|11 17 : 19
|358.7603357
|7.5912942
|
|
|
|
|841.2396643
|-
|13:16
|1v0
|8
|13:16
|359.4723382
|0.7120025
|
|
|
|
|840.5276618
|-
|2176:2717
|3v1
|1
|(11*13*19)/(17*2^7)
|384.4032108
|24.9308726
|4:5
|386.313714 -
|386.313714 -
|
|815.5967892
|-
|176:221
|2v1
|2
|11 : 13 17
|394.1651289
|9.7619181
|
|
|
|
|805.8348711
|-
|256:323
|2v0
|4
|17 19
|402.4684256
|8.3032967
|
|
|
|
|797.5315744
|-
|323:416
|3v0
|2
|17 19 13
|438.0592361
|35.5908105
|7:9, 25:32
|435.0840953 -, 427.3725723 -
|435.0840953 -, 427.3725723 -
|
|761.9407639
|-
|17:22
|1v1
|4
|17 11
|446.3625329
|8.3032968
|
|
|
|
|753.6374671
|-
|209:272
|2v1
|2
|17 : 11 19
|456.124451
|9.7619181
|
|
|
|
|743.875549
|-
|13:17
|1v1
|4
|
|464.4277477
|8.3032967
|16:21, 49:64
|470.7809073 -, 462.3481871-
|470.7809073 -, 462.3481871-
|
|735.5722523
|-
|187:247
|2v2
|1
|11 17 : 13 19
|481.767326
|17.3395783
|
|
|
|
|718.232674
|-
|2048:2717
|2v1
|2
|11 13 19
|489.3586203
|7.5912943
|3:4
|498.044999 -
|498.044999 -
|
|710.6413797
|-
|4199:5632
|3v1
|1
|(11*2^9)/(13*17*19)
|508.321855
|18.9632347
|
|
|
|
|691.678145
|-
|19:26
|1v1
|4
|19:26
|543.0146456
|34.6927906
|
|
|
|
|656.9853544
|-
|187:256
|2v0
|4
|11 17
|543.7266481
|0.7120025
|35:48
|546.8153805 -
|546.8153805 -
|
|656.2733519
|-
|8:11
|1v0
|8
|8:11
|551.3179424
|7.5912943
|
|
|
|
|648.6820576
|-
|221:304
|2v1
|2
|13 17 : 19
|552.0299449
|0.7120025
|
|
|
|
|647.9700551
|-
|176:247
|2v1
|2
|13 19
|586.7227355
|34.6927906
|5:7
|582.512193 -
|582.512193 -
|
|613.2772645
|-
|32768:46189
|4v0
|1
|<nowiki>[ 15 0 0 0 1 1 1 1 ></nowiki>
|594.3140298
|7.5912943
|
|
|
|
|605.6859702
|}


== Scala file ==
== Scala file ==
 
<pre>
   ! blumeyer_ji.scl
   ! yer.scl
   !
   !
   Blumeyer JI scale, combination of CPS's of 11, 13, 17, 19
   Yer, EFG of 11, 13, 17, 19
  16
  16
   !
   !
Line 334: Line 1,225:
  247/128
  247/128
  2/1
  2/1
 
</pre>
== Video explanation ==
== Video explanation ==
More details are presented in this video: [https://vimeo.com/184148765 Yer (pitch system)]
More details are presented in this video: [https://vimeo.com/184148765 Yer (pitch system)]


== Listening ==
== Listening ==
* [https://soundcloud.com/cmloegcmluin/tsraxcfaubdj Douglas Blumeyer - Tsraxcfaubdj]
* [https://soundcloud.com/cmloegcmluin/blumeyer-comma-ji-unpump Douglas Blumeyer - Blumeyer Comma JI Unpump]
* [https://chrisvaisvil.com/the-figment-for-woodwinds-muted-strings-and-choir-in-just-intonation-tuning Chris Vaisvil - The Figment]
* [http://chrisvaisvil.com/now-yer-talkin-ji-piano Chris Vaisvil - Now Yer Talkin']
== Tritave-Based Yer ==
One could imagine a tritave-repeating variation of Yer, where 2's are out but 3's are in (the EFG of 11, 13, 17, 19 remains but is tritave-reduced instead).
== Yer as a temperament ==
If you [[temper out]] only the Blumeyer comma, you get this 2.11.13.17.19 [[subgroup]] [[mapping]], which naturally should be called "Blumeyer [[regular temperament|temperament]]":
[ ⟨ 1 0 0 0 -7 ]
⟨ 0 1 0 0 1 ]
⟨ 0 0 1 0 1 ]
⟨ 0 0 0 1 1 ] ⟩
Expressed as a [[map-merging|map-merge]] of [[ET]]s, that's 13&amp;113&amp;137&amp;194. So one ~19 is up one each of the ~11, ~13, and ~17 here.
Now if you temper out the Blumeyer comma and the yama comma (and therefore also the blume comma; in fact, the [[canonical form]] of the [[comma basis]] appears to be blume and yama, with the Blumeyer comma being a linear combination of them), then you get a [[rank-3 temperament]], with mapping:
[ ⟨ 1 0 0 11 4 ]
⟨ 0 1 0 -2 -1 ]
⟨ 0 0 1 0 1 ] ⟩
Again that's still in the 2.11.13.17.19 subgroup. And this one is the map-merge of 13&24&33, with [[generators]] 1200.4457, 4149.8305, 4442.6991 (that 1st generator modulo the [[period]] is 548.4934 and the 2nd generator is 841.362). So one ~17 is down two ~11, and one ~19 is down one ~11 and up a ~13. Cool! This you'd call "Yer temperament" then. It has been registered here: [[No-threes subgroup temperaments#Yer (rank 3)]]


[https://soundcloud.com/cmloegcmluin/tsraxcfaubdj Douglas Blumeyer - Tsraxcfaubdj]
A 13-note scale with only three scale step sizes is possible here. It's 7L 4M 2s, with the pattern LMLLMLsLMLMLs, where L ≈ 103.4¢, M ≈ 85.9¢, and s ≈ 66.2¢:


[https://soundcloud.com/cmloegcmluin/blumeyer-comma-ji-unpump Douglas Blumeyer - Blumeyer Comma JI Unpump]
<pre>
  ! yerMV3.scl
  !
  yer temperament 13-note scale
13
  !
103.4
189.3
292.7
396.1
482.0
585.4
651.7
755.1
841.0
944.4
1030.3
1133.7
(2/1)
</pre>


[https://chrisvaisvil.com/the-figment-for-woodwinds-muted-strings-and-choir-in-just-intonation-tuning Chris Vaisvil - The Figment]
However this is not quite an [[MV3]] scale. It is a rotation of the "LLMLsLMLMLsLM" pattern which is identified [[User:Xenoindex/Tridecatonic_MV3|here]] as "Conditional (MM=Ls)", and while it's quite close, L + s = 103.4 + 66.2 = 169.6 but 169.6/2 = 84.8, not 85.9 which is the M here. 


[http://chrisvaisvil.com/now-yer-talkin-ji-piano Chris Vaisvil - Now Yer Talkin']
The lattice for this scale looks like:
 
{| class="wikitable"
|+
| -
|(-2,2)
 
11·13·17·19
 
13·19/11
 
585.4¢
|(-1,2)
13·19
 
 
 
1133.7¢
|(0,2)
11·13·19
 
 
 
482.0¢
|(1,2)
11·13²
 
 
 
1030.3¢
|-
|(-3,1)
17·19
 
 
 
396.1¢
|(-2,1)
13·17
 
11·17·19
 
944.4¢
|(-1,1)
11·13·17
 
19
 
292.7¢
|(0,1)
13
 
11·19
 
841.0¢
|(1,1)
11·13
 
 
 
189.3¢
|-
|(-3,0)
17·19/13
 
17/11
 
755.1¢
|(-2,0)
17
 
 
 
103.4¢
|(-1,0)
11·17
 
 
 
651.7¢
|(0,0)
1
 
11²·17
 
0¢
| -
|}


== See also ==
== See also ==
[https://musical-patterns.douglasblumeyer.com Musical Patterns] - select Tsraxcfaubdj (GitHub page is here to explore the programmatic reasoning behind how this piece uses Yer: https://github.com/MusicalPatterns/pattern-tsraxcfaubdj)
* [https://musical-patterns.douglasblumeyer.com Musical Patterns] - select Tsraxcfaubdj (GitHub page is here to explore the programmatic reasoning behind how this piece uses Yer: https://github.com/MusicalPatterns/pattern-tsraxcfaubdj)
* [[Hilim13]], another tridecatonic tuning in the [[2.11.13.17.19 subgroup]], [[Just intonation|JI]], designed by [[Gene Ward Smith]]
* [[Gjaeck]], another tuning in the [[2.11.13.17.19 subgroup]], a tempered [[MOS scale|MOS]] of [[57edo|57ed2]], also designed by [[Douglas Blumeyer]]
 
[[Category:Euler-Fokker genera]]
[[Category:Tables]]
[[Category:19-limit]]
[[Category:Just intonation scales]]
[[Category:Blumeyer]]
[[Category:Combination product sets]]
[[Category:Comma pump]]
Retrieved from "https://en.xen.wiki/w/Yer"