- The first list contains ratios with an accurate representation in the Olympian set
- The second list contains other important or potentially useful ratios. Since there are no corresponding accidentals in the Olympian set, the list contains slightly inaccurate accidentals that usually represent different ratios.
When working with temperament notations, it's important to use accidentals that work with any possible tuning for the temperament. So in that case, it is strongly recommended to use accurate accidentals from the first list. The inaccurate ratios from the second list should only be considered if it is clear which ratio they are refering to.
This post is work-in-progress, and currently incomplete. Here is my todo list:
- Add systematic comma names
- Maybe extend the second list
odd-limit ratio monzo cents sys. interval name accidentals (up / down)
-------------------------------------------------------------------------------------------------------------------------
9 9/8 [-3 2> 203.91 major tone ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
.X\ & 'Y/
15 16/15 [4 -1 -1> 111.73 5-limit diat. semitone ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
./||\ & '\!!/
15/14 [-1 1 1 -1> 119.44 sept. diat. semitone
|||( & !!!(
21 21/20 [-2 1 -1 1> 84.47 sept. chrom. semitone ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
.||) & '!!)
25 25/24 [-3 -1 2> 70.67 5-limit chrom. semitone
)||( & )!!(
27 28/27 [2 -3 0 1> 62.96 7L sept. third-tone ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
.(|\ & '(!/
27/25 [0 3 -2> 133.24 large limma ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
./||| & '\!!!
35 36/35 [2 2 -1 -1> 48.77 35M sept. quarter-tone
/|) & \!)
35/32 [-5 0 1 1> 155.14 sept. neutral second ![bo :.:](./images/smilies/bo.png)
![sharp phai or sharp fai :/ /|||:](./images/smilies/sharp_phai.png)
![bi :':](./images/smilies/bi.png)
![flat phao or flat fao :\ \!!!:](./images/smilies/flat_phao.png)
.//|||' & '\\!!!.
49 50/49 [1 0 2 -2> 34.98 25:49S jubilisma ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'(| & .(!
49/48 [-4 -1 0 2> 35.70 slendro diesis
~|) & ~!)
54/49 [1 3 0 -2> 168.21 mujannab
(/||| & (\!!!
63 64/63 [6 -2 0 -1> 27.26 7C Archytas comma
|) & !)
81 81/80 [-4 4 -1> 21.51 5C syntonic comma
/| & \!
125 126/125 [1 2 -3 1> 13.79 7:125C starling comma ![sanai :~|(:](./images/smilies/sanai.png)
![bo :.:](./images/smilies/bo.png)
![sanao :~!(:](./images/smilies/sanao.png)
![bi :':](./images/smilies/bi.png)
~|(.. & ~!(''
128/125 [7 0 -3> 41.06 125S augmented comma ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
.//| & '\\!
135 135/128 [-7 3 1> 92.18 pelogic comma
||\ & !!/
175 192/175 [6 1 -2 -1> 160.50 - ![sharp phrai or sharp frai :)/ /|||:](./images/smilies/sharp_phrai.png)
![flat phrao or flat frao :)\ \!!!:](./images/smilies/flat_phrao.png)
)//|||' & )\\!!!.
189 200/189 [3 -3 2 -1> 97.94 - ![sharp zao :/||~:](./images/smilies/sharp_zao.png)
![bi :':](./images/smilies/bi.png)
![flat zai :\!!~:](./images/smilies/flat_zai.png)
![bo :.:](./images/smilies/bo.png)
/||~'' & \!!~..
225 225/224 [-5 2 2 -1> 7.71 7:25k septimal kleisma ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'|( & .!(
243 250/243 [1 -5 3> 49.17 125M porcupine comma ![patai or gai :/|):](./images/smilies/patai.png)
![patao or gao :\!):](./images/smilies/patao.png)
/|)' & \!).
256/243 [8 -5> 90.22 Pythagorean limma ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
.||\ & '!!/
243/224 [-5 5 0 -1> 140.95 -
|||) & !!!)
245 245/243 [0 -5 1 2> 14.19 245C sensamagic comma ![sanai :~|(:](./images/smilies/sanai.png)
![sanao :~!(:](./images/smilies/sanao.png)
~|(. & ~!('
256/245 [8 0 -1 -2> 76.03 - ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
.~~|| & '~~!!
343 343/320 [-6 0 -1 3> 120.16 - ![sharp nai :|||(:](./images/smilies/sharp_nai.png)
![flat nao :!!!(:](./images/smilies/flat_nao.png)
|||(' & !!!(.
405 405/392 [-3 4 1 -2> 56.48 5:49M greenwoodma
)/|\ & )\!/
525 525/512 [-9 1 2 1> 43.41 175S Avicenna ![phai or fai :/ /|:](./images/smilies/phai.png)
![phao or fao :\ \!:](./images/smilies/phao.png)
//|' & \\!.
567 567/512 [-9 4 0 1> 176.65 - ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
.(|||\ & '(!!!/
625 648/625 [3 4 -4> 62.57 625L diminished comma ![bo :.:](./images/smilies/bo.png)
![jakai or dai :(|\:](./images/smilies/jakai.png)
![bi :':](./images/smilies/bi.png)
![jakao or dao :(!/:](./images/smilies/jakao.png)
.(|\. & '(!/'
625/576 [-6 -2 4> 141.34 2x 5-lim. chr. semitone ![sharp tai :|||):](./images/smilies/sharp_tai.png)
![flat tao :!!!):](./images/smilies/flat_tao.png)
|||)' & !!!).
729 729/700 [-2 6 -2 -1> 70.28 - ![sharp phao :)||(:](./images/smilies/sharp_phao.png)
![flat phai :)!!(:](./images/smilies/flat_phai.png)
)||(. & )!!('
729/686 [-1 6 0 -3> 105.25 - ![sharp sao :)/ /||:](./images/smilies/sharp_sao.png)
![flat sai :)\ \!!:](./images/smilies/flat_sai.png)
)//||' & )\\!!.
729/640 [-7 6 -1> 225.42 - ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
./X\ '\Y/
875 875/864 [-5 -3 3 1> 21.90 keema ![pai :/|:](./images/smilies/pai.png)
![pao :\!:](./images/smilies/pao.png)
/|' & \!.
945 1024/945 [10 -3 -1 -1> 139.00 - ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
.|||) & '!!!)
1029 1029/1024 [-10 1 0 3> 8.43 343k gamelisma ![sai :~|:](./images/smilies/sai.png)
![sao :~!:](./images/smilies/sao.png)
~|. & ~!'
1125 1125/1024 [-10 2 3> 162.85 enipucrop ![sharp patai or sharp gai :/|||):](./images/smilies/sharp_patai.png)
![flat patao or flat gao :\!!!):](./images/smilies/flat_patao.png)
/|||)' & \!!!).
1225 1296/1225 [4 4 -2 -2> 97.54 2x sept. quarter-tone ![sharp zao :/||~:](./images/smilies/sharp_zao.png)
![flat zai :\!!~:](./images/smilies/flat_zai.png)
/||~' & \!!~.
1323 1323/1280 [-8 3 -1 2> 57.20 5:49L - ![prakai or vrai :)/|\:](./images/smilies/prakai.png)
![prakao or vrao :)\!/:](./images/smilies/prakao.png)
)/|\' & )\!/.
1701 1701/1600 [-6 5 -2 1> 105.97 - ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
./||) & '\!!)
2025 2048/2025 [11 -4 -2> 19.55 25C diaschisma ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
./| & '\!
2025/1792 [-8 4 2 -1> 211.62 - ![double sharp zao :/X~:](./images/smilies/double_sharp_zao.png)
![bi :':](./images/smilies/bi.png)
![double flat zai :\Y~:](./images/smilies/double_flat_zai.png)
![bo :.:](./images/smilies/bo.png)
/X~'' & \Y~..
2187 2240/2187 [6 -7 1 1> 41.45 35S - ![bo :.:](./images/smilies/bo.png)
![phai or fai :/ /|:](./images/smilies/phai.png)
![bi :':](./images/smilies/bi.png)
![phao or fao :\ \!:](./images/smilies/phao.png)
.//|' & '\\!.
2187/2048 [-11 7> 113.69 apotome
/||\ & \!!/
2187/2000 [-4 7 -3> 154.74 gorgo ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
.//||| & '\\!!!
2187/1960 [-3 7 -1 -2> 189.72 - ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
.~~X & '~~Y
2205 2205/2048 [-11 2 1 2> 127.88 - ![sharp sanai :~|||(:](./images/smilies/sharp_sanai.png)
![flat sanao :~!!!(:](./images/smilies/flat_sanao.png)
~|||(. & ~!!!('
3125 3125/3072 [-10 -1 5> 29.61 magic comma ![bi :':](./images/smilies/bi.png)
![tai :|):](./images/smilies/tai.png)
![bo :.:](./images/smilies/bo.png)
![tao :!):](./images/smilies/tao.png)
'|)' & .!).
3645 3645/3584 [-9 6 1 -1> 29.22 5:7C - ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'|) & .!)
3969 4096/3969 [12 -4 0 -2> 54.53 49M 2x Archytas comma
(/| & (\!
4375 4608/4375 [9 2 -4 -1> 89.83 - ![bo :.:](./images/smilies/bo.png)
![sharp pao :||\:](./images/smilies/sharp_pao.png)
![bi :':](./images/smilies/bi.png)
![flat pai :!!/:](./images/smilies/flat_pai.png)
.||\. & '!!/'
5103 5120/5103 [10 -6 1 -1> 5.76 5:7k hemifamity comma
|( & !(
6561 6561/6400 [-8 8 -2> 43.01 25S 2x syntonic comma
//| & \\!
6561/6272 [-7 8 0 -2> 77.99 large deep red comma
~~|| & ~~!!
6561/6250 [-1 8 -5 > 84.07 ripple ![bo :.:](./images/smilies/bo.png)
![sharp tao :||):](./images/smilies/sharp_tao.png)
![bi :':](./images/smilies/bi.png)
![flat tai :!!):](./images/smilies/flat_tai.png)
.||). & '!!)'
8505 8505/8192 [-13 5 1 1> 64.91 35L -
(|\ & (!/
8575 9216/8575 [10 2 -2 -3> 124.80 - ![bi :':](./images/smilies/bi.png)
![sharp ranai :)|||(:](./images/smilies/sharp_ranai.png)
![bo :.:](./images/smilies/bo.png)
![flat ranao :)!!!(:](./images/smilies/flat_ranao.png)
')|||(. & .)!!!('
10935 10976/10935 [5 -7 -1 3> 6.48 hemimage ![nai :|(:](./images/smilies/nai.png)
![nao :!(:](./images/smilies/nao.png)
|(' & !(.
16875 16875/16384 [-14 3 4> 51.12 625M negri comma ![bi :':](./images/smilies/bi.png)
![patai or gai :/|):](./images/smilies/patai.png)
![bo :.:](./images/smilies/bo.png)
![patao or gao :\!):](./images/smilies/patao.png)
'/|)' & .\!).
18225 18225/16384 [-14 6 2> 184.36 2x pelogic comma
)X( & )Y(
19683 20000/19683 [5 -9 4> 27.66 tetracot comma ![tai :|):](./images/smilies/tai.png)
![tao :!):](./images/smilies/tao.png)
|)' & !).
19683/17920 [-9 9 -1 -1> 162.46 -
/|||) & \!!!)
19683/17500 [-2 9 -4 -1> 203.51 - ![bo :.:](./images/smilies/bo.png)
![double sharp pao :X\:](./images/smilies/double_sharp_pao.png)
![bi :':](./images/smilies/bi.png)
![double flat pai :Y/:](./images/smilies/double_flat_pai.png)
.X\. & 'Y/'
30375 30375/28672 [-12 5 3 -1> 99.89 - ![sharp sanao :(||(:](./images/smilies/sharp_sanao.png)
![bi :':](./images/smilies/bi.png)
![flat sanai :(!!(:](./images/smilies/flat_sanai.png)
![bo :.:](./images/smilies/bo.png)
(||('' & (!!(..
32805 32805/32768 [-15 8 1> 1.95 5s schisma ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'| & .!
33075 33075/32768 [-15 3 2 2> 16.14 mirwomo comma ![zai :|~:](./images/smilies/zai.png)
![zao :!~:](./images/smilies/zao.png)
|~. & !~'
35721 35721/32768 [-15 6 0 2> 149.38 -
~|||) & ~!!!)
45927 45927/40960 [-13 8 -1 1> 198.15 - ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
.X) & 'Y)
54675 54675/50176 [-10 7 2 -2> 148.66 - ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'(||| & .(!!!
59049 59049/57344 [-13 10 0 -1> 50.72 7M Harrison's comma ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'/|) & .\!)
59049/56000 [-6 10 -3 -1> 91.78 - ![sharp pao :||\:](./images/smilies/sharp_pao.png)
![flat pai :!!/:](./images/smilies/flat_pai.png)
||\. & !!/'
70875 70875/65536 [-16 4 3 1> 135.59 - ![sharp pai :/|||:](./images/smilies/sharp_pai.png)
![flat pao :\!!!:](./images/smilies/flat_pao.png)
/|||' & \!!!.
107163 107163/102400 [-12 7 -2 2> 78.71 - ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
.)||~ & ')!!~
127575 131072/127575 [17 -6 -2 -1> 46.82 - ![phrai or frai :)/ /|:](./images/smilies/phrai.png)
![phrao or frao :)\ \!:](./images/smilies/phrao.png)
)//|' & )\\!.
137781 137781/131072 [-17 9 0 1> 86.42 -
||) & !!)
137781/128000 [-10 9 -3 1> 127.48 - ![sharp sanai :~|||(:](./images/smilies/sharp_sanai.png)
![bo :.:](./images/smilies/bo.png)
![flat sanao :~!!!(:](./images/smilies/flat_sanao.png)
![bi :':](./images/smilies/bi.png)
~|||(.. & ~!!!(''
177147 177147/163840 [-15 11 -1> 135.19 -
/||| & \!!!
177147/160000 [-8 11 -4> 176.25 - ![bo :.:](./images/smilies/bo.png)
![sharp jakai or sharp dai :(|||\:](./images/smilies/sharp_jakai.png)
![bi :':](./images/smilies/bi.png)
![flat jakao or flat dao :(!!!/:](./images/smilies/flat_jakao.png)
.(|||\. & '(!!!/'
177147/156800 [-7 11 -2 -2> 211.23 - ![double sharp zao :/X~:](./images/smilies/double_sharp_zao.png)
![double flat zai :\Y~:](./images/smilies/double_flat_zai.png)
/X~' & \Y~.
273375 273375/262144 [-18 7 3> 72.63 - ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
')||( & .)!!(
295245 295245/262144 [-18 10 1> 205.86 -
X\ & Y/
413343 413343/409600 [-14 10 -2 1> 15.75 7:25C - ![zai :|~:](./images/smilies/zai.png)
![bo :.:](./images/smilies/bo.png)
![zao :!~:](./images/smilies/zao.png)
![bi :':](./images/smilies/bi.png)
|~.. & !~''
492075 492075/458752 [-16 9 2 -1> 121.40 - ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'|||( & .!!!(
531441 531441/524288 [-19 12> 23.46 1C Pythagorean comma ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'/| & .\!
531441/512000 [-12 12 -3> 64.52 125L 3x syntonic comma ![jakai or dai :(|\:](./images/smilies/jakai.png)
![jakao or dao :(!/:](./images/smilies/jakao.png)
(|\. & (!/'
531441/501760 [-11 12 -1 -2> 99.49 - ![sharp sanao :(||(:](./images/smilies/sharp_sanao.png)
![flat sanai :(!!(:](./images/smilies/flat_sanai.png)
(||(' & (!!(.
535815 535815/524288 [-19 7 1 2> 37.65 245S - ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'~|) & .~!)
885735 885735/802816 [-14 11 1 -2> 170.17 -
)/|||\ & )\!!!/
1063125 1063125/1048576 [-20 5 4 1> 23.86 - ![bi :':](./images/smilies/bi.png)
![pai :/|:](./images/smilies/pai.png)
![bo :.:](./images/smilies/bo.png)
![pao :\!:](./images/smilies/pao.png)
'/|' & .\!.
1148175 1148175/1048576 [-20 8 2 1> 157.09 - ![sharp phai or sharp fai :/ /|||:](./images/smilies/sharp_phai.png)
![flat phao or flat fao :\ \!!!:](./images/smilies/flat_phao.png)
//|||' & \\!!!.
1594323 1594323/1433600 [-13 13 -2 -1> 183.96 - ![double sharp phao :)X(:](./images/smilies/double_sharp_phao.png)
![double flat phai :)Y(:](./images/smilies/double_flat_phai.png)
)X(. & )Y('
1594323/1404928 [-12 13 0 -3> 218.94 - ![double sharp sao :)/ /X:](./images/smilies/double_sharp_sao.png)
![double flat sai :)\ \Y:](./images/smilies/double_flat_sai.png)
)//X' & )\\Y.
2066715 2097152/2066715 [21 -10 -1 -1> 25.31 35C - ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
.|) & '!)
2083725 2097152/2083725 [21 -5 -2 -3> 11.12 - ![bi :':](./images/smilies/bi.png)
![ranai :)|(:](./images/smilies/ranai.png)
![bo :.:](./images/smilies/bo.png)
![ranao :)!(:](./images/smilies/ranao.png)
')|(. .)!('
2250423 2250423/2097152 [-21 8 0 3> 122.12 - ![sharp sai :~|||:](./images/smilies/sharp_sai.png)
![flat sao :~!!!:](./images/smilies/flat_sao.png)
~|||. & ~!!!'
2278125 2278125/2097152 [-21 6 5> 143.30 - ![bi :':](./images/smilies/bi.png)
![sharp tai :|||):](./images/smilies/sharp_tai.png)
![bo :.:](./images/smilies/bo.png)
![flat tao :!!!):](./images/smilies/flat_tao.png)
'|||)' & .!!!).
2893401 2893401/2621440 [-19 10 -1 2> 170.89 - ![sharp prakai or sharp vrai :)/|||\:](./images/smilies/sharp_prakai.png)
![flat prakao or flat vrao :)\!!!/:](./images/smilies/flat_prakao.png)
)/|||\' & )\!!!/.
3720087 3720087/3276800 [-17 12 -2 1> 219.66 - ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
./X) & '\Y)
4428675 4428675/4194304 [-22 11 2> 94.13 - ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'||\ & .!!/
4782969 4782969/4587520 [-17 14 -1 -1> 72.23 - ![bi :':](./images/smilies/bi.png)
![sharp phao :)||(:](./images/smilies/sharp_phao.png)
![bo :.:](./images/smilies/bo.png)
![flat phai :)!!(:](./images/smilies/flat_phai.png)
')||(. & .)!!('
4782969/4194304 [-22 14> 227.37 2x apotome
/X\ & \Y/
7971615 7971615/7340032 [-20 13 1 -1> 142.90 - ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'|||) & .!!!)
8680203 8680203/8388608 [-23 11 0 2> 59.16 49L -
|\) & !/)
11160261 11160261/10485760 [-21 13 -1 1> 107.93 -
/||) & \!!)
14348907 14348907/13107200 [-19 15 -2> 156.70 -
//||| & \\!!!
14348907/12845056 [-18 15 0 -2> 191.67 -
~~X & ~~Y
14348907/12800000 [-12 15 -5> 197.76 - ![bo :.:](./images/smilies/bo.png)
![double sharp tao :X):](./images/smilies/double_sharp_tao.png)
![bi :':](./images/smilies/bi.png)
![double flat tai :Y):](./images/smilies/double_flat_tai.png)
.X). & 'Y)'
18600435 18600435/16777216 [-24 12 1 1> 178.60 -
(|||\ & (!!!/
23914845 23914845/22478848 [-16 14 1 -3> 107.21 - ![sharp nao :/||):](./images/smilies/sharp_nao.png)
![flat nai :\!!):](./images/smilies/flat_nai.png)
/||). & \!!)'
36905625 36905625/33554432 [-25 10 4> 164.80 - ![bi :':](./images/smilies/bi.png)
![sharp patai or sharp gai :/|||):](./images/smilies/sharp_patai.png)
![bo :.:](./images/smilies/bo.png)
![flat patao or flat gao :\!!!):](./images/smilies/flat_patao.png)
'/|||)' & .\!!!).
43046721 43046721/40960000 [-16 16 -4> 86.03 4x syntonic comma ![sharp tao :||):](./images/smilies/sharp_tao.png)
![flat tai :!!):](./images/smilies/flat_tai.png)
||). & !!)'
66430125 66430125/58720256 [-23 12 3 -1> 213.58 - ![double sharp sanao :(X(:](./images/smilies/double_sharp_sanao.png)
![bi :':](./images/smilies/bi.png)
![double flat sanai :(Y(:](./images/smilies/double_flat_sanai.png)
![bo :.:](./images/smilies/bo.png)
(X('' & (Y(..
71744535 71744535/67108864 [-26 15 1> 115.64 - ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'/||\ & .\!!/
72335025 72335025/67108864 [-26 10 2 2> 129.83 (8505/8192)^2 ![sharp zai :|||~:](./images/smilies/sharp_zai.png)
![flat zao :!!!~:](./images/smilies/flat_zao.png)
|||~. & !!!~'
129140163 129140163/114688000 [-17 17 -3 -1> 205.47 - ![double sharp pao :X\:](./images/smilies/double_sharp_pao.png)
![double flat pai :Y/:](./images/smilies/double_flat_pai.png)
X\. & Y/'
129140163/117440512 [-24 17 0 -1> 164.41 - ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'/|||) & .\!!!)
234365481 234365481/209715200 [-23 14 -2 2> 192.39 - ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
.)X~ & ')Y~
279006525 279006525/268435456 [-28 13 2 1> 66.87 - ![rakhai or rachai :)|\ \:](./images/smilies/rakhai.png)
![rakhao or rachao :)!/ /:](./images/smilies/rakhao.png)
)|\\. & )!//'
301327047 301327047/268435456 [-28 16 0 1> 200.11 -
X) & Y)
597871125 597871125/536870912 [-29 14 3> 186.31 - ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
')X( & .)Y(
903981141 903981141/838860800 [-25 17 -2 1> 129.43 - ![sharp zai :|||~:](./images/smilies/sharp_zai.png)
![bo :.:](./images/smilies/bo.png)
![flat zao :!!!~:](./images/smilies/flat_zao.png)
![bi :':](./images/smilies/bi.png)
|||~.. & !!!~''
1162261467 1162261467/1027604480 [-22 19 -1 -2> 213.18 - ![double sharp sanao :(X(:](./images/smilies/double_sharp_sanao.png)
![double flat sanai :(Y(:](./images/smilies/double_flat_sanai.png)
(X(' & (Y(.
1162261467/1048576000 [-23 19 -3> 178.20 - ![sharp jakai or sharp dai :(|||\:](./images/smilies/sharp_jakai.png)
![flat jakao or flat dao :(!!!/:](./images/smilies/flat_jakao.png)
(|||\. & (!!!/'
1162261467/1073741824 [-30 19> 137.15 19-comma ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'/||| & .\!!!
1171827405 1171827405/1073741824 [-30 14 1 2> 151.34 - ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'~|||) & .~!!!)
2325054375 2325054375/2147483648 [-31 12 4 1> 137.54 - ![bi :':](./images/smilies/bi.png)
![sharp pai :/|||:](./images/smilies/sharp_pai.png)
![bo :.:](./images/smilies/bo.png)
![flat pao :\!!!:](./images/smilies/flat_pao.png)
'/|||' & .\!!!.
4519905705 4519905705/4294967296 [-32 17 1 1> 88.37 - ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'||) & .!!)
4557106575 4557106575/4294967296 [-32 12 2 3> 102.57 - ![bo :.:](./images/smilies/bo.png)
![sharp ranao :/ /||:](./images/smilies/sharp_ranao.png)
![bi :':](./images/smilies/bi.png)
![flat ranai :\ \!!:](./images/smilies/flat_ranai.png)
.//||' & '\\!!.
9685512225 9685512225/8589934592 [-33 18 2> 207.82 - ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'X\ & .Y/
10460353203 10460353203/9395240960 [-28 21 -1 -1> 185.92 - ![bi :':](./images/smilies/bi.png)
![double sharp phao :)X(:](./images/smilies/double_sharp_phao.png)
![bo :.:](./images/smilies/bo.png)
![double flat phai :)Y(:](./images/smilies/double_flat_phai.png)
')X(. & .)Y(.
18983603961 18983603961/17179869184 [-34 18 0 2> 172.84 (137781/131072)^2
|||\) & !!!/)
24407490807 24407490807/21474836480 [-32 20 -1 1> 221.61 -
/X) & \Y)
52301766015 52301766015/46036680704 [-27 21 1 -3> 220.89 - ![double sharp nao :/X):](./images/smilies/double_sharp_nao.png)
![double flat nai :\Y):](./images/smilies/double_flat_nai.png)
/X). & \Y)'
94143178827 94143178827/83886080000 [-27 23 -4> 199.71 - ![double sharp tao :X):](./images/smilies/double_sharp_tao.png)
![double flat tai :Y):](./images/smilies/double_flat_tai.png)
X). & Y)'
610187270175 610187270175/549755813888 [-39 20 2 1> 180.55 - ![sharp rakhai or sharp rachai :)|||\ \:](./images/smilies/sharp_rakhai.png)
![flat rakhao or flat rachao :)!!!/ /:](./images/smilies/flat_rakhao.png)
)|||\\. & )!!!//'
... 9885033776835/8796093022208 [-43 24 1 1> 202.06 - ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'X) & .Y)
... 9966392079525/8796093022208 [-43 19 2 3> 216.25 - ![bo :.:](./images/smilies/bo.png)
![double sharp ranao :/ /X:](./images/smilies/double_sharp_ranao.png)
![bi :':](./images/smilies/bi.png)
![double flat ranai :\ \Y:](./images/smilies/double_flat_ranai.png)
.//X' & '\\Y.
Some 7-limit ratios without an accurate representation in the Olympian set:
odd-limit ratio monzo cents sys. interval name accidentals (inaccur.) possible compositions
-------------------------------------------------------------------------------------------------------------------------------------------
9 10/9 [1 -2 1> 182.40 minor tone ![bo :.:](./images/smilies/bo.png)
![bi :':](./images/smilies/bi.png)
.)X( & ')Y( 9/8 * 80/81, 16/15 * 25/24
25 28/25 [2 0 -2 1> 196.20 middle second ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
')X~ & .)Y~ 27/25 * 28/27, 9/8 * 224/225
49 49/45 [0 -2 -1 2> 147.43 swetismic neutr. sec. ![sharp jai :(|||:](./images/smilies/sharp_jai.png)
![flat jao :(!!!:](./images/smilies/flat_jao.png)
(|||' & (!!!. 16/15 * 49/48
125 125/112 [-4 0 3 -1> 190.12 - ... 9/8 * 125/126, 15/14 * 25/24
147 160/147 [5 -1 1 -2> 146.71 -
(||| & (!!! 15/14 * 64/63, 16/15 * 50/49,
256/245 * 25/24, 54/49 * 80/81
175 175/162 [-1 -4 2 1> 133.63 -
)|||~ & )!!!~ 25/24 * 28/27, 35/32 * 80/81
343 360/343 [3 2 1 -3> 83.75 - ![bo :.:](./images/smilies/bo.png)
![sharp tao :||):](./images/smilies/sharp_tao.png)
![bi :':](./images/smilies/bi.png)
![flat tai :!!):](./images/smilies/flat_tai.png)
.||). & '!!)' 36/35 * 50/49, 15/14 * 48/49
384/343 [7 1 0 -3> 195.48 -
/X & \Y 15/14 * 256/245
375 392/375 [3 -1 -3 2> 76.76 - ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'~||( & .~!!( 28/27 * 126/125, 16/15 * 49/50
625 672/625 [5 1 -4 1> 125.53 - ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
')|||( & .)!!!( 21/20 * 128/125, 28/25 * 24/25
675 686/675 [1 -3 -2 3> 27.99 senga ![tai :|):](./images/smilies/tai.png)
![tao :!):](./images/smilies/tao.png)
|)' & !). 28/27 * 49/50, 49/48 * 224/225
1029 1029/1000 [-3 1 -3 3> 49.49 keega ![patai or gai :/|):](./images/smilies/patai.png)
![patao or gao :\!):](./images/smilies/patao.png)
/|)' & \!). 21/20 * 49/50, 49/48 * 126/125
1225 1225/1152 [-7 -2 2 2> 106.37 - ![bo :.:](./images/smilies/bo.png)
![sharp nao :/||):](./images/smilies/sharp_nao.png)
![bi :':](./images/smilies/bi.png)
![flat nai :\!!):](./images/smilies/flat_nai.png)
./||)' & '\!!). 25/24 * 49/48, 35/32 * 35/36
1715 1728/1715 [6 3 -1 -3> 13.07 orwellisma ![bo :.:](./images/smilies/bo.png)
![sanai :~|(:](./images/smilies/sanai.png)
![bi :':](./images/smilies/bi.png)
![sanao :~!(:](./images/smilies/sanao.png)
.~|(' & '~!(. 36/35 * 48/49, 256/245 * 27/28,
54/49 * 32/35
1715/1536 [-9 -1 1 3> 190.84 - ![double sharp satao :~~X:](./images/smilies/double_sharp_satao.png)
![bo :.:](./images/smilies/bo.png)
![double flat satai :~~Y:](./images/smilies/double_flat_satai.png)
![bi :':](./images/smilies/bi.png)
~~X.. & ~~Y'' 35/32 * 49/48
2401 2401/2400 [-5 -1 -2 4> 0.72 breedsma ![ai :|:](./images/smilies/ai.png)
![ao :!:](./images/smilies/ao.png)
|' & !. 49/48 * 49/50
2430/2401 [1 5 1 -4> 20.79 nuwell comma ![pai :/|:](./images/smilies/pai.png)
![pao :\!:](./images/smilies/pao.png)
/|. & \!' 50/49 * 243/245
2500/2401 [2 0 4 -4> 69.95 (50/49)^2 ![sharp phao :)||(:](./images/smilies/sharp_phao.png)
![bo :.:](./images/smilies/bo.png)
![flat phai :)!!(:](./images/smilies/flat_phai.png)
![bi :':](./images/smilies/bi.png)
)||(.. & )!!('' 50/49 * 50/49
3087 3200/3087 [7 -2 2 -3> 62.24 - ![bi :':](./images/smilies/bi.png)
![bo :.:](./images/smilies/bo.png)
'(|) & .(!) 50/49 * 64/63
3125 3136/3125 [6 0 -5 2> 6.08 hemimean comma
|( & !( 128/125 * 49/50, 126/125 * 224/225
3125/3087 [0 -2 5 -3> 21.18 gariboh comma ![pai :/|:](./images/smilies/pai.png)
![pao :\!:](./images/smilies/pao.png)
/|. & \!' 50/49 * 125/126
3125/3024 [-4 -3 5 -1> 56.88 - ![prakai or vrai :)/|\:](./images/smilies/prakai.png)
![prakao or vrao :)\!/:](./images/smilies/prakao.png)
)/|\' & )\!/. 25/24 * 125/126, 200/189 * 125/128,
15/14 * 625/648
3969 4000/3969 [5 -4 3 -2> 13.47 octagar comma ![sanai :~|(:](./images/smilies/sanai.png)
![bo :.:](./images/smilies/bo.png)
![sanao :~!(:](./images/smilies/sanao.png)
![bi :':](./images/smilies/bi.png)
~|(.. & ~!('' 64/63 * 125/126, 50/49 * 80/81
4375 4375/4374 [-1 -7 4 1> 0.40 ragisma ![ai :|:](./images/smilies/ai.png)
![ao :!:](./images/smilies/ao.png)
|' & !. 250/243 * 35/36
5103 5103/5000 [-3 6 -4 1> 35.30 - ![satai :~|):](./images/smilies/satai.png)
![satao :~!):](./images/smilies/satao.png)
~|). & ~!)' 81/80 * 126/125, 648/625 * 63/64,
27/25 * 189/200
5625 5625/5488 [-4 2 4 -3> 42.69 - ![phai or fai :/ /|:](./images/smilies/phai.png)
![phao or fao :\ \!:](./images/smilies/phao.png)
//|. & \\!' 50/49 * 225/224
6125 6144/6125 [11 1 -3 -2> 5.36 porwell comma ![nai :|(:](./images/smilies/nai.png)
![nao :!(:](./images/smilies/nao.png)
|(. & !(' 256/245 * 24/25, 128/125 * 48/49
-----------------------------------------------------------------------
I have edited this post a lot, so here's the text from the previous version, for reference:
List of 7-prime limit / 6125-odd limit accidentals
When working a lot with MOS notations, I find it vital to have a list of simple 7-limit accidentals. So I started to compile a list, and decided to post it here so I can share it with others, or ask for help to complete / improve it.
The ratios in the list are 7-prime limit, 6125-odd limit, and not larger than a 9/8 major tone. This list isn't complete, and only contains ratios that I find either important, or potentially useful. I list accidental symbols / characters where I could find them.
Note: This list isn't intended as a reference for standard Sagittal notation. A lot of the accidentals in this list aren't required in standard Sagittal notation, and using more different types of accidentals than necessary can make it difficult for others to read a score.
On the contrary, MOS notations (or other non-standard notations) sometimes require unusual accidentals, and can become unwieldy if multiple accidentals have to be used for a single chroma, or if stacks of chromas are common, and can't be reduced to fewer accidentals (for example, there are considerable chroma stacks when notating 87-edo with a Rodan[5] notation).
Edit note: I started to compile a list of 7-limit ratios with representations in the Olympian set, and moved ratios that are not contained in this set into a second list. This is work in progress, so the lists are currently incomplete!
Edit: The following points have already been solved or addressed:
One accidental pair which I'd be interested in is for 256:245, the difference between 8/7 and 35/32. I'm working a lot with scales where the major third 5/4 is divided into 35/32 and 8/7. Examples where 256:245 accidentals can become useful are:
For Moments of Symmetry based on pentatonic scales, it would be very useful to have accidentals for 16:15 and 256:243.
- Consider a notation based on the 6-note MOS 1/1 28/25 5/4 7/5 25/16 7/4 for the 2.5.7 temperament that tempers out the hemimean comma 3136/3125 (i.e. 5/4 is split into two 28/25 steps). Two chromas could be expressed as 256/245. This notation could be used as a basis for a hemithirds, hemiwürschmidt, or roulette notation. A roulette[6] notation could be used as a 2.5.7.11.13 notation (no-fifths 13-limit) for 37-edo, without having to decide for a fifth (i.e. between 37p and 37b), and it shares the same mapping with 74- and 111-edo.
- I'm currently experimenting with a 7-limit JI notation based on the following pair of max-variety-4 scales: 1/1 35/32 5/4 21/16 3/2 105/64 15/8, and 1/1 35/32 5/4 21/16 3/2 105/64 7/4. Both scales are mirror symmetric, and can be obtained by removing a pitch from 1/1 35/32 5/4 21/16 3/2 105/64 7/4 15/8, which is the octave-reduced version of the complete chord 1:3:5:7:15:21:35:105. Having a single accidental for 256/245 would be extremely useful for this notation. I'm planning to use it as a basis for several temperaments that are supported by 31-edo, including the planar temperaments hewuermity, gamelan and marvel, and the linear temperaments mohajira, orwell, valentine, miracle and mothra, probably more. If it works out, this notation should help me considerably with 31-based compositions with modulations between scales of different temperaments.
I think I'd prefer to use two accidentals for neutral or major second-sized ratios like 160/147 or 9/8, as an indicator for large (or uncommon) alterations. For example, 160/147 could be expressed as 15/14 * 64/63, or 25/24 * 256/245.
A question: Herman Miller mentioned accidentals for 21:20 that I couldn't find anywhere else. Is there a list of Sagittal accidentals that I don't know about?
P.S.: Apparently, some Sagittal accidentals aren't implemented as emoticons yet, or don't display for other reasons.