A shruti list
- Original article by ma1937, on the Yahoo tuning forum, is quoted here:
The listing of the srutis of Indian classical music given below is based on decades of study of the srutis, study with several masters of Indian classical music, pitch analysis of recordings by several masters of raga performance, and the following quote by Ali Akbar Khan:
"I am still learning about the srutis. They reach to your heart and help you feel the ragas and the notes. In old theory, they say that there are twenty-two in number, but right now I feel that there are more like twenty-three and a half. There is only one sa and one pa. Komal re, komal ga, and komal dha all have three. Shuddha ma, tivra ma, shuddha dha, and komal ni each have two. And shuddha re, shuddha ga, and shuddha ni each have one and a half." - Ali Akbar Khan
This quotation yields many insights... Below I have just listed the twenty-three and a half srutis he is referring to.
In brief summary, Khansahib's list is basically the usually-given twenty-two srutis plus the three "ati ati komals" (ati ati komal re; ati ati komal ga; and ati ati komal dha). Though not on the usual list of 22 srutis, it is well-known that these notes do appear is some ragas. So really there are twenty-five notes on Khansahib's list. It's reduced to twenty-three and half because he gives "half" status to three notes that are usually considered srutis -- the lesser-used versions of shuddha re, shuddha ga, and shuddha ni. I think this is the most illuminating aspect of his comment.
With each set of srutis associated with a given note, the principal sruti is listed first, the others in descending order of significance. Most ratios given are exact. Cent values given are rounded to the nearest whole cent:
| Principal | Shruti | Ratio | ¢
(approx) |
Remarks |
|---|---|---|---|---|
| Sa | 1/1 | 000 | ||
| komal re | komal re | 16/15 | 112 | |
| ati komal re | 256/243 | 090 | ||
| ati ati komal re | 25/24 | 070 | ||
| Re | shuddha re | 9/8 | 204 | |
| "half"-status shuddha re | 10/9 | 182 | ||
| komal ga | komal ga | 6/5 | 316 | |
| ati komal ga | 32/27 | 294 | ||
| ati ati komal ga | 75/64 | 274 | inverse ekasruti shuddha dha:[~256/219, 273] is the schismatic tuning of this shruti | |
| Ga | shuddha ga | 5/4 | 386 | inverse "half"-status shuddha ga/"half"-status shuddha ga [384] is the schismatic tuning of this shruti |
| "half"-status shuddha ga | 81/64 | 408 | inverse "half"-status shuddha ga/shuddha ga [512/405; 406] is the schismatic tuning of this shruti | |
| (inverse ati ati komal dha) | 32/25 | 428 | ||
| Ma | shuddha Ma | 4/3 | 498 | |
| ekasruti Ma | 27/20 | 520 | ||
| tivra Ma | tivra(tar) Ma | 45/32 | 590 | (these two essentially inverses; maybe not entirely a true priority) |
| 729/512 | 612 | |||
| (inverse ekasruti Ma) | 40/27 | 680 | ||
| Pa | 3/2 | 702 | ||
| komal dha | komal dha | 8/5 | 814 | "half"-status shuddha ga/"half"-status shuddha ga [816] is the schismatic tuning of this shruti |
| ati komal dha | 128/81 | 792 | "half"-status shuddha ga/shuddha ga [405/256; 794] is the schismatic tuning of this shruti | |
| ati ati komal dha | 25/16 | 772 | ||
| Dha | shuddha dha | 5/3 | 884 | (these two hard to prioritize; maybe a toss-up) |
| 27/16 | 906 | |||
| (inverse ati ati komal ga) | 128/75 | 926 | ekasruti shuddha dha:[~219/128, 927] is the schismatic tuning of this shruti | |
| komal ni | komal ni | 9/5 | 1018 | (these two hard to prioritize; maybe a toss-up) |
| 16/9 | 996 | |||
| Ni | shuddha ni | 15/8 | 1088 | |
| "half"-status shuddha ni | 243/128 | 1110 | ||
| (inverse ati ati komal re) | 48/25 | 1130 | ||
Secondary functions and "artifact shrutis" introduced by using 19 or 22 or 23 or 25 or 26 or 29 (out of n) edo to simulate ragas
komal-ardha re (1): [250/243; 48]: 22, 23. 25, 26, 29
ekasruti komal re (1 3/4), ati ati komal re/ati ati komal re: [27/25; 133], [~13/12; 138], [625/576; 141]: 25, 26
inverse ekasruti komal ni, inverse ekasruti Ma/ekasruti Ma: [800/729; 160]: 22, 23, 29
inverse ati ati komal ga/Pa, komal re/komal re, inverse komal-ardha ni: [256/225; 224], [729/640; 226]: 22, 26
komal-ardha ga (1 3/4): [144/125; 246], [125/108; 252]: 19, 25*, 29
ekasruti komal ga: [243/200; 338]: 25, 29
inverse inverse ati ati komal dha/inverse ati ati komal dha: [625/512; 344]: 25
inverse ekasruti komal dha, "half"-status shuddha re/"half"-status shuddha re [100/81; 365]; 23, 26, 29
inverse komal-ardha dha [162/125; 449]: 19, 29
(ati) ati komal re/shuddha ga, inverse komal re/tivratar Ma, inverse ekasruti Pa: [~13/10; 454], [320/243; 476]: 25, 29
inverse ati ati komal re/tivra(tar) Ma [512/375, 539; ~82/61, 518]: 22, 23, 25
ati ati komal ga/ati ati komal ga: [~56/41; 548]: 22
inverse komal ga/komal ga; [25/18; 569]: 19
komal ga/komal ga; [36/25; 631]: 19
inverse ati ati komal ga/ati ati komal ga: [~820/563; 652]: 22
ati ati komal re/tivra(tar) Ma [375/256, 661; ~61/41, 682]: 22, 23, 25
inverse (ati) ati komal re/shuddha ga, komal re/tivratar Ma, ekasruti Pa: [~20/13; 746], [243/160; 724]: 25, 29
komal-ardha dha [125/81; 751]: 19
ekasruti komal dha, inverse "half"-status shuddha re/"half"-status shuddha re [81/50; 835]: 23, 26, 29
inverse ati ati komal dha/inverse ati ati komal dha: [1024/625; 856]
inverse ekasruti komal ga: [400/243; 862]: 25, 29
komal-ardha ga (1 3/4): [125/72; 954], [216/125; 948]: 19, 25*, 29
ati ati komal ga/Pa, inverse komal re/komal re, komal-ardha ni: [225/128; 976], [1280/729; 974]: 22, 26
ekasruti komal ni, ekasruti Ma/ekasruti Ma: [729/400; 1040]: 22, 23
inverse ekasruti komal re (1 3/4), inverse ati ati komal re/ati ati komal re: [50/27; 1067], [~24/13; 1062], [1152/625; 1059]: 26
inverse komal-ardha re (1): [243/125; 1152]: 22, 23, 26
Regular temperaments of the shrutis
Note:
- generators in (bold) italics will generate a 19/23 (diatonic) or 22/25/26/29 tone (superdiatonic) set which is too weakly tonal for serious practice
- all 23, 25 and 29 tone temperaments given in italics due to either not necessarily possessing "real" Ma/Pa or counting "half" status shrutis as full status, thus messing up what the 25 and 29 tone temperaments should technically be
Underlying full status shrutis
Excluding inverses
| Large-small numbers | Status | Generator range | Boundaries of propriety, maximum expressiveness, diatonicity | Large step+Small step |
|---|---|---|---|---|
| 1L 18s | "half" | 18\19 < g < 1 | g = 19\20, 20\21, 21\22 | 18g-17+1-g = 17g-16 |
| 2L 17s | full | 9\19 < g < 1\2 | g = 10\21, 11\23, 12\25 | 17g-8+1-2g = 15g-7 |
| 3L 16s | 6\19 < g < 1\3 | g = 7\22, 8\25, 10\31 | 16g-5+1-3g = 13g-4 | |
| 4L 15s | 14\19 < g < 3\4 | g = 17\23, 20\27, 23\31 | 15g-11+3-4g = 11g-8 | |
| 5L 14s | 15\19 < g < 4\5 | g = 19\24, 23\29, 27\34 | 14g-11+4-5g = 9g-7 | |
| 6L 13s | 3\19 < g < 1\6 | g = 4\25, 5\31, 6/37 | 13g-2+1-6g = 7g-1 | |
| 7L 12s | 8\19 < g < 3\7 | g = 11\26, 14\33, 17\40 | 12g-5+3-7g = 5g-2 | |
| 8L 11s | 7\19 < g < 3\8 | g = 10\27, 13\35, 16\43 | 11g-4+3-8g = 3g-1 | |
| 9L 10s | 2\19 < g < 1\9 | g = 3\28, 4\37, 5\46 | 10g-1+1-9g = g | |
| 10L 9s | 17\19 < g < 9\10 | g = 26\29, 35\39, 44\49 | 9g-8+9-10g = 1-g | |
| 11L 8s | 12\19 < g < 7\11 | g = 19\30, 26\41, 33\52 | 8g-5+7-11g = 2-3g | |
| 12L 7s | 11\19 < g < 7\12 | g = 18\31, 25\43, 32\55 | 7g-4+7-12g = 3-5g | |
| 13L 6s | 16\19 < g < 11\13 | g = 27\32, 38\45, 49\58 | 6g-5+11-13g = 6-7g | |
| 14L 5s | 4\19 < g < 3\14 | g = 7\33, 10\47, 13\61 | 5g-1+3-14g = 2-9g | |
| 15L 4s | 5\19 < g < 4\15 | g = 9\34, 13\49, 17\64 | 4g-1+4-15g = 3-11g | |
| 16L 3s | 13\19 < g < 11\16 | g = 24\35, 35\51, 46\67 | 3g-2+11-16g = 9-13g | |
| 17L 2s | 10\19 < g < 9\17 | g = 19\36, 28\53, 37\70 | 2g-1+9-17g = 8-15g | |
| 18L 1s | "half" | 1\19 < g < 1\18 | g = 2\37, 3\55, 4\73 | g+1-18g = 1-17g |
Including inverses
| Large-small numbers | Status | Generator range | Boundaries of propriety, maximum expressiveness, diatonicity | Large step+Small step |
|---|---|---|---|---|
| 1L22s | "half" | 22\23 < g < 1 | g = 23\24, 24\25, 25\26 | 22g-21+1-g = 21g-20 |
| 2L21s | full | 11\23 < g < 1\2 | g = 12\25, 13\27, 14\29 | 21g-10+1-2g = 19g-9 |
| 3L20s | 15\23 < g < 2\3 | g = 17\26, 19\29, 21\32 | 20g-13+1-3g = 17g-12 | |
| 4L19s | 17\23 < g < 3\4 | g = 20\27, 23\31, 26\35 | 19g-14+3-4g = 15g-11 | |
| 5L18s | 9\23 < g < 2\5 | g = 11\28, 13\33, 15\38 | 18g-7+2-5g = 13g-5 | |
| 6L17s | 19\23 < g < 5\6 | g = 24\29, 29\35, 34\41 | 17g-15+1-6g = 11g-14 | |
| 7L16s | 13\23 < g < 4\7 | g = 17\30, 21\37, 25\44 | 16g-9+4-7g = 9g-5 | |
| 8L15s | 20\23 < g < 7\8 | g = 27\31, 34\39, 41\47 | 15g-13+7-8g = 7g-6 | |
| 9L14s | 5\23 < g < 2\9 | g = 7\32, 9\41, 11\50 | 14g-7+2-9g = 5g-5 | |
| 10L13s | 16\23 < g < 7\10 | g = 23\33, 30\43, 37\53 | 13g-9+7-10g = 3g-2 | |
| 11L12s | 2\23 < g < 1\11 | g = 3\34, 4\45, 5\56 | 12g-1+1-11g = g | |
| 12L11s | 21\23 < g < 11\12 | g = 32\35, 43\47, 54\59 | 11g-10+11-12g = 1-g | |
| 13L10s | 7\23 < g < 4\13 | g = 11\36, 15\49, 19\62 | 10g-3+4-13g =1-3g | |
| 14L9s | 18\23 < g < 11\14 | g = 29\37, 40\51, 51\65 | 9g-7+11-14g = 4-5g | |
| 15L8s | 3\23 < g < 2\15 | g = 5\38, 7\53, 9\68 | 8g-1+2-15g = 1-7g | |
| 16L7s | 10\23 < g < 7\16 | g = 17\39, 24\55, 31\71 | 7g-3+7-16g = 4-9g | |
| 17L6s | 4\23 < g < 3\17 | g = 7\40, 10\57, 13\74 | 6g-1+3-17g = 2-11g | |
| 18L5s | 14\23 < g < 11\18 | g = 25\41, 36\59, 47\77 | 5g-4+11-18g = 7-13g | |
| 19L4s | 6\23 < g < 5\19 | g = 11\42, 16\61, 21\80 | 4g-1+5-19g = 4-15g | |
| 20L3s | 8\23 < g < 7\20 | g = 15\43, 22\63, 29\83 | 3g-1+13-20g = 12-17g | |
| 21L2s | 12\23 < g < 11\21 | g = 23\44, 34\65, 45\86 | 2g-1+11-21g = 10-19g | |
| 22L1s | "half" | 1\23 < g < 1\22 | g = 2\45, 3\67, 4\89 | g+1-22g = 1-221 |
Quoted
Excluding "half" status shrutis
Excluding inverses
| Large-small numbers | Status | Generator range | Boundaries of propriety, maximum expressiveness, diatonicity | Large step+Small step |
|---|---|---|---|---|
| 1L 21s | "half" | 21\22 < g < 1 | g = 22\23, 23\24, 24\25 | 21g-20+1-g = 20g-19 |
| 2L 20s | "3/4" | 10\22 < g < 1\2 | g = 11\24, 12\26, 13\28 | 10g-9\2+1\2-g = 9g-4 |
| 3L 19s | full | 7\22 < g < 1\3 | g = 8\25, 9\28, 10\31 | 19g-6+1-3g = 16g-5 |
| 4L 18s | "3/4" | 5\22 < g < 1\4 | g = 6\26, 7\30, 8\34 | 9g-2+1\2-2g = 7g-3\2 |
| 5L 17s | full | 13\22 < g < 3\5 | g = 16\27, 19\32, 22\37 | 17g-10+3-5g = 12g-7 |
| 6L 16s | "3/4" | 7\22 < g < 2\6 | g = 9\28, 11\34, 13\40 | 8g-5\2+1-3g = 5g-2 |
| 7L 15s | full | 3\22 < g < 1\7 | g = 4\29, 5\36, 6\43 | 15g-2+1-7g = 8g-1 |
| 8L 14s | "3/4" | 8\22 < g < 3\8 | g = 11\30, 14\38, 17\46 | 7g-5\2+3\2-4g = 3g-2 |
| 9L 13s | full | 17\22 < g < 7\9 | g = 24\31, 31\40, 38\49 | 13g-10+7-9g = 4g-3 |
| 10L 12s | "3/4" | 2\22 < g < 1\10 | g = 3\32, 4\42, 5\52 | 6g-1\2+1\2-5g = g |
| 11L 11s | "7/8" | 1\22 < g < 1\11 | g = 2\33, 3\44, 4\55 | g + 1\11-g = 1\11 |
| 12L 10s | "3/4" | 9\22 < g < 5\12 | g = 14\34, 19\46, 24\58 | 5g-2+5\2-6g = 1\2-g |
| 13L 9s | full | 5\22 < g < 3\13 | g = 8\35, 11\48, 14\61 | 9g-2+3-13g = 1-4g |
| 14L 8s | "3/4" | 3\22 < g < 2\14 | g = 5\36, 7\50, 9\64 | 4g-1\2+1-7g = 1\2-3g |
| 15L 7s | full | 19\22 < g < 13\15 | g = 32\37, 45\52, 58\67 | 7g-6+13-15g = 7-8g |
| 16L 6s | "3/4" | 4\22 < g < 3\16 | g = 7\38, 10\54, 13\70 | 3g-1\2+3\2-8g = 1-5g |
| 17L 5s | full | 9\22 < g < 7\17 | g = 16\39, 23\56, 30\73 | 5g-2+7-17g = 5-12g |
| 18L 4s | "3/4" | 6\22 < g < 5\18 | g = 11\40, 16\58, 21\76 | 2g-1\2+5\2-9g = 2-7g |
| 19L 3s | full | 15\22 < g < 13\19 | g = 28\41, 41\60, 54\79 | 3g-2+13-19g = 11-16g |
| 20L 2s | "3/4" | 1\22 < g < 1\20 | g = 2\42, 3\62, 4\72 | g+1\2-10g = 1\2-9g |
| 21L 1s | "half" | 1\22 < g < 1\21 | g = 2\43, 3\64, 4\85 | g+1-21g = 1-20g |
Including inverses
| Large-small numbers | Status | Generator range | Boundaries of propriety, maximum expressiveness, diatonicity | Large step+Small step |
|---|---|---|---|---|
| 1L25s | "half" | 25\26 < g < 1 | g = 26\27, 27\28, 28\29 | 25g-24+1-g = 24g-23 |
| 2L24s | "3/4" | 12\26 < g < 1\2 | g = 13\28, 14\30, 15\32 | 12g-11\2+1\2-g = 11g-5 |
| 3L23s | full | 17\26 < g < 2\3 | g = 19\29, 21\32, 23\35 | 23g-15+2-3g = 20g-13 |
| 4L22s | "3/4" | 6\26 < g < 1\4 | g = 7\30, 8\34, 9\38 | 11g-5\2+1\2-2g = 9g-2 |
| 5L21s | full | 5\26 < g < 1\5 | g = 6\31, 7\36, 8\41 | 21g-4+1-5g = 16g-3 |
| 6L20s | "3/4" | 4\26 < g < 1\6 | g = 5\32, 6\38, 7\44 | 10g-3\2+1\2-3g = 7g-1 |
| 7L19s | full | 11\26 < g < 3\7 | g = 14\33, 17\40, 20\47 | 19g-8+3-7g = 12g-5 |
| 8L18s | "3/4" | 3\26 < g < 1\8 | g = 4\34, 5\42, 6\50 | 9g-1+1\2-4g = 5g-1\2 |
| 9L17s | full | 23\26 < g < 8\9 | g = 31\35, 39\44, 47\53 | 17g-15+8-9g = 8g-7 |
| 10L16s | "3/4" | 5\26 < g < 2\10 | g = 7\36, 9\46, 11\56 | 8g-3\2+1-5g = 3g-1\2 |
| 11L15s | full | 7\26 < g < 3\11 | g = 10\37, 13\48, 16\59 | 15g-4+3-11g = 4g-1 |
| 12L14s | "3/4" | 2\26 < g < 1\12 | g = 3\38, 4\50, 5\62 | 7g-1\2+1\2-6g = g |
| 13L13s | "7/8" | 1\26 < g < 1\13 | g = 2\39, 3\52, 4\65 | g+1\13-g = 1\13 |
| 14L12s | "3/4" | 11\26 < g < 6\14 | g = 17\40, 23\54, 29\68 | 6g-5\2+3-7g = 1\2-g |
| 15L11s | full | 19\26 < g < 11\15 | g = 30\41, 41\56, 52\71 | 11g-8+11-15g = 3-4g |
| 16L10s | "3/4" | 8\26 < g < 5\16 | g = 13\42, 18\58, 23\74 | 5g-3\2+5\2-8g = 1-3g |
| 17L9s | full | 3\26 < g < 2\17 | g = 5\43, 7\60, 9\77 | 9g-1+2-17g = 1-8g |
| 18L8s | "3/4" | 10\26 < g < 7\18 | g = 17\44, 24\62, 31\80 | 4g-7\2+7-9g = 7\2-5g |
| 19L7s | full | 15\26 < g < 11\19 | g = 26\45, 37\64, 48\83 | 7g-4+11-19g = 7-12g |
| 20L6s | "3/4" | 9\26 < g < 7\20 | g = 16\46, 23\66, 30\86 | 3g-1+7\2-10g = 5\2-7g |
| 21L5s | full | 21\26 < g < 17\21 | g = 38\47, 55\68, 72\89 | 5g-4+16-21g = 12-16g |
| 22L4s | "3/4" | 7\26 < g < 6\22 | g = 13\48, 19\70, 25\92 | 2g-1\2+3-11g = 5\2-9g |
| 23L3s | full | 9\26 < g < 8\23 | g = 17\49, 25\72, 33/95 | 3g-1+8-23g = 7-20g |
| 24L2s | "3/4" | 1\26 < g < 1\24 | g = 2\50, 3\74, 4\98 | g+1\2-12g = 1\2-11g |
| 25L1s | "half" | 1\26 < g < 1\25 | g = 2\51, 3\76, 4\101 | g+1-25g = 1-24g |
Including "half" status shrutis
Excluding inverses
| Large-small numbers | Status | Generator range | Boundaries of propriety, maximum expressiveness, diatonicity | Large step+Small step |
|---|---|---|---|---|
| 1L24s | "half" | 24\25 < g < 1 | g = 25\26, 26\27, 27\28 | 24g-23+1-g = 23g-22 |
| 2L23s | full | 12\25 < g < 1\2 | g = 13\27, 14\29, 15\31 | 23g-11+1-2g = 21g-10 |
| 3L22s | 8\25 < g < 1\3 | g = 9\28, 10\31, 11\34 | 22g-7+1-3g = 19g-6 | |
| 4L21s | 6\25 < g < 1\4 | g = 7\29, 8\33, 9\37 | 21g-5+1-4g = 17g-4 | |
| 5L20s | "7/8" | 4\25 < g < 1\5 | g = 5\30, 6\35, 7\40 | 4g-3\5+1\5-g = 3g-2\5 |
| 6L19s | full | 4\25 < g < 1\6 | g = 5\31, 6\37, 7\43 | 19g-3+1-6g = 13g-2 |
| 7L18s | 7\25 < g < 2\7 | g = 9\32, 11\39, 13\46 | 18g-5+2-7g = 11g-3 | |
| 8L17s | 3\25 < g < 1\8 | g = 4\33, 5\41, 6\47 | 17g-2+1-8g = 9g-1 | |
| 9L16s | 11\25 < g < 4\9 | g = 15\34, 19\43, 23\52 | 16g-7+4-9g = 3-7g | |
| 10L15s | "7/8" | 2\25 < g < 1\10 | g = 3\35, 4\45, 5\55 | 3g-1\5+1\5-2g = g |
| 11L14s | full | 9\25 < g < 4\11 | g = 13\36, 17\47, 21\58 | 14g-5+4-11g = 3g-1 |
| 12L13s | 2\25 < g < 1\12 | g = 3\37, 4\49, 5\61 | 13g-1+1-12g = g | |
| 13L12s | 23\25 < g < 12\13 | g = 35\38, 47\51, 59\64 | 12g-11+12-13g = 1-g | |
| 14L11s | 16\25 < g < 9\14 | g = 25\39, 34\53, 43\67 | 11g-7+9-14g = 2-3g | |
| 15L10s | "7/8" | 3\25 < g < 2\15 | g = 5\40, 7\55, 9\70 | 2g-1\5+2\5-3g = 1\5-g |
| 16L9s | full | 14\25 < g < 9\16 | g = 23\41, 32\57, 41\73 | 9g-5+9-16g = 4-7g |
| 17L8s | 22\25 < g < 15\17 | g = 37\42, 52\59, 67\76 | 8g-7+15-17g = 8-9g | |
| 18L7s | 18\25 < g < 13\18 | g = 31\43, 44\61, 57\79 | 7g-5+13-18g = 8-11g | |
| 19L6s | 21\25 < g < 16\19 | g = 37\44, 53\63, 69\82 | 6g-5+16-19g = 11-13g | |
| 20L5s | "7/8" | 1\25 < g < 1\20 | g = 2\45, 3\65, 4\85 | g+1\5-4g = 1\5-3g |
| 21L4s | full | 16\21 < g < 19\25 | g = 35\46, 51\67, 71\88 | 4g-3+16-21g = 13-17g |
| 22L3s | 17\25 < g < 15\22 | g = 32\47, 47\69, 62\91 | 3g-2+15-22g = 13-19g | |
| 23L2s | 13\25 < g < 12\23 | g = 25\48, 37\71, 49\94 | 2g-1+11-23g = 10-21g | |
| 24L1s | "half" | 1\25 < g < 1\24 | g = 2\49, 3\73, 4\97 | g+1-24g = 1-23g |
Including inverses
| Large-small numbers | Status | Generator range | Boundaries of propriety, maximum expressiveness, diatonicity | Large step+Small step |
|---|---|---|---|---|
| 1L28s | "half" | 28\29 < g < 1 | g = 29\30, 30\31, 31\32 | 28g-27+1-g = 27g-26 |
| 2L27s | full | 14\29 < g < 1\2 | g = 15\31, 16\33, 17\35 | 27g-13+1-2g = 25g-12 |
| 3L26s | 19\29 < g < 2\3 | g = 21\32, 23\35, 25\38 | 26g-17+2-3g = 23g-15 | |
| 4L25s | 7\29 < g < 1\4 | g = 8\33, 9\37, 10\41 | 25g-6+1-4g = 21g-5 | |
| 5L24s | 23\29 < g < 4\5 | g = 27\34, 31\39, 35\44 | 24g-19+4-5g = 19g-15 | |
| 6L23s | 24\29 < g < 5\6 | g = 29\35, 34\41, 39\47 | 23g-19+5-6g = 17g-14 | |
| 7L22s | 4\29 < g < 1\7 | g = 5\36, 6\43, 7\50 | 22g-3+1-7g = 15g-2 | |
| 8L21s | 18\29 < g < 5\8 | g = 23\37, 28\45, 33\53 | 21g-13+5-8g = 13g-8 | |
| 9L20s | 16\29 < g < 5\9 | g = 21\38, 26\47, 31\56 | 20g-11+5-9g = 11g-6 | |
| 10L19s | 26\29 < g < 9\10 | g = 35\39, 44\49, 53\59 | 19g-17+9-10g = 9g-8 | |
| 11L18s | 21\29 < g < 8\11 | g = 29\40, 37\51, 45\62 | 18g-13+8-11g = 7g-2 | |
| 12L17s | 12\29 < g < 5\12 | g = 17\41, 22\53, 27\65 | 17g-7+5-12g = 5g-2 | |
| 13L16s | 20\29 < g < 9\13 | g = 29\42, 38\55, 47\68 | 16g+11+9-13g = 3g-2 | |
| 14L15s | 2\29 < g < 1\14 | g = 3\43, 4\57, 5\71 | 15g-1+1-14g = g | |
| 15L14s | 27\29 < g < 14\15 | g = 41\44, 55\59, 69\74 | 14g-13+14-15g = 1-g | |
| 16L13s | 9\29 < g < 5\16 | g = 14\45, 19\61, 24\77 | 13g-4+5-16g = 1-3g | |
| 17L12s | 17\29 < g < 10\17 | g = 27\46, 37\63, 47\80 | 12g-5+7-17g = 2-5g | |
| 18L11s | 8\29 < g < 5\18 | g = 13\47, 18\65, 23\83 | 11g-3+5-18g = 2-7g | |
| 19L10s | 3\29 < g < 2\19 | g = 5\48, 7\67, 9\86 | 10g-1+2-19g = 1-9g | |
| 20L9s | 13\29 < g < 9\20 | g = 22\49, 31\69, 40\89 | 9g-5+9-20g = 4-11g | |
| 21L8s | 11\29 < g < 8\21 | g = 19\50, 27\71, 35\92 | 8g-3+8-21g = 5-13g | |
| 22L7s | 25\29 < g < 19\22 | g = 44\51, 63\73, 82\95 | 7g-6+9-22g = 3-16g | |
| 23L6s | 5\29 < g < 4\23 | g = 9\52, 13\75, 17\98 | 6g-1+4-23g = 3-17g | |
| 24L5s | 6\29 < g < 5\24 | g = 11\53, 16\77, 21\101 | 5g-9+5-24g = 4-19g | |
| 25L4s | 22\29 < g < 19\25 | g = 41\54, 60\79, 79\104 | 4g-3+19-25g = 16-21g | |
| 26L3s | 10\29 < g < 9\26 | g = 19\55, 28\81, 37\107 | 3g-1+9-26g = 8-23g | |
| 27L2s | 15\29 < g < 14\27 | g = 29\56, 43\83, 57\110 | 2g-1+17-27g = 16-25g | |
| 28L1s | "half" | 1\29 < g < 1\28 | g = 2\57, 3\85, 4\113 | g+1-28g = 1-27g |