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| <h2>IMPORTED REVISION FROM WIKISPACES</h2> | | <span style="font-size: 150%;">'''Tuning systems for the qanun'''</span> |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | __FORCETOC__ |
| : This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-08-25 06:03:42 UTC</tt>.<br>
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| : The original revision id was <tt>248375203</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"><span style="font-size: 150%;">**Tuning systems for the qanun**</span>
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| [[toc]]
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| Julien Jalaleddine Weiss, used with permission. | | Julien Jalaleddine Weiss, used with permission. |
| Reference: Pohlit, Stefan. 2011. Julien Jalâl Ed-Dine Weiss – A New Qānūn System: Its Application in the Performance
| |
| Practice of the Ensemble “Al-Kindi” and in Contemporary Western Music. PhD Thesis, MIAM/Istanbul Technical
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| University.
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| The tuning tables on this page are specifically designed for the tuning system of the [[qanun]] (see the link for details on the system of tuning and playing a qanun with mandals/orabs).
| | Reference: Pohlit, Stefan. 2011. Julien Jalâl Ed-Dine Weiss – A New Qānūn System: Its Application in the Performance Practice of the Ensemble “Al-Kindi” and in Contemporary Western Music. PhD Thesis, MIAM/Istanbul Technical University. |
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| The possible pitches of a string obtained via raising/lowering the mandals lie within one [[2187_2048|apotome (2187/2048, 113.7 cents)]]. All systems use 8 mandals, allowing a subdivision of the apotome into 7 parts.
| | Online version of Stefan Pohlit's dissertation: see [http://stefanpohlit.com/dissertation.engl..htm http://stefanpohlit.com/dissertation.engl..htm] |
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| The first rough subdivision of the apotome is always into one [[81_80|syntonic comma (81/80, 21.5 cents)]], one [[25_24|Zarlinian semitone (25/24, 70.7 cents)]] and another syntonic comma. The middle part (25/24, Zarlinian semitone) is then further subdivided into 5 (unequal or equal) parts. The various systems differ mainly in the division of the middle part. | | The tuning tables on this page are specifically designed for the tuning system of the [[qanun|qanun]] (see the link for details on the system of tuning and playing a qanun with mandals/orabs). The logic behind the systems is as follows: |
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| =Explanation of the tuning tables=
| | The empty strings of the qanun are tuned to a pythagorean diatonic scale, with a major third of [[81/64|81/64]], a major sixth of [[27/16|27/16]] and a major seventh of [[243/128|243/128]]. |
| The whole table covers roughly the range of a fourth (the range where the ajnas - maqam tetrachords - reside). Each row corresponds to one string of the qanun. In the first column stands the basic (relative) tuning of a string while the following columns indicate possible intervals that can be reached via raising/lowering the mandals. | |
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| The first table contains the cent values and the second the just intervals, sometimes differing between ascending and descending ratios. The middle part (which is the one that is mainly varying among the systems) is marked in **bold**. | | The possible pitches of a string obtained via raising/lowering the mandals lie within two [[2187/2048|apotomes (2187/2048, 113.7 cents)]]. The base note is assumed in the middle. The mandals allow raising and lowering this note by maximally one apotome. |
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| |
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| Any given configuration of mandal positions, resulting in a certain set of pitches that can be played at a given time
| | Each apotome is divided into 7 unequal parts, which requires 14 mandals per string. The first rough subdivision of the apotome is always into one [[81/80|syntonic comma (81/80, 21.5 cents)]], one [[25/24|Zarlinian semitone (25/24, 70.7 cents)]] and another syntonic comma. The middle part (25/24, Zarlinian semitone) is then further subdivided into 5 (unequal or equal) parts. The various systems differ mainly in the division of the middle part. |
| (base for a maqam tetrachord) is represented by a choice of one cell in each row.
| | |
| | The tuning systems are all described by a series of cent values, which describe the subdivision of one apotome. According to the system sketched above, the first and the last value are always 22 cents (or 21.5 cents). This subdivision pattern occurs twice on each string, altogether 14 times per octave. This is followed by listings of some important rational intervals that are possible in this tuning, mainly in the range of a fourth (the range where the ajnas - maqam [[tetrachord|tetrachords]] - reside), |
| | |
| | An notable property (of all systems) is that the second-highest mandal position of, say, the C string is 114-22=92 cents (the [[135/128|major limma]]), while the lowest mandal position on the following string (D in the example) is 214 (one wholetone above C) - 114 = 90 cents (the [[256/243|pythagorean limma]], the same interval as between E and F) - we have two notes differing by one [[32805/32768|schisma (2 cents)]]. So the interval of the schisma is present and can be played on a qanun in any of the tuning systems described here. |
| | |
| | =Notation= |
| | The notes without accidentals stand for the pythagorean intervals of the base tuning of the qanun. Raising a pitch by an apotome is notated with "#", lowering a pitch by the same amount is notated with "b". Sharps are higher than flats (unlike in [[Meantone|meantone]] systems): C# is one apotome (114 cents) above C, while Db is 9/8 (214 cents) minus one apotome = 90 cents. Both properties indicate that the framework is essentially pythagorean. The tuning system as a whole, however, is not. |
| | |
| | For the steps in between, additional symbols are used - altogether 7 symbols for raising pitches and 7 for lowering pitches. |
| | |
| | This gives 15 potential different pitches per base note, corresponding to the mandals. Seven base notes (C, D, E, F, G, A, B or Do, Re, Mi, Fa, Sol, La, Si), corresponding to the strings, lead to a notation system of 7*15=105 pitches, in accordance with the real playing capabilities of the qanun. See the following document, which also gives all the pitches in one octave (in ratios and cents) that can be played by system 1 and 2. |
| | |
| | [[:File:Tableaux_JJW_VIII-2011.pdf|Tableaux JJW VIII-2011.pdf]] |
| | |
| | (used with permission J. J. Weiss/S. Pohlit) |
| | |
| | =System 1= |
| | © J.J.Weiss. Luthier: Ejder Gulec. |
|
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|
| =Older systems=
| |
| ==First System J.J.Weiss==
| |
| Luthier: Ejder Gulec.
| |
| Subdivision of 25/24 into 65/64 (26 cents), 144/143 (12 cents) and 55/54 (32 cents). | | Subdivision of 25/24 into 65/64 (26 cents), 144/143 (12 cents) and 55/54 (32 cents). |
| 65/64 and 55/54 are each split into two.
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|
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|
| This gives the following interval positions of the mandals: 22, 13, 13, 12, 16, 16, 22 cents. | | 65/64 and 55/54 are each split into two roughly equal parts. |
| | |
| | This gives the following rational intervals between the mandals: |
| | |
| | 81/80, 245/243, 3159/3136, 144/143, 121/120, 100/99, 81/80 |
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| Interval table (cents):
| | In cents (approximations): |
| || 0 || 22 || **35** || **48** || **60** || **76** || 92 || 114 ||
| |
| || 90 || 112 || **125** || **138** || **150** || **166** || 182 || 204 ||
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| || 294 || 316 || **330** || **342** || **354** || **370** || 386 || 408 ||
| |
| || 384 || 406 || **420** || **432** || **444** || **460** || 476 || 498 ||
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| || 498 || 520 || **533** || **546** || **558** || **574** || 590 || 612 ||
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|
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| Interval table of just intervals (ascending, descending):
| | 22, 13, 13, 12, 16, 16, 22 |
| || 1/1 || 81/80 || **49/48** || **1053/1024** || **729/704** || **2673/2560** || 135/128 || 2187/2048 ||
| |
| || 256/243 || 16/15 || **784/729, 128/119, 43/40 (asc.)**
| |
| **320/297 (desc.)** || **13/12 (asc.)**
| |
| **88/81 (desc.)** || **12/11 (asc.)**
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| **128/117 (desc.)** || **11/10, 208/189 (asc.)**
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| **54/49 (desc.)** || 10/9 || 9/8 ||
| |
| || 32/27 || 6/5 || **98/81 (asc.)**
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| **40/33 (desc.)** || **39/32 (asc.)**
| |
| **11/9 (desc.)** || **27/22 (asc.)**
| |
| **16/13 (desc.)** || **99/80, 26/21 (asc.)**
| |
| **243/196, XXX (desc.)** || 5/4 || 81/64 ||
| |
| || 8192/6581 || XXX || **25088/19683** || **104/81** || **XXX** || **176/135** || 320/243 || 4/3 ||
| |
| || XXX || 27/20 || **351/258** || **XXX** || **243/176** || **891/640** || 45/32 || 729/512 ||
| |
|
| |
|
| ==Variant with 128/119==
| | Rational intervals each string can be detuned (approximations in cents in parentheses): |
| 128/119: 126.2 cents
| |
| XXX
| |
|
| |
|
| ==Variant with 128/119 ascending/descending==
| | 81/80 (22), 49/48 (35), 1053/1024 (48), 729/704 (60), 2673/2560 (76), 135/128 (92), 2187/2048 (114) |
| XXX
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|
| |
|
| ==Variant with 43/40 ascending/descending==
| | Intervals ratios, ascending from C: |
| 43/40: 125.2 cents
| |
| XXX
| |
|
| |
|
| =Newer systems=
| | <ul><li>On the D string (from Db to D): |
| ==System 2, better suited for ottoman maqams==
| |
| J.J. Weiss, Qanun no. 9. Luthier: Kenan Ozten.
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|
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|
| Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22|16|15|12|13|14|22</span>
| | 256/243 (90), 16/15 (112), '''784/729 (126)''', '''13/12 (138)''', '''12/11 (150)''', '''11/10 (166)''', 10/9 (182), 9/8 (204)</li><li>On the E string (from Eb to E): |
|
| |
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| Interval table (cents):
| | 32/27 (294), 6/5 (316), '''98/81 (330)''', '''39/32 (342)''', '''27/22 (354)''', '''99/80 (370)''', 5/4 (386), 81/64 (408)</li></ul> |
| || 0 || 22 || **38** || **54** || **66** || **78** || 92 || 114 ||
| |
| || 90 || 112 || **128** || **143** || **156** || **168** || 182 || 204 ||
| |
| || 294 || 316 || **332** || **348** || **360** || **372** || 386 || 408 ||
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|
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| Interval table of just intervals: | | Interval ratios, descending from F: |
| XXX
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|
| |
|
| ==Symmetrical and super-symmetrical models==
| | <ul><li>On the E string (from Eb to E): |
| XXX
| |
| Characteristics of super-symmetric systems: no difference between ascending and descending ratios.
| |
|
| |
|
| ===Symmetrical model===
| | 9/8 (204), 10/9 (182), '''54/49 (169)''', '''128/117 (156)''', '''88/81 (144)''', '''320/297 (129)''', 16/15 (112), 256/243 (90)</li><li>On the D string (from Db to D): |
| J.J. Weiss
| |
| Advantage: marked contrast between Segah of Ushaq and Segah of arabic Rast.
| |
| Equal division of 65/54 (320.98 cents)
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|
| |
|
| Mandal positions (cents): <span style="font-family: Tahoma;">22|13|13|1</span><span style="color: #00000a; font-family: Tahoma;">8</span><span style="font-family: Tahoma;">|13|13|22</span>
| | 81/64 (408), 5/4 (386), '''243/196 (372)''', '''16/13 (360)''', '''11/9 (348)''', '''40/33 (333)''', 6/5 (316), 32/27 (294)</li></ul> |
|
| |
|
| Interval table (cents):
| | A complete list of all intervals available within one octave can be found in the above-mentioned [[:File:Tableaux_JJW_VIII-2011.pdf|document]] (on the first page). |
| || 0 || 22 || **35** || **48** || **66** || **79** || 92 || 114 ||
| |
| || 90 || 112 || **125** || **138** || **156** || **169** || 182 || 204 ||
| |
| || 294 || 316 || **329** || **342** || **360** || **373** || 386 || 408 ||
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|
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| Interval table of just intervals:
| | =System 2, better suited for ottoman maqams= |
| XXX
| | © J.J. Weiss. Qanun no. 9, luthier: Kenan Ozten. |
|
| |
|
| ===Super-symmetrical model===
| | Mandal positions in ratios: |
| J.J. Weiss
| |
| Equal division of 65/54 (320.98 cents) and its inverse. (XXX Is that formulation correct?)
| |
| Ascending/descending with 43/40 (125.2 cents).
| |
|
| |
|
| Mandal positions (cents): <span style="font-family: Tahoma;">22|13|13|1</span><span style="color: #00000a; font-family: Tahoma;">8</span><span style="font-family: Tahoma;">|13|13</span><span style="color: #00000a; font-family: Tahoma;">|</span><span style="font-family: Tahoma;">22</span>
| | 81/80, 105/104, 572/567, 144/143, 1547/1536, 120/119, 81/80 |
|
| |
|
| Interval table (cents): same as in previous model.
| | In cents (approximations): |
|
| |
|
| Interval table of just intervals:
| | <span style="color: #00000a; font-family: Tahoma;">22|16|15|12|13|14|22</span> |
| XXX
| |
|
| |
|
| ===Unequal division of 65/54===
| | Rational intervals each string can be detuned (approximations in cents in parentheses): |
| J.J. Weiss
| |
|
| |
|
| XXX
| | 81/80 (22), 1701/1664 (38), 33/32 (54), 27/26 (66), 243/232 (78), 135/128 (92), 2187/2048 (114) |
|
| |
|
| ===Equal division of the Zarlinian semitone=== | | Intervals ratios, ascending from C: |
| J.J. Weiss | | |
| This is the simplest variant for luthiers...
| | <ul><li>On the D string (from Db to D): |
| | |
| | 256/243 (90), 16/15 (112), '''14/13 (128), 88/81 (144), 128/117 or 35/32 (156), 119/108 (168)''', 10/9 (182), 9/8 (204)</li><li>On the E string (from Eb to E): |
| | |
| | 32/27 (294), 6/5 (316), '''63/52 (332), 11/9 (348), 16/13 or 315/256 (360), 119/96 (372)''', 5/4 (386), 81/64 (408)</li></ul> |
| | |
| | Interval ratios descending from F: |
| | |
| | <ul><li>On the E string (from Eb to E): |
| | |
| | 9/8 (204), 10/9 (182), '''208/189 (166), 12/11 (150), 13/12 (138), 128/119 (126)''', 16/15 (112), 256/243 (90)</li><li>On the D string (from Db to D): |
| | |
| | 81/64 (408), 5/4 (386), '''26/21 (370), 27/22 (354), 39/32 (342), 144/119 (330)''', 6/5 (316), 32/27 (294)</li></ul> |
| | |
| | A complete list of all intervals available within one octave can be found in the above-mentioned [[:File:Tableaux_JJW_VIII-2011.pdf|document]] (on the second page). |
| | |
| | =Other models= |
| | Julien Weiss has developed a number of other systems besides the two described above. A notable class of these are so-called super-symmetrical systems, which have the property that the intervals ascending from C and the intervals descending from F (which show slight differences in the previous two systems, marked in '''bold''' above) are the same. |
| | |
| | 3 examples are described below. For more and detailed descriptions see chapter 3.4 and appendix I in [http://stefanpohlit.com/dissertation.engl..htm Stefan Pohlit's dissertation] . |
| | |
| | ==Super-symmetric model with non-aliquot division of 65/64== |
| | © J.J. Weiss |
| | |
| | Similar to [[tuning_systems_for_qanun#System 1|system 1]], but with 65/64 (26.84 cents) divided into two non-equal parts (14 and 12 cents instead of 13 and 13). |
| | |
| | Mandal positions in ratios: |
| | |
| | 81/80 (22), 120/119 (14), 1547/1536 (12), <span style="color: #00000a; font-family: 'Tahoma','sans-serif';">512/507 (17), </span>1547/1536 (12), 120/119 (14)<span style="color: #00000a; font-family: 'Tahoma','sans-serif';">, 81/80 (22)</span> |
| | |
| | Table of pitches from C to F (approximations in cents): |
|
| |
|
| Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22|14|14|14|14|14|22</span>
| | {| class="wikitable" |
| | |- |
| | ! | String |
| | ! | b |
| | ! | |
| | ! | |
| | ! | |
| | ! | |
| | ! | |
| | ! | |
| | ! | Base note |
| | ! | |
| | ! | |
| | ! | |
| | ! | |
| | ! | |
| | ! | |
| | ! | # |
| | |- |
| | ! | C |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | style="text-align:center;" | 0 |
| | | | 22 |
| | | | 36 |
| | | | 48 |
| | | | 66 |
| | | | 78 |
| | | | 92 |
| | | | 114 |
| | |- |
| | ! | D |
| | | | 90 |
| | | | 112 |
| | | | 126 |
| | | | 138 |
| | | | 156 |
| | | | 168 |
| | | | 182 |
| | | style="text-align:center;" | 204 |
| | | | 226 |
| | | | 240 |
| | | | 252 |
| | | | 270 |
| | | | 282 |
| | | | 296 |
| | | | 318 |
| | |- |
| | ! | E |
| | | | 294 |
| | | | 316 |
| | | | 330 |
| | | | 342 |
| | | | 360 |
| | | | 372 |
| | | | 386 |
| | | style="text-align:center;" | 408 |
| | | | 430 |
| | | | 444 |
| | | | 456 |
| | | | 474 |
| | | | 486 |
| | | | 500 |
| | | | 522 |
| | |- |
| | ! | F |
| | | | 384 |
| | | | 406 |
| | | | 420 |
| | | | 433 |
| | | | 450 |
| | | | 462 |
| | | | 476 |
| | | style="text-align:center;" | 498 |
| | | | 520 |
| | | | 534 |
| | | | 546 |
| | | | 464 |
| | | | 476 |
| | | | 590 |
| | | | 612 |
| | |} |
|
| |
|
| Interval table (cents): | | Interval ratios, ascending from C: |
| || 0 || 22 || **36** || **50** || **64** || **76** || 92 || 114 ||
| |
| || 90 || 112 || **126** || **140** || **154** || **168** || 182 || 204 ||
| |
| || 294 || 316 || **330** || **344** || **358** || **372** || 386 || 408 ||
| |
|
| |
|
| Interval table of just intervals (ascending, descending):
| | <ul><li>On the D string (from Db to D): |
| || 1/1 || 81/80 || || || || || 135/128 || 2187/2048 ||
| |
| || 256/243 || 16/15 || **100/93 (asc.)**
| |
| **242/225 (desc.)** || **400/369 (asc.)**
| |
| **248/225 (desc.)** || **200/183 (asc.)**
| |
| **82/75 (desc.)** || **400/363 (asc.)**
| |
| **248/225 (desc.)** || 10/9 || 9/8 ||
| |
| || 32/27 || 6/5 || **75/62 (asc.)**
| |
| **121/100 (desc.)** || **50/41 (asc.)**
| |
| **61/50 (desc.)** || **75/61 (asc.)**
| |
| **49/40 (desc.)** || **150/121 (asc.)**
| |
| **31/25 (desc.)** || 5/4 || 81/64 ||
| |
|
| |
|
| ===Ascending/descending with 54/49===
| | 245/243 (90), 16/15 (112), '''128/119 (126), 13/12 (138), 128/117 (156), 119/108 (168),''' 10/9 (182), 9/8 (204) |
| J.J. Weiss
| |
| [[54_49|54/49]]: 168.2 cents, Zalzal's mujannab (Al Farabi)
| |
|
| |
|
| Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22|14|17|13|13|13|22</span>
| | On the E string (from Eb to E): |
| <span style="color: #00000a; font-family: Tahoma;">XXX</span> There seem to be typos here...
| |
|
| |
|
| Interval table (cents):
| | 32/27 (294), 6/5 (316), '''144/119 (330), 39/32 (342), 16/13 (360), 119/96 (372),''' 5/4 (386), 81/64 (408)</li></ul> |
| || 0 || 22 || **36** || **48** || **60** || **76** || 92 || 114 ||
| |
| || 90 || 112 || **126** || **138** || **156** || **168** || 182 || 204 ||
| |
| || 294 || 316 || **330** || || **360** || **374** || 386 || 408 ||
| |
|
| |
|
| Interval table of just intervals XXX | | Interval ratios, descending from F: |
|
| |
|
| ===Ascending/descending with 14/13===
| | <ul><li>On the E string (from Eb to E): |
| J.J. Weiss
| |
| <span style="color: #00000a; font-family: Tahoma;">14/13: 128.3 cents</span>
| |
| <span style="color: #00000a; font-family: Tahoma;">XXX</span>
| |
|
| |
|
| Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22, 16, 13, 12, 13, 16, 22</span>
| | 9(8 /204), 10/9 (182), '''119/108 (168), 128/117 (156), 13/12 (138), 128/119 (126)''', 16/15 (112), 256/243 (90)</li><li>On the D string (from Db to D) |
|
| |
|
| Interval table XXX
| | X81/64 (408), 5/4 (386), '''119/96 (372), 16/13 (360), 39/32 (342), 144/119 (330)''', 6/5 (316), 32/27 (294)</li></ul> |
|
| |
|
| ===Ascending/descending with 11/10===
| | Ascending and descending intervals are indeed the same, which is what "super-symmetrical" means in this context. |
| J.J. Weiss
| |
| 11/10: 165 cents
| |
|
| |
|
| Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22|16|15|13|13|13|22</span>
| | ==Equal division of the Zarlinian semitone== |
| | © J.J. Weiss |
|
| |
|
| Interval table XXX
| | This is the simplest variant for luthiers... |
|
| |
|
| ===Ascending/descending with 35/32=== | | Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;"> 22|14|14|14|14|14|22 </span> |
| J.J. Weiss
| |
| 35/32: 155.14 cents
| |
| XXX
| |
|
| |
|
| ==System Jacques Dudon (2006)==
| | Mandal positions in ratios: |
|
| |
|
| Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">21,5 | 14,5 | 14,5 | 14,5 | 15 | 12 | 21,5</span>
| | <span style="color: #00000a; font-family: Tahoma;">81/80, 125/124, 124/123, 123/122, 122/121, 121/120, 81/80</span> |
|
| |
|
| Interval table (cents):
| | Table of pitches from C to F (approximations in cents): |
|
| |
|
| Interval table of just intervals XXX
| | {| class="wikitable" |
| | |- |
| | ! | String |
| | ! | b |
| | ! | |
| | ! | |
| | ! | |
| | ! | |
| | ! | |
| | ! | |
| | ! | Base note |
| | ! | |
| | ! | |
| | ! | |
| | ! | |
| | ! | |
| | ! | |
| | ! | # |
| | |- |
| | ! | C |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | style="text-align:center;" | 0 |
| | | | 22 |
| | | | 36 |
| | | | 50 |
| | | | 64 |
| | | | 78 |
| | | | 92 |
| | | | 114 |
| | |- |
| | ! | D |
| | | | 90 |
| | | | 112 |
| | | | 126 |
| | | | 140 |
| | | | 154 |
| | | | 168 |
| | | | 182 |
| | | style="text-align:center;" | 204 |
| | | | 226 |
| | | | 240 |
| | | | 254 |
| | | | 268 |
| | | | 282 |
| | | | 296 |
| | | | 318 |
| | |- |
| | ! | E |
| | | | 294 |
| | | | 316 |
| | | | 330 |
| | | | 344 |
| | | | 358 |
| | | | 372 |
| | | | 386 |
| | | style="text-align:center;" | 408 |
| | | | 430 |
| | | | 444 |
| | | | 458 |
| | | | 472 |
| | | | 486 |
| | | | 500 |
| | | | 522 |
| | |- |
| | ! | F |
| | | | 384 |
| | | | 406 |
| | | | 420 |
| | | | 434 |
| | | | 448 |
| | | | 462 |
| | | | 476 |
| | | style="text-align:center;" | 498 |
| | | | 520 |
| | | | 534 |
| | | | 548 |
| | | | 562 |
| | | | 576 |
| | | | 590 |
| | | | 612 |
| | |} |
|
| |
|
| ===Arithmetic system===
| | Interval ratios, ascending from C: |
|
| |
|
| Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">21,5 | 12 | 15 | 14,5 | 14,5 | 14,5 | 21,5 </span>
| | <ul><li>On the D string (from Db to D): |
|
| |
|
| | 256/243 (90), 16/15 (112), '''100/93 (126), 400/369 (140), 200/183 (153.78), 400/363 (168).''' 10/9 (182), 9/8 (204)</li><li>On the E string (from Eb to E): |
|
| |
|
| Interval table XXX</pre></div>
| | 32/27 (294), 6/5 (316), '''75/62 (329.54), 50/41 (343.56), 75/61 (357.69), 150/121 (371.94),''' 5/4 (386), 81/64 (408)</li></ul> |
| <h4>Original HTML content:</h4>
| |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>tuning systems for qanun</title></head><body><span style="font-size: 150%;"><strong>Tuning systems for the qanun</strong></span><br />
| |
| <!-- ws:start:WikiTextTocRule:38:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 1em;"><a href="#Explanation of the tuning tables">Explanation of the tuning tables</a></div>
| |
| <!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><div style="margin-left: 1em;"><a href="#Older systems">Older systems</a></div>
| |
| <!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: 2em;"><a href="#Older systems-First System J.J.Weiss">First System J.J.Weiss</a></div>
| |
| <!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: 2em;"><a href="#Older systems-Variant with 128/119">Variant with 128/119</a></div>
| |
| <!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><div style="margin-left: 2em;"><a href="#Older systems-Variant with 128/119 ascending/descending">Variant with 128/119 ascending/descending</a></div>
| |
| <!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 2em;"><a href="#Older systems-Variant with 43/40 ascending/descending">Variant with 43/40 ascending/descending</a></div>
| |
| <!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 1em;"><a href="#Newer systems">Newer systems</a></div>
| |
| <!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><div style="margin-left: 2em;"><a href="#Newer systems-System 2, better suited for ottoman maqams">System 2, better suited for ottoman maqams</a></div>
| |
| <!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><div style="margin-left: 2em;"><a href="#Newer systems-Symmetrical and super-symmetrical models">Symmetrical and super-symmetrical models</a></div>
| |
| <!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 3em;"><a href="#Newer systems-Symmetrical and super-symmetrical models-Symmetrical model">Symmetrical model</a></div>
| |
| <!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 3em;"><a href="#Newer systems-Symmetrical and super-symmetrical models-Super-symmetrical model">Super-symmetrical model</a></div>
| |
| <!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 3em;"><a href="#Newer systems-Symmetrical and super-symmetrical models-Unequal division of 65/54">Unequal division of 65/54</a></div>
| |
| <!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 3em;"><a href="#Newer systems-Symmetrical and super-symmetrical models-Equal division of the Zarlinian semitone">Equal division of the Zarlinian semitone</a></div>
| |
| <!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><div style="margin-left: 3em;"><a href="#Newer systems-Symmetrical and super-symmetrical models-Ascending/descending with 54/49">Ascending/descending with 54/49</a></div>
| |
| <!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><div style="margin-left: 3em;"><a href="#Newer systems-Symmetrical and super-symmetrical models-Ascending/descending with 14/13">Ascending/descending with 14/13</a></div>
| |
| <!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><div style="margin-left: 3em;"><a href="#Newer systems-Symmetrical and super-symmetrical models-Ascending/descending with 11/10">Ascending/descending with 11/10</a></div>
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| <!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><div style="margin-left: 3em;"><a href="#Newer systems-Symmetrical and super-symmetrical models-Ascending/descending with 35/32">Ascending/descending with 35/32</a></div>
| |
| <!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><div style="margin-left: 2em;"><a href="#Newer systems-System Jacques Dudon (2006)">System Jacques Dudon (2006)</a></div>
| |
| <!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><div style="margin-left: 3em;"><a href="#Newer systems-System Jacques Dudon (2006)-Arithmetic system">Arithmetic system</a></div>
| |
| <!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --></div>
| |
| <!-- ws:end:WikiTextTocRule:58 -->Julien Jalaleddine Weiss, used with permission.<br />
| |
| Reference: Pohlit, Stefan. 2011. Julien Jalâl Ed-Dine Weiss – A New Qānūn System: Its Application in the Performance<br />
| |
| Practice of the Ensemble “Al-Kindi” and in Contemporary Western Music. PhD Thesis, MIAM/Istanbul Technical<br />
| |
| University.<br />
| |
| <br />
| |
| The tuning tables on this page are specifically designed for the tuning system of the <a class="wiki_link" href="/qanun">qanun</a> (see the link for details on the system of tuning and playing a qanun with mandals/orabs).<br />
| |
| <br />
| |
| The possible pitches of a string obtained via raising/lowering the mandals lie within one <a class="wiki_link" href="/2187_2048">apotome (2187/2048, 113.7 cents)</a>. All systems use 8 mandals, allowing a subdivision of the apotome into 7 parts.<br />
| |
| <br />
| |
| The first rough subdivision of the apotome is always into one <a class="wiki_link" href="/81_80">syntonic comma (81/80, 21.5 cents)</a>, one <a class="wiki_link" href="/25_24">Zarlinian semitone (25/24, 70.7 cents)</a> and another syntonic comma. The middle part (25/24, Zarlinian semitone) is then further subdivided into 5 (unequal or equal) parts. The various systems differ mainly in the division of the middle part.<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Explanation of the tuning tables"></a><!-- ws:end:WikiTextHeadingRule:0 -->Explanation of the tuning tables</h1>
| |
| The whole table covers roughly the range of a fourth (the range where the ajnas - maqam tetrachords - reside). Each row corresponds to one string of the qanun. In the first column stands the basic (relative) tuning of a string while the following columns indicate possible intervals that can be reached via raising/lowering the mandals.<br />
| |
| <br />
| |
| The first table contains the cent values and the second the just intervals, sometimes differing between ascending and descending ratios. The middle part (which is the one that is mainly varying among the systems) is marked in <strong>bold</strong>.<br />
| |
| <br />
| |
| Any given configuration of mandal positions, resulting in a certain set of pitches that can be played at a given time<br />
| |
| (base for a maqam tetrachord) is represented by a choice of one cell in each row.<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Older systems"></a><!-- ws:end:WikiTextHeadingRule:2 -->Older systems</h1>
| |
| <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Older systems-First System J.J.Weiss"></a><!-- ws:end:WikiTextHeadingRule:4 -->First System J.J.Weiss</h2>
| |
| Luthier: Ejder Gulec.<br />
| |
| Subdivision of 25/24 into 65/64 (26 cents), 144/143 (12 cents) and 55/54 (32 cents).<br />
| |
| 65/64 and 55/54 are each split into two.<br />
| |
| <br />
| |
| This gives the following interval positions of the mandals: 22, 13, 13, 12, 16, 16, 22 cents.<br />
| |
| <br />
| |
| Interval table (cents):<br />
| |
|
| |
|
| | Interval ratios descending from F: |
|
| |
|
| <table class="wiki_table">
| | <ul><li>On the E string (from Eb to E): |
| <tr>
| |
| <td>0<br />
| |
| </td>
| |
| <td>22<br />
| |
| </td>
| |
| <td><strong>35</strong><br />
| |
| </td>
| |
| <td><strong>48</strong><br />
| |
| </td>
| |
| <td><strong>60</strong><br />
| |
| </td>
| |
| <td><strong>76</strong><br />
| |
| </td>
| |
| <td>92<br />
| |
| </td>
| |
| <td>114<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>90<br />
| |
| </td>
| |
| <td>112<br />
| |
| </td>
| |
| <td><strong>125</strong><br />
| |
| </td>
| |
| <td><strong>138</strong><br />
| |
| </td>
| |
| <td><strong>150</strong><br />
| |
| </td>
| |
| <td><strong>166</strong><br />
| |
| </td>
| |
| <td>182<br />
| |
| </td>
| |
| <td>204<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>294<br />
| |
| </td>
| |
| <td>316<br />
| |
| </td>
| |
| <td><strong>330</strong><br />
| |
| </td>
| |
| <td><strong>342</strong><br />
| |
| </td>
| |
| <td><strong>354</strong><br />
| |
| </td>
| |
| <td><strong>370</strong><br />
| |
| </td>
| |
| <td>386<br />
| |
| </td>
| |
| <td>408<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>384<br />
| |
| </td>
| |
| <td>406<br />
| |
| </td>
| |
| <td><strong>420</strong><br />
| |
| </td>
| |
| <td><strong>432</strong><br />
| |
| </td>
| |
| <td><strong>444</strong><br />
| |
| </td>
| |
| <td><strong>460</strong><br />
| |
| </td>
| |
| <td>476<br />
| |
| </td>
| |
| <td>498<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>498<br />
| |
| </td>
| |
| <td>520<br />
| |
| </td>
| |
| <td><strong>533</strong><br />
| |
| </td>
| |
| <td><strong>546</strong><br />
| |
| </td>
| |
| <td><strong>558</strong><br />
| |
| </td>
| |
| <td><strong>574</strong><br />
| |
| </td>
| |
| <td>590<br />
| |
| </td>
| |
| <td>612<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| <br />
| | 9/8, 10/9, '''248/225 (168.49), 82/75 (154.47), 244/225 (140.34), 242/225 (126.09'''), 16/15, 256/243or approximating ratios: XXX</li><li>On the D string (from Db to D): |
| Interval table of just intervals (ascending, descending):<br />
| |
|
| |
|
| | 81/64, 5/4, '''31/25 (372.40), 49/40 (351.33), 61/50 (344.25), 121/100 (330)''', 6/5, 32/27Or approximatiing ratios: XXX</li></ul> |
|
| |
|
| <table class="wiki_table">
| | ==Super-symmetrical model with 14/13== |
| <tr>
| | © J.J. Weiss |
| <td>1/1<br />
| |
| </td>
| |
| <td>81/80<br />
| |
| </td>
| |
| <td><strong>49/48</strong><br />
| |
| </td>
| |
| <td><strong>1053/1024</strong><br />
| |
| </td>
| |
| <td><strong>729/704</strong><br />
| |
| </td>
| |
| <td><strong>2673/2560</strong><br />
| |
| </td>
| |
| <td>135/128<br />
| |
| </td>
| |
| <td>2187/2048<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>256/243<br />
| |
| </td>
| |
| <td>16/15<br />
| |
| </td>
| |
| <td><strong>784/729, 128/119, 43/40 (asc.)</strong><br />
| |
| <strong>320/297 (desc.)</strong><br />
| |
| </td>
| |
| <td><strong>13/12 (asc.)</strong><br />
| |
| <strong>88/81 (desc.)</strong><br />
| |
| </td>
| |
| <td><strong>12/11 (asc.)</strong><br />
| |
| <strong>128/117 (desc.)</strong><br />
| |
| </td>
| |
| <td><strong>11/10, 208/189 (asc.)</strong><br />
| |
| <strong>54/49 (desc.)</strong><br />
| |
| </td>
| |
| <td>10/9<br />
| |
| </td>
| |
| <td>9/8<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>32/27<br />
| |
| </td>
| |
| <td>6/5<br />
| |
| </td>
| |
| <td><strong>98/81 (asc.)</strong><br />
| |
| <strong>40/33 (desc.)</strong><br />
| |
| </td>
| |
| <td><strong>39/32 (asc.)</strong><br />
| |
| <strong>11/9 (desc.)</strong><br />
| |
| </td>
| |
| <td><strong>27/22 (asc.)</strong><br />
| |
| <strong>16/13 (desc.)</strong><br />
| |
| </td>
| |
| <td><strong>99/80, 26/21 (asc.)</strong><br />
| |
| <strong>243/196, XXX (desc.)</strong><br />
| |
| </td>
| |
| <td>5/4<br />
| |
| </td>
| |
| <td>81/64<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>8192/6581<br />
| |
| </td>
| |
| <td>XXX<br />
| |
| </td>
| |
| <td><strong>25088/19683</strong><br />
| |
| </td>
| |
| <td><strong>104/81</strong><br />
| |
| </td>
| |
| <td><strong>XXX</strong><br />
| |
| </td>
| |
| <td><strong>176/135</strong><br />
| |
| </td>
| |
| <td>320/243<br />
| |
| </td>
| |
| <td>4/3<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>XXX<br />
| |
| </td>
| |
| <td>27/20<br />
| |
| </td>
| |
| <td><strong>351/258</strong><br />
| |
| </td>
| |
| <td><strong>XXX</strong><br />
| |
| </td>
| |
| <td><strong>243/176</strong><br />
| |
| </td>
| |
| <td><strong>891/640</strong><br />
| |
| </td>
| |
| <td>45/32<br />
| |
| </td>
| |
| <td>729/512<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| <br />
| | The idea behind this system is as follows: |
| <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Older systems-Variant with 128/119"></a><!-- ws:end:WikiTextHeadingRule:6 -->Variant with 128/119</h2>
| |
| 128/119: 126.2 cents<br />
| |
| XXX<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Older systems-Variant with 128/119 ascending/descending"></a><!-- ws:end:WikiTextHeadingRule:8 -->Variant with 128/119 ascending/descending</h2>
| |
| XXX<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Older systems-Variant with 43/40 ascending/descending"></a><!-- ws:end:WikiTextHeadingRule:10 -->Variant with 43/40 ascending/descending</h2>
| |
| 43/40: 125.2 cents<br />
| |
| XXX<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Newer systems"></a><!-- ws:end:WikiTextHeadingRule:12 -->Newer systems</h1>
| |
| <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Newer systems-System 2, better suited for ottoman maqams"></a><!-- ws:end:WikiTextHeadingRule:14 -->System 2, better suited for ottoman maqams</h2>
| |
| J.J. Weiss, Qanun no. 9. Luthier: Kenan Ozten.<br />
| |
| <br />
| |
| Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22|16|15|12|13|14|22</span><br />
| |
| <br />
| |
| Interval table (cents):<br />
| |
|
| |
|
| | Dividing the apotome (114 cents) into 3 equal parts gives 38 cents, and adding this to the pythagorean limma (90 cents) gives 128 cents, which is an approximation for [[14/13|14/13]] (two-third tone, a favorite interval of [http://en.wikipedia.org/wiki/Avicenna Avicenna/Ibn Sina]). |
|
| |
|
| <table class="wiki_table">
| | The division of the apotome derived from this combines the known basic division into apotome, Zarlinian semitone and apotome with an equal division into 3 parts, which yields the following mandal positions (cents): |
| <tr>
| |
| <td>0<br />
| |
| </td>
| |
| <td>22<br />
| |
| </td>
| |
| <td><strong>38</strong><br />
| |
| </td>
| |
| <td><strong>54</strong><br />
| |
| </td>
| |
| <td><strong>66</strong><br />
| |
| </td>
| |
| <td><strong>78</strong><br />
| |
| </td>
| |
| <td>92<br />
| |
| </td>
| |
| <td>114<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>90<br />
| |
| </td>
| |
| <td>112<br />
| |
| </td>
| |
| <td><strong>128</strong><br />
| |
| </td>
| |
| <td><strong>143</strong><br />
| |
| </td>
| |
| <td><strong>156</strong><br />
| |
| </td>
| |
| <td><strong>168</strong><br />
| |
| </td>
| |
| <td>182<br />
| |
| </td>
| |
| <td>204<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>294<br />
| |
| </td>
| |
| <td>316<br />
| |
| </td>
| |
| <td><strong>332</strong><br />
| |
| </td>
| |
| <td><strong>348</strong><br />
| |
| </td>
| |
| <td><strong>360</strong><br />
| |
| </td>
| |
| <td><strong>372</strong><br />
| |
| </td>
| |
| <td>386<br />
| |
| </td>
| |
| <td>408<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| <br />
| | 22, 16, 13, 12, 13, 16, 22 |
| Interval table of just intervals:<br />
| |
| XXX<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Newer systems-Symmetrical and super-symmetrical models"></a><!-- ws:end:WikiTextHeadingRule:16 -->Symmetrical and super-symmetrical models</h2>
| |
| XXX<br />
| |
| Characteristics of super-symmetric systems: no difference between ascending and descending ratios.<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="Newer systems-Symmetrical and super-symmetrical models-Symmetrical model"></a><!-- ws:end:WikiTextHeadingRule:18 -->Symmetrical model</h3>
| |
| J.J. Weiss<br />
| |
| Advantage: marked contrast between Segah of Ushaq and Segah of arabic Rast.<br />
| |
| Equal division of 65/54 (320.98 cents)<br />
| |
| <br />
| |
| Mandal positions (cents): <span style="font-family: Tahoma;">22|13|13|1</span><span style="color: #00000a; font-family: Tahoma;">8</span><span style="font-family: Tahoma;">|13|13|22</span><br />
| |
| <br />
| |
| Interval table (cents):<br />
| |
|
| |
|
| | (Observe that 22+16 = 38, as well as 13+12+13.) |
|
| |
|
| <table class="wiki_table">
| | Mandal positions in ratios: |
| <tr>
| |
| <td>0<br />
| |
| </td>
| |
| <td>22<br />
| |
| </td>
| |
| <td><strong>35</strong><br />
| |
| </td>
| |
| <td><strong>48</strong><br />
| |
| </td>
| |
| <td><strong>66</strong><br />
| |
| </td>
| |
| <td><strong>79</strong><br />
| |
| </td>
| |
| <td>92<br />
| |
| </td>
| |
| <td>114<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>90<br />
| |
| </td>
| |
| <td>112<br />
| |
| </td>
| |
| <td><strong>125</strong><br />
| |
| </td>
| |
| <td><strong>138</strong><br />
| |
| </td>
| |
| <td><strong>156</strong><br />
| |
| </td>
| |
| <td><strong>169</strong><br />
| |
| </td>
| |
| <td>182<br />
| |
| </td>
| |
| <td>204<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>294<br />
| |
| </td>
| |
| <td>316<br />
| |
| </td>
| |
| <td><strong>329</strong><br />
| |
| </td>
| |
| <td><strong>342</strong><br />
| |
| </td>
| |
| <td><strong>360</strong><br />
| |
| </td>
| |
| <td><strong>373</strong><br />
| |
| </td>
| |
| <td>386<br />
| |
| </td>
| |
| <td>408<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| <br />
| | 81/80, 1701/1664, 416/413, 3456/3481, 416/413, 1701/1664, 81/80 |
| Interval table of just intervals:<br />
| |
| XXX<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Newer systems-Symmetrical and super-symmetrical models-Super-symmetrical model"></a><!-- ws:end:WikiTextHeadingRule:20 -->Super-symmetrical model</h3>
| |
| J.J. Weiss<br />
| |
| Equal division of 65/54 (320.98 cents) and its inverse. (XXX Is that formulation correct?)<br />
| |
| Ascending/descending with 43/40 (125.2 cents).<br />
| |
| <br />
| |
| Mandal positions (cents): <span style="font-family: Tahoma;">22|13|13|1</span><span style="color: #00000a; font-family: Tahoma;">8</span><span style="font-family: Tahoma;">|13|13</span><span style="color: #00000a; font-family: Tahoma;">|</span><span style="font-family: Tahoma;">22</span><br />
| |
| <br />
| |
| Interval table (cents): same as in previous model.<br />
| |
| <br />
| |
| Interval table of just intervals:<br />
| |
| XXX<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Newer systems-Symmetrical and super-symmetrical models-Unequal division of 65/54"></a><!-- ws:end:WikiTextHeadingRule:22 -->Unequal division of 65/54</h3>
| |
| J.J. Weiss<br />
| |
| <br />
| |
| XXX<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Newer systems-Symmetrical and super-symmetrical models-Equal division of the Zarlinian semitone"></a><!-- ws:end:WikiTextHeadingRule:24 -->Equal division of the Zarlinian semitone</h3>
| |
| J.J. Weiss<br />
| |
| This is the simplest variant for luthiers...<br />
| |
| <br />
| |
| Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22|14|14|14|14|14|22</span><br />
| |
| <br />
| |
| Interval table (cents):<br />
| |
|
| |
|
| | Since the pythagorean limma appears prominently in the basic framework anyway (as semitone from E to F and from B to C as well as one apotome minus a syntonic comma several times on each string), 14/13 also appears at various positions. |
|
| |
|
| <table class="wiki_table">
| | Table of pitches from C to F (approximations in cents): |
| <tr>
| |
| <td>0<br />
| |
| </td>
| |
| <td>22<br />
| |
| </td>
| |
| <td><strong>36</strong><br />
| |
| </td>
| |
| <td><strong>50</strong><br />
| |
| </td>
| |
| <td><strong>64</strong><br />
| |
| </td>
| |
| <td><strong>76</strong><br />
| |
| </td>
| |
| <td>92<br />
| |
| </td>
| |
| <td>114<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>90<br />
| |
| </td>
| |
| <td>112<br />
| |
| </td>
| |
| <td><strong>126</strong><br />
| |
| </td>
| |
| <td><strong>140</strong><br />
| |
| </td>
| |
| <td><strong>154</strong><br />
| |
| </td>
| |
| <td><strong>168</strong><br />
| |
| </td>
| |
| <td>182<br />
| |
| </td>
| |
| <td>204<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>294<br />
| |
| </td>
| |
| <td>316<br />
| |
| </td>
| |
| <td><strong>330</strong><br />
| |
| </td>
| |
| <td><strong>344</strong><br />
| |
| </td>
| |
| <td><strong>358</strong><br />
| |
| </td>
| |
| <td><strong>372</strong><br />
| |
| </td>
| |
| <td>386<br />
| |
| </td>
| |
| <td>408<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| <br />
| | {| class="wikitable" |
| Interval table of just intervals (ascending, descending):<br />
| | |- |
| | ! | String |
| | ! | b |
| | ! | |
| | ! | |
| | ! | |
| | ! | |
| | ! | |
| | ! | |
| | ! | Base note |
| | ! | |
| | ! | |
| | ! | |
| | ! | |
| | ! | |
| | ! | |
| | ! | # |
| | |- |
| | ! | C |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | style="text-align:center;" | 0 |
| | | | 22 |
| | | | 38 |
| | | | 51 |
| | | | 63 |
| | | | 76 |
| | | | 92 |
| | | | 114 |
| | |- |
| | ! | D |
| | | | 90 |
| | | | 112 |
| | | | 128 |
| | | | 141 |
| | | | 153 |
| | | | 166 |
| | | | 182 |
| | | style="text-align:center;" | 204 |
| | | | 226 |
| | | | 242 |
| | | | 255 |
| | | | 267 |
| | | | 280 |
| | | | 296 |
| | | | 318 |
| | |- |
| | ! | E |
| | | | 294 |
| | | | 316 |
| | | | 329 |
| | | | 341 |
| | | | 354 |
| | | | 370 |
| | | | 386 |
| | | style="text-align:center;" | 408 |
| | | | 430 |
| | | | 446 |
| | | | 459 |
| | | | 471 |
| | | | 484 |
| | | | 500 |
| | | | 522 |
| | |- |
| | ! | F |
| | | | 384 |
| | | | 406 |
| | | | 422 |
| | | | 435 |
| | | | 447 |
| | | | 460 |
| | | | 476 |
| | | style="text-align:center;" | 498 |
| | | | 520 |
| | | | 536 |
| | | | 549 |
| | | | 561 |
| | | | 574 |
| | | | 590 |
| | | | 612 |
| | |} |
|
| |
|
| | Interval ratios, ascending from C: |
|
| |
|
| <table class="wiki_table">
| | <ul><li>On the D string (from Db to D): |
| <tr>
| |
| <td>1/1<br />
| |
| </td>
| |
| <td>81/80<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>135/128<br />
| |
| </td>
| |
| <td>2187/2048<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>256/243<br />
| |
| </td>
| |
| <td>16/15<br />
| |
| </td>
| |
| <td><strong>100/93 (asc.)</strong><br />
| |
| <strong>242/225 (desc.)</strong><br />
| |
| </td>
| |
| <td><strong>400/369 (asc.)</strong><br />
| |
| <strong>248/225 (desc.)</strong><br />
| |
| </td>
| |
| <td><strong>200/183 (asc.)</strong><br />
| |
| <strong>82/75 (desc.)</strong><br />
| |
| </td>
| |
| <td><strong>400/363 (asc.)</strong><br />
| |
| <strong>248/225 (desc.)</strong><br />
| |
| </td>
| |
| <td>10/9<br />
| |
| </td>
| |
| <td>9/8<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>32/27<br />
| |
| </td>
| |
| <td>6/5<br />
| |
| </td>
| |
| <td><strong>75/62 (asc.)</strong><br />
| |
| <strong>121/100 (desc.)</strong><br />
| |
| </td>
| |
| <td><strong>50/41 (asc.)</strong><br />
| |
| <strong>61/50 (desc.)</strong><br />
| |
| </td>
| |
| <td><strong>75/61 (asc.)</strong><br />
| |
| <strong>49/40 (desc.)</strong><br />
| |
| </td>
| |
| <td><strong>150/121 (asc.)</strong><br />
| |
| <strong>31/25 (desc.)</strong><br />
| |
| </td>
| |
| <td>5/4<br />
| |
| </td>
| |
| <td>81/64<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| <br />
| | 256/243 (90), 16/15/112), 14/13 (128), 64/59 (141), 59/54 /153), 209/189 (166), 10(9 (182), 9/8 (204)</li><li>On the E string (from Eb to E): |
| <!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Newer systems-Symmetrical and super-symmetrical models-Ascending/descending with 54/49"></a><!-- ws:end:WikiTextHeadingRule:26 -->Ascending/descending with 54/49</h3>
| |
| J.J. Weiss<br />
| |
| <a class="wiki_link" href="/54_49">54/49</a>: 168.2 cents, Zalzal's mujannab (Al Farabi)<br />
| |
| <br />
| |
| Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22|14|17|13|13|13|22</span><br />
| |
| <span style="color: #00000a; font-family: Tahoma;">XXX</span> There seem to be typos here...<br />
| |
| <br />
| |
| Interval table (cents):<br />
| |
|
| |
|
| | 32/27 (294), 6/5 (316), 63/52 (332), 72/59 (345), 59/48 (357), 26/21 (370), 5/4 (386), 81/64 (408)</li></ul> |
|
| |
|
| <table class="wiki_table">
| | {{Navbox notation}} |
| <tr>
| |
| <td>0<br />
| |
| </td>
| |
| <td>22<br />
| |
| </td>
| |
| <td><strong>36</strong><br />
| |
| </td>
| |
| <td><strong>48</strong><br />
| |
| </td>
| |
| <td><strong>60</strong><br />
| |
| </td>
| |
| <td><strong>76</strong><br />
| |
| </td>
| |
| <td>92<br />
| |
| </td>
| |
| <td>114<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>90<br />
| |
| </td>
| |
| <td>112<br />
| |
| </td>
| |
| <td><strong>126</strong><br />
| |
| </td>
| |
| <td><strong>138</strong><br />
| |
| </td>
| |
| <td><strong>156</strong><br />
| |
| </td>
| |
| <td><strong>168</strong><br />
| |
| </td>
| |
| <td>182<br />
| |
| </td>
| |
| <td>204<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>294<br />
| |
| </td>
| |
| <td>316<br />
| |
| </td>
| |
| <td><strong>330</strong><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><strong>360</strong><br />
| |
| </td>
| |
| <td><strong>374</strong><br />
| |
| </td>
| |
| <td>386<br />
| |
| </td>
| |
| <td>408<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| <br />
| | {{Todo| cleanup }} |
| Interval table of just intervals XXX<br />
| | [[Category:Arabic music]] |
| <br />
| | [[Category:Persian music]] |
| <!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Newer systems-Symmetrical and super-symmetrical models-Ascending/descending with 14/13"></a><!-- ws:end:WikiTextHeadingRule:28 -->Ascending/descending with 14/13</h3>
| | [[Category:Turkish music]] |
| J.J. Weiss<br />
| | [[Category:Qanun]] |
| <span style="color: #00000a; font-family: Tahoma;">14/13: 128.3 cents</span><br />
| |
| <span style="color: #00000a; font-family: Tahoma;">XXX</span><br />
| |
| <br />
| |
| Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22, 16, 13, 12, 13, 16, 22</span><br />
| |
| <br />
| |
| Interval table XXX<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:30:&lt;h3&gt; --><h3 id="toc15"><a name="Newer systems-Symmetrical and super-symmetrical models-Ascending/descending with 11/10"></a><!-- ws:end:WikiTextHeadingRule:30 -->Ascending/descending with 11/10</h3>
| |
| J.J. Weiss<br />
| |
| 11/10: 165 cents<br />
| |
| <br />
| |
| Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22|16|15|13|13|13|22</span><br />
| |
| <br />
| |
| Interval table XXX<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="Newer systems-Symmetrical and super-symmetrical models-Ascending/descending with 35/32"></a><!-- ws:end:WikiTextHeadingRule:32 -->Ascending/descending with 35/32</h3>
| |
| J.J. Weiss<br />
| |
| 35/32: 155.14 cents<br />
| |
| XXX<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Newer systems-System Jacques Dudon (2006)"></a><!-- ws:end:WikiTextHeadingRule:34 -->System Jacques Dudon (2006)</h2>
| |
| <br />
| |
| Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">21,5 | 14,5 | 14,5 | 14,5 | 15 | 12 | 21,5</span><br />
| |
| <br />
| |
| Interval table (cents):<br />
| |
| <br />
| |
| Interval table of just intervals XXX<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:36:&lt;h3&gt; --><h3 id="toc18"><a name="Newer systems-System Jacques Dudon (2006)-Arithmetic system"></a><!-- ws:end:WikiTextHeadingRule:36 -->Arithmetic system</h3>
| |
| <br />
| |
| Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">21,5 | 12 | 15 | 14,5 | 14,5 | 14,5 | 21,5 </span><br />
| |
| <br />
| |
| <br />
| |
| Interval table XXX</body></html></pre></div>
| |
Tuning systems for the qanun
Julien Jalaleddine Weiss, used with permission.
Reference: Pohlit, Stefan. 2011. Julien Jalâl Ed-Dine Weiss – A New Qānūn System: Its Application in the Performance Practice of the Ensemble “Al-Kindi” and in Contemporary Western Music. PhD Thesis, MIAM/Istanbul Technical University.
Online version of Stefan Pohlit's dissertation: see http://stefanpohlit.com/dissertation.engl..htm
The tuning tables on this page are specifically designed for the tuning system of the qanun (see the link for details on the system of tuning and playing a qanun with mandals/orabs). The logic behind the systems is as follows:
The empty strings of the qanun are tuned to a pythagorean diatonic scale, with a major third of 81/64, a major sixth of 27/16 and a major seventh of 243/128.
The possible pitches of a string obtained via raising/lowering the mandals lie within two apotomes (2187/2048, 113.7 cents). The base note is assumed in the middle. The mandals allow raising and lowering this note by maximally one apotome.
Each apotome is divided into 7 unequal parts, which requires 14 mandals per string. The first rough subdivision of the apotome is always into one syntonic comma (81/80, 21.5 cents), one Zarlinian semitone (25/24, 70.7 cents) and another syntonic comma. The middle part (25/24, Zarlinian semitone) is then further subdivided into 5 (unequal or equal) parts. The various systems differ mainly in the division of the middle part.
The tuning systems are all described by a series of cent values, which describe the subdivision of one apotome. According to the system sketched above, the first and the last value are always 22 cents (or 21.5 cents). This subdivision pattern occurs twice on each string, altogether 14 times per octave. This is followed by listings of some important rational intervals that are possible in this tuning, mainly in the range of a fourth (the range where the ajnas - maqam tetrachords - reside),
An notable property (of all systems) is that the second-highest mandal position of, say, the C string is 114-22=92 cents (the major limma), while the lowest mandal position on the following string (D in the example) is 214 (one wholetone above C) - 114 = 90 cents (the pythagorean limma, the same interval as between E and F) - we have two notes differing by one schisma (2 cents). So the interval of the schisma is present and can be played on a qanun in any of the tuning systems described here.
Notation
The notes without accidentals stand for the pythagorean intervals of the base tuning of the qanun. Raising a pitch by an apotome is notated with "#", lowering a pitch by the same amount is notated with "b". Sharps are higher than flats (unlike in meantone systems): C# is one apotome (114 cents) above C, while Db is 9/8 (214 cents) minus one apotome = 90 cents. Both properties indicate that the framework is essentially pythagorean. The tuning system as a whole, however, is not.
For the steps in between, additional symbols are used - altogether 7 symbols for raising pitches and 7 for lowering pitches.
This gives 15 potential different pitches per base note, corresponding to the mandals. Seven base notes (C, D, E, F, G, A, B or Do, Re, Mi, Fa, Sol, La, Si), corresponding to the strings, lead to a notation system of 7*15=105 pitches, in accordance with the real playing capabilities of the qanun. See the following document, which also gives all the pitches in one octave (in ratios and cents) that can be played by system 1 and 2.
Tableaux JJW VIII-2011.pdf
(used with permission J. J. Weiss/S. Pohlit)
System 1
© J.J.Weiss. Luthier: Ejder Gulec.
Subdivision of 25/24 into 65/64 (26 cents), 144/143 (12 cents) and 55/54 (32 cents).
65/64 and 55/54 are each split into two roughly equal parts.
This gives the following rational intervals between the mandals:
81/80, 245/243, 3159/3136, 144/143, 121/120, 100/99, 81/80
In cents (approximations):
22, 13, 13, 12, 16, 16, 22
Rational intervals each string can be detuned (approximations in cents in parentheses):
81/80 (22), 49/48 (35), 1053/1024 (48), 729/704 (60), 2673/2560 (76), 135/128 (92), 2187/2048 (114)
Intervals ratios, ascending from C:
- On the D string (from Db to D):
256/243 (90), 16/15 (112), 784/729 (126), 13/12 (138), 12/11 (150), 11/10 (166), 10/9 (182), 9/8 (204)
- On the E string (from Eb to E):
32/27 (294), 6/5 (316), 98/81 (330), 39/32 (342), 27/22 (354), 99/80 (370), 5/4 (386), 81/64 (408)
Interval ratios, descending from F:
- On the E string (from Eb to E):
9/8 (204), 10/9 (182), 54/49 (169), 128/117 (156), 88/81 (144), 320/297 (129), 16/15 (112), 256/243 (90)
- On the D string (from Db to D):
81/64 (408), 5/4 (386), 243/196 (372), 16/13 (360), 11/9 (348), 40/33 (333), 6/5 (316), 32/27 (294)
A complete list of all intervals available within one octave can be found in the above-mentioned document (on the first page).
System 2, better suited for ottoman maqams
© J.J. Weiss. Qanun no. 9, luthier: Kenan Ozten.
Mandal positions in ratios:
81/80, 105/104, 572/567, 144/143, 1547/1536, 120/119, 81/80
In cents (approximations):
22|16|15|12|13|14|22
Rational intervals each string can be detuned (approximations in cents in parentheses):
81/80 (22), 1701/1664 (38), 33/32 (54), 27/26 (66), 243/232 (78), 135/128 (92), 2187/2048 (114)
Intervals ratios, ascending from C:
- On the D string (from Db to D):
256/243 (90), 16/15 (112), 14/13 (128), 88/81 (144), 128/117 or 35/32 (156), 119/108 (168), 10/9 (182), 9/8 (204)
- On the E string (from Eb to E):
32/27 (294), 6/5 (316), 63/52 (332), 11/9 (348), 16/13 or 315/256 (360), 119/96 (372), 5/4 (386), 81/64 (408)
Interval ratios descending from F:
- On the E string (from Eb to E):
9/8 (204), 10/9 (182), 208/189 (166), 12/11 (150), 13/12 (138), 128/119 (126), 16/15 (112), 256/243 (90)
- On the D string (from Db to D):
81/64 (408), 5/4 (386), 26/21 (370), 27/22 (354), 39/32 (342), 144/119 (330), 6/5 (316), 32/27 (294)
A complete list of all intervals available within one octave can be found in the above-mentioned document (on the second page).
Other models
Julien Weiss has developed a number of other systems besides the two described above. A notable class of these are so-called super-symmetrical systems, which have the property that the intervals ascending from C and the intervals descending from F (which show slight differences in the previous two systems, marked in bold above) are the same.
3 examples are described below. For more and detailed descriptions see chapter 3.4 and appendix I in Stefan Pohlit's dissertation .
Super-symmetric model with non-aliquot division of 65/64
© J.J. Weiss
Similar to system 1, but with 65/64 (26.84 cents) divided into two non-equal parts (14 and 12 cents instead of 13 and 13).
Mandal positions in ratios:
81/80 (22), 120/119 (14), 1547/1536 (12), 512/507 (17), 1547/1536 (12), 120/119 (14), 81/80 (22)
Table of pitches from C to F (approximations in cents):
| String
|
b
|
|
|
|
|
|
|
Base note
|
|
|
|
|
|
|
#
|
| C
|
|
|
|
|
|
|
|
0
|
22
|
36
|
48
|
66
|
78
|
92
|
114
|
| D
|
90
|
112
|
126
|
138
|
156
|
168
|
182
|
204
|
226
|
240
|
252
|
270
|
282
|
296
|
318
|
| E
|
294
|
316
|
330
|
342
|
360
|
372
|
386
|
408
|
430
|
444
|
456
|
474
|
486
|
500
|
522
|
| F
|
384
|
406
|
420
|
433
|
450
|
462
|
476
|
498
|
520
|
534
|
546
|
464
|
476
|
590
|
612
|
Interval ratios, ascending from C:
- On the D string (from Db to D):
245/243 (90), 16/15 (112), 128/119 (126), 13/12 (138), 128/117 (156), 119/108 (168), 10/9 (182), 9/8 (204)
On the E string (from Eb to E):
32/27 (294), 6/5 (316), 144/119 (330), 39/32 (342), 16/13 (360), 119/96 (372), 5/4 (386), 81/64 (408)
Interval ratios, descending from F:
- On the E string (from Eb to E):
9(8 /204), 10/9 (182), 119/108 (168), 128/117 (156), 13/12 (138), 128/119 (126), 16/15 (112), 256/243 (90)
- On the D string (from Db to D)
X81/64 (408), 5/4 (386), 119/96 (372), 16/13 (360), 39/32 (342), 144/119 (330), 6/5 (316), 32/27 (294)
Ascending and descending intervals are indeed the same, which is what "super-symmetrical" means in this context.
Equal division of the Zarlinian semitone
© J.J. Weiss
This is the simplest variant for luthiers...
Mandal positions (cents): 22|14|14|14|14|14|22
Mandal positions in ratios:
81/80, 125/124, 124/123, 123/122, 122/121, 121/120, 81/80
Table of pitches from C to F (approximations in cents):
| String
|
b
|
|
|
|
|
|
|
Base note
|
|
|
|
|
|
|
#
|
| C
|
|
|
|
|
|
|
|
0
|
22
|
36
|
50
|
64
|
78
|
92
|
114
|
| D
|
90
|
112
|
126
|
140
|
154
|
168
|
182
|
204
|
226
|
240
|
254
|
268
|
282
|
296
|
318
|
| E
|
294
|
316
|
330
|
344
|
358
|
372
|
386
|
408
|
430
|
444
|
458
|
472
|
486
|
500
|
522
|
| F
|
384
|
406
|
420
|
434
|
448
|
462
|
476
|
498
|
520
|
534
|
548
|
562
|
576
|
590
|
612
|
Interval ratios, ascending from C:
- On the D string (from Db to D):
256/243 (90), 16/15 (112), 100/93 (126), 400/369 (140), 200/183 (153.78), 400/363 (168). 10/9 (182), 9/8 (204)
- On the E string (from Eb to E):
32/27 (294), 6/5 (316), 75/62 (329.54), 50/41 (343.56), 75/61 (357.69), 150/121 (371.94), 5/4 (386), 81/64 (408)
Interval ratios descending from F:
- On the E string (from Eb to E):
9/8, 10/9, 248/225 (168.49), 82/75 (154.47), 244/225 (140.34), 242/225 (126.09), 16/15, 256/243or approximating ratios: XXX
- On the D string (from Db to D):
81/64, 5/4, 31/25 (372.40), 49/40 (351.33), 61/50 (344.25), 121/100 (330), 6/5, 32/27Or approximatiing ratios: XXX
Super-symmetrical model with 14/13
© J.J. Weiss
The idea behind this system is as follows:
Dividing the apotome (114 cents) into 3 equal parts gives 38 cents, and adding this to the pythagorean limma (90 cents) gives 128 cents, which is an approximation for 14/13 (two-third tone, a favorite interval of Avicenna/Ibn Sina).
The division of the apotome derived from this combines the known basic division into apotome, Zarlinian semitone and apotome with an equal division into 3 parts, which yields the following mandal positions (cents):
22, 16, 13, 12, 13, 16, 22
(Observe that 22+16 = 38, as well as 13+12+13.)
Mandal positions in ratios:
81/80, 1701/1664, 416/413, 3456/3481, 416/413, 1701/1664, 81/80
Since the pythagorean limma appears prominently in the basic framework anyway (as semitone from E to F and from B to C as well as one apotome minus a syntonic comma several times on each string), 14/13 also appears at various positions.
Table of pitches from C to F (approximations in cents):
| String
|
b
|
|
|
|
|
|
|
Base note
|
|
|
|
|
|
|
#
|
| C
|
|
|
|
|
|
|
|
0
|
22
|
38
|
51
|
63
|
76
|
92
|
114
|
| D
|
90
|
112
|
128
|
141
|
153
|
166
|
182
|
204
|
226
|
242
|
255
|
267
|
280
|
296
|
318
|
| E
|
294
|
316
|
329
|
341
|
354
|
370
|
386
|
408
|
430
|
446
|
459
|
471
|
484
|
500
|
522
|
| F
|
384
|
406
|
422
|
435
|
447
|
460
|
476
|
498
|
520
|
536
|
549
|
561
|
574
|
590
|
612
|
Interval ratios, ascending from C:
- On the D string (from Db to D):
256/243 (90), 16/15/112), 14/13 (128), 64/59 (141), 59/54 /153), 209/189 (166), 10(9 (182), 9/8 (204)
- On the E string (from Eb to E):
32/27 (294), 6/5 (316), 63/52 (332), 72/59 (345), 59/48 (357), 26/21 (370), 5/4 (386), 81/64 (408)