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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:JosephRuhf|JosephRuhf]] and made on <tt>2012-06-01 13:55:47 UTC</tt>.<br>
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| : The original revision id was <tt>341822682</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. |
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| 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. | | 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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| Online version of Stefan Pohlit's dissertation: see [[http://stefanpohlit.com/dissertation.engl..htm]] | | 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 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 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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| 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 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]]. |
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| 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. | | 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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| 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. | | 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. |
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| 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), | | 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), |
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| 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. | | 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. |
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| =Notation= | | =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. | | 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. |
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| For the steps in between, additional symbols are used - altogether 7 symbols for raising pitches and 7 for lowering pitches. | | For the steps in between, additional symbols are used - altogether 7 symbols for raising pitches and 7 for lowering pitches. |
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| This gives 15 potential different pitches per base note, correpsonding 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. | | 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]] | | |
| | [[:File:Tableaux_JJW_VIII-2011.pdf|Tableaux JJW VIII-2011.pdf]] |
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| (used with permission J. J. Weiss/S. Pohlit) | | (used with permission J. J. Weiss/S. Pohlit) |
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| =System 1= | | =System 1= |
| © J.J.Weiss. Luthier: Ejder Gulec. | | © J.J.Weiss. Luthier: Ejder Gulec. |
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| 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). |
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| 65/64 and 55/54 are each split into two roughly equal parts. | | 65/64 and 55/54 are each split into two roughly equal parts. |
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| This gives the following rational intervals between the mandals: | | This gives the following rational intervals between the mandals: |
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| 81/80, 245/243, 3159/3136, 144/143, 121/120, 100/99, 81/80 | | 81/80, 245/243, 3159/3136, 144/143, 121/120, 100/99, 81/80 |
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| In cents (approximations): | | In cents (approximations): |
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| 22, 13, 13, 12, 16, 16, 22 | | 22, 13, 13, 12, 16, 16, 22 |
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| Rational intervals each string can be detuned (approximations in cents in parentheses): | | Rational intervals each string can be detuned (approximations in cents in parentheses): |
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| 81/80 (22), 49/48 (35), 1053/1024 (48), 729/704 (60), 2673/2560 (76), 135/128 (92), 2187/2048 (114) | | 81/80 (22), 49/48 (35), 1053/1024 (48), 729/704 (60), 2673/2560 (76), 135/128 (92), 2187/2048 (114) |
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| Intervals ratios, ascending from C: | | Intervals ratios, ascending from C: |
| * On the D string (from Db to D):
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| > 256/243 (90), 16/15 (112), **784/729 (126)**, **13/12 (138)**, **12/11 (150)**, **11/10 (166)**, 10/9 (182), 9/8 (204)
| | <ul><li>On the D string (from Db to D): |
| * On the E string (from Eb to E):
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| > 32/27 (294), 6/5 (316), **98/81 (330)**, **39/32 (342)**, **27/22 (354)**, **99/80 (370)**, 5/4 (386), 81/64 (408)
| | 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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| | 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> |
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| Interval ratios, descending from F: | | Interval ratios, descending from F: |
| * On the E string (from Eb to E):
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| > 9/8 (204), 10/9 (182), **54/49 (169)**, **128/117 (156)**, **88/81 (144)**, **320/297 (129)**, 16/15 (112), 256/243 (90)
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| * On the D string (from Db to D):
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| > 81/64 (408), 5/4 (386), **243/196 (372)**, **16/13 (360)**, **11/9 (348)**, **40/33 (333)**, 6/5 (316), 32/27 (294)
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| A complete list of all intervals available within one octave can be found in the above-mentioned [[file:xenharmonic/Tableaux JJW VIII-2011.pdf|document]] (on the first page).
| | <ul><li>On the E string (from Eb to E): |
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| | 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): |
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| =System 2, better suited for ottoman maqams= | | 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> |
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| | 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). |
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| | =System 2, better suited for ottoman maqams= |
| © J.J. Weiss. Qanun no. 9, luthier: Kenan Ozten. | | © J.J. Weiss. Qanun no. 9, luthier: Kenan Ozten. |
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| Mandal positions in ratios: | | Mandal positions in ratios: |
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| 81/80, 105/104, 572/567, 144/143, 1547/1536, 120/119, 81/80 | | 81/80, 105/104, 572/567, 144/143, 1547/1536, 120/119, 81/80 |
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| In cents (approximations): | | In cents (approximations): |
| <span style="color: #00000a; font-family: Tahoma;">22|16|15|12|13|14|22</span>
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| | <span style="color: #00000a; font-family: Tahoma;">22|16|15|12|13|14|22</span> |
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| Rational intervals each string can be detuned (approximations in cents in parentheses): | | Rational intervals each string can be detuned (approximations in cents in parentheses): |
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| 81/80 (22), 1701/1664 (38), 33/32 (54), 27/26 (66), 243/232 (78), 135/128 (92), 2187/2048 (114) | | 81/80 (22), 1701/1664 (38), 33/32 (54), 27/26 (66), 243/232 (78), 135/128 (92), 2187/2048 (114) |
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| Intervals ratios, ascending from C: | | Intervals ratios, ascending from C: |
| * On the D string (from Db to D):
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| > 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)
| | <ul><li>On the D string (from Db to D): |
| * On the E string (from Eb to E):
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| > 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)
| | 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): |
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| | 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> |
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| Interval ratios descending from F: | | Interval ratios descending from F: |
| * On the E string (from Eb to E):
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| > 9/8 (204), 10/9 (182), **208/189 (166), 12/11 (150), 13/12 (138), 128/119 (126)**, 16/15 (112), 256/243 (90)
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| * On the D string (from Db to D):
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| > 81/64 (408), 5/4 (386), **26/21 (370), 27/22 (354), 39/32 (342), 144/119 (330)**, 6/5 (316), 32/27 (294)
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| A complete list of all intervals available within one octave can be found in the above-mentioned [[file:xenharmonic/Tableaux JJW VIII-2011.pdf|document]] (on the second page). | | <ul><li>On the E string (from Eb to E): |
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| | 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): |
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| | 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> |
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| | 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). |
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| =Other models= | | =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. | | 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. |
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| 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]] . | | 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] . |
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| ==Super-symmetric model with non-aliquot division of 65/64== | | ==Super-symmetric model with non-aliquot division of 65/64== |
| © J.J. Weiss | | © J.J. Weiss |
| Similar to [[tuning systems for qanun#System%201|system 1]], but with 65/64 (26.84 cents) divided into two non-equal parts (14 and 12 cents instead of 13 and 13). | | |
| | 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). |
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| Mandal positions in ratios: | | 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> | | |
| | 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> |
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| Table of pitches from C to F (approximations in cents): | | Table of pitches from C to F (approximations in cents): |
| ||~ String ||~ b ||~ ||~ ||~ ||~ ||~ ||~ ||~ Base note ||~ ||~ ||~ ||~ ||~ ||~ ||~ # || | | |
| ||~ C || || || || || || || ||= 0 || 22 || 36 || 48 || 66 || 78 || 92 || 114 || | | {| class="wikitable" |
| ||~ D || 90 || 112 || 126 || 138 || 156 || 168 || 182 ||= 204 || 226 || 240 || 252 || 170 || 182 || 196 || 318 || | | |- |
| ||~ E || 294 || 316 || 330 || 342 || 360 || 372 || 386 ||= 408 || 430 || 444 || 456 || 374 || 386 || 400 || 422 || | | ! | String |
| ||~ F || 384 || 406 || 420 || 433 || 450 || 462 || 476 ||= 498 || 520 || 534 || 546 || 464 || 476 || 590 || 612 || | | ! | 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 |
| | |} |
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| Interval ratios, ascending from C: | | Interval ratios, ascending from C: |
| * On the D string (from Db to D):
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| > 245/243 (90), 16/15 (112), **128/119 (126), 13/12 (138), 128/117 (156), 119/108 (168),** 10/9 (182), 9/8 (204)
| | <ul><li>On the D string (from Db to D): |
| > 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)
| | 245/243 (90), 16/15 (112), '''128/119 (126), 13/12 (138), 128/117 (156), 119/108 (168),''' 10/9 (182), 9/8 (204) |
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| | On the E string (from Eb to E): |
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| | 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> |
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| Interval ratios, descending from F: | | Interval ratios, descending from F: |
| * On the E string (from Eb to E):
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| > 9(8 /204), 10/9 (182), **119/108 (168), 128/117 (156), 13/12 (138), 128/119 (126)**, 16/15 (112), 256/243 (90)
| | <ul><li>On the E string (from Eb to E): |
| * 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)
| | 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) |
| | |
| | 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> |
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| Ascending and descending intervals are indeed the same, which is what "super-symmetrical" means in this context. | | Ascending and descending intervals are indeed the same, which is what "super-symmetrical" means in this context. |
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| ==Equal division of the Zarlinian semitone== | | ==Equal division of the Zarlinian semitone== |
| © J.J. Weiss | | © J.J. Weiss |
| | |
| This is the simplest variant for luthiers... | | This is the simplest variant for luthiers... |
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| Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;"> 22|14|14|14|14|14|22 </span> | | Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;"> 22|14|14|14|14|14|22 </span> |
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| Mandal positions in ratios: | | Mandal positions in ratios: |
| <span style="color: #00000a; font-family: Tahoma;">81/80, 125/124, 124/123, 123/122, 122/121, 121/120, 81/80</span>
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| | <span style="color: #00000a; font-family: Tahoma;">81/80, 125/124, 124/123, 123/122, 122/121, 121/120, 81/80</span> |
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| Table of pitches from C to F (approximations in cents): | | Table of pitches from C to F (approximations in cents): |
| ||~ String ||~ b ||~ ||~ ||~ ||~ ||~ ||~ ||~ Base note ||~ ||~ ||~ ||~ ||~ ||~ ||~ # || | | |
| ||~ C || || || || || || || ||= 0 || 22 || 36 || 50 || 64 || 78 || 92 || 114 || | | {| class="wikitable" |
| ||~ 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 || | | ! | String |
| ||~ F || 384 || 406 || 420 || 434 || 448 || 462 || 476 ||= 498 || 520 || 534 || 548 || 562 || 576 || 590 || 612 || | | ! | 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 |
| | |} |
|
| |
|
| Interval ratios, ascending from C: | | 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)
| | <ul><li>On the D string (from Db to D): |
| * 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)
| | 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): |
| | |
| | 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> |
|
| |
|
| Interval ratios descending from F: | | 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== | | <ul><li>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</li><li>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</li></ul> |
| | |
| | ==Super-symmetrical model with 14/13== |
| © J.J. Weiss | | © J.J. Weiss |
| | |
| The idea behind this system is as follows: | | 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|14/13]] (two-third tone, a favorite interval of [[http://en.wikipedia.org/wiki/Avicenna|Avicenna/Ibn Sina]]). | | |
| | 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]). |
|
| |
|
| 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): | | 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): |
| Line 163: |
Line 355: |
|
| |
|
| Mandal positions in ratios: | | Mandal positions in ratios: |
| | |
| 81/80, 1701/1664, 416/413, 3456/3481, 416/413, 1701/1664, 81/80 | | 81/80, 1701/1664, 416/413, 3456/3481, 416/413, 1701/1664, 81/80 |
|
| |
|
| Line 168: |
Line 361: |
|
| |
|
| Table of pitches from C to F (approximations in cents): | | Table of pitches from C to F (approximations in cents): |
| ||~ String ||~ b ||~ ||~ ||~ ||~ ||~ ||~ ||~ Base note ||~ ||~ ||~ ||~ ||~ ||~ ||~ # || | | |
| ||~ C || || || || || || || ||= 0 || 22 || 38 || 51 || 63 || 76 || 92 || 114 || | | {| class="wikitable" |
| ||~ 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 || | | ! | String |
| ||~ F || 384 || 406 || 422 || 435 || 447 || 460 || 476 ||= 498 || 520 || 536 || 549 || 561 || 574 || 590 || 612 || | | ! | 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: | | Interval ratios, ascending from C: |
| * On the D string (from Db to D):
| |
| > 256/243 (90), [[tel:16/15/112|16/15/112]]), 14/13 (128), 64/59 (141), [[tel:59/54 /153|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)</pre></div>
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| <h4>Original HTML 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;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 />
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| <!-- ws:start:WikiTextTocRule:14:&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:14 --><!-- ws:start:WikiTextTocRule:15: --><div style="margin-left: 1em;"><a href="#Notation">Notation</a></div>
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| <!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --><div style="margin-left: 1em;"><a href="#System 1">System 1</a></div>
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| <!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><div style="margin-left: 1em;"><a href="#System 2, better suited for ottoman maqams">System 2, better suited for ottoman maqams</a></div>
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| <!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><div style="margin-left: 1em;"><a href="#Other models">Other models</a></div>
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| <!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><div style="margin-left: 2em;"><a href="#Other models-Super-symmetric model with non-aliquot division of 65/64">Super-symmetric model with non-aliquot division of 65/64</a></div>
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| <!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --><div style="margin-left: 2em;"><a href="#Other models-Equal division of the Zarlinian semitone">Equal division of the Zarlinian semitone</a></div>
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| <!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><div style="margin-left: 2em;"><a href="#Other models-Super-symmetrical model with 14/13">Super-symmetrical model with 14/13</a></div>
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| <!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --></div>
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| <!-- ws:end:WikiTextTocRule:22 -->Julien Jalaleddine Weiss, used with permission.<br />
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| 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.<br />
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| <br />
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| Online version of Stefan Pohlit's dissertation: see <a class="wiki_link_ext" href="http://stefanpohlit.com/dissertation.engl..htm" rel="nofollow">http://stefanpohlit.com/dissertation.engl..htm</a><br />
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| <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). The logic behind the systems is as follows:<br />
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| <br />
| |
| The empty strings of the qanun are tuned to a pythagorean diatonic scale, with a major third of <a class="wiki_link" href="/81_64">81/64</a>, a major sixth of <a class="wiki_link" href="/27_16">27/16</a> and a major seventh of <a class="wiki_link" href="/243_128">243/128</a>.<br />
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| <br />
| |
| The possible pitches of a string obtained via raising/lowering the mandals lie within two <a class="wiki_link" href="/2187_2048">apotomes (2187/2048, 113.7 cents)</a>. The base note is assumed in the middle. The mandals allow raising and lowering this note by maximally one apotome.<br />
| |
| <br />
| |
| 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 <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 />
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| <br />
| |
| 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 <a class="wiki_link" href="/tetrachord">tetrachords</a> - reside),<br />
| |
| <br />
| |
| An notable property (of all systems) is that the second-highest mandal position of, say, the C string is 114-22=92 cents (the <a class="wiki_link" href="/135_128">major limma</a>), while the lowest mandal position on the following string (D in the example) is 214 (one wholetone above C) - 114 = 90 cents (the <a class="wiki_link" href="/256_243">pythagorean limma</a>, the same interval as between E and F) - we have two notes differing by one <a class="wiki_link" href="/32805_32768">schisma (2 cents)</a>. So the interval of the schisma is present and can be played on a qanun in any of the tuning systems described here.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Notation"></a><!-- ws:end:WikiTextHeadingRule:0 -->Notation</h1>
| |
| 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 &quot;#&quot;, lowering a pitch by the same amount is notated with &quot;b&quot;. Sharps are higher than flats (unlike in <a class="wiki_link" href="/meantone">meantone</a> 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.<br />
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| <br />
| |
| For the steps in between, additional symbols are used - altogether 7 symbols for raising pitches and 7 for lowering pitches.<br />
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| <br />
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| This gives 15 potential different pitches per base note, correpsonding 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.<br />
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| <!-- ws:start:WikiTextFileRule:644:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file/Tableaux%20JJW%20VIII-2011.pdf?h=52&amp;w=320&quot; class=&quot;WikiFile&quot; id=&quot;wikitext@@file@@Tableaux JJW VIII-2011.pdf&quot; title=&quot;File: Tableaux JJW VIII-2011.pdf&quot; width=&quot;320&quot; height=&quot;52&quot; /&gt; --><div class="objectEmbed"><a href="/file/view/Tableaux%20JJW%20VIII-2011.pdf/253043932/Tableaux%20JJW%20VIII-2011.pdf" onclick="ws.common.trackFileLink('/file/view/Tableaux%20JJW%20VIII-2011.pdf/253043932/Tableaux%20JJW%20VIII-2011.pdf');"><img src="http://www.wikispaces.com/i/mime/32/application/pdf.png" height="32" width="32" alt="Tableaux JJW VIII-2011.pdf" /></a><div><a href="/file/view/Tableaux%20JJW%20VIII-2011.pdf/253043932/Tableaux%20JJW%20VIII-2011.pdf" onclick="ws.common.trackFileLink('/file/view/Tableaux%20JJW%20VIII-2011.pdf/253043932/Tableaux%20JJW%20VIII-2011.pdf');" class="filename" title="Tableaux JJW VIII-2011.pdf">Tableaux JJW VIII-2011.pdf</a><br /><ul><li><a href="/file/detail/Tableaux%20JJW%20VIII-2011.pdf">Details</a></li><li><a href="/file/view/Tableaux%20JJW%20VIII-2011.pdf/253043932/Tableaux%20JJW%20VIII-2011.pdf">Download</a></li><li style="color: #666">130 KB</li></ul></div></div><!-- ws:end:WikiTextFileRule:644 --><br />
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| <br />
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| (used with permission J. J. Weiss/S. Pohlit)<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="System 1"></a><!-- ws:end:WikiTextHeadingRule:2 -->System 1</h1>
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| © J.J.Weiss. Luthier: Ejder Gulec.<br />
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| 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 roughly equal parts.<br />
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| <br />
| |
| This gives the following rational intervals between the mandals:<br />
| |
| 81/80, 245/243, 3159/3136, 144/143, 121/120, 100/99, 81/80<br />
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| <br />
| |
| In cents (approximations):<br />
| |
| 22, 13, 13, 12, 16, 16, 22<br />
| |
| <br />
| |
| Rational intervals each string can be detuned (approximations in cents in parentheses):<br />
| |
| 81/80 (22), 49/48 (35), 1053/1024 (48), 729/704 (60), 2673/2560 (76), 135/128 (92), 2187/2048 (114)<br />
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| <br />
| |
| Intervals ratios, ascending from C:<br />
| |
| <ul><li>On the D string (from Db to D):<br />
| |
| 256/243 (90), 16/15 (112), <strong>784/729 (126)</strong>, <strong>13/12 (138)</strong>, <strong>12/11 (150)</strong>, <strong>11/10 (166)</strong>, 10/9 (182), 9/8 (204)</li><li>On the E string (from Eb to E):<br />
| |
| 32/27 (294), 6/5 (316), <strong>98/81 (330)</strong>, <strong>39/32 (342)</strong>, <strong>27/22 (354)</strong>, <strong>99/80 (370)</strong>, 5/4 (386), 81/64 (408)</li></ul><br />
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| Interval ratios, descending from F:<br />
| |
| <ul><li>On the E string (from Eb to E):<br />
| |
| 9/8 (204), 10/9 (182), <strong>54/49 (169)</strong>, <strong>128/117 (156)</strong>, <strong>88/81 (144)</strong>, <strong>320/297 (129)</strong>, 16/15 (112), 256/243 (90)</li><li>On the D string (from Db to D):<br />
| |
| 81/64 (408), 5/4 (386), <strong>243/196 (372)</strong>, <strong>16/13 (360)</strong>, <strong>11/9 (348)</strong>, <strong>40/33 (333)</strong>, 6/5 (316), 32/27 (294)</li></ul><br />
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| A complete list of all intervals available within one octave can be found in the above-mentioned <a href="http://xenharmonic.wikispaces.com/file/view/Tableaux%20JJW%20VIII-2011.pdf/253043932/Tableaux%20JJW%20VIII-2011.pdf" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/Tableaux%20JJW%20VIII-2011.pdf/253043932/Tableaux%20JJW%20VIII-2011.pdf');">document</a> (on the first page).<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="System 2, better suited for ottoman maqams"></a><!-- ws:end:WikiTextHeadingRule:4 -->System 2, better suited for ottoman maqams</h1>
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| © J.J. Weiss. Qanun no. 9, luthier: Kenan Ozten.<br />
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| <br />
| |
| Mandal positions in ratios:<br />
| |
| 81/80, 105/104, 572/567, 144/143, 1547/1536, 120/119, 81/80<br />
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| <br />
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| In cents (approximations):<br />
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| <span style="color: #00000a; font-family: Tahoma;">22|16|15|12|13|14|22</span><br />
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| <br />
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| Rational intervals each string can be detuned (approximations in cents in parentheses):<br />
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| 81/80 (22), 1701/1664 (38), 33/32 (54), 27/26 (66), 243/232 (78), 135/128 (92), 2187/2048 (114)<br />
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| <br />
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| Intervals ratios, ascending from C:<br />
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| <ul><li>On the D string (from Db to D):<br />
| |
| 256/243 (90), 16/15 (112), <strong>14/13 (128), 88/81 (144), 128/117 or 35/32 (156), 119/108 (168)</strong>, 10/9 (182), 9/8 (204)</li><li>On the E string (from Eb to E):<br />
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| 32/27 (294), 6/5 (316), <strong>63/52 (332), 11/9 (348), 16/13 or 315/256 (360), 119/96 (372)</strong>, 5/4 (386), 81/64 (408)</li></ul><br />
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| Interval ratios descending from F:<br />
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| <ul><li>On the E string (from Eb to E):<br />
| |
| 9/8 (204), 10/9 (182), <strong>208/189 (166), 12/11 (150), 13/12 (138), 128/119 (126)</strong>, 16/15 (112), 256/243 (90)</li><li>On the D string (from Db to D):<br />
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| 81/64 (408), 5/4 (386), <strong>26/21 (370), 27/22 (354), 39/32 (342), 144/119 (330)</strong>, 6/5 (316), 32/27 (294)</li></ul><br />
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| A complete list of all intervals available within one octave can be found in the above-mentioned <a href="http://xenharmonic.wikispaces.com/file/view/Tableaux%20JJW%20VIII-2011.pdf/253043932/Tableaux%20JJW%20VIII-2011.pdf" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/Tableaux%20JJW%20VIII-2011.pdf/253043932/Tableaux%20JJW%20VIII-2011.pdf');">document</a> (on the second page).<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Other models"></a><!-- ws:end:WikiTextHeadingRule:6 -->Other models</h1>
| |
| 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 <strong>bold</strong> above) are the same.<br />
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| <br />
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| 3 examples are described below. For more and detailed descriptions see chapter 3.4 and appendix I in <a class="wiki_link_ext" href="http://stefanpohlit.com/dissertation.engl..htm" rel="nofollow">Stefan Pohlit's dissertation</a> .<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Other models-Super-symmetric model with non-aliquot division of 65/64"></a><!-- ws:end:WikiTextHeadingRule:8 -->Super-symmetric model with non-aliquot division of 65/64</h2>
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| © J.J. Weiss<br />
| |
| Similar to <a class="wiki_link" href="/tuning%20systems%20for%20qanun#System%201">system 1</a>, but with 65/64 (26.84 cents) divided into two non-equal parts (14 and 12 cents instead of 13 and 13).<br />
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| <br />
| |
| Mandal positions in ratios:<br />
| |
| 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><br />
| |
| <br />
| |
| Table of pitches from C to F (approximations in cents):<br />
| |
|
| |
|
| |
| <table class="wiki_table">
| |
| <tr>
| |
| <th>String<br />
| |
| </th>
| |
| <th>b<br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th>Base note<br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th>#<br />
| |
| </th>
| |
| </tr>
| |
| <tr>
| |
| <th>C<br />
| |
| </th>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td style="text-align: center;">0<br />
| |
| </td>
| |
| <td>22<br />
| |
| </td>
| |
| <td>36<br />
| |
| </td>
| |
| <td>48<br />
| |
| </td>
| |
| <td>66<br />
| |
| </td>
| |
| <td>78<br />
| |
| </td>
| |
| <td>92<br />
| |
| </td>
| |
| <td>114<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <th>D<br />
| |
| </th>
| |
| <td>90<br />
| |
| </td>
| |
| <td>112<br />
| |
| </td>
| |
| <td>126<br />
| |
| </td>
| |
| <td>138<br />
| |
| </td>
| |
| <td>156<br />
| |
| </td>
| |
| <td>168<br />
| |
| </td>
| |
| <td>182<br />
| |
| </td>
| |
| <td style="text-align: center;">204<br />
| |
| </td>
| |
| <td>226<br />
| |
| </td>
| |
| <td>240<br />
| |
| </td>
| |
| <td>252<br />
| |
| </td>
| |
| <td>170<br />
| |
| </td>
| |
| <td>182<br />
| |
| </td>
| |
| <td>196<br />
| |
| </td>
| |
| <td>318<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <th>E<br />
| |
| </th>
| |
| <td>294<br />
| |
| </td>
| |
| <td>316<br />
| |
| </td>
| |
| <td>330<br />
| |
| </td>
| |
| <td>342<br />
| |
| </td>
| |
| <td>360<br />
| |
| </td>
| |
| <td>372<br />
| |
| </td>
| |
| <td>386<br />
| |
| </td>
| |
| <td style="text-align: center;">408<br />
| |
| </td>
| |
| <td>430<br />
| |
| </td>
| |
| <td>444<br />
| |
| </td>
| |
| <td>456<br />
| |
| </td>
| |
| <td>374<br />
| |
| </td>
| |
| <td>386<br />
| |
| </td>
| |
| <td>400<br />
| |
| </td>
| |
| <td>422<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <th>F<br />
| |
| </th>
| |
| <td>384<br />
| |
| </td>
| |
| <td>406<br />
| |
| </td>
| |
| <td>420<br />
| |
| </td>
| |
| <td>433<br />
| |
| </td>
| |
| <td>450<br />
| |
| </td>
| |
| <td>462<br />
| |
| </td>
| |
| <td>476<br />
| |
| </td>
| |
| <td style="text-align: center;">498<br />
| |
| </td>
| |
| <td>520<br />
| |
| </td>
| |
| <td>534<br />
| |
| </td>
| |
| <td>546<br />
| |
| </td>
| |
| <td>464<br />
| |
| </td>
| |
| <td>476<br />
| |
| </td>
| |
| <td>590<br />
| |
| </td>
| |
| <td>612<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
| <br />
| |
| Interval ratios, ascending from C:<br />
| |
| <ul><li>On the D string (from Db to D):<br />
| |
| 245/243 (90), 16/15 (112), <strong>128/119 (126), 13/12 (138), 128/117 (156), 119/108 (168),</strong> 10/9 (182), 9/8 (204)<br />
| |
| On the E string (from Eb to E):<br />
| |
| 32/27 (294), 6/5 (316), <strong>144/119 (330), 39/32 (342), 16/13 (360), 119/96 (372),</strong> 5/4 (386), 81/64 (408)</li></ul><br />
| |
| Interval ratios, descending from F:<br />
| |
| <ul><li>On the E string (from Eb to E):<br />
| |
| 9(8 /204), 10/9 (182), <strong>119/108 (168), 128/117 (156), 13/12 (138), 128/119 (126)</strong>, 16/15 (112), 256/243 (90)</li><li>On the D string (from Db to D)<br />
| |
| X81/64 (408), 5/4 (386), <strong>119/96 (372), 16/13 (360), 39/32 (342), 144/119 (330)</strong>, 6/5 (316), 32/27 (294)</li></ul><br />
| |
| Ascending and descending intervals are indeed the same, which is what &quot;super-symmetrical&quot; means in this context.<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Other models-Equal division of the Zarlinian semitone"></a><!-- ws:end:WikiTextHeadingRule:10 -->Equal division of the Zarlinian semitone</h2>
| |
| © 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 />
| |
| Mandal positions in ratios:<br />
| |
| <span style="color: #00000a; font-family: Tahoma;">81/80, 125/124, 124/123, 123/122, 122/121, 121/120, 81/80</span><br />
| |
| <br />
| |
| Table of pitches from C to F (approximations in cents):<br />
| |
|
| |
|
| |
|
| <table class="wiki_table">
| | <ul><li>On the D string (from Db to D): |
| <tr>
| |
| <th>String<br />
| |
| </th>
| |
| <th>b<br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th>Base note<br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th>#<br />
| |
| </th>
| |
| </tr>
| |
| <tr>
| |
| <th>C<br />
| |
| </th>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td style="text-align: center;">0<br />
| |
| </td>
| |
| <td>22<br />
| |
| </td>
| |
| <td>36<br />
| |
| </td>
| |
| <td>50<br />
| |
| </td>
| |
| <td>64<br />
| |
| </td>
| |
| <td>78<br />
| |
| </td>
| |
| <td>92<br />
| |
| </td>
| |
| <td>114<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <th>D<br />
| |
| </th>
| |
| <td>90<br />
| |
| </td>
| |
| <td>112<br />
| |
| </td>
| |
| <td>126<br />
| |
| </td>
| |
| <td>140<br />
| |
| </td>
| |
| <td>154<br />
| |
| </td>
| |
| <td>168<br />
| |
| </td>
| |
| <td>182<br />
| |
| </td>
| |
| <td style="text-align: center;">204<br />
| |
| </td>
| |
| <td>226<br />
| |
| </td>
| |
| <td>240<br />
| |
| </td>
| |
| <td>254<br />
| |
| </td>
| |
| <td>268<br />
| |
| </td>
| |
| <td>282<br />
| |
| </td>
| |
| <td>296<br />
| |
| </td>
| |
| <td>318<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <th>E<br />
| |
| </th>
| |
| <td>294<br />
| |
| </td>
| |
| <td>316<br />
| |
| </td>
| |
| <td>330<br />
| |
| </td>
| |
| <td>344<br />
| |
| </td>
| |
| <td>358<br />
| |
| </td>
| |
| <td>372<br />
| |
| </td>
| |
| <td>386<br />
| |
| </td>
| |
| <td style="text-align: center;">408<br />
| |
| </td>
| |
| <td>430<br />
| |
| </td>
| |
| <td>444<br />
| |
| </td>
| |
| <td>458<br />
| |
| </td>
| |
| <td>472<br />
| |
| </td>
| |
| <td>486<br />
| |
| </td>
| |
| <td>500<br />
| |
| </td>
| |
| <td>522<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <th>F<br />
| |
| </th>
| |
| <td>384<br />
| |
| </td>
| |
| <td>406<br />
| |
| </td>
| |
| <td>420<br />
| |
| </td>
| |
| <td>434<br />
| |
| </td>
| |
| <td>448<br />
| |
| </td>
| |
| <td>462<br />
| |
| </td>
| |
| <td>476<br />
| |
| </td>
| |
| <td style="text-align: center;">498<br />
| |
| </td>
| |
| <td>520<br />
| |
| </td>
| |
| <td>534<br />
| |
| </td>
| |
| <td>548<br />
| |
| </td>
| |
| <td>562<br />
| |
| </td>
| |
| <td>576<br />
| |
| </td>
| |
| <td>590<br />
| |
| </td>
| |
| <td>612<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): |
| Interval ratios, ascending from C:<br />
| |
| <ul><li>On the D string (from Db to D):<br />
| |
| 256/243 (90), 16/15 (112), <strong>100/93 (126), 400/369 (140), 200/183 (153.78), 400/363 (168).</strong> 10/9 (182), 9/8 (204)</li><li>On the E string (from Eb to E):<br /> | |
| 32/27 (294), 6/5 (316), <strong>75/62 (329.54), 50/41 (343.56), 75/61 (357.69), 150/121 (371.94),</strong> 5/4 (386), 81/64 (408)</li></ul><br />
| |
| Interval ratios descending from F:<br />
| |
| <ul><li>On the E string (from Eb to E):<br />
| |
| 9/8, 10/9, <strong>248/225 (168.49), 82/75 (154.47), 244/225 (140.34), 242/225 (126.09</strong>), 16/15, 256/243or approximating ratios: XXX</li><li>On the D string (from Db to D):<br />
| |
| 81/64, 5/4, <strong>31/25 (372.40), 49/40 (351.33), 61/50 (344.25), 121/100 (330)</strong>, 6/5, 32/27Or approximatiing ratios: XXX</li></ul><br />
| |
| <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Other models-Super-symmetrical model with 14/13"></a><!-- ws:end:WikiTextHeadingRule:12 -->Super-symmetrical model with 14/13</h2>
| |
| © J.J. Weiss<br />
| |
| The idea behind this system is as follows:<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 <a class="wiki_link" href="/14_13">14/13</a> (two-third tone, a favorite interval of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Avicenna" rel="nofollow">Avicenna/Ibn Sina</a>).<br />
| |
| <br />
| |
| 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):<br />
| |
| <br />
| |
| 22, 16, 13, 12, 13, 16, 22<br />
| |
| <br />
| |
| (Observe that 22+16 = 38, as well as 13+12+13.)<br />
| |
| <br />
| |
| Mandal positions in ratios:<br />
| |
| 81/80, 1701/1664, 416/413, 3456/3481, 416/413, 1701/1664, 81/80<br />
| |
| <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.<br />
| |
| <br />
| |
| Table of pitches from C to F (approximations in 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>
| |
| <th>String<br />
| |
| </th>
| |
| <th>b<br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th>Base note<br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th><br />
| |
| </th>
| |
| <th>#<br />
| |
| </th>
| |
| </tr>
| |
| <tr>
| |
| <th>C<br />
| |
| </th>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td style="text-align: center;">0<br />
| |
| </td>
| |
| <td>22<br />
| |
| </td>
| |
| <td>38<br />
| |
| </td>
| |
| <td>51<br />
| |
| </td>
| |
| <td>63<br />
| |
| </td>
| |
| <td>76<br />
| |
| </td>
| |
| <td>92<br />
| |
| </td>
| |
| <td>114<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <th>D<br />
| |
| </th>
| |
| <td>90<br />
| |
| </td>
| |
| <td>112<br />
| |
| </td>
| |
| <td>128<br />
| |
| </td>
| |
| <td>141<br />
| |
| </td>
| |
| <td>153<br />
| |
| </td>
| |
| <td>166<br />
| |
| </td>
| |
| <td>182<br />
| |
| </td>
| |
| <td style="text-align: center;">204<br />
| |
| </td>
| |
| <td>226<br />
| |
| </td>
| |
| <td>242<br />
| |
| </td>
| |
| <td>255<br />
| |
| </td>
| |
| <td>267<br />
| |
| </td>
| |
| <td>280<br />
| |
| </td>
| |
| <td>296<br />
| |
| </td>
| |
| <td>318<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <th>E<br />
| |
| </th>
| |
| <td>294<br />
| |
| </td>
| |
| <td>316<br />
| |
| </td>
| |
| <td>329<br />
| |
| </td>
| |
| <td>341<br />
| |
| </td>
| |
| <td>354<br />
| |
| </td>
| |
| <td>370<br />
| |
| </td>
| |
| <td>386<br />
| |
| </td>
| |
| <td style="text-align: center;">408<br />
| |
| </td>
| |
| <td>430<br />
| |
| </td>
| |
| <td>446<br />
| |
| </td>
| |
| <td>459<br />
| |
| </td>
| |
| <td>471<br />
| |
| </td>
| |
| <td>484<br />
| |
| </td>
| |
| <td>500<br />
| |
| </td>
| |
| <td>522<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <th>F<br />
| |
| </th>
| |
| <td>384<br />
| |
| </td>
| |
| <td>406<br />
| |
| </td>
| |
| <td>422<br />
| |
| </td>
| |
| <td>435<br />
| |
| </td>
| |
| <td>447<br />
| |
| </td>
| |
| <td>460<br />
| |
| </td>
| |
| <td>476<br />
| |
| </td>
| |
| <td style="text-align: center;">498<br />
| |
| </td>
| |
| <td>520<br />
| |
| </td>
| |
| <td>536<br />
| |
| </td>
| |
| <td>549<br />
| |
| </td>
| |
| <td>561<br />
| |
| </td>
| |
| <td>574<br />
| |
| </td>
| |
| <td>590<br />
| |
| </td>
| |
| <td>612<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| <br />
| | {{Todo| cleanup }} |
| Interval ratios, ascending from C:<br />
| | [[Category:Arabic music]] |
| <ul><li>On the D string (from Db to D):<br />
| | [[Category:Persian music]] |
| 256/243 (90), [[tel:16/15/112|16/15/112]]), 14/13 (128), 64/59 (141), [[tel:59/54 /153|59/54 /153]]), 209/189 (166), 10(9 (182), 9/8 (204)</li><li>On the E string (from Eb to E):<br />
| | [[Category:Turkish music]] |
| 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></body></html></pre></div>
| | [[Category:Qanun]] |