Starling temperaments
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This page discusses some of the temperaments tempering out 126/125, the starling comma or septimal semicomma. Since (6/5)^3 = 126/125 * 12/7, these temperaments tend to have a relatively small complexity for 6/5. They also possess 6/5-6/5-6/5-7/6 versions of the diminished seventh chord. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before [[12edo]] established itself as the standard tuning, it is arguably more authentic to tune it as three stacked minor thirds and an augmented second, which is what it is in meantone, than as the modern version of four stacked very flat minor thirds. ==Myna temperament== In addition to 126/125, myna tempers out 1728/1715, the orwell comma, and 2401/2400, the breedsma. It can also be described as the 27&31 temperament, or in terms of its wedgie <<10 9 7 -9 -17 -9||. It has 6/5 as a generator, and [[58edo]] can be used as a tuning, with [[89edo]] being a better one, and fans of round amounts in cents may like [[120edo]]. It is also possible to tune myna with pure fifths by taking 6^(1/10) as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits. [[Comma|Commas]]: 126/125, 1728/1715 7 and 9 limit minimax [|1 0 0 0>, |0 1 0 0 >, |9/10 9/10 0 0>, |17/10 7/10 0 0>] [[Eigenmonzo|Eigenmonzos]]: 2, 3 [[POTE tuning|POTE generator]]: 310.146 Map: [<1 9 9 8|, <0 -10 -9 -7|] [[Generator|Generators]]: 2, 5/3 EDOs: 27, 31, 58, 89 Badness: 0.0270 ===11-limit=== Commas: 126/125, 176/175, 243/242 [[POTE tuning|POTE generator]]: 310.144 Map: [<1 9 9 8 22|, <0 -10 -9 -7 -25|]: 310.146 EDOs: 31, 58, 89 Badness: 0.0168 ==Sensi temperament== Sensi tempers out 686/675, 245/243 and 4375/4374 in addition to 126/125, and can be described as the 19&27 temperament. It has as a generator half of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 13-limit sensi tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and MOS of size 11, 19 and 27 are available. [[Comma|Commas]]: 126/125, 245/243 7-limit minimax [|1 0 0 0>, |1/13 0 0 7/13>, |5/13 0 0 9/13>, |0 0 0 1>] [[Eigenmonzo|Eigenmonzos]]: 2, 7 9-limit minimax [|1 0 0 0>, |2/5 14/5 -7/5 0>, |4/5 18/5 -9/5 0>, |3/5 26/5 -13/5 0>] [[Eigenmonzo|Eigenmonzos]]: 2, 9/5 [[POTE tuning|POTE generator]]: 443.383 Algebraic generator: Calista, the [[Algebraic number|real root]] of x^7-2x^2-1, at 340.6467 cents. Map: [<1 6 8 11|, <0 -7 -9 -13|] [[Generator|Generators]]: 2, 14/9 EDOs: 19, 27, 46, 249, 295 Badness: 0.0256 ===Sensis=== Commas: 56/55, 100/99, 245/243 [[POTE tuning|POTE generator]]: 443.962 Map: [<1 6 8 11 6|, <0 -7 -9 -13 -4|] EDOs: 19, 27, 73, 100 Badness: 0.0287 ====13-limit==== Commas: 56/55, 78/77, 91/90, 100/99 [[POTE tuning|POTE generator]]: 443.945 Map: [<1 6 8 11 6 10|, <0 -7 -9 -13 -4 -10|] EDOs: 19, 27, 73, 100 Badness: 0.0200 ==Valentine temperament== Valentine tempers out 1029/1024 and 6144/6125 as well as 126/125, so it also fits under the heading of the gamelismic clan. It has a generator of 21/20, which can be stripped of its 2 and taken as 3*7/5. In this respect it resembles miracle, with a generator of 3*5/7, and casablanca, with a generator of 5*7/3. These three generators are the simplest in terms of the relationship of tetrads in the [[The Seven Limit Symmetrical Lattices|lattice of 7-limit tetrads]]. Valentine can also be described as the 31&46 temperament, and [[77edo]], [[108edo]] or [[185edo]] make for excellent tunings, which also happen to be excellent tunings for starling temperament, the 126/125 planar temperament. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)^(1/9) as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit as <<9 5 -3 7 ... ||, tempering out 121/120 and 441/440; 46et has a valentine generator 3/46 which is only 0.0117 cents sharp of the minimax generator, (11/7)^(1/10). Valentine is very closely related to [[Carlos Alpha]], the rank one nonoctave temperament of Wendy Carlos, as the generator chain of valentine is the same thing as Carlos Alpha. Indeed, the way Carlos uses Alpha in //Beauty in the Beast// suggests that she really intended Alpha to be the same thing as valentine, and that it is misdescribed as a rank one temperament. Carlos tells us that "The melodic motions of Alpha are amazingly exotic and fresh, like you've never heard before", and since Alpha lives inside valentine this comment carries over and applies to it if you stick close melodically to generator steps, which is almost impossible not to do since the generator step is so small. MOS of 15, 16, 31 and 46 notes are available to explore these exotic and fresh melodies, or the less exotic ones you might cook up otherwise. [[Comma|Commas]]: 1029/1024, 126/125 [[Minimax tuning]]: 7-limit: [|1 0 0 0>, |5/2 3/4 0 -3/4>, |17/6 5/12 0 -5/12>, [5/2 -1/4 0 1/4>] [[Eigenmonzo|Eigenmonzos]]: 2, 7/6 9-limit: [|1 0 0 0>, |10/7 6/7 0 -3/7>, |47/21 10/21 0 -5/21>, |20/7 -2/7 0 1/7>] [[Eigenmonzo|Eigenmonzos]]: 2, 9/7 [[POTE tuning|POTE generator]]: 77.864 Algebraic generator: [[Algebraic number|smaller root]] of x^2-89x+92, or (89-sqrt(7553))/2, at 77.8616 cents. Map: [<1 1 2 3|, <0 9 5 -3|] [[Generator|Generators]]: 2, 21/20 EDOs: 15, 31, 46, 77, 185, 262 Badness: 0.0311 ===11-limit=== [[Comma|Commas]]: 121/120, 126/125, 176/175 [[Minimax tuning]]: [|1 0 0 0 0>, |1 0 0 -9/10 9/10>, |2 0 0 -1/2 1/2>, |3 0 0 3/10 -3/10>, |3 0 0 -7/10 7/10>] [[Eigenmonzo|Eigenmonzos]]: 2, 11/7 Minimax generator: (11/7)^(1/10) = 78.249 [[POTE tuning|POTE generator]]: 77.881 Algebraic generator: Gontrand2, the smallest positive root of 4x^7-8x^6+5, at 77.9989 cents. Map: [<1 1 2 3 3|, <0 9 5 -3 7|] [[Edo|Edos]]: [[15edo|15]], [[31edo|31]], [[46edo|46]], [[77edo|77]], [[108edo|108]], [[185edo|185]] Badness: 0.0167 ==Casablanca temperament== Aside from 126/125, casablanca tempers out the no-threes comma 823543/819200 and also 589824/588245, and may also be described by its wedgie, <<19 14 4 -22 -47 -30||, or as 31&73. 74/135 or 91/166 supply good tunings for the generator, and 20 and 31 note MOS are available. It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the 35/24 generator is particularly interesting; like 15/14 and 21/20, it represents an interval between one vertex of a [[Hexany|hexany]] and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads. For another, if we add 385/384 to the list of commas, 35/24 is identified with 16/11, and casablanca is revealed as an 11-limit temperament with a very low complexity for 11 and not too high a one for 7; we might compare 1, 4, 14, 19, the generator steps to 11, 7, 5 and 3 respectively, with 1, 4, 10, 18, the steps to 3, 5, 7 and 11 in 11-limit meantone. ==Nusecond temperament== Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31&70, or in terms of its wedgie as <<11 13 17 -5 -4 3||. It has a neutral second generator of 49/45, two of which make up a 6/5 minor third since 2430/2401 is tempered out. [[31edo]] can be used as a tuning, or [[132edo]] with a val which is the sum of the [[Patent val|patent vals]] for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. MOS of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note MOS might also be considered from the melodic point of view. [[Comma|Commas]]: 126/125, 2430/2401 7-limit minimax [|1 0 0 0>, |-5/13 0 11/13 0>, |0 0 1 0>, |-3/13 0 17/13 0>] [[Eigenmonzo|Eigenmonzos]]: 2, 5 9-limit minimax [|1 0 0 0>, |0 1 0 0>, |5/11 13/11 0 0>, |4/11 17/11 0 0>] [[Eigenmonzo|Eigenmonzos]]: 2, 3 [[POTE tuning|POTE generator]]: 154.579 Map: [<1 3 4 5|, <0 -11 -13 -17|] [[Generator|Generators]]: 2, 49/45 EDOs: 7, 8, 31, 101, 132, 163 Badness: 0.0504 ===11-limit=== [[Comma|Commas]]: 99/98, 121/120, 126/125 11-limit minimax [|1 0 0 0 0>, |19/10 11/5 0 0 -11/10>, |27/10 13/5 0 0 -13/10>, |33/10 17/5 0 0 -17/10>, |19/5 12/5 0 0 -6/5>] [[Eigenmonzo|Eigenmonzos]]: 2, 11/9 [[POTE tuning|POTE generator]]: 154.645 Algebraic generator: [[Algebraic number|positive root]] of 15x^2-10x-7, or (5+sqrt(130))/15, at 154.6652 cents. The recurrence converges very quickly. Map: [<1 3 4 5 5|, <0 -11 -13 -17 -12|] [[Generator|Generators]]: 2, 11/10 EDOs: 7, 8, 31, 101, 194 Badness: 0.0256
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<html><head><title>Starling temperaments</title></head><body>This page discusses some of the temperaments tempering out 126/125, the starling comma or septimal semicomma. Since (6/5)^3 = 126/125 * 12/7, these temperaments tend to have a relatively small complexity for 6/5. They also possess 6/5-6/5-6/5-7/6 versions of the diminished seventh chord. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before <a class="wiki_link" href="/12edo">12edo</a> established itself as the standard tuning, it is arguably more authentic to tune it as three stacked minor thirds and an augmented second, which is what it is in meantone, than as the modern version of four stacked very flat minor thirds.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Myna temperament"></a><!-- ws:end:WikiTextHeadingRule:0 -->Myna temperament</h2> In addition to 126/125, myna tempers out 1728/1715, the orwell comma, and 2401/2400, the breedsma. It can also be described as the 27&31 temperament, or in terms of its wedgie <<10 9 7 -9 -17 -9||. It has 6/5 as a generator, and <a class="wiki_link" href="/58edo">58edo</a> can be used as a tuning, with <a class="wiki_link" href="/89edo">89edo</a> being a better one, and fans of round amounts in cents may like <a class="wiki_link" href="/120edo">120edo</a>. It is also possible to tune myna with pure fifths by taking 6^(1/10) as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.<br /> <br /> <a class="wiki_link" href="/Comma">Commas</a>: 126/125, 1728/1715<br /> <br /> 7 and 9 limit minimax<br /> [|1 0 0 0>, |0 1 0 0 >, |9/10 9/10 0 0>, |17/10 7/10 0 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 3<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 310.146<br /> <br /> Map: [<1 9 9 8|, <0 -10 -9 -7|]<br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 5/3<br /> EDOs: 27, 31, 58, 89<br /> Badness: 0.0270<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="x-Myna temperament-11-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->11-limit</h3> Commas: 126/125, 176/175, 243/242<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 310.144<br /> <br /> Map: [<1 9 9 8 22|, <0 -10 -9 -7 -25|]: 310.146<br /> EDOs: 31, 58, 89<br /> Badness: 0.0168<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="x-Sensi temperament"></a><!-- ws:end:WikiTextHeadingRule:4 -->Sensi temperament</h2> Sensi tempers out 686/675, 245/243 and 4375/4374 in addition to 126/125, and can be described as the 19&27 temperament. It has as a generator half of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 13-limit sensi tempers out 91/90. 22/17, in the middle, is even closer to the generator. <a class="wiki_link" href="/46edo">46edo</a> is an excellent sensi tuning, and MOS of size 11, 19 and 27 are available.<br /> <br /> <a class="wiki_link" href="/Comma">Commas</a>: 126/125, 245/243<br /> <br /> 7-limit minimax<br /> [|1 0 0 0>, |1/13 0 0 7/13>, |5/13 0 0 9/13>, |0 0 0 1>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 7<br /> <br /> 9-limit minimax<br /> [|1 0 0 0>, |2/5 14/5 -7/5 0>, <br /> |4/5 18/5 -9/5 0>, |3/5 26/5 -13/5 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 9/5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 443.383<br /> Algebraic generator: Calista, the <a class="wiki_link" href="/Algebraic%20number">real root</a> of x^7-2x^2-1, at 340.6467 cents. <br /> <br /> Map: [<1 6 8 11|, <0 -7 -9 -13|]<br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 14/9<br /> EDOs: 19, 27, 46, 249, 295<br /> Badness: 0.0256<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="x-Sensi temperament-Sensis"></a><!-- ws:end:WikiTextHeadingRule:6 -->Sensis</h3> Commas: 56/55, 100/99, 245/243<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 443.962<br /> <br /> Map: [<1 6 8 11 6|, <0 -7 -9 -13 -4|]<br /> EDOs: 19, 27, 73, 100<br /> Badness: 0.0287<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h4> --><h4 id="toc4"><a name="x-Sensi temperament-Sensis-13-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->13-limit</h4> Commas: 56/55, 78/77, 91/90, 100/99<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 443.945<br /> <br /> Map: [<1 6 8 11 6 10|, <0 -7 -9 -13 -4 -10|]<br /> EDOs: 19, 27, 73, 100<br /> Badness: 0.0200<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="x-Valentine temperament"></a><!-- ws:end:WikiTextHeadingRule:10 -->Valentine temperament</h2> Valentine tempers out 1029/1024 and 6144/6125 as well as 126/125, so it also fits under the heading of the gamelismic clan. It has a generator of 21/20, which can be stripped of its 2 and taken as 3*7/5. In this respect it resembles miracle, with a generator of 3*5/7, and casablanca, with a generator of 5*7/3. These three generators are the simplest in terms of the relationship of tetrads in the <a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices">lattice of 7-limit tetrads</a>. Valentine can also be described as the 31&46 temperament, and <a class="wiki_link" href="/77edo">77edo</a>, <a class="wiki_link" href="/108edo">108edo</a> or <a class="wiki_link" href="/185edo">185edo</a> make for excellent tunings, which also happen to be excellent tunings for starling temperament, the 126/125 planar temperament. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)^(1/9) as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit as <<9 5 -3 7 ... ||, tempering out 121/120 and 441/440; 46et has a valentine generator 3/46 which is only 0.0117 cents sharp of the minimax generator, (11/7)^(1/10).<br /> <br /> Valentine is very closely related to <a class="wiki_link" href="/Carlos%20Alpha">Carlos Alpha</a>, the rank one nonoctave temperament of Wendy Carlos, as the generator chain of valentine is the same thing as Carlos Alpha. Indeed, the way Carlos uses Alpha in <em>Beauty in the Beast</em> suggests that she really intended Alpha to be the same thing as valentine, and that it is misdescribed as a rank one temperament. Carlos tells us that "The melodic motions of Alpha are amazingly exotic and fresh, like you've never heard before", and since Alpha lives inside valentine this comment carries over and applies to it if you stick close melodically to generator steps, which is almost impossible not to do since the generator step is so small. MOS of 15, 16, 31 and 46 notes are available to explore these exotic and fresh melodies, or the less exotic ones you might cook up otherwise.<br /> <br /> <a class="wiki_link" href="/Comma">Commas</a>: 1029/1024, 126/125<br /> <br /> <a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a>:<br /> 7-limit: [|1 0 0 0>, |5/2 3/4 0 -3/4>, <br /> |17/6 5/12 0 -5/12>, [5/2 -1/4 0 1/4>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 7/6<br /> <br /> 9-limit: [|1 0 0 0>, |10/7 6/7 0 -3/7>, <br /> |47/21 10/21 0 -5/21>, |20/7 -2/7 0 1/7>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 9/7<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 77.864<br /> <br /> Algebraic generator: <a class="wiki_link" href="/Algebraic%20number">smaller root</a> of x^2-89x+92, or (89-sqrt(7553))/2, at 77.8616 cents.<br /> <br /> Map: [<1 1 2 3|, <0 9 5 -3|]<br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 21/20<br /> EDOs: 15, 31, 46, 77, 185, 262<br /> Badness: 0.0311<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h3> --><h3 id="toc6"><a name="x-Valentine temperament-11-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->11-limit</h3> <a class="wiki_link" href="/Comma">Commas</a>: 121/120, 126/125, 176/175<br /> <br /> <a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a>:<br /> [|1 0 0 0 0>, |1 0 0 -9/10 9/10>, <br /> |2 0 0 -1/2 1/2>, |3 0 0 3/10 -3/10>, |3 0 0 -7/10 7/10>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 11/7<br /> <br /> Minimax generator: (11/7)^(1/10) = 78.249<br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 77.881<br /> <br /> Algebraic generator: Gontrand2, the smallest positive root of 4x^7-8x^6+5, at 77.9989 cents.<br /> <br /> Map: [<1 1 2 3 3|, <0 9 5 -3 7|]<br /> <a class="wiki_link" href="/Edo">Edos</a>: <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/46edo">46</a>, <a class="wiki_link" href="/77edo">77</a>, <a class="wiki_link" href="/108edo">108</a>, <a class="wiki_link" href="/185edo">185</a><br /> Badness: 0.0167<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="x-Casablanca temperament"></a><!-- ws:end:WikiTextHeadingRule:14 -->Casablanca temperament</h2> Aside from 126/125, casablanca tempers out the no-threes comma 823543/819200 and also 589824/588245, and may also be described by its wedgie, <<19 14 4 -22 -47 -30||, or as 31&73. 74/135 or 91/166 supply good tunings for the generator, and 20 and 31 note MOS are available.<br /> <br /> It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the 35/24 generator is particularly interesting; like 15/14 and 21/20, it represents an interval between one vertex of a <a class="wiki_link" href="/Hexany">hexany</a> and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads. For another, if we add 385/384 to the list of commas, 35/24 is identified with 16/11, and casablanca is revealed as an 11-limit temperament with a very low complexity for 11 and not too high a one for 7; we might compare 1, 4, 14, 19, the generator steps to 11, 7, 5 and 3 respectively, with 1, 4, 10, 18, the steps to 3, 5, 7 and 11 in 11-limit meantone.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h2> --><h2 id="toc8"><a name="x-Nusecond temperament"></a><!-- ws:end:WikiTextHeadingRule:16 -->Nusecond temperament</h2> Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31&70, or in terms of its wedgie as <<11 13 17 -5 -4 3||. It has a neutral second generator of 49/45, two of which make up a 6/5 minor third since 2430/2401 is tempered out. <a class="wiki_link" href="/31edo">31edo</a> can be used as a tuning, or <a class="wiki_link" href="/132edo">132edo</a> with a val which is the sum of the <a class="wiki_link" href="/Patent%20val">patent vals</a> for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. MOS of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note MOS might also be considered from the melodic point of view.<br /> <br /> <a class="wiki_link" href="/Comma">Commas</a>: 126/125, 2430/2401<br /> <br /> 7-limit minimax<br /> [|1 0 0 0>, |-5/13 0 11/13 0>, |0 0 1 0>, |-3/13 0 17/13 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 5<br /> <br /> 9-limit minimax<br /> [|1 0 0 0>, |0 1 0 0>, |5/11 13/11 0 0>, |4/11 17/11 0 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 3<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 154.579<br /> <br /> Map: [<1 3 4 5|, <0 -11 -13 -17|]<br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 49/45<br /> EDOs: 7, 8, 31, 101, 132, 163<br /> Badness: 0.0504<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h3> --><h3 id="toc9"><a name="x-Nusecond temperament-11-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->11-limit</h3> <a class="wiki_link" href="/Comma">Commas</a>: 99/98, 121/120, 126/125<br /> <br /> 11-limit minimax<br /> [|1 0 0 0 0>, |19/10 11/5 0 0 -11/10>, <br /> |27/10 13/5 0 0 -13/10>, |33/10 17/5 0 0 -17/10>, <br /> |19/5 12/5 0 0 -6/5>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 11/9<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 154.645<br /> Algebraic generator: <a class="wiki_link" href="/Algebraic%20number">positive root</a> of 15x^2-10x-7, or (5+sqrt(130))/15, at 154.6652 cents. The recurrence converges very quickly.<br /> <br /> Map: [<1 3 4 5 5|, <0 -11 -13 -17 -12|]<br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 11/10<br /> EDOs: 7, 8, 31, 101, 194<br /> Badness: 0.0256</body></html>