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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
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| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | | de = Untergruppe der reinen Stimmung |
| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-03 14:03:20 UTC</tt>.<br>
| | | en = Just intonation subgroup |
| : The original revision id was <tt>216626854</tt>.<br>
| | | es = |
| : The revision comment was: <tt></tt><br>
| | | ja = 純正律部分群 |
| 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>
| | A '''just intonation subgroup''' is a {{w|free abelian group|group}} generated by a finite set of positive rational numbers via arbitrary multiplications and divisions. Using subgroups implies a way to organize [[just intonation]] intervals such that they form a lattice. Therefore it is closely related to [[regular temperament theory]]. |
| <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">By a just intonation subgroup is meant a [[http://en.wikipedia.org/wiki/Free_abelian_group|group]] generated by a finite set of positive rational numbers via arbitrary multiplications and divisions. Any such group will be contained in a [[Harmonic Limit|p-limit]] group for some minimal choice of prime p, which is the prime limit of the subgroup.
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| It is only when the group in question is not the entire p-limit group that we have a just intonation subgroup in the strict sense. Such subgroups come in two flavors: finite [[http://en.wikipedia.org/wiki/Index_of_a_subgroup|index]] and infinite index, where intuitively speaking the index measures the relative size of the subgroup within the entire p-limit group. For example, the subgroups generated by 4 and 3, by 2 and 9, and by 4 and 6 all have index 2 in the full [[3-limit]] (Pythagorean) group. Half of the 3-limit intervals will belong to any one of them, and half will not, and all three groups are distinct. On the other hand, the group generated by 2, 3, and 7 is of infinite index in the full 7-limit group, which is generated by 2, 3, 5 and 7. The index can be computed by taking the determinant of the matrix whose rows are the [[monzos]] of the generators.
| | Just intonation subgroups can be described by listing their [[generator]]s with full stops between them; we use said convention below. In standard mathematical notation, let ''c''<sub>1</sub>, …, ''c''<sub>''r''</sub> be positive reals, and suppose ''v''<sub>''k''</sub> is the musical interval of log<sub>2</sub>(''c''<sub>''k''</sub>) octaves. Then |
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| A canonical naming system for just intonation subgroups is to give a [[Normal lists|normal interval list]] for the generators of the group, which will also show the [[http://en.wikipedia.org/wiki/Rank_of_an_abelian_group|rank]] of the group by the number of generators in the list. Below we give some of the more interesting subgroup systems. If a scale is given with the system, it means the subgroup is generated by the notes of the scale. Just intonation subgroups can be described by listing their generators with dots between them; the purpose of using dots is to flag the fact that it is a subgroup which is being referred to. This naming convention is employed below.
| | <math>c_1.c_2.\cdots.c_r := \operatorname{span}_\mathbb{Z} \{v_1, \cdots, v_k\}.</math> |
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| ===7-limit subgroups===
| | There are three categories of subgroups: |
| | * ''Prime subgroups'' (e.g. 2.3.7) contain only primes |
| | * ''Composite subgroups'' (e.g. 2.9.5) contain composite numbers and perhaps prime numbers too |
| | * ''Fractional subgroups'' (e.g. 2.3.7/5) contain fractional numbers and perhaps prime and/or composite numbers too |
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| 2.3.7 | | For composite and fractional subgroups, not all combinations of numbers are mathematically valid [[basis|bases]] for subgroups. For example, 2.3.9 has a redundant generator 9, and both 2.3.15 and 2.3.5/3 can be simplified to 2.3.5. |
| Ets: 5, 31, 36, 135, 571
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| Archytas Diatonic [8/7, 32/27, 4/3, 3/2, 12/7, 16/9, 2/1]
| | A prime subgroup that does not omit any primes < ''p'' (e.g. 2.3.5, 2.3.5.7, 2.3.5.7.11, etc. but not 2.3.7 or 3.5.7) is simply called [[harmonic limit|''p''-limit JI]]. It is customary of just intonation subgroups to refer only to prime subgroups that do omit such primes, as well as the other two categories. |
| Safi al-Din Septimal [8/7, 9/7, 4/3, 32/21, 12/7, 16/9, 2/1]
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| 2.5.7
| | The following terminology has been proposed for streamlining pedagogy: Given a subgroup written as generated by a fixed (non-redundant) set: ''a''.''b''.''c''.[…].''d'', call any member of this set a '''basis element''', '''structural prime''', or "'''formal prime'''".<ref>The meaning of "formal" this term is using is "of external form or structure, rather than nature or content", which is to say that a formal prime is not necessarily ''actually'' a prime, but we treat them as if they were. The original coiner of this term, [[Inthar]], has recommended its disuse, in favor of the mathematically accurate and generic "basis element", or possibly something else which indicates the co-uniqueness of the elements.</ref> For example, if the group is written 2.5/3.7/3, the basis elements are 2, 5/3 and 7/3. |
| Ets: 6, 25, 31, 171, 239, 379, 410, 789
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| 2.5.7/5
| | == Normalization == |
| Ets: 10, 29, 31, 41, 70, 171, 241, 412
| | A canonical naming system for just intonation subgroups is to give a [[normal forms #Normal forms for commas|normal form]] for the generators of the group, which will also show the [[Wikipedia: Rank of an abelian group|rank]] of the group by the number of generators in the list (the [[Hermite normal form]] should be used here, not the [[canonical form]], because in the case of subgroups, [[enfactoring]] is sometimes desirable, such as in the subgroup 2.9.7 which should not be reduced to 2.3.7 by subgroup canonicalization). Below we give some of the more interesting subgroup systems. If a scale is given with the system, it means the subgroup is generated by the notes of the scale. |
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| 2.5/3.7
| | == Index == |
| Ets: 12, 15, 42, 57, 270, 327
| | {{See also| Wikipedia: Index of a subgroup }} |
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| 2.5.7/3
| | Intuitively speaking, the '''index''' measures the relative size of the subgroup within another subgroup, which is usually the ''p''-limit. |
| Ets: 9, 31, 40, 50, 81, 90, 171, 261
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| 2.5/3.7/3 | | Subgroups in the strict sense come in two flavors: finite index and infinite index. For example, the subgroups generated by 4 and 3, by 2 and 9, and by 4 and 6 all have index 2 in the full [[3-limit]] (Pythagorean) group. Half of the 3-limit intervals will belong to any one of them, and half will not, and all three groups are distinct. On the other hand, the group generated by 2, 3, and 7 is of infinite index in the full [[7-limit]] group, which is generated by 2, 3, 5 and 7. The index can be computed by taking the determinant of the [[subgroup basis matrix]], whose columns are the [[monzo]]s of the generators. |
| Ets: 27, 68, 72, 99, 171, 517
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| 2.27/25.7/3 | | == Generalization == |
| Ets: 9
| | Non-JI intervals can also be used as basis elements, when the subgroup in question contains non-JI intervals. For example, 2.sqrt(3/2) (sometimes written 2.2ed3/2) is the group generated by [[2/1]] and [[sqrt(3/2)]] (a neutral third which is exactly one half of 3/2, 350.978 [[cent]]s). This is closely related to the [[3L 4s]] mos tuning with neutral third generator sqrt(3/2). |
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| In effect, equivalent to 9EDO, which has a 7-limit version given by [27/25, 7/6, 63/50, 49/36, 72/49, 100/63, 12/7, 50/27, 2]
| | == List of selected subgroups == |
| | === 7-limit subgroups === |
| | {{See also| 2.3.7 subgroup }} |
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| 2.9/5.9/7 | | ; 2.3.7: |
| Ets: 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558
| | * {{EDOs|legend=1| 5, 17, 31, 36, 135, 571 }} |
| | * Archytas Diatonic [8/7, 32/27, 4/3, 3/2, 12/7, 16/9, 2/1] |
| | * Safi al-Din Septimal [8/7, 9/7, 4/3, 32/21, 12/7, 16/9, 2/1] |
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| |
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| The [[Chromatic pairs|Terrain temperament]] subgroup.
| | ; 2.5.7: |
| | * {{EDOs|legend=1| 6, 25, 31, 35, 47, 171, 239, 379, 410, 789 }} |
|
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| ===11-limit subgroups=== | | ; 2.3.7/5: |
| | * {{EDOs|legend=1| 10, 29, 31, 41, 70, 171, 241, 412 }} |
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| 2.3.11 | | ; 2.5/3.7: |
| Ets: 7, 15, 17, 24, 159, 494, 518, 653
| | * {{EDOs|legend=1| 12, 15, 42, 57, 270, 327 }} |
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| Zalzal, al-Farabi's version [9/8, 27/22, 4/3, 3/2, 18/11, 16/9, 2/1]
| | ; 2.5.7/3: |
| | * {{EDOs|legend=1| 9, 31, 40, 50, 81, 90, 171, 261 }} |
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| 2.5.11 | | ; 2.5/3.7/3: |
| Ets: 6, 7, 9, 13, 15, 22, 37, 87, 320
| | * {{EDOs|legend=1| 27, 68, 72, 99, 171, 517 }} |
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| 2.7.11 | | ; 2.27/25.7/3: |
| Ets: 6, 9, 11, 20, 26, 135, 161, 296
| | * {{EDOs|legend=1| 9 }} |
| | * In effect, equivalent to 9EDO, which has a 7-limit version given by [27/25, 7/6, 63/50, 49/36, 72/49, 100/63, 12/7, 50/27, 2] |
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| |
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| 2.3.5.11 | | ; 2.9/5.9/7: |
| Ets: 7, 15, 22, 31, 65, 72, 87, 270, 342, 407, 494
| | * {{EDOs|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }} |
| | * ''Terrain temperament'' subgroup, see [[Subgroup temperaments #Terrain]] |
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| 2.3.7.11
| | ; 3.5.7: |
| Ets: 9, 17, 26, 31, 41, 46, 63, 72, 135
| | * Does not have octaves, commonly used for non-octave [[EDT]]s |
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| |
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| The [[Chromatic pairs|Radon temperament]] subgroup, generated by the Ptolemy Intense Chromatic [22/21, 8/7, 4/3, 3/2, 11/7, 12/7, 2/1]
| | === 11-limit subgroups === |
| | {{See also| 2.3.7.11 subgroup }} |
| | {{See also| Alpharabian tuning }} |
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| |
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| 2.5.7.11 | | ; 2.3.11: |
| Ets: 6, 15, 31, 35, 37, 109, 618, 960
| | * {{EDOs|legend=1| 7, 15, 17, 24, 159, 494, 518, 653 }} |
| | * Zalzal, al-Farabi's version [9/8, 27/22, 4/3, 3/2, 18/11, 16/9, 2/1] |
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| 2.5/3.7/3.11/3 | | ; 2.5.11: |
| Ets: 33, 41, 49, 57, 106, 204, 253
| | * {{EDOs|legend=1| 6, 7, 9, 13, 15, 22, 37, 87, 320 }} |
|
| |
|
| The [[Chromatic pairs|Indium temperament]] subgroup.
| | ; 2.7.11: |
| | * {{EDOs|legend=1| 6, 9, 11, 20, 26, 135, 161, 296 }} |
|
| |
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| ===13-limit subgroups | | ; 2.3.5.11: |
| | * {{EDOs|legend=1| 7, 15, 22, 31, 65, 72, 87, 270, 342, 407, 494 }} |
|
| |
|
| 2.3.13 | | ; 2.3.7.11: |
| Ets: 7, 10, 17, 60, 70, 130, 147, 277, 424
| | * {{EDOs|legend=1| 9, 17, 26, 31, 41, 46, 63, 72, 135 }} |
| | * The [[Gamelismic clan#Radon|Radon temperament]] subgroup, generated by the Ptolemy Intense Chromatic [22/21, 8/7, 4/3, 3/2, 11/7, 12/7, 2/1] |
| | * See: [[Gallery of 2.3.7.11 Subgroup Scales]] |
|
| |
|
| Mustaqim mode, Ibn Sina [9/8, 39/32, 4/3, 3/2, 13/8, 16/9, 2/1]
| | ; 2.5.7.11: |
| | * {{EDOs|legend=1| 6, 15, 31, 35, 37, 109, 618, 960 }} |
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| |
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| 2.3.5.13 | | ; 2.5/3.7/3.11/3: |
| Ets: 15, 19, 34, 53, 87, 130, 140, 270
| | * {{EDOs|legend=1| 33, 41, 49, 57, 106, 204, 253 }} |
| | * The [[Subgroup temperaments#Indium|Indium temperament]] subgroup. |
|
| |
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| The [[The Archipelago|Trinidad]] and [[The Archipelago|Parizekmic]] temperaments subgroup.
| | === 13-limit subgroups === |
|
| |
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| 2.3.7.13 | | ; 2.3.13: |
| Ets: 10, 26, 27, 36, 77, 94, 104, 130, 234
| | * {{EDOs|legend=1| 7, 10, 17, 60, 70, 130, 147, 277, 424 }} |
| | * Mustaqim mode, Ibn Sina [9/8, 39/32, 4/3, 3/2, 13/8, 16/9, 2/1] |
|
| |
|
| Buzurg [14/13, 16/13, 4/3, 56/39, 3/2]
| | ; 2.3.5.13: |
| Safi al-Din tuning [8/7, 16/13, 4/3, 32/21, 64/39, 16/9, 2/1]
| | * {{EDOs|legend=1| 15, 19, 34, 53, 87, 130, 140, 246, 270 }} |
| Ibn Sina tuning [14/13, 7/6, 4/3, 3/2, 21/13, 7/4, 2]
| | * The [[Kleismic family#Cata|Cata]], [[The Archipelago#Trinidad|Trinidad]] and [[The Archipelago#Parizekmic|Parizekmic]] temperaments subgroup. |
|
| |
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| 2.3.13/5 | | ; 2.3.7.13: |
| Ets: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362
| | * {{EDOs|legend=1| 10, 26, 27, 36, 77, 94, 104, 130, 234 }} |
| | * Buzurg [14/13, 16/13, 4/3, 56/39, 3/2] |
| | * Safi al-Din tuning [8/7, 16/13, 4/3, 32/21, 64/39, 16/9, 2/1] |
| | * Ibn Sina tuning [14/13, 7/6, 4/3, 3/2, 21/13, 7/4, 2] |
|
| |
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| The [[The Archipelago|Barbados temperament]] subgroup. | | ; 2.5.7.13: |
| | * {{EDOs|legend=1| 7, 10, 17, 27, 37, 84, 121, 400 }} |
| | * The [[No-threes subgroup temperaments#Huntington|Huntington temperament]] subgroup. |
|
| |
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| 2.3.11/5.13/5 | | ; 2.5.7.11.13: |
| 5, 9, 14, 19, 24, 29
| | * {{EDOs|legend=1| 6, 7, 13, 19, 25, 31, 37 }} |
| | * The [[Hemimean clan#Roulette|Roulette temperament]] subgroup |
|
| |
|
| The [[Chromatic pairs|Bridgetown temperament]] subgroup.</pre></div>
| | ; 2.3.13/5: |
| <h4>Original HTML content:</h4>
| | * {{EDOs|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }} |
| <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>Just intonation subgroups</title></head><body>By a just intonation subgroup is meant a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow">group</a> generated by a finite set of positive rational numbers via arbitrary multiplications and divisions. Any such group will be contained in a <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> group for some minimal choice of prime p, which is the prime limit of the subgroup. <br />
| | * The [[The Archipelago#Barbados|Barbados temperament]] subgroup. |
| <br />
| | |
| It is only when the group in question is not the entire p-limit group that we have a just intonation subgroup in the strict sense. Such subgroups come in two flavors: finite <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Index_of_a_subgroup" rel="nofollow">index</a> and infinite index, where intuitively speaking the index measures the relative size of the subgroup within the entire p-limit group. For example, the subgroups generated by 4 and 3, by 2 and 9, and by 4 and 6 all have index 2 in the full <a class="wiki_link" href="/3-limit">3-limit</a> (Pythagorean) group. Half of the 3-limit intervals will belong to any one of them, and half will not, and all three groups are distinct. On the other hand, the group generated by 2, 3, and 7 is of infinite index in the full 7-limit group, which is generated by 2, 3, 5 and 7. The index can be computed by taking the determinant of the matrix whose rows are the <a class="wiki_link" href="/monzos">monzos</a> of the generators.<br />
| | ; 2.3.11/5.13/5: |
| <br />
| | * {{EDOs|legend=1| 5, 9, 14, 19, 24, 29 }} |
| A canonical naming system for just intonation subgroups is to give a <a class="wiki_link" href="/Normal%20lists">normal interval list</a> for the generators of the group, which will also show the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rank_of_an_abelian_group" rel="nofollow">rank</a> of the group by the number of generators in the list. Below we give some of the more interesting subgroup systems. If a scale is given with the system, it means the subgroup is generated by the notes of the scale. Just intonation subgroups can be described by listing their generators with dots between them; the purpose of using dots is to flag the fact that it is a subgroup which is being referred to. This naming convention is employed below.<br />
| | * The [[Subgroup temperaments#Bridgetown|Bridgetown temperament]] subgroup. |
| <br />
| | |
| <!-- ws:start:WikiTextHeadingRule:0:&lt;h3&gt; --><h3 id="toc0"><a name="x--7-limit subgroups"></a><!-- ws:end:WikiTextHeadingRule:0 -->7-limit subgroups</h3>
| | ; 2.3.11/7.13/7: |
| <br />
| | * {{EDOs|legend=1| 5, 7, 12, 17, 29, 46, 75, 196, 271 }} |
| 2.3.7<br />
| | * The [[Subgroup temperaments#Pepperoni|Pepperoni temperament]] subgroup. |
| Ets: 5, 31, 36, 135, 571<br />
| | |
| <br />
| | ; 2.7/5.11/5.13/5: |
| Archytas Diatonic [8/7, 32/27, 4/3, 3/2, 12/7, 16/9, 2/1]<br />
| | * {{EDOs|legend=1| 5, 8, 21, 29, 37, 66, 169, 235 }} |
| Safi al-Din Septimal [8/7, 9/7, 4/3, 32/21, 12/7, 16/9, 2/1]<br />
| | * The [[Subgroup temperaments#Tridec|Tridec temperament]] subgroup. |
| <br />
| | |
| 2.5.7<br />
| | === Higher-limit subgroups === |
| Ets: 6, 25, 31, 171, 239, 379, 410, 789<br />
| | * [[2.11.13.17.19 subgroup]] |
| <br />
| | * [[2.17/13.19/13 subgroup]] |
| 2.5.7/5<br /> | | |
| Ets: 10, 29, 31, 41, 70, 171, 241, 412<br />
| | ; 8.9.5.7.11.13.17.23: |
| <br />
| | * [[143ed11]] |
| 2.5/3.7<br />
| | |
| Ets: 12, 15, 42, 57, 270, 327<br />
| | === Irrational subgroups === |
| <br />
| | * [[Hemipyth]] (√2.√3 subgroup) |
| 2.5.7/3<br />
| | * [[Hemipent]] (√2.√3.√5 subgroup) |
| Ets: 9, 31, 40, 50, 81, 90, 171, 261<br />
| | |
| <br />
| | == See also == |
| 2.5/3.7/3<br />
| | * [[Subgroup basis matrix]] – a formal discussion on matrix representations of subgroup bases |
| Ets: 27, 68, 72, 99, 171, 517<br />
| | |
| <br />
| | == Notes == |
| 2.27/25.7/3<br />
| | |
| Ets: 9<br />
| | [[Category:Subgroup| ]] <!-- main article --> |
| <br />
| | [[Category:Just intonation]] |
| In effect, equivalent to 9EDO, which has a 7-limit version given by [27/25, 7/6, 63/50, 49/36, 72/49, 100/63, 12/7, 50/27, 2]<br />
| |
| <br />
| |
| 2.9/5.9/7<br />
| |
| Ets: 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558<br />
| |
| <br />
| |
| The <a class="wiki_link" href="/Chromatic%20pairs">Terrain temperament</a> subgroup.<br /> | |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x--11-limit subgroups"></a><!-- ws:end:WikiTextHeadingRule:2 -->11-limit subgroups</h3>
| |
| <br />
| |
| 2.3.11<br /> | |
| Ets: 7, 15, 17, 24, 159, 494, 518, 653<br />
| |
| <br />
| |
| Zalzal, al-Farabi's version [9/8, 27/22, 4/3, 3/2, 18/11, 16/9, 2/1]<br />
| |
| <br />
| |
| 2.5.11<br />
| |
| Ets: 6, 7, 9, 13, 15, 22, 37, 87, 320<br />
| |
| <br />
| |
| 2.7.11<br />
| |
| Ets: 6, 9, 11, 20, 26, 135, 161, 296<br />
| |
| <br />
| |
| 2.3.5.11<br />
| |
| Ets: 7, 15, 22, 31, 65, 72, 87, 270, 342, 407, 494<br />
| |
| <br />
| |
| 2.3.7.11<br />
| |
| Ets: 9, 17, 26, 31, 41, 46, 63, 72, 135<br />
| |
| <br />
| |
| The <a class="wiki_link" href="/Chromatic%20pairs">Radon temperament</a> subgroup, generated by the Ptolemy Intense Chromatic [22/21, 8/7, 4/3, 3/2, 11/7, 12/7, 2/1]<br /> | |
| <br />
| |
| 2.5.7.11<br />
| |
| Ets: 6, 15, 31, 35, 37, 109, 618, 960<br />
| |
| <br />
| |
| 2.5/3.7/3.11/3<br /> | |
| Ets: 33, 41, 49, 57, 106, 204, 253<br />
| |
| <br />
| |
| The <a class="wiki_link" href="/Chromatic%20pairs">Indium temperament</a> subgroup.<br /> | |
| <br />
| |
| ===13-limit subgroups<br /> | |
| <br />
| |
| 2.3.13<br /> | |
| Ets: 7, 10, 17, 60, 70, 130, 147, 277, 424<br />
| |
| <br />
| |
| Mustaqim mode, Ibn Sina [9/8, 39/32, 4/3, 3/2, 13/8, 16/9, 2/1]<br />
| |
| <br />
| |
| 2.3.5.13<br />
| |
| Ets: 15, 19, 34, 53, 87, 130, 140, 270<br />
| |
| <br />
| |
| The <a class="wiki_link" href="/The%20Archipelago">Trinidad</a> and <a class="wiki_link" href="/The%20Archipelago">Parizekmic</a> temperaments subgroup.<br />
| |
| <br />
| |
| 2.3.7.13<br />
| |
| Ets: 10, 26, 27, 36, 77, 94, 104, 130, 234<br />
| |
| <br />
| |
| Buzurg [14/13, 16/13, 4/3, 56/39, 3/2]<br />
| |
| Safi al-Din tuning [8/7, 16/13, 4/3, 32/21, 64/39, 16/9, 2/1]<br />
| |
| Ibn Sina tuning [14/13, 7/6, 4/3, 3/2, 21/13, 7/4, 2]<br />
| |
| <br />
| |
| 2.3.13/5<br />
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| Ets: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362<br />
| |
| <br />
| |
| The <a class="wiki_link" href="/The%20Archipelago">Barbados temperament</a> subgroup.<br />
| |
| <br />
| |
| 2.3.11/5.13/5<br />
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| 5, 9, 14, 19, 24, 29<br />
| |
| <br />
| |
| The <a class="wiki_link" href="/Chromatic%20pairs">Bridgetown temperament</a> subgroup.</body></html></pre></div>
| |