Table of rank-3 temperament lattice bases: Difference between revisions

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**Imported revision 238769589 - Original comment: **
m FloraC moved page Temperament lattice bases to Table of rank-3 temperament lattice bases without leaving a redirect: Adopt a more precise title
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Todo|inline=1|improve synopsis|clarify|link|comments=Expand the intro to more thoroughly explain what this is. Link to here from other relevant pages.}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-06-26 02:02:43 UTC</tt>.<br>
: The original revision id was <tt>238769589</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=Rank Three=


|| Comma || Name || Gen 1 || Length || Gen 2 || Length || Angle ||
Below are tabulated bases (temperamental Minkowski bases) of minimal temperamental complexity for the pitch classes of various regular temperaments.
|| 686/675 || Keema || 15/14 || 0.6977 || 3/2 || 1.4208 || 103.732 ||
|| 64/63 || Archytas || 3/2 || 1.0508 || 5/4 || 2.3219 || 90.000 ||
|| 875/864 || Keema || 6/5 || 0.8879 || 3/2 || 1.3391 || 77.834 ||
|| 81/80 || Didymus || 3/2 || 0.5451 || 8/7 || 2.8074 || 90.000 ||
|| 3125/3087 || Gariboh || 25/21 || 0.5928 || 21/20 || 1.2018 || 80.810 ||
|| 4000/3969 || Octagar || 63/50 || 0.8966 || 21/20 || 1.0605 || 97.743 ||
|| 245/243 || Sensamagic || 3/2 || 0.9644 || 9/7 || 1.0807 || 96.914 ||
|| 126/125 || Starling || 6/5 || 1.068 || 5/4 || 1.206 || 100.364 ||
|| 1728/1715 || Orwellisma || 7/6 || 0.7526 || 3/2 || 1.3931 || 97.508 ||
|| 1029/1024 || Gamelisma || 8/7 || 0.5192 || 5/4 || 2.3219 || 90.000 ||
|| 225/224 || Marvel || 16/15 || 1.256 || 3/2 || 1.369 || 106.958 ||
|| 3136/3125 || Hemimean || 28/25 || 0.5055 || 3/2 || 1.5849 || 90.000 ||
|| 5120/5103 || Hemifamity || 3/2 || 0.5670 || 10/9 || 1.8063 || 82.112 ||
|| 6144/6125 || Porwell || 35/32 || 0.9584 || 35/24 || 1.1877 || 92.681 ||
|| 65625/65536 || Horwell || 5/4 || 0.6213 || 16/15 || 1.3863 || 64.638 ||
|| 703125/702464 || Meter || 75/56 || 0.7305 || 375/224 || 0.7675 || 97.289 ||
|| 420175 / 419904 || Wizma || 245/216 || 0.5541 || 324/245 || 0.9731 || 79.663 ||
|| 2401/2400 || Breedsma || 49/40 || 0.7858 || 8/7 || 1.1241 || 107.367 ||
|| 4375/4374 || Ragisma || 10/9 || 0.7890 || 6/5 || 0.9208 || 105.299 ||


</pre></div>
=Rank three=
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Temperament lattice bases&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Rank Three"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Rank Three&lt;/h1&gt;
&lt;br /&gt;


==Seven limit==


&lt;table class="wiki_table"&gt;
{| class="wikitable"
    &lt;tr&gt;
|-
        &lt;td&gt;Comma&lt;br /&gt;
! | Comma
&lt;/td&gt;
! | Name
        &lt;td&gt;Name&lt;br /&gt;
! | Gen 1
&lt;/td&gt;
! | Length
        &lt;td&gt;Gen 1&lt;br /&gt;
! | Gen 2
&lt;/td&gt;
! | Length
        &lt;td&gt;Length&lt;br /&gt;
! | Angle
&lt;/td&gt;
|-
        &lt;td&gt;Gen 2&lt;br /&gt;
| | 686/675
&lt;/td&gt;
| | [[senga]]
        &lt;td&gt;Length&lt;br /&gt;
| | 15/14
&lt;/td&gt;
| | 0.6977
        &lt;td&gt;Angle&lt;br /&gt;
| | 3/2
&lt;/td&gt;
| | 1.4208
    &lt;/tr&gt;
| | 103.732
    &lt;tr&gt;
|-
        &lt;td&gt;686/675&lt;br /&gt;
| | 64/63
&lt;/td&gt;
| | [[Archytas]]'
        &lt;td&gt;Keema&lt;br /&gt;
| | 3/2
&lt;/td&gt;
| | 1.0508
        &lt;td&gt;15/14&lt;br /&gt;
| | 5/4
&lt;/td&gt;
| | 2.3219
        &lt;td&gt;0.6977&lt;br /&gt;
| | 90.000
&lt;/td&gt;
|-
        &lt;td&gt;3/2&lt;br /&gt;
| | 875/864
&lt;/td&gt;
| | [[keema]]
        &lt;td&gt;1.4208&lt;br /&gt;
| | 6/5
&lt;/td&gt;
| | 0.8879
        &lt;td&gt;103.732&lt;br /&gt;
| | 3/2
&lt;/td&gt;
| | 1.3391
    &lt;/tr&gt;
| | 77.834
    &lt;tr&gt;
|-
        &lt;td&gt;64/63&lt;br /&gt;
| | 81/80
&lt;/td&gt;
| | [[Didymus]]'
        &lt;td&gt;Archytas&lt;br /&gt;
| | 3/2
&lt;/td&gt;
| | 0.5451
        &lt;td&gt;3/2&lt;br /&gt;
| | 8/7
&lt;/td&gt;
| | 2.8074
        &lt;td&gt;1.0508&lt;br /&gt;
| | 90.000
&lt;/td&gt;
|-
        &lt;td&gt;5/4&lt;br /&gt;
| | 3125/3087
&lt;/td&gt;
| | [[gariboh]]
        &lt;td&gt;2.3219&lt;br /&gt;
| | 25/21
&lt;/td&gt;
| | 0.5928
        &lt;td&gt;90.000&lt;br /&gt;
| | 21/20
&lt;/td&gt;
| | 1.2018
    &lt;/tr&gt;
| | 80.810
    &lt;tr&gt;
|-
        &lt;td&gt;875/864&lt;br /&gt;
| | 2430/2401
&lt;/td&gt;
| | [[nuwell]]
        &lt;td&gt;Keema&lt;br /&gt;
| | 7/6
&lt;/td&gt;
| | 0.6247
        &lt;td&gt;6/5&lt;br /&gt;
| | 9/7
&lt;/td&gt;
| | 1.1894
        &lt;td&gt;0.8879&lt;br /&gt;
| | 94.044
&lt;/td&gt;
|-
        &lt;td&gt;3/2&lt;br /&gt;
| | 50421/50000
&lt;/td&gt;
| | [[trimyna]]
        &lt;td&gt;1.3391&lt;br /&gt;
| | 7/5
&lt;/td&gt;
| | 0.3158
        &lt;td&gt;77.834&lt;br /&gt;
| | 5/4
&lt;/td&gt;
| | 1.7938
    &lt;/tr&gt;
| | 94.101
    &lt;tr&gt;
|-
        &lt;td&gt;81/80&lt;br /&gt;
| | 245/243
&lt;/td&gt;
| | [[sensamagic]]
        &lt;td&gt;Didymus&lt;br /&gt;
| | 3/2
&lt;/td&gt;
| | 0.9644
        &lt;td&gt;3/2&lt;br /&gt;
| | 9/7
&lt;/td&gt;
| | 1.0807
        &lt;td&gt;0.5451&lt;br /&gt;
| | 96.914
&lt;/td&gt;
|-
        &lt;td&gt;8/7&lt;br /&gt;
| | 126/125
&lt;/td&gt;
| | [[starling]]
        &lt;td&gt;2.8074&lt;br /&gt;
| | 6/5
&lt;/td&gt;
| | 1.068
        &lt;td&gt;90.000&lt;br /&gt;
| | 5/4
&lt;/td&gt;
| | 1.206
    &lt;/tr&gt;
| | 100.364
    &lt;tr&gt;
|-
        &lt;td&gt;3125/3087&lt;br /&gt;
| | 4000/3969
&lt;/td&gt;
| | [[octagar]]
        &lt;td&gt;Gariboh&lt;br /&gt;
| | 63/50
&lt;/td&gt;
| | 0.8966
        &lt;td&gt;25/21&lt;br /&gt;
| | 21/20
&lt;/td&gt;
| | 1.0605
        &lt;td&gt;0.5928&lt;br /&gt;
| | 97.743
&lt;/td&gt;
|-
        &lt;td&gt;21/20&lt;br /&gt;
| | 1728/1715
&lt;/td&gt;
| | [[orwellisma]]
        &lt;td&gt;1.2018&lt;br /&gt;
| | 7/6
&lt;/td&gt;
| | 0.7526
        &lt;td&gt;80.810&lt;br /&gt;
| | 3/2
&lt;/td&gt;
| | 1.3931
    &lt;/tr&gt;
| | 97.508
    &lt;tr&gt;
|-
        &lt;td&gt;4000/3969&lt;br /&gt;
| | 1029/1024
&lt;/td&gt;
| | [[gamelisma]]
        &lt;td&gt;Octagar&lt;br /&gt;
| | 8/7
&lt;/td&gt;
| | 0.5192
        &lt;td&gt;63/50&lt;br /&gt;
| | 5/4
&lt;/td&gt;
| | 2.3219
        &lt;td&gt;0.8966&lt;br /&gt;
| | 90.000
&lt;/td&gt;
|-
        &lt;td&gt;21/20&lt;br /&gt;
| | 225/224
&lt;/td&gt;
| | [[marvel]]
        &lt;td&gt;1.0605&lt;br /&gt;
| | 16/15
&lt;/td&gt;
| | 1.256
        &lt;td&gt;97.743&lt;br /&gt;
| | 3/2
&lt;/td&gt;
| | 1.369
    &lt;/tr&gt;
| | 106.958
    &lt;tr&gt;
|-
        &lt;td&gt;245/243&lt;br /&gt;
| | 3136/3125
&lt;/td&gt;
| | [[hemimean]]
        &lt;td&gt;Sensamagic&lt;br /&gt;
| | 28/25
&lt;/td&gt;
| | 0.5055
        &lt;td&gt;3/2&lt;br /&gt;
| | 3/2
&lt;/td&gt;
| | 1.5849
        &lt;td&gt;0.9644&lt;br /&gt;
| | 90.000
&lt;/td&gt;
|-
        &lt;td&gt;9/7&lt;br /&gt;
| | 5120/5103
&lt;/td&gt;
| | [[hemifamity]]
        &lt;td&gt;1.0807&lt;br /&gt;
| | 3/2
&lt;/td&gt;
| | 0.5670
        &lt;td&gt;96.914&lt;br /&gt;
| | 10/9
&lt;/td&gt;
| | 1.8063
    &lt;/tr&gt;
| | 82.112
    &lt;tr&gt;
|-
        &lt;td&gt;126/125&lt;br /&gt;
| | 6144/6125
&lt;/td&gt;
| | [[porwell]]
        &lt;td&gt;Starling&lt;br /&gt;
| | 35/32
&lt;/td&gt;
| | 0.9584
        &lt;td&gt;6/5&lt;br /&gt;
| | 35/24
&lt;/td&gt;
| | 1.1877
        &lt;td&gt;1.068&lt;br /&gt;
| | 92.681
&lt;/td&gt;
|-
        &lt;td&gt;5/4&lt;br /&gt;
| | 65625/65536
&lt;/td&gt;
| | [[horwell]]
        &lt;td&gt;1.206&lt;br /&gt;
| | 5/4
&lt;/td&gt;
| | 0.6213
        &lt;td&gt;100.364&lt;br /&gt;
| | 16/15
&lt;/td&gt;
| | 1.3863
    &lt;/tr&gt;
| | 64.638
    &lt;tr&gt;
|-
        &lt;td&gt;1728/1715&lt;br /&gt;
| | 703125/702464
&lt;/td&gt;
| | [[meter]]
        &lt;td&gt;Orwellisma&lt;br /&gt;
| | 75/56
&lt;/td&gt;
| | 0.7305
        &lt;td&gt;7/6&lt;br /&gt;
| | 375/224
&lt;/td&gt;
| | 0.7675
        &lt;td&gt;0.7526&lt;br /&gt;
| | 97.289
&lt;/td&gt;
|-
        &lt;td&gt;3/2&lt;br /&gt;
| | 420175/419904
&lt;/td&gt;
| | [[wizma]]
        &lt;td&gt;1.3931&lt;br /&gt;
| | 245/216
&lt;/td&gt;
| | 0.5541
        &lt;td&gt;97.508&lt;br /&gt;
| | 324/245
&lt;/td&gt;
| | 0.9731
    &lt;/tr&gt;
| | 79.663
    &lt;tr&gt;
|-
        &lt;td&gt;1029/1024&lt;br /&gt;
| | 2401/2400
&lt;/td&gt;
| | [[breedsma]]
        &lt;td&gt;Gamelisma&lt;br /&gt;
| | 49/40
&lt;/td&gt;
| | 0.7858
        &lt;td&gt;8/7&lt;br /&gt;
| | 8/7
&lt;/td&gt;
| | 1.1241
        &lt;td&gt;0.5192&lt;br /&gt;
| | 107.367
&lt;/td&gt;
|-
        &lt;td&gt;5/4&lt;br /&gt;
| | 4375/4374
&lt;/td&gt;
| | [[ragisma]]
        &lt;td&gt;2.3219&lt;br /&gt;
| | 10/9
&lt;/td&gt;
| | 0.7890
        &lt;td&gt;90.000&lt;br /&gt;
| | 6/5
&lt;/td&gt;
| | 0.9208
    &lt;/tr&gt;
| | 105.299
    &lt;tr&gt;
|}
        &lt;td&gt;225/224&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Marvel&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1.256&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1.369&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;106.958&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3136/3125&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Hemimean&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;28/25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.5055&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1.5849&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;90.000&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5120/5103&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Hemifamity&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.5670&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1.8063&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;82.112&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6144/6125&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Porwell&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;35/32&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.9584&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;35/24&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1.1877&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;92.681&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;65625/65536&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Horwell&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.6213&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1.3863&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;64.638&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;703125/702464&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Meter&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;75/56&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.7305&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;375/224&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.7675&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;97.289&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;420175 / 419904&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Wizma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;245/216&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.5541&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;324/245&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.9731&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;79.663&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2401/2400&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Breedsma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;49/40&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.7858&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1.1241&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;107.367&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4375/4374&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Ragisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.7890&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.9208&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;105.299&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
==Eleven limit==
 
{| class="wikitable"
|-
! | Commas
! | Name
! | Gen 1
! | Length
! | Gen 2
! | Length
! | Angle
|-
| | 225/224, 385/384
| | [[marvel]]
| | 16/15
| | 1.0364
| | 5/4
| | 1.0759
| | 104.028
|-
| | 225/224, 441/440 <!-- was 441/400, presumed incorrect -->
| | [[prodigy]]
| | 16/15
| | 0.9111
| | 3/2
| | 0.9477
| | 65.933
|-
| | 99/98, 176/175
| | [[minerva]]
| | 16/15
| | 0.8997
| | 5/4
| | 1.0457
| | 98.604
|-
| | 126/125, 176/175
| | [[thrush]]
| | 5/4
| | 0.8576
| | 6/5
| | 0.9314
| | 74.624
|-
| | 56/55, 100/99
| | [[thrasher]]
| | 6/5
| | 0.9089
| | 5/4
| | 1.2007
| | 98.845
|}
 
[[Category:Lattice]]
[[Category:Lists of temperaments]]
[[Category:Tables]]

Latest revision as of 18:21, 16 December 2024

Todo: improve synopsis, clarify, link

Below are tabulated bases (temperamental Minkowski bases) of minimal temperamental complexity for the pitch classes of various regular temperaments.

Rank three

Seven limit

Comma Name Gen 1 Length Gen 2 Length Angle
686/675 senga 15/14 0.6977 3/2 1.4208 103.732
64/63 Archytas' 3/2 1.0508 5/4 2.3219 90.000
875/864 keema 6/5 0.8879 3/2 1.3391 77.834
81/80 Didymus' 3/2 0.5451 8/7 2.8074 90.000
3125/3087 gariboh 25/21 0.5928 21/20 1.2018 80.810
2430/2401 nuwell 7/6 0.6247 9/7 1.1894 94.044
50421/50000 trimyna 7/5 0.3158 5/4 1.7938 94.101
245/243 sensamagic 3/2 0.9644 9/7 1.0807 96.914
126/125 starling 6/5 1.068 5/4 1.206 100.364
4000/3969 octagar 63/50 0.8966 21/20 1.0605 97.743
1728/1715 orwellisma 7/6 0.7526 3/2 1.3931 97.508
1029/1024 gamelisma 8/7 0.5192 5/4 2.3219 90.000
225/224 marvel 16/15 1.256 3/2 1.369 106.958
3136/3125 hemimean 28/25 0.5055 3/2 1.5849 90.000
5120/5103 hemifamity 3/2 0.5670 10/9 1.8063 82.112
6144/6125 porwell 35/32 0.9584 35/24 1.1877 92.681
65625/65536 horwell 5/4 0.6213 16/15 1.3863 64.638
703125/702464 meter 75/56 0.7305 375/224 0.7675 97.289
420175/419904 wizma 245/216 0.5541 324/245 0.9731 79.663
2401/2400 breedsma 49/40 0.7858 8/7 1.1241 107.367
4375/4374 ragisma 10/9 0.7890 6/5 0.9208 105.299

Eleven limit

Commas Name Gen 1 Length Gen 2 Length Angle
225/224, 385/384 marvel 16/15 1.0364 5/4 1.0759 104.028
225/224, 441/440 prodigy 16/15 0.9111 3/2 0.9477 65.933
99/98, 176/175 minerva 16/15 0.8997 5/4 1.0457 98.604
126/125, 176/175 thrush 5/4 0.8576 6/5 0.9314 74.624
56/55, 100/99 thrasher 6/5 0.9089 5/4 1.2007 98.845
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