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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>
''50edo'' divides the [[Octave|octave]] into 50 equal parts of precisely 24 [[cent|cent]]s each. In the [[5-limit|5-limit]], it tempers out 81/80, making it a [[Meantone|meantone]] system, and in that capacity has historically has drawn some notice. In [http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf "Harmonics or the Philosophy of Musical Sounds"] (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts - 50edo, in one word. Later, W.S.B. Woolhouse noted it was fairly close to the [[Target_tunings|least squares]] tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher limit point of view. While [[31edo|31edo]] extends meantone with a [[7/4|7/4]] which is nearly pure, 50 has a flat 7/4 but both [[11/8|11/8]] and [[13/8|13/8]] are nearly pure.
: This revision was by author [[User:iamcamtaylor|iamcamtaylor]] and made on <tt>2017-03-14 21:06:49 UTC</tt>.<br>
: The original revision id was <tt>608843897</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">[[toc]]
//50edo// divides the [[octave]] into 50 equal parts of precisely 24 [[cent]]s each. In the [[5-limit]], it tempers out 81/80, making it a [[meantone]] system, and in that capacity has historically has drawn some notice. In [[http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf|"Harmonics or the Philosophy of Musical Sounds"]] (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts - 50edo, in one word. Later, W.S.B. Woolhouse noted it was fairly close to the [[Target tunings|least squares]] tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher limit point of view. While [[31edo]] extends meantone with a [[7_4|7/4]] which is nearly pure, 50 has a flat 7/4 but both [[11_8|11/8]] and [[13_8|13/8]] are nearly pure.


50 tempers out 126/125, 225/224 and 3136/3125 in the [[7-limit]], indicating it supports septimal meantone; 245/242, 385/384 and 540/539 in the [[11-limit]] and 105/104, 144/143 and 196/195 in the [[13-limit]], and can be used for even higher limits. Aside from meantone and its extension meanpop, it can be used to advantage for the 15&amp;50 temperament ([[http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&amp;limit=11|Coblack]]), and provides the optimal patent val for 11 and 13 limit [[Meantone family#Septimal%20meantone-Bimeantone|bimeantone]]. It is also the unique equal temperament tempering out both 81/80 and the [[vishnuzma]], |23 6 -14&gt;, so that in 50et seven chromatic semitones are a perfect fourth. In 12et by comparison this gives a fifth, in 31et a doubly diminished fifth, and in 19et a diminished fourth.
50 tempers out 126/125, 225/224 and 3136/3125 in the [[7-limit|7-limit]], indicating it supports septimal meantone; 245/242, 385/384 and 540/539 in the [[11-limit|11-limit]] and 105/104, 144/143 and 196/195 in the [[13-limit|13-limit]], and can be used for even higher limits. Aside from meantone and its extension meanpop, it can be used to advantage for the 15&amp;50 temperament ([http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&limit=11 Coblack]), and provides the optimal patent val for 11 and 13 limit [[Meantone_family#Septimal meantone-Bimeantone|bimeantone]]. It is also the unique equal temperament tempering out both 81/80 and the [[vishnuzma|vishnuzma]], |23 6 -14&gt;, so that in 50et seven chromatic semitones are a perfect fourth. In 12et by comparison this gives a fifth, in 31et a doubly diminished fifth, and in 19et a diminished fourth.


=Relations=  
=Relations=
The 50edo system is related to [[7edo]], [[12edo]], [[19edo]], [[31edo]] as the next approximation to the "Golden Tone System" ([[Das Goldene Tonsystem]]) of Thorvald Kornerup (and similarly as the next step from 31edo in Joseph Yasser's "[[http://books.google.com.au/books/about/A_theory_of_evolving_tonality.html?id=-XUsAAAAMAAJ&amp;redir_esc=y|A Theory of Evolving Tonality]]").
The 50edo system is related to [[7edo|7edo]], [[12edo|12edo]], [[19edo|19edo]], [[31edo|31edo]] as the next approximation to the "Golden Tone System" ([[Das_Goldene_Tonsystem|Das Goldene Tonsystem]]) of Thorvald Kornerup (and similarly as the next step from 31edo in Joseph Yasser's "[http://books.google.com.au/books/about/A_theory_of_evolving_tonality.html?id=-XUsAAAAMAAJ&redir_esc=y A Theory of Evolving Tonality]").


=Intervals=  
=Intervals=
||~ Degrees of 50edo ||~ Cents value ||~ Ratios* ||~ Generator for* ||
 
|| 0 || 0 || 1/1 ||   ||
{| class="wikitable"
|| 1 || 24 || 45/44, 49/48, 56/55, 65/64, 66/65, 78/77, 91/90, 99/98, 100/99, 121/120, 169/168 || [[xenharmonic/Hemimean clan#Sengagen|Sengagen]] ||
|-
|| 2 || 48 || 33/32, 36/35, 50/49, 55/54, 64/63 ||   ||
! | Degrees of 50edo
|| 3 || 72 || 21/20, 25/24, 26/25, 27/26, 28/27 || [[xenharmonic/Vishnuzmic family#Vishnu|Vishnu]] (2/oct), [[http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&amp;limit=11|Coblack]] (5/oct) ||
! | Cents value
|| 4 || 96 || 22/21 || [[xenharmonic/Meantone family#Injera|Injera]] (50d val, 2/oct) ||
! | Ratios*
|| 5 || 120 || 16/15, 15/14, 14/13 ||   ||
! | Generator for*
|| 6 || 144 || 13/12, 12/11 ||   ||
|-
|| 7 || 168 || 11/10 ||   ||
| | 0
|| 8 || 192 || 9/8, 10/9 ||   ||
| | 0
|| 9 || 216 || 25/22 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=50%2661p&amp;limit=2.3.5.11.13|Tremka]], [[xenharmonic/Subgroup temperaments#x2.9.7.11-Machine|Machine]] (50b val) ||
| | 1/1
|| 10 || 240 || 8/7, 15/13 ||   ||
| |  
|| 11 || 264 || 7/6 || [[xenharmonic/Marvel temperaments#Septimin-13-limit|Septimin (13-limit)]] ||
|-
|| 12 || 288 || 13/11 ||   ||
| | 1
|| 13 || 312 || 6/5 ||   ||
| | 24
|| 14 || 336 || 27/22, 39/32, 40/33, 49/40 ||   ||
| | 45/44, 49/48, 56/55, 65/64, 66/65, 78/77, 91/90, 99/98, 100/99, 121/120, 169/168
|| 15 || 360 || 16/13, 11/9 ||   ||
| | [[Hemimean_clan#Sengagen|Sengagen]]
|| 16 || 384 || 5/4 || [[xenharmonic/Marvel temperaments#Wizard-11-limit|Wizard]] (2/oct) ||
|-
|| 17 || 408 || 14/11 || [[xenharmonic/Ditonmic family|Ditonic]] ||
| | 2
|| 18 || 432 || 9/7 || [[xenharmonic/Porcupine family#Hedgehog|Hedgehog]] (50cc val, 2/oct) ||
| | 48
|| 19 || 456 || 13/10 || [[xenharmonic/Starling temperaments#Bisemidim|Bisemidim]] (2/oct) ||
| | 33/32, 36/35, 50/49, 55/54, 64/63
|| 20 || 480 || 33/25, 55/42, 64/49 ||   ||
| |  
|| 21 || 504 || 4/3 || [[xenharmonic/Meantone|Meantone]]/[[xenharmonic/Meanpop|Meanpop]] ||
|-
|| 22 || 528 || 15/11 ||   ||
| | 3
|| 23 || 552 || 11/8, 18/13 || [[xenharmonic/Chromatic pairs#Barton|Barton]], [[xenharmonic/Hemimean clan#Emka|Emka]] ||
| | 72
|| 24 || 576 || 7/5 ||   ||
| | 21/20, 25/24, 26/25, 27/26, 28/27
|| 25 || 600 || 63/44, 88/63, 78/55, 55/39 ||   ||
| | [[Vishnuzmic_family#Vishnu|Vishnu]] (2/oct), [http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&limit=11 Coblack] (5/oct)
|| 26 || 624 || 10/7 ||   ||
|-
|| 27 || 648 || 16/11, 13/9 ||   ||
| | 4
|| 28 || 672 || 22/15 ||   ||
| | 96
|| 29 || 696 || 3/2 ||   ||
| | 22/21
|| 30 || 720 || 50/33, 84/55, 49/32 ||   ||
| | [[Meantone_family#Injera|Injera]] (50d val, 2/oct)
|| 31 || 744 || 20/13 ||   ||
|-
|| 32 || 768 || 14/9 ||   ||
| | 5
|| 33 || 792 || 11/7 ||   ||
| | 120
|| 34 || 816 || 8/5 ||   ||
| | 16/15, 15/14, 14/13
|| 35 || 840 || 13/8, 18/11 ||   ||
| |  
|| 36 || 864 || 44/27, 64/39, 33/20, 80/49 ||   ||
|-
|| 37 || 888 || 5/3 ||   ||
| | 6
|| 38 || 912 || 22/13 ||   ||
| | 144
|| 39 || 936 || 12/7 ||   ||
| | 13/12, 12/11
|| 40 || 960 || 7/4 ||   ||
| |  
|| 41 || 984 || 44/25 ||   ||
|-
|| 42 || 1008 || 16/9, 9/5 ||   ||
| | 7
|| 43 || 1032 || 20/11 ||   ||
| | 168
|| 44 || 1056 || 24/13, 11/6 ||   ||
| | 11/10
|| 45 || 1080 || 15/8, 28/15, 13/7 ||   ||
| |  
|| 46 || 1104 || 21/11 ||   ||
|-
|| 47 || 1128 || 40/21, 48/25, 25/13, 52/27, 27/14 ||   ||
| | 8
|| 48 || 1152 || 64/33, 35/18, 49/25, 108/55, 63/32 ||   ||
| | 192
|| 49 || 1176 ||   ||   ||
| | 9/8, 10/9
| |  
|-
| | 9
| | 216
| | 25/22
| | [http://x31eq.com/cgi-bin/rt.cgi?ets=50%2661p&limit=2.3.5.11.13 Tremka], [[Subgroup_temperaments#x2.9.7.11-Machine|Machine]] (50b val)
|-
| | 10
| | 240
| | 8/7, 15/13
| |  
|-
| | 11
| | 264
| | 7/6
| | [[Marvel_temperaments#Septimin-13-limit|Septimin (13-limit)]]
|-
| | 12
| | 288
| | 13/11
| |  
|-
| | 13
| | 312
| | 6/5
| |  
|-
| | 14
| | 336
| | 27/22, 39/32, 40/33, 49/40
| |  
|-
| | 15
| | 360
| | 16/13, 11/9
| |  
|-
| | 16
| | 384
| | 5/4
| | [[Marvel_temperaments#Wizard-11-limit|Wizard]] (2/oct)
|-
| | 17
| | 408
| | 14/11
| | [[Ditonmic_family|Ditonic]]
|-
| | 18
| | 432
| | 9/7
| | [[Porcupine_family#Hedgehog|Hedgehog]] (50cc val, 2/oct)
|-
| | 19
| | 456
| | 13/10
| | [[Starling_temperaments#Bisemidim|Bisemidim]] (2/oct)
|-
| | 20
| | 480
| | 33/25, 55/42, 64/49
| |  
|-
| | 21
| | 504
| | 4/3
| | [[Meantone|Meantone]]/[[Meanpop|Meanpop]]
|-
| | 22
| | 528
| | 15/11
| |  
|-
| | 23
| | 552
| | 11/8, 18/13
| | [[Chromatic_pairs#Barton|Barton]], [[Hemimean_clan#Emka|Emka]]
|-
| | 24
| | 576
| | 7/5
| |  
|-
| | 25
| | 600
| | 63/44, 88/63, 78/55, 55/39
| |  
|-
| | 26
| | 624
| | 10/7
| |  
|-
| | 27
| | 648
| | 16/11, 13/9
| |  
|-
| | 28
| | 672
| | 22/15
| |  
|-
| | 29
| | 696
| | 3/2
| |  
|-
| | 30
| | 720
| | 50/33, 84/55, 49/32
| |  
|-
| | 31
| | 744
| | 20/13
| |  
|-
| | 32
| | 768
| | 14/9
| |  
|-
| | 33
| | 792
| | 11/7
| |  
|-
| | 34
| | 816
| | 8/5
| |  
|-
| | 35
| | 840
| | 13/8, 18/11
| |  
|-
| | 36
| | 864
| | 44/27, 64/39, 33/20, 80/49
| |  
|-
| | 37
| | 888
| | 5/3
| |  
|-
| | 38
| | 912
| | 22/13
| |  
|-
| | 39
| | 936
| | 12/7
| |  
|-
| | 40
| | 960
| | 7/4
| |  
|-
| | 41
| | 984
| | 44/25
| |  
|-
| | 42
| | 1008
| | 16/9, 9/5
| |  
|-
| | 43
| | 1032
| | 20/11
| |  
|-
| | 44
| | 1056
| | 24/13, 11/6
| |  
|-
| | 45
| | 1080
| | 15/8, 28/15, 13/7
| |  
|-
| | 46
| | 1104
| | 21/11
| |  
|-
| | 47
| | 1128
| | 40/21, 48/25, 25/13, 52/27, 27/14
| |  
|-
| | 48
| | 1152
| | 64/33, 35/18, 49/25, 108/55, 63/32
| |  
|-
| | 49
| | 1176
| |  
| |  
|}
*using the 13-limit patent val except as noted
*using the 13-limit patent val except as noted


==Selected just intervals by error==  
==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 50edo (ordered by absolute error).
The following table shows how [[Just-24|some prominent just intervals]] are represented in 50edo (ordered by absolute error).
|| **Interval, complement** || **Error (abs., in [[cent|cents]])** ||
||= [[16_13|16/13]], [[13_8|13/8]] ||= 0.528 ||
||= [[15_14|15/14]], [[28_15|28/15]] ||= 0.557 ||
||= [[11_8|11/8]], [[16_11|16/11]] ||= 0.682 ||
||= [[13_11|13/11]], [[22_13|22/13]] ||= 1.210 ||
||= [[13_10|13/10]], [[20_13|20/13]] ||= 1.786 ||
||= [[5_4|5/4]], [[8_5|8/5]] ||= 2.314 ||
||= [[7_6|7/6]], [[12_7|12/7]] ||= 2.871 ||
||= [[11_10|11/10]], [[20_11|20/11]] ||= 2.996 ||
||= [[9_7|9/7]], [[14_9|14/9]] ||= 3.084 ||
||= [[6_5|6/5]], [[5_3|5/3]] ||= 3.641 ||
||= [[13_12|13/12]], [[24_13|24/13]] ||= 5.427 ||
||= [[4_3|4/3]], [[3_2|3/2]] ||= 5.955 ||
||= [[7_5|7/5]], [[10_7|10/7]] ||= 6.512 ||
||= [[12_11|12/11]], [[11_6|11/6]] ||= 6.637 ||
||= [[15_13|15/13]], [[26_15|26/15]] ||= 7.741 ||
||= [[16_15|16/15]], [[15_8|15/8]] ||= 8.269 ||
||= [[14_13|14/13]], [[13_7|13/7]] ||= 8.298 ||
||= [[8_7|8/7]], [[7_4|7/4]] ||= 8.826 ||
||= [[15_11|15/11]], [[22_15|22/15]] ||= 8.951 ||
||= [[14_11|14/11]], [[11_7|11/7]] ||= 9.508 ||
||= [[10_9|10/9]], [[9_5|9/5]] ||= 9.596 ||
||= [[18_13|18/13]], [[13_9|13/9]] ||= 11.382 ||
||= [[11_9|11/9]], [[18_11|18/11]] ||= 11.408 ||
||= [[9_8|9/8]], [[16_9|16/9]] ||= 11.910 ||


=Commas=  
{| class="wikitable"
|-
| | '''Interval, complement'''
| | '''Error (abs., in [[cent|cents]])'''
|-
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| style="text-align:center;" | 0.528
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| style="text-align:center;" | 0.557
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| style="text-align:center;" | 0.682
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| style="text-align:center;" | 1.210
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| style="text-align:center;" | 1.786
|-
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| style="text-align:center;" | 2.314
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| style="text-align:center;" | 2.871
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| style="text-align:center;" | 2.996
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| style="text-align:center;" | 3.084
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| style="text-align:center;" | 3.641
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| style="text-align:center;" | 5.427
|-
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| style="text-align:center;" | 5.955
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| style="text-align:center;" | 6.512
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| style="text-align:center;" | 6.637
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| style="text-align:center;" | 7.741
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| style="text-align:center;" | 8.269
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| style="text-align:center;" | 8.298
|-
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| style="text-align:center;" | 8.826
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| style="text-align:center;" | 8.951
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| style="text-align:center;" | 9.508
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| style="text-align:center;" | 9.596
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| style="text-align:center;" | 11.382
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| style="text-align:center;" | 11.408
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| style="text-align:center;" | 11.910
|}
 
=Commas=
50 EDO tempers out the following commas. (Note: This assumes the val &lt; 50 79 116 140 173 185 204 212 226 |, comma values in cents rounded to 2 decimal places.) This list is not all-inclusive, and is based on the interval table from Scala version 2.2.
50 EDO tempers out the following commas. (Note: This assumes the val &lt; 50 79 116 140 173 185 204 212 226 |, comma values in cents rounded to 2 decimal places.) This list is not all-inclusive, and is based on the interval table from Scala version 2.2.
||~ Monzo ||~ Cents ||~ Ratio ||~ Name 1 ||~ Name 2 ||
|| | -4 4 -1 &gt; ||&gt; 21.51 ||= 81/80 || Syntonic comma || Didymus comma ||
|| | -27 -2 13 &gt; ||&gt; 18.17 ||=  || Ditonma ||  ||
|| | 23 6 -14 &gt; ||&gt; 3.34 ||=  || Vishnu comma ||  ||
|| | 1 2 -3 1 &gt; ||&gt; 13.79 ||= 126/125 || Starling comma || Small septimal comma ||
|| | -5 2 2 -1 &gt; ||&gt; 7.71 ||= 225/224 || Septimal kleisma || Marvel comma ||
|| | 6 0 -5 2 &gt; ||&gt; 6.08 ||= 3136/3125 || Hemimean || Middle second comma ||
|| | -6 -8 2 5 &gt; ||&gt; 1.12 ||=  || Wizma ||  ||
|| |-11 2 7 -3 &gt; ||&gt; 1.63 ||=  || Meter ||  ||
|| | 11 -10 -10 10 &gt; ||&gt; 5.57 ||=  || Linus ||  ||
|| |-13 10 0 -1 &gt; ||&gt; 50.72 ||= 59049/57344 || Harrison's comma ||  ||
|| | 2 3 1 -2 -1 &gt; ||&gt; 3.21 ||= 540/539 || Swets' comma || Swetisma ||
|| | -3 4 -2 -2 2 &gt; ||&gt; 0.18 ||= 9801/9800 || Kalisma || Gauss' comma ||
|| | 5 -1 3 0 -3 &gt; ||&gt; 3.03 ||= 4000/3993 || Wizardharry || Undecimal schisma ||
|| | -7 -1 1 1 1 &gt; ||&gt; 4.50 ||= 385/384 || Keenanisma || Undecimal kleisma ||
|| | -1 0 1 2 -2 &gt; ||&gt; 21.33 ||= 245/242 || Cassacot ||  ||
|| | 2 -1 0 1 -2 1 &gt; ||&gt; 4.76 ||= 364/363 || Gentle comma ||  ||
|| | 2 -1 -1 2 0 -1 &gt; ||&gt; 8.86 ||= 196/195 || Mynucuma ||  ||
|| | 2 3 0 -1 1 -2 &gt; ||&gt; 7.30 ||= 1188/1183 || Kestrel Comma ||  ||
|| | 3 0 2 0 1 -3 &gt; ||&gt; 2.36 ||= 2200/2197 || Petrma || Parizek comma ||
|| | -3 1 1 1 0 -1 &gt; ||&gt; 16.57 ||= 105/104 || Animist comma || Small tridecimal comma ||  ||
|| | 4 2 0 0 -1 -1 &gt; ||&gt; 12.06 ||= 144/143 || Grossma ||  ||
|| | 3 -2 0 1 -1 -1 0 0 1 &gt; ||&gt; 1.34 ||= 1288/1287 || Triaphonisma ||  ||


=Music=  
{| class="wikitable"
[[http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3|Twinkle canon – 50 edo]] by [[http://soonlabel.com/xenharmonic/archives/573|Claudi Meneghin]]
|-
[[@http://soonlabel.com/xenharmonic/archives/1118|Fantasia Catalana by Claudi Meneghin]]
! | Monzo
[[http://soonlabel.com/xenharmonic/archives/1929|Fugue on the Dragnet theme by Claudi Meneghin]]
! | Cents
[[https://soundcloud.com/camtaylor-1/sets/the-late-little-xmas-album|the late little xmas album by Cam Taylor]]
! | Ratio
[[https://soundcloud.com/cam-taylor-2-1/harpsichord-meantone|Harpsichord meantone improvisation 1 in 50EDO by Cam Taylor]]
! | Name 1
[[https://soundcloud.com/cam-taylor-2-1/long-improvisation-2-in-50edo|Long improvisation 2 in 50EDO by Cam Taylor]]
! | Name 2
[[https://soundcloud.com/camtaylor-1/chord-sequence-for-difference|Chord sequence for Difference tones in 50EDO by Cam Taylor]]
|-
[[https://soundcloud.com/camtaylor-1/enharmonic-modulations-in|Enharmonic Modulations in 50EDO by Cam Taylor]]
| | | -4 4 -1 &gt;
[[https://soundcloud.com/cam-taylor-2-1/harmonic-clusters-on-50edo-harpsichord-bosanquet-axis-through-pianoteq|Harmonic Clusters on 50EDO Harpsichord by Cam Taylor]]
| style="text-align:right;" | 21.51
[[https://soundcloud.com/camtaylor-1/fragment-in-fifty|Fragment in Fifty]] by Cam Taylor
| style="text-align:center;" | 81/80
| | Syntonic comma
| | Didymus comma
|-
| | | -27 -2 13 &gt;
| style="text-align:right;" | 18.17
| style="text-align:center;" |
| | Ditonma
| |
|-
| | | 23 6 -14 &gt;
| style="text-align:right;" | 3.34
| style="text-align:center;" |
| | Vishnu comma
| |
|-
| | | 1 2 -3 1 &gt;
| style="text-align:right;" | 13.79
| style="text-align:center;" | 126/125
| | Starling comma
| | Small septimal comma
|-
| | | -5 2 2 -1 &gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | 225/224
| | Septimal kleisma
| | Marvel comma
|-
| | | 6 0 -5 2 &gt;
| style="text-align:right;" | 6.08
| style="text-align:center;" | 3136/3125
| | Hemimean
| | Middle second comma
|-
| | | -6 -8 2 5 &gt;
| style="text-align:right;" | 1.12
| style="text-align:center;" |
| | Wizma
| |
|-
| | |-11 2 7 -3 &gt;
| style="text-align:right;" | 1.63
| style="text-align:center;" |
| | Meter
| |
|-
| | | 11 -10 -10 10 &gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" |
| | Linus
| |
|-
| | |-13 10 0 -1 &gt;
| style="text-align:right;" | 50.72
| style="text-align:center;" | 59049/57344
| | Harrison's comma
| |
|-
| | | 2 3 1 -2 -1 &gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" | 540/539
| | Swets' comma
| | Swetisma
|-
| | | -3 4 -2 -2 2 &gt;
| style="text-align:right;" | 0.18
| style="text-align:center;" | 9801/9800
| | Kalisma
| | Gauss' comma
|-
| | | 5 -1 3 0 -3 &gt;
| style="text-align:right;" | 3.03
| style="text-align:center;" | 4000/3993
| | Wizardharry
| | Undecimal schisma
|-
| | | -7 -1 1 1 1 &gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" | 385/384
| | Keenanisma
| | Undecimal kleisma
|-
| | | -1 0 1 2 -2 &gt;
| style="text-align:right;" | 21.33
| style="text-align:center;" | 245/242
| | Cassacot
| |
|-
| | | 2 -1 0 1 -2 1 &gt;
| style="text-align:right;" | 4.76
| style="text-align:center;" | 364/363
| | Gentle comma
| |
|-
| | | 2 -1 -1 2 0 -1 &gt;
| style="text-align:right;" | 8.86
| style="text-align:center;" | 196/195
| | Mynucuma
| |
|-
| | | 2 3 0 -1 1 -2 &gt;
| style="text-align:right;" | 7.30
| style="text-align:center;" | 1188/1183
| | Kestrel Comma
| |
|-
| | | 3 0 2 0 1 -3 &gt;
| style="text-align:right;" | 2.36
| style="text-align:center;" | 2200/2197
| | Petrma
| | Parizek comma
|-
| | | -3 1 1 1 0 -1 &gt;
| style="text-align:right;" | 16.57
| style="text-align:center;" | 105/104
| | Animist comma
| | Small tridecimal comma
| |
|-
| | | 4 2 0 0 -1 -1 &gt;
| style="text-align:right;" | 12.06
| style="text-align:center;" | 144/143
| | Grossma
| |
|-
| | | 3 -2 0 1 -1 -1 0 0 1 &gt;
| style="text-align:right;" | 1.34
| style="text-align:center;" | 1288/1287
| | Triaphonisma
| |
|}
 
=Music=
[http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 Twinkle canon – 50 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]
 
[http://soonlabel.com/xenharmonic/archives/1118 Fantasia Catalana by Claudi Meneghin]
 
[http://soonlabel.com/xenharmonic/archives/1929 Fugue on the Dragnet theme by Claudi Meneghin]
 
[https://soundcloud.com/camtaylor-1/sets/the-late-little-xmas-album the late little xmas album by Cam Taylor]


=Additional reading=
[https://soundcloud.com/cam-taylor-2-1/harpsichord-meantone Harpsichord meantone improvisation 1 in 50EDO by Cam Taylor]
[[http://www.archive.org/details/harmonicsorphilo00smit|Robert Smith's book online]]
[[http://www.music.ed.ac.uk/russell/conference/robertsmithkirckman.html|More information about Robert Smith's temperament]]


[[https://www.dropbox.com/sh/4x81rzpkot32qzk/MQ3cJljjkh|50EDO Theory - Intervals, Chords and Scales in 50EDO by Cam Taylor]]
[https://soundcloud.com/cam-taylor-2-1/long-improvisation-2-in-50edo Long improvisation 2 in 50EDO by Cam Taylor]
[[http://iamcamtaylor.wordpress.com/|iamcamtaylor - Blog on 50EDO and extended meantone theory by Cam Taylor]]</pre></div>
<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;50edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:12:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Relations"&gt;Relations&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Intervals"&gt;Intervals&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Intervals-Selected just intervals by error"&gt;Selected just intervals by error&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Commas"&gt;Commas&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Music"&gt;Music&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Additional reading"&gt;Additional reading&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;em&gt;50edo&lt;/em&gt; divides the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; into 50 equal parts of precisely 24 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s each. In the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;, it tempers out 81/80, making it a &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; system, and in that capacity has historically has drawn some notice. In &lt;a class="wiki_link_ext" href="http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf" rel="nofollow"&gt;&amp;quot;Harmonics or the Philosophy of Musical Sounds&amp;quot;&lt;/a&gt; (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts - 50edo, in one word. Later, W.S.B. Woolhouse noted it was fairly close to the &lt;a class="wiki_link" href="/Target%20tunings"&gt;least squares&lt;/a&gt; tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher limit point of view. While &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; extends meantone with a &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt; which is nearly pure, 50 has a flat 7/4 but both &lt;a class="wiki_link" href="/11_8"&gt;11/8&lt;/a&gt; and &lt;a class="wiki_link" href="/13_8"&gt;13/8&lt;/a&gt; are nearly pure.&lt;br /&gt;
&lt;br /&gt;
50 tempers out 126/125, 225/224 and 3136/3125 in the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt;, indicating it supports septimal meantone; 245/242, 385/384 and 540/539 in the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; and 105/104, 144/143 and 196/195 in the &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt;, and can be used for even higher limits. Aside from meantone and its extension meanpop, it can be used to advantage for the 15&amp;amp;50 temperament (&lt;a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&amp;amp;limit=11" rel="nofollow"&gt;Coblack&lt;/a&gt;), and provides the optimal patent val for 11 and 13 limit &lt;a class="wiki_link" href="/Meantone%20family#Septimal%20meantone-Bimeantone"&gt;bimeantone&lt;/a&gt;. It is also the unique equal temperament tempering out both 81/80 and the &lt;a class="wiki_link" href="/vishnuzma"&gt;vishnuzma&lt;/a&gt;, |23 6 -14&amp;gt;, so that in 50et seven chromatic semitones are a perfect fourth. In 12et by comparison this gives a fifth, in 31et a doubly diminished fifth, and in 19et a diminished fourth.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Relations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Relations&lt;/h1&gt;
The 50edo system is related to &lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; as the next approximation to the &amp;quot;Golden Tone System&amp;quot; (&lt;a class="wiki_link" href="/Das%20Goldene%20Tonsystem"&gt;Das Goldene Tonsystem&lt;/a&gt;) of Thorvald Kornerup (and similarly as the next step from 31edo in Joseph Yasser's &amp;quot;&lt;a class="wiki_link_ext" href="http://books.google.com.au/books/about/A_theory_of_evolving_tonality.html?id=-XUsAAAAMAAJ&amp;amp;redir_esc=y" rel="nofollow"&gt;A Theory of Evolving Tonality&lt;/a&gt;&amp;quot;).&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Intervals&lt;/h1&gt;


&lt;table class="wiki_table"&gt;
[https://soundcloud.com/camtaylor-1/chord-sequence-for-difference Chord sequence for Difference tones in 50EDO by Cam Taylor]
    &lt;tr&gt;
        &lt;th&gt;Degrees of 50edo&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Cents value&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Ratios*&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Generator for*&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;24&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;45/44, 49/48, 56/55, 65/64, 66/65, 78/77, 91/90, 99/98, 100/99, 121/120, 169/168&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Hemimean%20clan#Sengagen"&gt;Sengagen&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;48&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;33/32, 36/35, 50/49, 55/54, 64/63&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;72&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;21/20, 25/24, 26/25, 27/26, 28/27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Vishnuzmic%20family#Vishnu"&gt;Vishnu&lt;/a&gt; (2/oct), &lt;a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&amp;amp;limit=11" rel="nofollow"&gt;Coblack&lt;/a&gt; (5/oct)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;96&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;22/21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Meantone%20family#Injera"&gt;Injera&lt;/a&gt; (50d val, 2/oct)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;120&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/15, 15/14, 14/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;144&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/12, 12/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;168&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;192&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/8, 10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;216&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;25/22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=50%2661p&amp;amp;limit=2.3.5.11.13" rel="nofollow"&gt;Tremka&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Subgroup%20temperaments#x2.9.7.11-Machine"&gt;Machine&lt;/a&gt; (50b val)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;240&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/7, 15/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;264&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Marvel%20temperaments#Septimin-13-limit"&gt;Septimin (13-limit)&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;288&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;312&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;336&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;27/22, 39/32, 40/33, 49/40&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;360&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/13, 11/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;384&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Marvel%20temperaments#Wizard-11-limit"&gt;Wizard&lt;/a&gt; (2/oct)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;17&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;408&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ditonmic%20family"&gt;Ditonic&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;18&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;432&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Porcupine%20family#Hedgehog"&gt;Hedgehog&lt;/a&gt; (50cc val, 2/oct)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;19&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;456&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Starling%20temperaments#Bisemidim"&gt;Bisemidim&lt;/a&gt; (2/oct)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;20&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;480&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;33/25, 55/42, 64/49&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;504&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Meantone"&gt;Meantone&lt;/a&gt;/&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Meanpop"&gt;Meanpop&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;528&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;15/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;23&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;552&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/8, 18/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chromatic%20pairs#Barton"&gt;Barton&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Hemimean%20clan#Emka"&gt;Emka&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;24&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;576&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;600&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;63/44, 88/63, 78/55, 55/39&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;26&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;624&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;648&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/11, 13/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;28&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;672&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;22/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;29&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;696&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;30&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;720&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;50/33, 84/55, 49/32&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;31&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;744&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;32&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;768&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;33&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;792&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;34&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;816&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;35&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;840&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/8, 18/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;36&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;864&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;44/27, 64/39, 33/20, 80/49&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;37&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;888&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;38&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;912&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;22/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;39&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;936&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;40&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;960&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;41&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;984&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;44/25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;42&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1008&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/9, 9/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;43&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1032&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;44&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1056&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;24/13, 11/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;45&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1080&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;15/8, 28/15, 13/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;46&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1104&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;21/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;47&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1128&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;40/21, 48/25, 25/13, 52/27, 27/14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;48&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1152&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;64/33, 35/18, 49/25, 108/55, 63/32&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;49&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1176&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


*using the 13-limit patent val except as noted&lt;br /&gt;
[https://soundcloud.com/camtaylor-1/enharmonic-modulations-in Enharmonic Modulations in 50EDO by Cam Taylor]
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Intervals-Selected just intervals by error"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Selected just intervals by error&lt;/h2&gt;
The following table shows how &lt;a class="wiki_link" href="/Just-24"&gt;some prominent just intervals&lt;/a&gt; are represented in 50edo (ordered by absolute error).&lt;br /&gt;


[https://soundcloud.com/cam-taylor-2-1/harmonic-clusters-on-50edo-harpsichord-bosanquet-axis-through-pianoteq Harmonic Clusters on 50EDO Harpsichord by Cam Taylor]


&lt;table class="wiki_table"&gt;
[https://soundcloud.com/camtaylor-1/fragment-in-fifty Fragment in Fifty] by Cam Taylor
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;Interval, complement&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;Error (abs., in &lt;a class="wiki_link" href="/cent"&gt;cents&lt;/a&gt;)&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/16_13"&gt;16/13&lt;/a&gt;, &lt;a class="wiki_link" href="/13_8"&gt;13/8&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;0.528&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/15_14"&gt;15/14&lt;/a&gt;, &lt;a class="wiki_link" href="/28_15"&gt;28/15&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;0.557&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/11_8"&gt;11/8&lt;/a&gt;, &lt;a class="wiki_link" href="/16_11"&gt;16/11&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;0.682&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/13_11"&gt;13/11&lt;/a&gt;, &lt;a class="wiki_link" href="/22_13"&gt;22/13&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1.210&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/13_10"&gt;13/10&lt;/a&gt;, &lt;a class="wiki_link" href="/20_13"&gt;20/13&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1.786&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;, &lt;a class="wiki_link" href="/8_5"&gt;8/5&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;2.314&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/7_6"&gt;7/6&lt;/a&gt;, &lt;a class="wiki_link" href="/12_7"&gt;12/7&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;2.871&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/11_10"&gt;11/10&lt;/a&gt;, &lt;a class="wiki_link" href="/20_11"&gt;20/11&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;2.996&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/9_7"&gt;9/7&lt;/a&gt;, &lt;a class="wiki_link" href="/14_9"&gt;14/9&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;3.084&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt;, &lt;a class="wiki_link" href="/5_3"&gt;5/3&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;3.641&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/13_12"&gt;13/12&lt;/a&gt;, &lt;a class="wiki_link" href="/24_13"&gt;24/13&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;5.427&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/4_3"&gt;4/3&lt;/a&gt;, &lt;a class="wiki_link" href="/3_2"&gt;3/2&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;5.955&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/7_5"&gt;7/5&lt;/a&gt;, &lt;a class="wiki_link" href="/10_7"&gt;10/7&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;6.512&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/12_11"&gt;12/11&lt;/a&gt;, &lt;a class="wiki_link" href="/11_6"&gt;11/6&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;6.637&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/15_13"&gt;15/13&lt;/a&gt;, &lt;a class="wiki_link" href="/26_15"&gt;26/15&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;7.741&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/16_15"&gt;16/15&lt;/a&gt;, &lt;a class="wiki_link" href="/15_8"&gt;15/8&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;8.269&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/14_13"&gt;14/13&lt;/a&gt;, &lt;a class="wiki_link" href="/13_7"&gt;13/7&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;8.298&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/8_7"&gt;8/7&lt;/a&gt;, &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;8.826&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/15_11"&gt;15/11&lt;/a&gt;, &lt;a class="wiki_link" href="/22_15"&gt;22/15&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;8.951&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/14_11"&gt;14/11&lt;/a&gt;, &lt;a class="wiki_link" href="/11_7"&gt;11/7&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;9.508&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/10_9"&gt;10/9&lt;/a&gt;, &lt;a class="wiki_link" href="/9_5"&gt;9/5&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;9.596&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/18_13"&gt;18/13&lt;/a&gt;, &lt;a class="wiki_link" href="/13_9"&gt;13/9&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;11.382&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/11_9"&gt;11/9&lt;/a&gt;, &lt;a class="wiki_link" href="/18_11"&gt;18/11&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;11.408&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/9_8"&gt;9/8&lt;/a&gt;, &lt;a class="wiki_link" href="/16_9"&gt;16/9&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;11.910&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
=Additional reading=
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Commas&lt;/h1&gt;
[http://www.archive.org/details/harmonicsorphilo00smit Robert Smith's book online]
50 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 50 79 116 140 173 185 204 212 226 |, comma values in cents rounded to 2 decimal places.) This list is not all-inclusive, and is based on the interval table from Scala version 2.2.&lt;br /&gt;


[http://www.music.ed.ac.uk/russell/conference/robertsmithkirckman.html More information about Robert Smith's temperament]


&lt;table class="wiki_table"&gt;
[https://www.dropbox.com/sh/4x81rzpkot32qzk/MQ3cJljjkh 50EDO Theory - Intervals, Chords and Scales in 50EDO by Cam Taylor]
    &lt;tr&gt;
        &lt;th&gt;Monzo&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Cents&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Ratio&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Name 1&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Name 2&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| -4 4 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;21.51&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;81/80&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Syntonic comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Didymus comma&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| -27 -2 13 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;18.17&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Ditonma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| 23 6 -14 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;3.34&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Vishnu comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| 1 2 -3 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;13.79&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;126/125&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Starling comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Small septimal comma&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| -5 2 2 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;7.71&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;225/224&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Septimal kleisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Marvel comma&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| 6 0 -5 2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;6.08&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;3136/3125&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Hemimean&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Middle second comma&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| -6 -8 2 5 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;1.12&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Wizma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|-11 2 7 -3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;1.63&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Meter&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| 11 -10 -10 10 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;5.57&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Linus&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;|-13 10 0 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;50.72&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;59049/57344&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Harrison's comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| 2 3 1 -2 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;3.21&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;540/539&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Swets' comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Swetisma&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| -3 4 -2 -2 2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;0.18&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;9801/9800&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Kalisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Gauss' comma&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| 5 -1 3 0 -3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;3.03&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;4000/3993&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Wizardharry&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Undecimal schisma&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| -7 -1 1 1 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;4.50&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;385/384&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Keenanisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Undecimal kleisma&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| -1 0 1 2 -2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;21.33&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;245/242&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Cassacot&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| 2 -1 0 1 -2 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;4.76&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;364/363&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Gentle comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| 2 -1 -1 2 0 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;8.86&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;196/195&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Mynucuma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| 2 3 0 -1 1 -2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;7.30&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1188/1183&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Kestrel Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| 3 0 2 0 1 -3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;2.36&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;2200/2197&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Petrma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Parizek comma&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| -3 1 1 1 0 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;16.57&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;105/104&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Animist comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Small tridecimal comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| 4 2 0 0 -1 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;12.06&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;144/143&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Grossma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;| 3 -2 0 1 -1 -1 0 0 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;1.34&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1288/1287&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Triaphonisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
[http://iamcamtaylor.wordpress.com/ iamcamtaylor - Blog on 50EDO and extended meantone theory by Cam Taylor]      [[Category:50edo]]
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Music&lt;/h1&gt;
[[Category:edo]]
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3" rel="nofollow"&gt;Twinkle canon – 50 edo&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/573" rel="nofollow"&gt;Claudi Meneghin&lt;/a&gt;&lt;br /&gt;
[[Category:golden]]
&lt;a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/1118" rel="nofollow" target="_blank"&gt;Fantasia Catalana by Claudi Meneghin&lt;/a&gt;&lt;br /&gt;
[[Category:intervals]]
&lt;a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/1929" rel="nofollow"&gt;Fugue on the Dragnet theme by Claudi Meneghin&lt;/a&gt;&lt;br /&gt;
[[Category:meantone]]
&lt;a class="wiki_link_ext" href="https://soundcloud.com/camtaylor-1/sets/the-late-little-xmas-album" rel="nofollow"&gt;the late little xmas album by Cam Taylor&lt;/a&gt;&lt;br /&gt;
[[Category:theory]]
&lt;a class="wiki_link_ext" href="https://soundcloud.com/cam-taylor-2-1/harpsichord-meantone" rel="nofollow"&gt;Harpsichord meantone improvisation 1 in 50EDO by Cam Taylor&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="https://soundcloud.com/cam-taylor-2-1/long-improvisation-2-in-50edo" rel="nofollow"&gt;Long improvisation 2 in 50EDO by Cam Taylor&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="https://soundcloud.com/camtaylor-1/chord-sequence-for-difference" rel="nofollow"&gt;Chord sequence for Difference tones in 50EDO by Cam Taylor&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="https://soundcloud.com/camtaylor-1/enharmonic-modulations-in" rel="nofollow"&gt;Enharmonic Modulations in 50EDO by Cam Taylor&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="https://soundcloud.com/cam-taylor-2-1/harmonic-clusters-on-50edo-harpsichord-bosanquet-axis-through-pianoteq" rel="nofollow"&gt;Harmonic Clusters on 50EDO Harpsichord by Cam Taylor&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="https://soundcloud.com/camtaylor-1/fragment-in-fifty" rel="nofollow"&gt;Fragment in Fifty&lt;/a&gt; by Cam Taylor&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Additional reading"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Additional reading&lt;/h1&gt;
&lt;a class="wiki_link_ext" href="http://www.archive.org/details/harmonicsorphilo00smit" rel="nofollow"&gt;Robert Smith's book online&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://www.music.ed.ac.uk/russell/conference/robertsmithkirckman.html" rel="nofollow"&gt;More information about Robert Smith's temperament&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link_ext" href="https://www.dropbox.com/sh/4x81rzpkot32qzk/MQ3cJljjkh" rel="nofollow"&gt;50EDO Theory - Intervals, Chords and Scales in 50EDO by Cam Taylor&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://iamcamtaylor.wordpress.com/" rel="nofollow"&gt;iamcamtaylor - Blog on 50EDO and extended meantone theory by Cam Taylor&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 00:00, 17 July 2018

50edo divides the octave into 50 equal parts of precisely 24 cents each. In the 5-limit, it tempers out 81/80, making it a meantone system, and in that capacity has historically has drawn some notice. In "Harmonics or the Philosophy of Musical Sounds" (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts - 50edo, in one word. Later, W.S.B. Woolhouse noted it was fairly close to the least squares tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher limit point of view. While 31edo extends meantone with a 7/4 which is nearly pure, 50 has a flat 7/4 but both 11/8 and 13/8 are nearly pure.

50 tempers out 126/125, 225/224 and 3136/3125 in the 7-limit, indicating it supports septimal meantone; 245/242, 385/384 and 540/539 in the 11-limit and 105/104, 144/143 and 196/195 in the 13-limit, and can be used for even higher limits. Aside from meantone and its extension meanpop, it can be used to advantage for the 15&50 temperament (Coblack), and provides the optimal patent val for 11 and 13 limit bimeantone. It is also the unique equal temperament tempering out both 81/80 and the vishnuzma, |23 6 -14>, so that in 50et seven chromatic semitones are a perfect fourth. In 12et by comparison this gives a fifth, in 31et a doubly diminished fifth, and in 19et a diminished fourth.

Relations

The 50edo system is related to 7edo, 12edo, 19edo, 31edo as the next approximation to the "Golden Tone System" (Das Goldene Tonsystem) of Thorvald Kornerup (and similarly as the next step from 31edo in Joseph Yasser's "A Theory of Evolving Tonality").

Intervals

Degrees of 50edo Cents value Ratios* Generator for*
0 0 1/1
1 24 45/44, 49/48, 56/55, 65/64, 66/65, 78/77, 91/90, 99/98, 100/99, 121/120, 169/168 Sengagen
2 48 33/32, 36/35, 50/49, 55/54, 64/63
3 72 21/20, 25/24, 26/25, 27/26, 28/27 Vishnu (2/oct), Coblack (5/oct)
4 96 22/21 Injera (50d val, 2/oct)
5 120 16/15, 15/14, 14/13
6 144 13/12, 12/11
7 168 11/10
8 192 9/8, 10/9
9 216 25/22 Tremka, Machine (50b val)
10 240 8/7, 15/13
11 264 7/6 Septimin (13-limit)
12 288 13/11
13 312 6/5
14 336 27/22, 39/32, 40/33, 49/40
15 360 16/13, 11/9
16 384 5/4 Wizard (2/oct)
17 408 14/11 Ditonic
18 432 9/7 Hedgehog (50cc val, 2/oct)
19 456 13/10 Bisemidim (2/oct)
20 480 33/25, 55/42, 64/49
21 504 4/3 Meantone/Meanpop
22 528 15/11
23 552 11/8, 18/13 Barton, Emka
24 576 7/5
25 600 63/44, 88/63, 78/55, 55/39
26 624 10/7
27 648 16/11, 13/9
28 672 22/15
29 696 3/2
30 720 50/33, 84/55, 49/32
31 744 20/13
32 768 14/9
33 792 11/7
34 816 8/5
35 840 13/8, 18/11
36 864 44/27, 64/39, 33/20, 80/49
37 888 5/3
38 912 22/13
39 936 12/7
40 960 7/4
41 984 44/25
42 1008 16/9, 9/5
43 1032 20/11
44 1056 24/13, 11/6
45 1080 15/8, 28/15, 13/7
46 1104 21/11
47 1128 40/21, 48/25, 25/13, 52/27, 27/14
48 1152 64/33, 35/18, 49/25, 108/55, 63/32
49 1176
  • using the 13-limit patent val except as noted

Selected just intervals by error

The following table shows how some prominent just intervals are represented in 50edo (ordered by absolute error).

Interval, complement Error (abs., in cents)
16/13, 13/8 0.528
15/14, 28/15 0.557
11/8, 16/11 0.682
13/11, 22/13 1.210
13/10, 20/13 1.786
5/4, 8/5 2.314
7/6, 12/7 2.871
11/10, 20/11 2.996
9/7, 14/9 3.084
6/5, 5/3 3.641
13/12, 24/13 5.427
4/3, 3/2 5.955
7/5, 10/7 6.512
12/11, 11/6 6.637
15/13, 26/15 7.741
16/15, 15/8 8.269
14/13, 13/7 8.298
8/7, 7/4 8.826
15/11, 22/15 8.951
14/11, 11/7 9.508
10/9, 9/5 9.596
18/13, 13/9 11.382
11/9, 18/11 11.408
9/8, 16/9 11.910

Commas

50 EDO tempers out the following commas. (Note: This assumes the val < 50 79 116 140 173 185 204 212 226 |, comma values in cents rounded to 2 decimal places.) This list is not all-inclusive, and is based on the interval table from Scala version 2.2.

Monzo Cents Ratio Name 1 Name 2
| -4 4 -1 > 21.51 81/80 Syntonic comma Didymus comma
| -27 -2 13 > 18.17 Ditonma
| 23 6 -14 > 3.34 Vishnu comma
| 1 2 -3 1 > 13.79 126/125 Starling comma Small septimal comma
| -5 2 2 -1 > 7.71 225/224 Septimal kleisma Marvel comma
| 6 0 -5 2 > 6.08 3136/3125 Hemimean Middle second comma
| -6 -8 2 5 > 1.12 Wizma
|-11 2 7 -3 > 1.63 Meter
| 11 -10 -10 10 > 5.57 Linus
|-13 10 0 -1 > 50.72 59049/57344 Harrison's comma
| 2 3 1 -2 -1 > 3.21 540/539 Swets' comma Swetisma
| -3 4 -2 -2 2 > 0.18 9801/9800 Kalisma Gauss' comma
| 5 -1 3 0 -3 > 3.03 4000/3993 Wizardharry Undecimal schisma
| -7 -1 1 1 1 > 4.50 385/384 Keenanisma Undecimal kleisma
| -1 0 1 2 -2 > 21.33 245/242 Cassacot
| 2 -1 0 1 -2 1 > 4.76 364/363 Gentle comma
| 2 -1 -1 2 0 -1 > 8.86 196/195 Mynucuma
| 2 3 0 -1 1 -2 > 7.30 1188/1183 Kestrel Comma
| 3 0 2 0 1 -3 > 2.36 2200/2197 Petrma Parizek comma
| -3 1 1 1 0 -1 > 16.57 105/104 Animist comma Small tridecimal comma
| 4 2 0 0 -1 -1 > 12.06 144/143 Grossma
| 3 -2 0 1 -1 -1 0 0 1 > 1.34 1288/1287 Triaphonisma

Music

Twinkle canon – 50 edo by Claudi Meneghin

Fantasia Catalana by Claudi Meneghin

Fugue on the Dragnet theme by Claudi Meneghin

the late little xmas album by Cam Taylor

Harpsichord meantone improvisation 1 in 50EDO by Cam Taylor

Long improvisation 2 in 50EDO by Cam Taylor

Chord sequence for Difference tones in 50EDO by Cam Taylor

Enharmonic Modulations in 50EDO by Cam Taylor

Harmonic Clusters on 50EDO Harpsichord by Cam Taylor

Fragment in Fifty by Cam Taylor

Additional reading

Robert Smith's book online

More information about Robert Smith's temperament

50EDO Theory - Intervals, Chords and Scales in 50EDO by Cam Taylor

iamcamtaylor - Blog on 50EDO and extended meantone theory by Cam Taylor

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