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User:Moremajorthanmajor/8L 3s (perfect twelfth-equivalent): Difference between revisions

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==Intervals==
==Intervals==
The table of Obikhodic intervals below takes the fifth as the generator. Given the size of the fifth generator ''g'', any Obikhodic interval can easily be found by noting what multiple of ''g'' it is, and multiplying the size by the number ''k'' of generators it takes to reach the interval and reducing mod 1900<ref>For relative cents</ref> if necessary (so you can use "''k''*''g'' % 1900" for search engines, for plugged-in values of ''k'' and ''g''). For example, since the major 3rd is reached by six fifth generators, [[27edt]]'s major 3rd is 6*703.7 mod 1900 = 4222.22 mod 1900 = 422.22r¢.
The table of Obikhodic intervals below takes the fifth as the generator.
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 1,444: Line 1,444:
{| class="wikitable"
{| class="wikitable"
|-
|-
| style="text-align:center;" |[[Modal UDP Notation|'''UDP''']]
| style="text-align:center;" |'''Mode'''
| style="text-align:center;" |'''Mode'''
| style="text-align:center;" |[[Modal UDP Notation|'''UDP''']]
| style="text-align:center;" |'''Name'''
| style="text-align:center;" |'''Name'''
|-
|-
| style="text-align:center;" |<nowiki>10|0</nowiki>
|LLLsLLLsLLs
|LLLsLLLsLLs
| style="text-align:center;" |<nowiki>10|0</nowiki>
|(Great) Lydian #8 (Tanagran)
|(Great) Lydian #8 (Tanagran)
|-
|-
| style="text-align:center;" |<nowiki>9|1</nowiki>
|LLLsLLsLLLs
|LLLsLLsLLLs
| style="text-align:center;" |<nowiki>9|1</nowiki>
|(Great) Lydian
|(Great) Lydian
|-
|-
| style="text-align:center;" |<nowiki>8|2</nowiki>
|LLsLLLsLLLs
|LLsLLLsLLLs
| style="text-align:center;" |<nowiki>8|2</nowiki>
|(Great) Lydian f4, Ionian #11 (Distomian)
|(Great) Lydian f4, Ionian #11 (Distomian)
|-
|-
| style="text-align:center;" |<nowiki>7|3</nowiki>
| |LLsLLLsLLsL
| |LLsLLLsLLsL
| style="text-align:center;" |<nowiki>7|3</nowiki>
| |(Great) Ionian
| |(Great) Ionian
|-
|-
| style="text-align:center;" |<nowiki>6|4</nowiki>
| |LLsLLsLLLsL
| |LLsLLsLLLsL
| style="text-align:center;" |<nowiki>6|4</nowiki>
| |(Great) Mixolydian
| |(Great) Mixolydian
|-
|-
| style="text-align:center;" |<nowiki>5|5</nowiki>
| |LsLLLsLLLsL
| |LsLLLsLLLsL
| style="text-align:center;" |<nowiki>5|5</nowiki>
| |(Great) Mixolydian f3, Dorian #10 (Livadeian)
| |(Great) Mixolydian f3, Dorian #10 (Livadeian)
|-
|-
| style="text-align:center;" |<nowiki>4|6</nowiki>
| |LsLLLsLLsLL
| |LsLLLsLLsLL
| style="text-align:center;" |<nowiki>4|6</nowiki>
| |(Great) Dorian
| |(Great) Dorian
|-
|-
| style="text-align:center;" |<nowiki>3|7</nowiki>
| |LsLLsLLLsLL
| |LsLLsLLLsLL
| style="text-align:center;" |<nowiki>3|7</nowiki>
| |(Great) Aeolian
| |(Great) Aeolian
|-
|-
| style="text-align:center;" |<nowiki>2|8</nowiki>
| |sLLLsLLLsLL
| |sLLLsLLLsLL
| style="text-align:center;" |<nowiki>2|8</nowiki>
| |(Great) Aeolian f2, Phrygian #9 (Theban)
| |(Great) Aeolian f2, Phrygian #9 (Theban)
|-
|-
| style="text-align:center;" |<nowiki>1|9</nowiki>
| |sLLLsLLsLLL
| |sLLLsLLsLLL
| style="text-align:center;" |<nowiki>1|9</nowiki>
| |(Great) Phrygian
| |(Great) Phrygian
|-
|-
| style="text-align:center;" |<nowiki>0|10</nowiki>
| |sLLsLLLsLLL
| |sLLsLLLsLLL
| style="text-align:center;" |<nowiki>0|10</nowiki>
| |(Great) Locrian
| |(Great) Locrian
|}
|}
This temperament is named Obikhodic because the Obikhod pitch set is the Mixolydian mode with the tenth flattened  
This temperament is named Obikhodic because the Obikhod pitch set is the Mixolydian mode with the tenth flattened or the Dorian mode with the third sharpened.
{| class="wikitable"
{| class="wikitable"
|+
|+
|'''Mode'''
|'''Mode'''
|[[Modal UDP Notation|'''UDP''']]
|[[Modal UDP Notation|'''UDP''']] 1
|'''Name'''
|[[Modal UDP Notation|'''UDP''']] 2
|'''Name 1'''
|'''Name 2'''
|-
|-
|LLsLLsLLsLL
|LLsLLsLLsLL
|<nowiki>6|4 b10</nowiki>
|<nowiki>6|4 b10</nowiki>
|<nowiki>4|6 #3</nowiki>
|(Great) Mixolydian f10
|(Great) Mixolydian f10
|(Great) Dorian #3
|-
|-
|LsLLLLsLLsL
|LLsLLsLLLLs
|<nowiki>5|5 b6</nowiki>
|<nowiki>8|2 b7</nowiki>
|(Great) Mixolydian f3 f6, Dorian f6 #10 (Livadeian f6)
|<nowiki>6|4 #11</nowiki>
|(Great) Lydian f4 f7, Ionian f7 #11 (Distomian Dominant)
|(Great) Mixolydian #11
|-
|LLsLLLLsLLs
|<nowiki>10|0 b4</nowiki>
|<nowiki>8|2 #8</nowiki>
|(Great) Lydian f4 #8 (Tanagran f4)
|(Great) Lydian f4 #8, Ionian #8 #11 (Distomian #8)
|-
|-
|sLLLLsLLsLL
|LLLLsLLsLLs
|<nowiki>4|6 b2</nowiki>
|<nowiki>1|9 *b1</nowiki>
|(Great) Dorian f2
|<nowiki>10|0 #5</nowiki>
|(Great) Phrygian *f1
|(Great) Lydian #5 #8 (Tanagran #5)
|-
|-
|LsLLsLLsLLL
|LsLLsLLsLLL
|<nowiki>3|7 b9</nowiki>
|<nowiki>3|7 b9</nowiki>
|<nowiki>1|9 #2</nowiki>
|(Great) Aeolian f9
|(Great) Aeolian f9
|-
|sLLsLLLLsLL
|<nowiki>2|8 b5</nowiki>
|(Great) Aeolian f2 f5, Phrygian f5 #9 (Theban f5)
|-
|LsLLsLLsLLL
|<nowiki>1|9 #2</nowiki>
|(Great) Phrygian #2
|(Great) Phrygian #2
|-
|-
|sLLsLLsLLLL
|LsLLsLLLLsL
|<nowiki>0|10 b8</nowiki>
|<nowiki>5|5 b6</nowiki>
|(Great) Locrian f8
|<nowiki>3|7 #10</nowiki>
|(Great) Mixolydian f3 f6, Dorian f6 #10 (Livadeian f6)
|(Great) Aeolian #10
|-
|-
|LLsLLLLsLLs
|sLLLLsLLsLL
|<nowiki>10|0 b4</nowiki>
|<nowiki>7|3 b3</nowiki>
|(Great) Lydian f4 #8 (Tanagran f4)
|<nowiki>5|5 #7</nowiki>
|(Great) Ionian f3
|(Great) Mixolydian f3 #7, Dorian #7 #10 (Livadeian #7)
|-
|-
|LLLsLLsLLsL
|LLLsLLsLLsL
|<nowiki>9|1 b11</nowiki>
|<nowiki>9|1 b11</nowiki>
|<nowiki>7|3 #4</nowiki>
|(Great) Lydian f11
|(Great) Lydian f11
|(Great) Ionian #4
|-
|-
|LLsLLsLLLLs
|sLLsLLsLLLL
|<nowiki>8|2 b7</nowiki>
|<nowiki>0|10 b8</nowiki>
|(Great) Lydian f4 f7, Ionian f7 #11 (Distomian Dominant)
|<nowiki>9|1 *#1</nowiki>
|(Great) Locrian f8
|(Great) Lydian *#1
|-
|sLLsLLLLsLL
|<nowiki>2|8 b5</nowiki>
|<nowiki>0|10 #9</nowiki>
|(Great) Aeolian f2 f5, Phrygian f5 #9 (Theban f5)
|(Great) Locrian #9
|-
|-
|LsLLLLsLLsL
|LsLLLLsLLsL
|<nowiki>7|3 b3</nowiki>
|<nowiki>4|6 b2</nowiki>
|(Great) Ionian f3
|<nowiki>2|8 #6</nowiki>
|(Great) Dorian f2
|(Great) Aeolian f2 #6, Phrygian #6 #9 (Theban #6)
|}
|}


Line 1,630: Line 1,654:
|LABCDEFfGHJKL
|LABCDEFfGHJKL
|LLsLLsLLLLs
|LLsLLsLLLLs
| style="text-align:center;" |<nowiki>8|2 #8</nowiki>
| style="text-align:center;" |<nowiki>8|2 b7</nowiki>
|(Great) Distomian Dominant
|(Great) Distomian Dominant
|-
|-
Line 1,666: Line 1,690:
|FfGHJKLABCDEFf
|FfGHJKLABCDEFf
|LsLLsLLsLLL
|LsLLsLLsLLL
| style="text-align:center;" |<nowiki>1|9 #2</nowiki>
| style="text-align:center;" |<nowiki>1|9 *b1</nowiki>
|(Great) Phrygian #2
|(Great) Phrygian *f1
|-
|-
|CDEFGHJKLABfC
|CDEFGHJKLABfC
Line 1,693: Line 1,717:
|(Great) Locrian f8
|(Great) Locrian f8
|}
|}
===Notes on Naming===
The modes of the Obikhodic scale are named after the existing modes, but contain the "Great" prefix (e.g. Great Ionian, Great Aeolian, etc.). The "Great" prefixes can be left in to explicitly distinguish which MOS's modes you're talking about, or can be omitted for convention.
Each Obikhodic mode contains its corresponding mode in the diatonic scale. This leads to a pattern: LLsLLLsLLLs and LLsLLLsLLsL both contain the meantone LLsLLLs Ionian mode. Additionally, sLLsLLLsLLL contains the diatonic sLLsLLL Locrian mode.
Since there are only seven diatonic modes, four of the superdiatonic modes need additional names and cannot reference any mode of the diatonic scale. These four modes present themselves as "altered" modes, which have an accidental the mode below them lacks, or vice versa. These are the only four modes to exhibit this behavior. They're interspersed on the ranking above and below Lydian, between Dorian and Mixolydian and between Aeolian and Phrygian and on the rotational continuum between Locrian and Ionian.
As were the original modes named after regions of ancient Greece, so are these new Obikhodic extensions. They are called after regions of Boeotia, set up so that the Locrian -> Distomian -> Livadeian -&gt; Theban -&gt; Tanagran -&gt; Ionian cyclic sequence will resemble the geography of ancient Greece.
==Scale tree==
{| class="wikitable"
{| class="wikitable"
|+
!Spelling 1
!Generator
!Spelling 2
!Normalized
!'''Mode'''
![[Modal UDP Notation|'''UDP''']]
!'''Name'''
|-
|GHJKLABC#DEFG
|LABCDEFG#HJKL
|LLsLLsLLLLs
|<nowiki>8|2 #8</nowiki>
|(Great) Distomian #8
|-
|HJKLABC#DEFGH
|ABCDEFG#HJKLA
|LsLLLLsLLsL
|<nowiki>5|5 #7</nowiki>
|(Great) Livadeian #7
|-
|JKLABC#DEFGHJ
|BCDEFG#HJKLAB
|sLLLLsLLsLL
|<nowiki>2|8 #6</nowiki>
|(Great) Theban #6
|-
|KLABC#DEFGHJK
|CDEFG#HJKLABC
|LLLLsLLsLLs
|<nowiki>10|0 #5</nowiki>
|(Great) Tanagran #5
|-
|LABC#DEFGHJKL
|DEFG#HJKLABCD
|LLLsLLsLLsL
|<nowiki>7|3 #4</nowiki>
|(Great) Ionian #4
|-
|ABC#DEFGHJKLA
|EFG#HJKLABCDE
|LLsLLsLLsLL
|<nowiki>4|6 #3</nowiki>
|(Great) Dorian #3
|-
|BC#DEFGHJKLAB
|FG#HJKLABCDEF
|LsLLsLLsLLL
|<nowiki>1|9 #2</nowiki>
|(Great) Phrygian #2
|-
|C#DEFGHJKLABC#
|G#HJKLABCDEFG#
|sLLsLLsLLLL
|<nowiki>9|1 *#1</nowiki>
|(Great) Lydian *#1
|-
|DEFGHJKLABC#D
|HJKLABCDEFG#H
|LLsLLsLLLLs
|<nowiki>6|4 #11</nowiki>
|(Great) Mixolydian #11
|-
|EFGHJKLABC#DE
|JKLABCDEFG#HJ
|LsLLsLLLLsL
|<nowiki>3|7 #10</nowiki>
|(Great) Aeolian #10
|-
|FGHJKLABC#DEF
|KLABCDEFG#HJK
|sLLsLLLLsLL
|<nowiki>0|10 #9</nowiki>
|(Great) Locrian #9
|}
===Notes on Naming===
The modes of the Obikhodic scale are named after the existing modes, but contain the "Great" prefix (e.g. Great Ionian, Great Aeolian, etc.). The "Great" prefixes can be left in to explicitly distinguish which MOS's modes you're talking about, or can be omitted for convention.
 
Each Obikhodic mode contains its corresponding mode in the diatonic scale. This leads to a pattern: LLsLLLsLLLs and LLsLLLsLLsL both contain the meantone LLsLLLs Ionian mode. Additionally, sLLsLLLsLLL contains the diatonic sLLsLLL Locrian mode.
 
Since there are only seven diatonic modes, four of the superdiatonic modes need additional names and cannot reference any mode of the diatonic scale. These four modes present themselves as "altered" modes, which have an accidental the mode below them lacks, or vice versa. These are the only four modes to exhibit this behavior. They're interspersed on the ranking above and below Lydian, between Dorian and Mixolydian and between Aeolian and Phrygian and on the rotational continuum between Locrian and Ionian.
 
As were the original modes named after regions of ancient Greece, so are these new Obikhodic extensions. They are called after regions of Boeotia, set up so that the Locrian -> Distomian -> Livadeian -&gt; Theban -&gt; Tanagran -&gt; Ionian cyclic sequence will resemble the geography of ancient Greece.
==Scale tree==
{| class="wikitable"
|+
!Generator
!Normalized
!Large step
!Small step
|-
|3\8
|<u>720.000</u>
|1\5, <u>240.000</u>
|0
|-
|19\51
|<u>712.500</u>
|6\51, <u>225.000</u>
|1\51, <u>37.500</u>
|-
|54\145
|<u>712.088</u>
|17\145, <u>224.175</u>
|3\145, <u>39.560</u>
|-
|35\94
|<u>711.864</u>
|11\94, <u>223.729</u>
|2\94, <u>40.678</u>
|-
|16\43
|<u>711.111</u>
|5\43, <u>222.222</u>
|1\43, <u>44.444</u>
|-
|45\121
|<u>710.526</u>
|14\121, <u>221.053</u>
|3\121, <u>47.368</u>
|-
|29\78
|<u>710.204</u>
|9\78, <u>220.408</u>
|2\78, <u>48.980</u>
|-
|42\113
|<u>709.859</u>
|13\113, <u>219.718</u>
|3\113, <u>50.704</u>
|-
|13\35
|<u>709.091</u>
|4\35, <u>218.182</u>
|1\35, <u>54.545</u>
|-
|49\132
|<u>708.434</u>
|15\132, <u>216.867</u>
|4\132, <u>57.831</u>
|-
|36\97
|<u>708.197</u>
|11\97, <u>216.393</u>
|3\97, <u>59.016</u>
|-
|23\62
|<u>707.692</u>
|7\62, <u>215.385</u>
|2\62, <u>61.538</u>
|-
|33\89
|<u>707.143</u>
|10\89, <u>214.286</u>
|3\89, <u>64.286</u>
|-
|43\116
|<u>706.849</u>
|13\116, <u>213.699</u>
|4\116, <u>65.753</u>
|-
|53\143
|<u>706.667</u>
|16\143, <u>213.333</u>
|5\143, <u>66.667</u>
|-
|63\170
|<u>706.542</u>
|19\170, <u>213.084</u>
|6\170, <u>67.290</u>
|-
|73\197
|<u>706.452</u>
|22\197, <u>212.903</u>
|7\197, <u>67.742</u>
|-
|10\27
|<u>705.882</u>
|3\27, <u>211.765</u>
|1\27, <u>70.588</u>
|-
|47\127
|<u>705.000</u>
|14\127, <u>210.000</u>
|5\127, <u>75.000</u>
|-
|37\100
|<u>704.762</u>
|11\100, <u>209.524</u>
|4\100, <u>76.190</u>
|-
|27\73
|<u>704.348</u>
|8\73, <u>208.696</u>
|3\27, <u>78.261</u>
|-
|17\46
|<u>703.448</u>
|5\46, <u>206.897</u>
|2\46, <u>82.759</u>
|-
|41\111
|<u>702.857</u>
|12\111, <u>205.714</u>
|5\111, <u>85.714</u>
|-
|24\65
|<u>702.439</u>
|7\65, <u>204.878</u>
|3\65, <u>87.805</u>
|-
|31\84
|<u>701.887</u>
|9\84, <u>203.774</u>
|4\84, <u>90.566</u>
|-
|-
|3\8
|…
|<u>720</u>
|…
|…
|…
|-
|-
|19\51
|59\160
|<u>712.5</u>
|<u>700.990</u>
|17\160, <u>201.980</u>
|8\160, <u>95.050</u>
|-
|-
|16\43
|…
|<u>711.111</u>
|…
|…
|…
|-
|-
|29\78
|122\331
|<u>710.204</u>
|<u>700.478</u>
|35\331, <u>200.960</u>
|18\334, <u>97.608</u>
|-
|-
|13\35
|7\19
|<u>709.091</u>
|<u>700.000</u>
|-
|2\19, <u>200.000</u>
|36\97
|1\19, <u>100.000</u>
|<u>708.197</u>
|-
|23\62
|<u>707.692</u>
|-
|33\89
|<u>707.143</u>
|-
|43\116
|<u>706.849</u>
|-
|10\27
|<u>705.882</u>
|-
|27\73
|<u>704.348</u>
|-
|17\46
|<u>703.448</u>
|-
|24\65
|<u>702.439</u>
|-
|-
|31\84
|123\334
|<u>701.887</u>
|<u>699.526</u>
|35\334, <u>199.052</u>
|18\334, <u>102.370</u>
|-
|-
|7\19
|…
|<u>700</u>
|…
|…
|…
|-
|-
|60\163
|63\163
|<u>699.029</u>
|<u>699.029</u>
|17\163, <u>198.058</u>
|9\163, <u>104.854</u>
|-
|-
|53\144
|…
|<u>698.901</u>
|…
|-
|…
|46\125
|…
|<u>698.734</u>
|-
|39\106
|<u>698.507</u>
|-
|-
|32\87
|32\87
|<u>698.182</u>
|<u>698.182</u>
|9\87, <u>196.364</u>
|5\87, <u>109.091</u>
|-
|-
|25\68
|25\68
|<u>697.674</u>
|<u>697.674</u>
|7\68, <u>195.349</u>
|4\68, <u>111.628</u>
|-
|43\117
|<u>697.297</u>
|12\117, <u>194.594</u>
|7\117, <u>113.514</u>
|-
|-
|18\49
|18\49
|<u>696.77</u>
|<u>696.774</u>
|5\49, <u>193.548</u>
|3\49, <u>116.129</u>
|-
|-
|29\79
|29\79
|<u>696</u>
|<u>696.000</u>
|8\79, <u>192.000</u>
|5\79, <u>120.000</u>
|-
|-
|40\109
|40\109
|<u>695.652</u>
|<u>695.652</u>
|11\109, <u>191.304</u>
|7\109, <u>121.739</u>
|-
|51\139
|<u>695.455</u>
|14\139, <u>190.909</u>
|9\139, <u>122.727</u>
|-
|-
|11\30
|11\30
|<u>694.737</u>
|<u>694.737</u>
|3\30, <u>189.474</u>
|2\30, <u>126.316</u>
|-
|59\161
|<u>694.118</u>
|16\161, <u>188.235</u>
|11\161, <u>129.412</u>
|-
|48\131
|<u>693.976</u>
|13\131, <u>187.952</u>
|9\131, <u>130.120</u>
|-
|37\101
|<u>693.750</u>
|10\101, <u>187.500</u>
|7\101, <u>131.250</u>
|-
|-
|26\71
|26\71
|<u>693.333</u>
|<u>693.333</u>
|7\71, <u>186.667</u>
|5\71, <u>133.333</u>
|-
|41\112
|<u>692.958</u>
|11\112, <u>185.915</u>
|8\112, <u>135.211</u>
|-
|-
|15\41
|15\41
|<u>692.308</u>
|<u>692.308</u>
|4\41, <u>184.615</u>
|3\41, <u>138.462</u>
|-
|-
|34\93
|34\93
|<u>691.525</u>
|<u>691.525</u>
|9\93, <u>183.051</u>
|7\93, <u>142.373</u>
|-
|53\145
|<u>691.304</u>
|14\145, <u>182.609</u>
|11\145, <u>143.478</u>
|-
|-
|19\52
|19\52
|<u>690.909</u>
|<u>690.909</u>
|5\52, <u>181.818</u>
|4\52, <u>145.455</u>
|-
|-
|23\63
|23\63
|<u>690</u>
|<u>690.000</u>
|6\63, <u>180.000</u>
|5\63, <u>150.000</u>
|-
|-
|4\11
|4\11
|<u>685.714</u>
|<u>685.714</u>
|1\11, <u>171.429</u>
|1\11, <u>171.429</u>
|}
|}


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