Ennealimmal: Difference between revisions
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{{URWTC}} | {{URWTC}} | ||
'''Ennealimmal temperament''' has a period of 1 | '''Ennealimmal temperament''' has a period of {{frac|1|9}} octave and tempers out [[2401/2400]] and [[4375/4374]]. EDOs that support ennealimmal include EDOs {{EDOs| 27, 45, 72, 99, 171, 270, 441, and 612 }}. | ||
See [[Ragismic microtemperaments #Ennealimmal]] for more technical data. | See [[Ragismic microtemperaments #Ennealimmal]] for more technical data. | ||
Ennealimmal scales are built from a ''period'' (which is exactly 1 | Ennealimmal scales are built from a ''period'' (which is exactly {{frac|1|9}} of an octave), and a ''generator'' (which is approximately 49 cents and represents several small intervals including 36/35). Depending on the size of the generator and the period in steps, the above listed EDOs make sense: | ||
{| class="wikitable | {| class="wikitable" | ||
! Period (steps) !! Generator (steps) !! Generator (cents) <br/> (pure octave) !! EDO | |- style="white-space: nowrap;" | ||
! Period (steps) !! Generator (steps) !! Generator (cents)<br />(pure octave) !! EDO | |||
|- | |- | ||
| 3 || 1 || 44.444 || 27 | | 3 || 1 || 44.444 || 27 | ||
| Line 25: | Line 26: | ||
== Interval chain == | == Interval chain == | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Generator based off 11-limit ennealimmal | |+ style="font-size: 105%;" | Generator based off 11-limit ennealimmal | ||
!Generator | |- style="white-space: nowrap;" | ||
!Period 1 | ! Generator | ||
!Period 2 | ! Period 1 | ||
!Period 3 | ! Period 2 | ||
!Period 4 | ! Period 3 | ||
!Period 5 | ! Period 4 | ||
!Period 6 | ! Period 5 | ||
!Period 7 | ! Period 6 | ||
!Period 8 | ! Period 7 | ||
!Period 9 (0) | ! Period 8 | ||
! Period 9 (0) | |||
|- | |- | ||
!0 | ! 0 | ||
|133.333 | | 133.333 | ||
|266.666 | | 266.666 | ||
|400.000 | | 400.000 | ||
|533.333 | | 533.333 | ||
|666.666 | | 666.666 | ||
|800.000 | | 800.000 | ||
|933.333 | | 933.333 | ||
|1066.666 | | 1066.666 | ||
|1200.000 | | 1200.000 | ||
|- | |- | ||
! | ! −1 | ||
|84.430 | | 84.430 | ||
|217.764 | | 217.764 | ||
|351.097 | | 351.097 | ||
|484.430 | | 484.430 | ||
|617.764 | | 617.764 | ||
|751.097 | | 751.097 | ||
|884.430 | | 884.430 | ||
|1017.764 | | 1017.764 | ||
|1151.097 | | 1151.097 | ||
|- | |- | ||
! | ! −2 | ||
|35.527 | | 35.527 | ||
|168.861 | | 168.861 | ||
|302.194 | | 302.194 | ||
|435.527 | | 435.527 | ||
|568.861 | | 568.861 | ||
|702.194 | | 702.194 | ||
|835.527 | | 835.527 | ||
|968.861 | | 968.861 | ||
|1102.194 | | 1102.194 | ||
|- | |- | ||
! | ! −3 | ||
|1186.624 | | 1186.624 | ||
|119.958 | | 119.958 | ||
|253.291 | | 253.291 | ||
|386.624 | | 386.624 | ||
|519.958 | | 519.958 | ||
|653.291 | | 653.291 | ||
|786.624 | | 786.624 | ||
|919.958 | | 919.958 | ||
|1053.291 | | 1053.291 | ||
|- | |- | ||
! | ! −4 | ||
|1137.721 | | 1137.721 | ||
|71.055 | | 71.055 | ||
|204.388 | | 204.388 | ||
|337.721 | | 337.721 | ||
|471.055 | | 471.055 | ||
|604.388 | | 604.388 | ||
|737.721 | | 737.721 | ||
|871.055 | | 871.055 | ||
|1004.388 | | 1004.388 | ||
|- | |- | ||
! | ! −5 | ||
|1088.818 | | 1088.818 | ||
|22.152 | | 22.152 | ||
|155.485 | | 155.485 | ||
|288.818 | | 288.818 | ||
|422.152 | | 422.152 | ||
|555.485 | | 555.485 | ||
|688.818 | | 688.818 | ||
|822.152 | | 822.152 | ||
|955.485 | | 955.485 | ||
|- | |- | ||
! | ! −6 | ||
|1039.915 | | 1039.915 | ||
|1173.249 | | 1173.249 | ||
|106.582 | | 106.582 | ||
|239.915 | | 239.915 | ||
|373.249 | | 373.249 | ||
|506.582 | | 506.582 | ||
|639.915 | | 639.915 | ||
|773.249 | | 773.249 | ||
|906.582 | | 906.582 | ||
|- | |- | ||
! | ! −7 | ||
|991.012 | | 991.012 | ||
|1124.346 | | 1124.346 | ||
|57.679 | | 57.679 | ||
|191.012 | | 191.012 | ||
|324.346 | | 324.346 | ||
|457.679 | | 457.679 | ||
|591.012 | | 591.012 | ||
|724.346 | | 724.346 | ||
|857.679 | | 857.679 | ||
|- | |- | ||
! | ! −8 | ||
|942.109 | | 942.109 | ||
|1075.443 | | 1075.443 | ||
|8.776 | | 8.776 | ||
|142.109 | | 142.109 | ||
|275.443 | | 275.443 | ||
|408.776 | | 408.776 | ||
|542.109 | | 542.109 | ||
|675.443 | | 675.443 | ||
|808.776 | | 808.776 | ||
|- | |- | ||
! | ! −9 | ||
|893.206 | | 893.206 | ||
|1026.540 | | 1026.540 | ||
|1159.873 | | 1159.873 | ||
|93.206 | | 93.206 | ||
|226.540 | | 226.540 | ||
|359.873 | | 359.873 | ||
|493.206 | | 493.206 | ||
|626.540 | | 626.540 | ||
|759.873 | | 759.873 | ||
|- | |- | ||
! | ! −10 | ||
|844.303 | | 844.303 | ||
|977.637 | | 977.637 | ||
|1110.970 | | 1110.970 | ||
|44.303 | | 44.303 | ||
|177.637 | | 177.637 | ||
|310.970 | | 310.970 | ||
|444.303 | | 444.303 | ||
|577.637 | | 577.637 | ||
|710.970 | | 710.970 | ||
|- | |- | ||
! | ! −11 | ||
|795.400 | | 795.400 | ||
|928.734 | | 928.734 | ||
|1062.067 | | 1062.067 | ||
|1195.400 | | 1195.400 | ||
|128.734 | | 128.734 | ||
|262.067 | | 262.067 | ||
|395.400 | | 395.400 | ||
|528.734 | | 528.734 | ||
|662.067 | | 662.067 | ||
|- | |- | ||
! | ! −12 | ||
|746.497 | | 746.497 | ||
|879.831 | | 879.831 | ||
|1013.164 | | 1013.164 | ||
|1146.497 | | 1146.497 | ||
|79.831 | | 79.831 | ||
|213.164 | | 213.164 | ||
|346.497 | | 346.497 | ||
|479.831 | | 479.831 | ||
|613.164 | | 613.164 | ||
|- | |- | ||
! | ! −13 | ||
|697.594 | | 697.594 | ||
|830.928 | | 830.928 | ||
|964.261 | | 964.261 | ||
|1097.594 | | 1097.594 | ||
|30.928 | | 30.928 | ||
|164.261 | | 164.261 | ||
|297.594 | | 297.594 | ||
|430.928 | | 430.928 | ||
|564.261 | | 564.261 | ||
|- | |- | ||
! | ! −14 | ||
|648.691 | | 648.691 | ||
|782.025 | | 782.025 | ||
|915.358 | | 915.358 | ||
|1048.691 | | 1048.691 | ||
|1182.025 | | 1182.025 | ||
|115.358 | | 115.358 | ||
|248.691 | | 248.691 | ||
|382.025 | | 382.025 | ||
|515.358 | | 515.358 | ||
|- | |- | ||
! | ! −15 | ||
|599.788 | | 599.788 | ||
|733.122 | | 733.122 | ||
|866.455 | | 866.455 | ||
|999.788 | | 999.788 | ||
|1133.122 | | 1133.122 | ||
|66.455 | | 66.455 | ||
|199.788 | | 199.788 | ||
|333.122 | | 333.122 | ||
|466.455 | | 466.455 | ||
|- | |- | ||
! | ! −16 | ||
|550.885 | | 550.885 | ||
|684.219 | | 684.219 | ||
|817.552 | | 817.552 | ||
|950.885 | | 950.885 | ||
|1084.219 | | 1084.219 | ||
|17.552 | | 17.552 | ||
|150.885 | | 150.885 | ||
|284.219 | | 284.219 | ||
|417.552 | | 417.552 | ||
|- | |- | ||
! | ! −17 | ||
|501.982 | | 501.982 | ||
|635.316 | | 635.316 | ||
|768.649 | | 768.649 | ||
|901.982 | | 901.982 | ||
|1035.316 | | 1035.316 | ||
|1168.649 | | 1168.649 | ||
|101.982 | | 101.982 | ||
|235.316 | | 235.316 | ||
|368.649 | | 368.649 | ||
|} | |} | ||
== Ennealimmal extensions == | == Ennealimmal extensions == | ||
Ennealimmal temperament has various extensions to the 11-limit. These are all members of the ennealimmal family, but in addition they are linear temperaments: | Ennealimmal temperament has various extensions to the 11-limit. These are all members of the ennealimmal family, but in addition they are linear temperaments: | ||
* Ennealimmal (99e & 270) | * Ennealimmal (99e & 270) – tempering out 2401/2400, 4375/4374, 5632/5625 | ||
* Ennealimmia (171 & 270) | * Ennealimmia (171 & 270) – tempering out 2401/2400, 4375/4374, 131072/130977 | ||
* Ennealimnic (72 & 99e) | * Ennealimnic (72 & 99e) – tempering out 243/242, 441/440, 4375/4356 | ||
* Ennealiminal (72 & 171e) | * Ennealiminal (72 & 171e) – tempering out 385/384, 1375/1372, 4375/4374 | ||
== Scales == | == Scales == | ||
* [[Ennealimmal27]] | * [[Ennealimmal27]] – proper [[18L 9s]]. Ninth-octave analog of haplotonic scale | ||
* [[Ennealimmal45]] | * [[Ennealimmal45]] – improper [[27L 18s]]. Ninth-octave analog of mega-haplotonic scale | ||
** [[Ennealimmal45trans]] | ** [[Ennealimmal45trans]] – symmetric 5-limit transversal version | ||
* [[Ennealimmal72]] | * [[Ennealimmal72]] – proper [[27L 45s]]. Ninth-octave analog of albitonic scale | ||
* [[Ennealimmal99]] | * [[Ennealimmal99]] – proper [[72L 27s]]. Ninth-octave analog of chromatic scale | ||
* [[Ennealimmal171]] | * [[Ennealimmal171]] – [[99L 72s]] scale. The boundary of propriety is [[270edo]]. | ||
[[Category:Ragismic microtemperaments]] | [[Category:Ragismic microtemperaments]] | ||
[[Category:Temperaments]] | [[Category:Temperaments]] | ||
[[Category:Ennealimmal| ]] | [[Category:Ennealimmal| ]] | ||
Revision as of 00:30, 26 October 2024
|
This page on a regular temperament, temperament collection, or aspect of regular temperament theory is being revised for clarity as part of WikiProject TempClean. |
Ennealimmal temperament has a period of 1⁄9 octave and tempers out 2401/2400 and 4375/4374. EDOs that support ennealimmal include EDOs 27, 45, 72, 99, 171, 270, 441, and 612.
See Ragismic microtemperaments #Ennealimmal for more technical data.
Ennealimmal scales are built from a period (which is exactly 1⁄9 of an octave), and a generator (which is approximately 49 cents and represents several small intervals including 36/35). Depending on the size of the generator and the period in steps, the above listed EDOs make sense:
| Period (steps) | Generator (steps) | Generator (cents) (pure octave) |
EDO |
|---|---|---|---|
| 3 | 1 | 44.444 | 27 |
| 11 | 4 | 48.485 | 99 |
| 30 | 11 | 48.889 | 270 |
| 19 | 7 | 49.123 | 171 |
| 8 | 3 | 50.000 | 72 |
| 5 | 2 | 53.333 | 45 |
Interval chain
| Generator | Period 1 | Period 2 | Period 3 | Period 4 | Period 5 | Period 6 | Period 7 | Period 8 | Period 9 (0) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 133.333 | 266.666 | 400.000 | 533.333 | 666.666 | 800.000 | 933.333 | 1066.666 | 1200.000 |
| −1 | 84.430 | 217.764 | 351.097 | 484.430 | 617.764 | 751.097 | 884.430 | 1017.764 | 1151.097 |
| −2 | 35.527 | 168.861 | 302.194 | 435.527 | 568.861 | 702.194 | 835.527 | 968.861 | 1102.194 |
| −3 | 1186.624 | 119.958 | 253.291 | 386.624 | 519.958 | 653.291 | 786.624 | 919.958 | 1053.291 |
| −4 | 1137.721 | 71.055 | 204.388 | 337.721 | 471.055 | 604.388 | 737.721 | 871.055 | 1004.388 |
| −5 | 1088.818 | 22.152 | 155.485 | 288.818 | 422.152 | 555.485 | 688.818 | 822.152 | 955.485 |
| −6 | 1039.915 | 1173.249 | 106.582 | 239.915 | 373.249 | 506.582 | 639.915 | 773.249 | 906.582 |
| −7 | 991.012 | 1124.346 | 57.679 | 191.012 | 324.346 | 457.679 | 591.012 | 724.346 | 857.679 |
| −8 | 942.109 | 1075.443 | 8.776 | 142.109 | 275.443 | 408.776 | 542.109 | 675.443 | 808.776 |
| −9 | 893.206 | 1026.540 | 1159.873 | 93.206 | 226.540 | 359.873 | 493.206 | 626.540 | 759.873 |
| −10 | 844.303 | 977.637 | 1110.970 | 44.303 | 177.637 | 310.970 | 444.303 | 577.637 | 710.970 |
| −11 | 795.400 | 928.734 | 1062.067 | 1195.400 | 128.734 | 262.067 | 395.400 | 528.734 | 662.067 |
| −12 | 746.497 | 879.831 | 1013.164 | 1146.497 | 79.831 | 213.164 | 346.497 | 479.831 | 613.164 |
| −13 | 697.594 | 830.928 | 964.261 | 1097.594 | 30.928 | 164.261 | 297.594 | 430.928 | 564.261 |
| −14 | 648.691 | 782.025 | 915.358 | 1048.691 | 1182.025 | 115.358 | 248.691 | 382.025 | 515.358 |
| −15 | 599.788 | 733.122 | 866.455 | 999.788 | 1133.122 | 66.455 | 199.788 | 333.122 | 466.455 |
| −16 | 550.885 | 684.219 | 817.552 | 950.885 | 1084.219 | 17.552 | 150.885 | 284.219 | 417.552 |
| −17 | 501.982 | 635.316 | 768.649 | 901.982 | 1035.316 | 1168.649 | 101.982 | 235.316 | 368.649 |
Ennealimmal extensions
Ennealimmal temperament has various extensions to the 11-limit. These are all members of the ennealimmal family, but in addition they are linear temperaments:
- Ennealimmal (99e & 270) – tempering out 2401/2400, 4375/4374, 5632/5625
- Ennealimmia (171 & 270) – tempering out 2401/2400, 4375/4374, 131072/130977
- Ennealimnic (72 & 99e) – tempering out 243/242, 441/440, 4375/4356
- Ennealiminal (72 & 171e) – tempering out 385/384, 1375/1372, 4375/4374
Scales
- Ennealimmal27 – proper 18L 9s. Ninth-octave analog of haplotonic scale
- Ennealimmal45 – improper 27L 18s. Ninth-octave analog of mega-haplotonic scale
- Ennealimmal45trans – symmetric 5-limit transversal version
- Ennealimmal72 – proper 27L 45s. Ninth-octave analog of albitonic scale
- Ennealimmal99 – proper 72L 27s. Ninth-octave analog of chromatic scale
- Ennealimmal171 – 99L 72s scale. The boundary of propriety is 270edo.