Septiennealimmal clan: Difference between revisions

* POTE: ~99/98 = 17.626

Ennealimmal tempers out the two smallest 7-limit [[superparticular]] commas, [[2401/2400]] and [[4375/4374]], leading to a temperament of unusual [[efficiency]]. It also tempers out the [[ennealimma]], {{monzo| 1 -27 18 }}, which leads to the identification of (27/25)⁹ with the [[octave]], and gives ennealimmal a [[period]] of 1/9 octave. Its [[pergen]] is (P8/9, P5/2), and [[ploidacot]] enneaploid dicot. While [[27/25]] is a 5-limit interval, a stack of two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit.  

7-limit ennealimmal's [[S-expression]]-based comma list is {[[4375/4374|S25/S27]], [[2401/2400|S49]]}. Interestingly, the [[landscape comma]] is equal to [[2401/2400|S49]]/([[4375/4374|S25/S27]]) while the [[wizma]] is equal to [[2401/2400|S49]]*[[4375/4374|S25/S27]].

Hemiennealimmal (72 & 198) has a period of 1/18 octave and tempers out the four smallest superparticular commas of the 11-limit JI, 2401/2400, 3025/3024, 4375/4374, and 9801/9800. Tempering out [[9801/9800]] leads to an octave split into two equal parts. Notably, every one of these commas is part of one or more known infinite comma families; see directly below.

Its [[S-expression]]-based comma list is {([[3025/3024|S22/S24 = S55 = S25/S27 * S99]],) [[4375/4374|S25/S27]], [[2401/2400|S49]], [[9801/9800|S33/S35 = S99]]}.

Optimal tuning (POTE): ~80/77 = 1\18, ~400/231 = 950.9553

{{Optimal ET sequence|legend=1| 72, 198, 270, 342, 612, 954, 1566 }}

Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837

{{Optimal ET sequence|legend=1| 72, 198, 270 }}

Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837

{{Optimal ET sequence|legend=1| 72, 198g, 270 }}

Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837

Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727

{{Optimal ET sequence|legend=1| 126, 144, 270, 684, 954 }}

Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727

{{Optimal ET sequence|legend=1| 270, 684, 954 }}

Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727

{{Optimal ET sequence|legend=1| 270, 684h, 954h, 1224 }}

Optimal tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3367

{{Optimal ET sequence|legend=1| 72, 369, 441 }}

Optimal tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3375

{{Optimal ET sequence|legend=1| 72, 297ef, 369f, 441 }}

Optimal tuning (POTE): ~27/25 = 1\9, ~25/22 = 221.0717

{{Optimal ET sequence|legend=1| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}

Optimal tuning (POTE): ~2744/2673 = 1\27, ~2352/1375 = 928.8000

{{Optimal ET sequence|legend=1| 27, 243, 270, 783, 1053, 1323 }}

Rhodium splits the ennealimmal period in five parts and thereby features a period of 9 × 5 = 45, thus the name is given after the 45th element.

* CTE: ~3072/3025 = 1\45, ~55/32 = 937.6658 (~385/384 = 4.3325)

* CWE: ~3072/3025 = 1\45, ~55/32 = 937.6630 (~385/384 = 4.3397)

Optimal ET sequence: {{Optimal ET sequence| 45, 225c, 270, 1125, 1395, 1665, 5265d }}

* CTE: ~66/65 = 1\45, ~55/32 = 937.6569 (~385/384 = 4.3236)

* CWE: ~66/65 = 1\45, ~55/32 = 937.6515 (~385/384 = 4.3182)

Optimal ET sequence: {{Optimal ET sequence| 45, 270, 855, 1125, 1395, 1665, 3060d, 4725df }}

Undecentic (99&198) has a period of 1/99 octave.

Subgroup: 2.3.5.7.11

[[Mapping]]: [{{val|99 157 230 278 0}}, {{val|0 0 0 0 1}}]

[[POTE generator]]: ~11/8 = 552.756

Mapping: [{{val|99 157 230 278 0 24}}, {{val|0 0 0 0 1 1}}]

POTE generator: ~11/8 = 552.024

{{Optimal ET sequence|legend=1| 99ef, 198 }}

Schisennealimmal (171&342) has a period of 1/171 octave. [[171edo|171EDO]] and its multiples are members of both [[Schismatic family|schismic]] and [[Ragismic microtemperaments #Ennealimmal|ennealimmal]], and from this it derives its name.

Subgroup: 2.3.5.7.11

[[Mapping]]: [{{val| 171 271 397 480 0 }}, {{val| 0 0 0 0 1 }}]

[[POTE generator]]: ~11/8 = 550.954

Mapping: [{{val| 171 271 397 480 0 633 }}, {{val| 0 0 0 0 1 0 }}]

POTE generator: ~11/8 = 551.322

{{Optimal ET sequence|legend=1| 171, 342, 855ff, 1197fff }}

Mapping: [{{val| 171 271 397 480 0 633 699 }}, {{val| 0 0 0 0 1 0 0 }}]

POTE generator: ~11/8 = 551.365

{{Optimal ET sequence|legend=1| 171, 342, 855ff, 1197fff }}

Mapping: [{{val| 171 271 397 480 0 41 }}, {{val| 0 0 0 0 1 1 }}]

POTE generator: ~11/8 = 551.625

Mapping: [{{val|171 271 397 480 0 41 699 }}, {{val|0 0 0 0 1 1 0 }}]

POTE generator: ~11/8 = 551.756

{{Optimal ET sequence|legend=1| 171, 342f, 513, 855f }}

Lunennealimmal (441&882) has has a period of 1/441 octave. [[441edo|441EDO]] and its multiples are members of both [[Luna family|luna]] and [[Ragismic microtemperaments #Ennealimmal|ennealimmal]], and from this it derives its name.

Subgroup: 2.3.5.7.11

[[Mapping]]: [{{val|441 699 1024 1238 1526}}, {{val|0 0 0 0 -1}}]

[[POTE generator]]: ~11/8 = 551.3584

Mapping: [{{val|441 699 1024 1238 1526 1632}}, {{val|0 0 0 0 -1 0}}]

POTE generator: ~11/8 = 551.4043

{{Optimal ET sequence|legend=1| 441, 882, 1323, 3528f, 4851ff, 6174dff }}

Mapping: [{{val|441 699 1024 1238 1526 1632 1803}}, {{val|0 0 0 0 -1 0 -1}}]

POTE generator: ~11/8 = 551.3688

{{Optimal ET sequence|legend=1| 441, 882, 1323, 2205f, 3528f }}