Ennealimmal

Ennealimmal

Subgroups	2.3.5.7
Comma basis	2401/2400, 4375/4374
Reduced mapping	⟨9; 2 3 2]
ET join	27 & 45
Generators (CWE)	~5/3 = 884.322 ¢
MOS scales	18L 9s, 27L 18s, 27L 45s
Ploidacot	enneaploid dicot
Pergen	(P8/9, P5/2)
Minimax error	9-odd-limit: 0.204 ¢; 7-limit 81-odd-limit: 0.408 ¢
Target scale size	9-odd-limit: 45 notes; 7-limit 81-odd-limit: 99 notes

Ennealimmal is a temperament with a period of 1⁄9 octave and tempers out 2401/2400 and 4375/4374. Edos that support ennealimmal include 27, 45, 72, 99, 171, 270, 441, and 612.

See Septiennealimmal clan #Ennealimmal for 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

Ennealimmal extends less well to the 11-limit. Extensions include enneabiotic (99e & 270), ennealympic (171 & 270), ennealimnic (72 & 99e), and ennealiminal (72 & 171e).

See Ennealimmal extensions for a discussion on 11-limit extensions.

Interval chain

In the following table, odd harmonics 1–9 are labeled in bold.

Period	Generator 0		Generator 1		Generator 2		Generator 3
	Cents*	Approx. ratios	Cents*	Approx. ratios	Cents*	Approx. ratios	Cents*	Approx. ratios
0	0.000	1/1
1	133.333	27/25	84.322	21/20	35.310	49/48, 50/49	1186.298	125/63
2	266.667	7/6	217.655	245/216	168.643	54/49	119.631	15/14
3	400.000	63/50	350.988	49/40, 60/49	301.976	25/21	252.965	81/70, 125/108
4	533.333	49/36	484.322	250/189	435.310	9/7	386.298	5/4
5	666.667	72/49	617.655	10/7	568.643	25/18	519.631	27/20
6	800.000	100/63	750.988	54/35	701.976	3/2	652.965	35/24
7	933.333	12/7	884.322	5/3	835.310	81/50	786.298	63/40
8	1066.667	50/27	1017.655	9/5	968.643	7/4	919.631	245/144
9	1200.000	2/1	1150.988	35/18	1101.976	189/100	1052.965	147/80

* In 7-limit CWE tuning, octave reduced

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.

Music

Gene Ward Smith

• The 45000 fingers of Dr. S (2003) – Ennealimmal[54] in TOP tuning