9edf

← 8edf

9edf

10edf →

Prime factorization	32
Step size	77.995¢
Octave	15\9edf (1169.93¢) (→5\3edf)
Twelfth	24\9edf (1871.88¢) (→8\3edf)
Consistency limit	2
Distinct consistency limit	2

9 equal divisions of the perfect fifth (abbreviated 9edf or 9ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 9 equal parts of about 78⁠ ⁠¢ each. Each step represents a frequency ratio of (3/2)^1/9, or the 9th root of 3/2.

Theory

9edf corresponds to 15.391524edo. It is closely related to Carlos Alpha, and can be used as a temperament of the 3/2.5/4.7/8.11/8 subgroup.

Harmonics

Approximation of harmonics in 9edf

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-30.1	-30.1	+17.8	+21.5	+17.8	-15.0	-12.2	+17.8	-8.6	-17.6	-12.2
	Relative (%)	-38.6	-38.6	+22.9	+27.6	+22.9	-19.3	-15.7	+22.9	-11.0	-22.5	-15.7
Steps (reduced)		15 (6)	24 (6)	31 (4)	36 (0)	40 (4)	43 (7)	46 (1)	49 (4)	51 (6)	53 (8)	55 (1)

Approximation of harmonics in 9edf (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	+5.2	+32.9	-8.6	+35.7	+8.7	-12.2	-27.8	-38.6	+32.9	+30.3	+31.4	+35.7
	Relative (%)	+6.7	+42.2	-11.0	+45.8	+11.2	-15.7	-35.7	-49.5	+42.2	+38.9	+40.2	+45.8
Steps (reduced)		57 (3)	59 (5)	60 (6)	62 (8)	63 (0)	64 (1)	65 (2)	66 (3)	68 (5)	69 (6)	70 (7)	71 (8)

Intervals

#	Cents	Approximate ratios	Neptunian notation using 7\9edf
0	0.0	1/1	C
1	78.0	21/20, 22/21, 25/24	C#
2	156.0	11/10, 12/11	Db
3	234.0	8/7	D
4	312.0	6/5	D#, Eb
5	390.0	5/4	E
6	468.0	21/16	E#, Fb
7	546.0	11/8, 15/11	F
8	624.0	10/7, 36/25	F#, Cb
9	701.9	3/2	C

Scales

Livewire (this is the original/default tuning)

• 77.995
• 701.955
• 779.950
• 1403.905
• 1481.905
• 2105.865
• 2183.860
• 2807.820
• 2885.815
• 3509.775
• 3587.770

Corrugated (this is the original/default tuning)

• 155.990
• 701.955
• 857.945
• 1403.910
• 1559.900
• 2105.865
• 2261.855
• 2807.820
• 2963.810
• 3509.775
• 3665.765
• 4211.730
• 4367.720
• 4913.685
• 5069.675
• 5615.640
• 5771.630
• 6317.595
• 6473.585
• 7019.550
• 7175.540
• 7721.505
• 7877.495
• 8423.460

Snowcone (this is the original/default tuning)

• 233.985
• 701.955
• 935.940
• 1403.910
• 1637.895
• 2105.865
• 2339.850
• 2807.820
• 3041.805
• 3509.775
• 3743.760
• 4211.730
• 4445.715
• 4913.685
• 5147.670
• 5615.640
• 5849.625
• 6317.595
• 6551.580
• 7019.550
• 7253.535
• 7721.505
• 7955.490
• 8423.460

Swan (this is the original/default tuning)

• 311.980
• 701.955
• 1013.935
• 1403.910
• 1715.890
• 2105.865
• 2417.845

Cloudscape (this is the original/default tuning)

• 389.975
• 701.955
• 1091.930
• 1403.910
• 1793.885
• 2105.865
• 2495.840
• 2807.820
• 3197.795
• 3509.775
• 3899.750
• 4211.730
• 4601.705
• 4913.685
• 5303.660
• 5615.640
• 6005.615

Pylon (this is the original/default tuning)

• 467.970
• 701.955
• 1169.925
• 1403.910
• 1871.880
• 2105.865
• 2573.835
• 2807.820
• 3275.790
• 3509.775
• 3977.745
• 4211.730
• 4679.700
• 4913.685
• 5381.655
• 5615.640
• 6083.610
• 6317.595
• 6785.565
• 7019.550
• 7487.520
• 7721.505
• 8189.475
• 8423.460

Quest (this is the original/default tuning)

• 545.965
• 701.955
• 1247.920
• 1403.910
• 1949.875
• 2105.865
• 2651.830
• 2807.820
• 3353.785
• 3509.775
• 4055.740
• 4211.730
• 4757.695
• 4913.685
• 5459.650
• 5615.640
• 6161.605
• 6317.595
• 6863.560
• 7019.550
• 7565.515
• 7721.505
• 8267.470
• 8423.460

Purgatory (this is the original/default tuning)

• 623.960
• 701.955
• 1325.915
• 1403.910
• 2027.870
• 2105.865
• 2729.825
• 2807.820
• 3431.780
• 3509.775
• 4133.735
• 4211.730
• 4835.690

Molten Pelog (this is the original/default tuning)

• 155.990
• 311.980
• 701.955
• 857.945
• 1247.920

Molten Slendro (exact tuning from 3edf)

• 233.985
• 467.970
• 701.955
• 935.940
• 1169.925

Music

See Carlos Alpha #Music. Many are tuned to exactly 9edf.