11-LIMIT JUST TUNINGS

Full 11-limit tuning: 2, 3, 5, 7, 11

The most complete 11-limit just tuning employs all smaller primes.

**2, 3, 5, 7, 11**
Interval name	Numerator	/	Denominator	Decimal	Diapasons	Cents
Perfect octave	2	/	1	2.000	1.000	1200
?	49	/	25	1.960	0.971	1165
?	27	/	14	1.929	0.948	1137
?	21	/	11	1.909	0.933	1119
Maj 7th	15	/	8	1.875	0.907	1088
Neutral 7th	11	/	6	1.833	0.874	1049
Large min 7th	9	/	5	1.800	0.848	1018
Small min 7th	16	/	9	1.778	0.830	996
Aug 6th	7	/	4	1.750	0.807	969
dim 7th	12	/	7	1.714	0.778	933
Maj 6th	5	/	3	1.667	0.737	884
Neutral 6th	18	/	11	1.636	0.710	853
min 6th	8	/	5	1.600	0.678	814
?	11	/	7	1.571	0.652	782
Perfect 5th	3	/	2	1.500	0.585	702
?	22	/	15	1.467	0.553	663
?	16	/	11	1.455	0.541	649
dim 5th	10	/	7	1.429	0.515	617
Tritone	7	/	5	1.400	0.485	583
?	11	/	8	1.375	0.459	551
Perfect 4th	4	/	3	1.333	0.415	498
dim 4th	9	/	7	1.286	0.363	435
?	14	/	11	1.273	0.348	418
Maj 3rd	5	/	4	1.250	0.322	386
Neutral 3rd	11	/	9	1.222	0.290	347
min 3rd	6	/	5	1.200	0.263	316
Aug 2nd	7	/	6	1.167	0.222	267
dim 2nd	8	/	7	1.143	0.193	231
Maj 2nd	9	/	8	1.125	0.082	204
Neutral 2nd	11	/	10	1.100	0.138	165
min 2nd	15	/	14	1.071	0.100	119
?	21	/	20	1.050	0.070	84
dim 2nd	50	/	49	1.020	0.029	35
Unison	1	/	1	1.000	0.000	0

2, 3, 5, 11

**2, 3, 5, 11**
Interval name	Numerator	/	Denominator	Decimal	Diapasons	Cents
Perfect octave	2	/	1	2.000	1.000	1200
?	64	/	33	1.939	0.956	1147
dim octave	48	/	25	1.920	0.941	1129
Maj 7th	15	/	8	1.875	0.907	1088
Grave Maj 7th	50	/	27	1.852	0.889	1067
Neutral 7th	11	/	6	1.833	0.874	1049
Large min 7th	9	/	5	1.800	0.848	1018
Small min 7th	16	/	9	1.778	0.830	996
?	27	/	16	1.688	0.755	906
Maj 6th	5	/	3	1.667	0.737	884
Neutral 6th	18	/	11	1.636	0.710	853
min 6th	8	/	5	1.600	0.678	814
Perfect 5th	3	/	2	1.500	0.585	702
?	16	/	11	1.455	0.541	649
?	11	/	8	1.375	0.459	551
Perfect 4th	4	/	3	1.333	0.415	498
Maj 3rd	5	/	4	1.250	0.322	386
Neutral 3rd	11	/	9	1.222	0.290	347
min 3rd	6	/	5	1.200	0.263	316
?	25	/	22	1.136	0.184	221
Maj 2nd	9	/	8	1.125	0.082	204
Neutral 2nd	11	/	10	1.100	0.138	165
Great limma	27	/	25	1.080	0.111	133
min 2nd	16	/	15	1.067	0.093	112
Small semitone	25	/	24	1.042	0.059	71
?	33	/	32	1.031	0.044	53
Unison	1	/	1	1.000	0.000	0

Pythagorean-mediant 11-limit tuning: 2, 3, 7, 11

This tuning omits the 5-limit thirds and sixths that are staples of Western harmony, replacing them with approximations.

**2, 3, 7, 11**
Interval name	Numerator	/	Denominator	Decimal	Diapasons	Cents
Perfect octave	2	/	1	2.000	1.000	1200
?	63	/	32	1.969	0.977	1173
dim octave	27	/	14	1.929	0.948	1137
?	21	/	11	1.909	0.933	1119
Neutral 7th	11	/	6	1.833	0.874	1049
Aug 6th	7	/	4	1.750	0.807	969
dim 7th	12	/	7	1.714	0.778	933
optional	27	/	16	1.688	0.755	906
?	81	/	49	1.653	0.725	870
Neutral 6th	18	/	11	1.636	0.710	853
?	11	/	7	1.571	0.652	782
Aug 5th	14	/	9	1.556	0.637	765
Perfect 5th	3	/	2	1.500	0.585	702
?	16	/	11	1.455	0.541	649
?	11	/	8	1.375	0.459	551
Perfect 4th	4	/	3	1.333	0.415	498
Aug 3rd	21	/	16	1.313	0.392	471
dim 4th	9	/	7	1.286	0.363	435
?	14	/	11	1.273	0.348	418
Neutral 3rd	11	/	9	1.222	0.290	347
min 3rd	32	/	27	1.185	0.245	294
Aug 2nd	7	/	6	1.167	0.222	267
dim 2nd	8	/	7	1.143	0.193	231
Maj 2nd	9	/	8	1.125	0.082	204
?	12	/	11	1.091	0.126	151
?	22	/	21	1.048	0.067	81
Aug unison	28	/	27	1.037	0.052	63
?	49	/	48	1021	0.030	36
Unison	1	/	1	1.000	0.000	0

Non-circular 2, 5, 7, 11

The most notable quality of this tuning is the absence of perfect fifths and fourths, the building blocks of simple harmony and complex harmonic progression.

**2, 5, 7, 11**
Interval name	Numerator	/	Denominator	Decimal	Diapasons	Cents
Perfect octave	2	/	1	2.000	1.000	1200
?	49	/	25	1.960	0.971	1165
?	20	/	11	1.818	0.862	1035
Aug 6th	7	/	4	1.750	0.807	969
dim 7th	12	/	7	1.714	0.778	933
Neutral 6th	18	/	11	1.636	0.710	853
min 6th	8	/	5	1.600	0.678	814
?	11	/	7	1.571	0.652	782
dim 6th	49	/	32	1.531	0.615	738
?	16	/	11	1.455	0.541	649
dim 5th	10	/	7	1.429	0.515	617
Tritone	7	/	5	1.400	0.485	583
?	11	/	8	1.375	0.459	551
Aug 3rd	64	/	49	1.306	0.385	462
dim 4th	9	/	7	1.286	0.363	435
?	14	/	11	1.273	0.348	418
Maj 3rd	5	/	4	1.250	0.322	386
?	27	/	22	1.227	0.295	355
dim 2nd	8	/	7	1.143	0.193	231
Maj 2nd	28	/	25	1.120	0.163	196
Neutral 2nd	11	/	10	1.100	0.138	165
dim 2nd	50	/	49	1.020	0.029	35
Unison	1	/	1	1.000	0.000	0

Non-octave odd-prime 11-limit tuning: 3, 5, 7, 11

This tuning dispenses with octave-equivalency. Symmetry is based instead on the larger interval of the twelfth.

Western fifths, fourths, thirds, and seconds are omitted. The familiar chords of close Western harmony are omitted.

Based on only odd harmonics, this tuning may lend itself to single-reed instruments such as clarinets, that produce terraced rather than sawtoothed sound waves.

**3, 5, 7, 11**
Interval name	Numerator	/	Denominator	Decimal	Diapasons	Cents
Perfect 12th	3	/	1	3.000	1.585	1902
?	133	/	45	2.956	1.563	1876
?	77	/	27	2.852	1.512	1814
Aug 11th	25	/	9	2.778	1.474	1769
Maj 10th	63	/	25	2.520	1.333	1600
?	27	/	11	2.455	1.295	1555
min 10th (Aug 9th)	7	/	3	2.333	1.222	1467
?	25	/	11	2.273	1.184	1421
?	11	/	5	2.200	1.138	1365
min 9th	15	/	7	2.143	1.100	1319
8th (Aug 7th)	49	/	25	1.960	0.971	1165
?	21	/	11	1.909	0.933	1119
min 7th	9	/	5	1.800	0.848	1018
Maj 6th	5	/	3	1.667	0.737	884
?	11	/	7	1.571	0.652	782
Rough 5th (Dim 6th)	75	/	49	1.531	0.614	737
Tritone	7	/	5	1.400	0.485	583
?	15	/	11	1.364	0.693	537
dim 4th	9	/	7	1.286	0.363	435
Neutral 3rd	11	/	9	1.222	0.290	347
min 3rd	25	/	21	1.190	0.252	302
Great limma	27	/	25	1.080	0.111	133
min 2nd	35	/	33	1.061	0.085	102
?	65	/	63	1.032	0.045	54
Unison	1	/	1	1.000	0.000	0

2, 7, 11

**2, 7, 11**
Interval name	Numerator	/	Denominator	Decimal	Diapasons	Cents
Octave	2	/	1	2.000	1.000	1200
?	1,331	/	686	1.940	0.956	1147
Maj 7th	121	/	64	1.891	0.919	1103
?	3,773	/	2,048	1.842	0.881	1058
?	616	/	343	1.796	0.845	1014
Aug 6th	7	/	4	1.750	0.807	969
dim 7th	2,401	/	1,408	1.705	0.770	924
Maj 6th	128	/	77	1.662	0.733	880
?	1,372	/	847	1.620	0.696	835
?	11	/	7	1.571	0.652	782
?	49	/	32	1.531	0.615	738
5th	512	/	343	1.493	0.578	694
?	16	/	11	1.455	0.541	649
Tritone	343	/	242	1.417	0.503	604
?	11	/	8	1.375	0.459	551
4th	343	/	256	1.340	0.422	506
?	64	/	49	1.306	0.385	462
Maj 3rd	14	/	11	1.273	0.348	418
?	847	/	686	1.235	0.304	365
?	77	/	64	1.203	0.267	320
Aug 2nd	2,401	/	2,048	1.172	0.229	275
?	8	/	7	1.143	0.193	231
?	539	/	484	1.114	0.155	186
min 2nd	128	/	121	1.058	0.081	97
?	3,872	/	3,773	1.026	0.037	45
Unison	1	/	1	1.000	0.000	0

2, 5, 11

**2, 5, 11**
Interval name	Numerator	/	Denominator	Decimal	Diapasons	Cents
Octave	2	/	1	2.000	1.000	1200
Superfluous 7th	125	/	64	1.953	0.966	1159
min 7th	121	/	64	1.891	0.919	1103
?	5,632	/	3,025	1.862	0.897	1076
?	20	/	11	1.818	0.862	1035
Aug 6th	44	/	25	1.760	0.816	979
?	55	/	32	1.719	0.781	938
Maj 6th	1,024	/	605	1.693	0.759	911
?	1,024	/	625	1.638	0.712	855
min 6th	8	/	5	1.600	0.678	814
Grave superfluous 5th	25	/	16	1.563	0.644	773
?	3,125	/	2,048	1.526	0.610	732
5th	2,048	/	1,375	1.489	0.575	690
?	16	/	11	1.455	0.541	649
Tritone	176	/	125	1.408	0.494	592
?	11	/	8	1.375	0.459	551
?	1,375	/	1,024	1.343	0.425	510
?	160	/	121	1.322	0.403	484
?	1,331	/	1,024	1.300	0.378	454
dim 4th	32	/	25	1.280	0.356	427
Maj 3rd	5	/	4	1.250	0.322	386
?	625	/	512	1.221	0.288	345
?	605	/	512	1.182	0.241	289
Aug 2nd	64	/	55	1.164	0.219	262
?	25	/	22	1.136	0.184	221
Maj 2nd	16,384	/	14,641	1.119	0.162	195
Neutral 2nd	11	/	10	1.100	0.138	165
?	16,384	/	15,125	1.083	0.115	138
min 2nd	128	/	121	1.058	0.081	97
?	125	/	121	1.033	0.047	56
?	128	/	125	1.024	0.034	41
Unison	1	/	1	1.000	0.000	0

7, 11

**7, 11**
Interval name	Numerator	/	Denominator	Decimal	Diapasons	Cents
?	11	/	1	11.000	3.459	4151
?	14,641	/	2,401	6.098	2.608	3130
19th	214,358,881	/	40,353,607	5.312	2.409	2891
?	823,543	/	161,051	5.114	2.354	2825
min 16th	49	/	11	4.455	2.155	2586
?	1,331	/	343	3.880	1.956	2347
Maj 13th	19,487,171	/	5,764,801	3.380	1.757	2109
?	5,764,801	/	1,771,561	3.254	1.702	2043
dim 15th	343	/	121	2.835	1.503	1804
?	121	/	49	2.469	1.304	1565
?	1,771,561	/	823,543	2.151	1.105	1326
?	40,353,607	/	19,487,171	2.071	1.050	1260
?	2,401	/	1,331	1.804	0.851	1021
?	161,051	/	117,649	1.369	0.453	544
?	282,475,249	/	214,358,881	1.318	0.398	478
min 3rd	2,357,947,691	/	1,977,326,743	1.192	0.254	305
?	16,807	/	14,641	1.148	0.199	239
Unison	1	/	1	1.000	0.000	0

3, 11

**3, 11**
Interval name	Numerator	/	Denominator	Decimal	Diapasons	Cents
?	161,051	/	59,049	2.727	1.448	1737
11th	4,782,969	/	1,771,561	2.700	1.433	1719
?	27	/	11	2.455	1.295	1555
Maj 9th	14,641	/	6,561	2.232	1.158	1390
?	43,046,721	/	19,487,171	2.209	1.143	1372
?	2,357,947,691	/	1,162,261,467	2.029	1.021	1225
Octave	243	/	121	2.008	1.006	1207
?	1,331	/	729	1.826	0.869	1042
?	387,420,489	/	214,358,881	1.807	0.854	1025
?	214,358,881	/	129,140,163	1.660	0.731	877
?	2,187	/	1,331	1.643	0.716	860
5th	121	/	81	1.494	0.579	695
?	19,487,171	/	14,348,907	1.358	0.442	530
?	19,683	/	14,641	1.344	0.427	512
Neutral 3rd	11	/	9	1.222	0.290	347
?	1,771,561	/	1,594,323	1.111	0.152	182
Neutral 2nd	177,147	/	161,051	1.100	0.137	165
Unison	1	/	1	1.000	0.000	0

2, 11

**2, 11**
Interval name	Numerator	/	Denominator	Decimal	Diapasons	Cents
Octave	2	/	1	2.000	1.000	1200
Maj 7th	121	/	64	1.891	0.919	1103
?	4,294,967,296	/	2,357,947,691	1.821	0.865	1038
min 7th	14,641	/	8,192	1.787	0.838	1005
?	33,554,432	/	19,487,171	1.722	0.784	941
Maj 6th	1,771,561	/	1,048,576	1.689	0.757	908
?	262,144	/	161,051	1.628	0.703	843
min 6th	214,358,881	/	134,217,728	1.597	0.675	811
?	2,048	/	1,331	1.539	0.622	746
dim 5th	16	/	11	1.455	0.541	649
Tritone	11	/	8	1.375	0.459	551
?	1,331	/	1,024	1.300	0.378	454
Maj 3rd	268,435,456	/	214,358,881	1.252	0.325	389
?	161,051	/	131,072	1.229	0.297	357
min 3rd	2,097,152	/	1,771,561	1.184	0.243	292
?	19,487,171	/	16,777,216	1.162	0.216	259
Maj 2nd	16,384	/	14,641	1.119	0.162	195
?	2,357,947,691	/	2,147,483,648	1.098	0.135	162
min 2nd	128	/	121	1.058	0.081	97
Unison	1	/	1	1.000	0.000	0

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Last revised: 16 August 2008
visitors since 30 May 1999