13-LIMIT JUST TUNINGS

Full 13-limit tuning: 2, 3, 5, 7, 11, 13

The most complete 13-limit just tuning employs all smaller primes.

**2, 3, 5, 7, 11, 13**
Interval name	Numerator	/	Denominator	Decimal	Diapasons	Cents
Perfect octave	2	/	1	2.000	1.000	1200
?	21	/	11	1.909	0.933	1119
Maj 7th	13	/	7	1.857	0.893	1072
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
?	22	/	13	1.692	0.759	911
Maj 6th	5	/	3	1.667	0.737	884
Neutral 6th	13	/	8	1.625	0.700	841
min 6th	8	/	5	1.600	0.678	814
Aug 5th	11	/	7	1.571	0.652	782
Perfect 5th	3	/	2	1.500	0.585	702
?	22	/	15	1.467	0.553	663
dim 5th	10	/	7	1.429	0.515	617
Tritone	7	/	5	1.400	0.485	583
?	11	/	8	1.375	0.459	551
?	27	/	20	1.350	0.433	520
Perfect 4th	4	/	3	1.333	0.415	498
dim 4th	9	/	7	1.286	0.363	435
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
?	13	/	11	1.182	0.241	289
Aug 2nd	7	/	6	1.167	0.222	267
dim 3rd	8	/	7	1.143	0.193	231
Maj 2nd	9	/	8	1.125	0.170	204
minor tone	10	/	9	1.111	0.152	182
?	21	/	20	1.050	0.070	84
Unison	1	/	1	1.000	0.000	0

2, 3, 5, 7, 13

**2, 3, 5, 7, 13**
Interval name	Numerator	/	Denominator	Decimal	Diapasons	Cents
Perfect octave	2	/	1	2.000	1.000	1200
?	25	/	13	1.923	0.943	1132
Maj 7th	13	/	7	1.857	0.893	1072
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
Pythagorean Maj 6th	27	/	16	1.688	0.755	906
Maj 6th	5	/	3	1.667	0.737	884
Neutral 6th	13	/	8	1.625	0.700	841
min 6th	8	/	5	1.600	0.678	814
Aug 5th	14	/	9	1.556	0.637	765
Perfect 5th	3	/	2	1.500	0.585	702
?	13	/	9	1.444	0.531	637
dim 5th	10	/	7	1.429	0.515	617
Tritone	7	/	5	1.400	0.485	583
?	18	/	13	1.385	0.469	563
Acute 4th	27	/	20	1.350	0.433	520
Perfect 4th	4	/	3	1.333	0.415	498
dim 4th	9	/	7	1.286	0.363	435
Maj 3rd	5	/	4	1.250	0.322	386
Neutral 3rd	16	/	13	1.231	0.300	359
min 3rd	6	/	5	1.200	0.263	316
Aug 2nd	7	/	6	1.167	0.222	267
dim 3rd	8	/	7	1.143	0.193	231
Maj 2nd	9	/	8	1.125	0.170	204
minor tone	10	/	9	1.111	0.152	182
?	13	/	12	1.083	0.115	139
?	21	/	20	1.050	0.070	84
Unison	1	/	1	1.000	0.000	0

2, 3, 5, 11, 13

**2, 3, 5, 11, 13**
Interval name	Numerator	/	Denominator	Decimal	Diapasons	Cents
Perfect octave	2	/	1	2.000	1.000	1200
?	39	/	20	1.950	0.963	1156
?	25	/	13	1.923	0.943	1132
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
?	22	/	13	1.692	0.759	911
Maj 6th	5	/	3	1.667	0.737	884
Neutral 6th	18	/	11	1.636	0.710	853
?	13	/	8	1.625	0.700	841
min 6th	8	/	5	1.600	0.678	814
Aug 5th	25	/	16	1.563	0.644	773
?	20	/	13	1.538	0.621	746
?	50	/	33	1.515	0.599	719
Perfect 5th	3	/	2	1.500	0.585	702
?	13	/	9	1.444	0.531	637
?	45	/	32	1.406	0.492	590
?	11	/	8	1.375	0.459	551
Acute 4th	27	/	20	1.350	0.433	520
Perfect 4th	4	/	3	1.333	0.415	498
Aug 3rd	13	/	10	1.300	0.379	454
dim 4th	32	/	25	1.280	0.356	427
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
?	13	/	11	1.182	0.241	289
?	15	/	13	1.154	0.206	248
Maj 2nd	9	/	8	1.125	0.170	204
minor tone	10	/	9	1.111	0.152	182
min 2nd	15	/	14	1.071	0.100	119
Aug unison	25	/	24	1.042	0.059	71
Unison	1	/	1	1.000	0.000	0

Pythagorean-mediant 13-limit tuning: 2, 3, 7, 11, 13

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

**2, 3, 7, 11, 13**
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
?	49	/	26	1.885	0.914	1097
Maj 7th	13	/	7	1.857	0.893	1072
Neutral 7th	11	/	6	1.833	0.874	1049
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
?	22	/	13	1.692	0.759	911
Neutral 6th	13	/	8	1.625	0.700	841
Aug 5th	11	/	7	1.571	0.652	782
?	32	/	21	1.524	0.608	729
Perfect 5th	3	/	2	1.500	0.585	702
?	49	/	33	1.485	0.570	684
?	13	/	9	1.444	0.531	637
dim 5th	10	/	7	1.429	0.515	617
?	11	/	8	1.375	0.459	551
Acute 4th	27	/	20	1.350	0.433	520
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
Maj 3rd	49	/	39	1.256	0.329	395
Neutral 3rd	11	/	9	1.222	0.290	347
?	13	/	11	1.182	0.241	289
Aug 2nd	7	/	6	1.167	0.222	267
dim 3rd	8	/	7	1.143	0.193	231
Maj 2nd	9	/	8	1.125	0.170	204
?	49	/	44	1.114	0.155	186
?	54	/	49	1.102	0.140	168
Neutral 2nd	12	/	11	1.091	0.126	151
min 2nd	52	/	49	1.061	0.086	103
?	22	/	21	1.048	0.067	81
?	27	/	26	1.038	0.054	65
dim 2nd	49	/	48	1.021	0.030	36
Unison	1	/	1	1.000	0.000	0

Non-circular 2, 5, 7, 11, 13

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, 13**
Interval name	Numerator	/	Denominator	Decimal	Diapasons	Cents
Perfect octave	2	/	1	2.000	1.000	1200
?	39	/	20	1.950	0.963	1156
?	25	/	13	1.923	0.943	1132
Large Maj 7th	49	/	26	1.885	0.914	1097
Small Maj 7th	13	/	7	1.857	0.893	1072
?	20	/	11	1.818	0.862	1035
min 7th	25	/	14	1.786	0.837	1004
Aug 6th	7	/	4	1.750	0.807	969
?	55	/	32	1.719	0.781	938
Maj 6th	22	/	13	1.692	0.759	911
Neutral 6th	13	/	8	1.625	0.700	841
min 6th	8	/	5	1.600	0.678	814
Aug 5th	11	/	7	1.571	0.652	782
?	20	/	13	1.538	0.621	746
5th	52	/	35	1.486	0.571	685
?	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
?	35	/	26	1.346	0.429	515
?	46	/	35	1.314	0.394	473
?	13	/	10	1.300	0.379	454
dim 4th	14	/	11	1.273	0.348	418
Maj 3rd	5	/	4	1.250	0.322	386
Neutral 3rd	16	/	13	1.231	0.300	359
?	13	/	11	1.182	0.241	289
Aug 2nd	64	/	55	1.164	0.219	262
dim 3rd	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
Great limma	27	/	25	1.080	0.111	133
Aug unison	26	/	25	1.040	0.057	68
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, 13

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, 13**
Interval name	Numerator	/	Denominator	Decimal	Diapasons	Cents
Perfect 12th	3	/	1	3.000	1.585	1902
Aug 11th	25	/	9	2.778	1.474	1769
?	35	/	13	2.692	1.429	1715
?	13	/	5	2.600	1.379	1654
?	33	/	13	2.538	1.344	1613
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
?	27	/	13	2.077	1.054	1265
?	55	/	27	2.037	1.026	1232
8th (Aug 7th)	49	/	25	1.960	0.971	1165
?	21	/	11	1.909	0.933	1119
Maj 7th	13	/	7	1.857	0.893	1072
Large min 7th	9	/	5	1.800	0.848	1018
Maj 6th	5	/	3	1.667	0.737	884
?	33	/	20	1.650	0.722	867
?	21	/	13	1.615	0.692	830
Aug 5th	11	/	7	1.571	0.652	782
5th	49	/	33	1.485	0.570	684
?	13	/	9	1.444	0.531	637
Tritone	7	/	5	1.400	0.485	583
?	15	/	11	1.364	0.447	537
4th	33	/	25	1.320	0.401	481
dim 4th	9	/	7	1.286	0.363	435
Maj 3rd	49	/	39	1.256	0.329	395
Neutral 3rd	11	/	9	1.222	0.290	347
Min 3rd	13	/	11	1.182	0.241	289
?	15	/	13	1.154	0.206	248
?	39	/	35	1.114	0.156	187
Great limma	27	/	25	1.080	0.111	133
Min 2nd	35	/	33	1.061	0.085	102
?	63	/	61	1.033	0.047	56
Unison	1	/	1	1.000	0.000	0

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Last revised: 16 August 2008
visitors since 15 June 1999