7-LIMIT JUST TUNINGS

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

The most complete 7-limit just tuning employs all smaller primes.

**2, 3, 5, 7**
Interval name	Numerator	/	Denominator	Decimal	Diapasons	Cents
Perfect octave	2	/	1	2.000	1.000	1200
?	27	/	14	1.929	0.948	1137
?	40	/	21	1.905	0.930	1116
Maj 7th	15	/	8	1.875	0.907	1088
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	80	/	49	1.633	0.707	849
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
dim 5th	10	/	7	1.429	0.515	617
Tritone	7	/	5	1.400	0.485	583
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	49	/	40	1.225	0.293	351
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
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, 7

This tuning omits the 5-limit thirds and sixths that are staples of Western harmony.

This tuning makes available the ratio 64/63, a 27-cent interval from unison. This small interval from unison is unusual among just tunings.

**2, 3, 7**
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
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
optional	27	/	16	1.688	0.755	906
?	81	/	49	1.653	0.725	870
Aug 5th	14	/	9	1.556	0.637	765
Perfect 5th	3	/	2	1.500	0.585	702
dim 5th	81	/	56	1.446	0.532	639
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
optional	81	/	64	1.266	0.340	408
min 3rd	32	/	27	1.185	0.245	294
Aug 2nd	7	/	6	1.167	0.222	267
dim 3rd	8	/	7	1.143	0.193	231
Maj 2nd	9	/	8	1.125	0.082	204
Neutral 2nd	35	/	32	1.094	0.129	155
Aug unison	28	/	27	1.037	0.052	63
?	64	/	63	1.016	0.023	27
Unison	1	/	1	1.000	0.000	0

Non-circular 2, 5, 7

This tuning omits the 3-limit fourths and fifths that are staples of harmony, leaving no close equivalents.

**2, 5, 7**
Interval name	Numerator	/	Denominator	Decimal	Diapasons	Cents
Perfect octave	2	/	1	2.000	1.000	1200
Aug 7th	49	/	25	1.960	0.971	1165
Neutral 7th	64	/	35	1.829	0.871	1045
Aug 6th	7	/	4	1.750	0.807	969
Neutral 6th	80	/	49	1.633	0.707	849
min 6th	8	/	5	1.600	0.678	814
Grave superfluous 5th	25	/	16	1.563	0.644	773
dim 6th	49	/	32	1.531	0.615	738
dim 5th	10	/	7	1.429	0.515	617
Tritone	7	/	5	1.400	0.485	583
Aug 3rd	64	/	49	1.306	0.385	462
dim 4th	32	/	25	1.280	0.356	427
Maj 3rd	5	/	4	1.250	0.322	386
Neutral 3rd	49	/	40	1.225	0.293	351
dim 3rd	8	/	7	1.143	0.193	231
Maj 2nd	28	/	25	1.120	0.163	196
Neutral 2nd	35	/	32	1.094	0.129	155
dim 2nd	50	/	49	1.020	0.029	35
Unison	1	/	1	1.000	0.000	0

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

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.

The consistent distance between adjacent pitches is an interesting characteristic. The largest such interval is only 27% larger than the smallest. Additional intervals fall so close to those already on the chart as to be superfluous.

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**
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
Maj 10th	63	/	25	2.520	1.333	1600
min 10th (Aug 9th)	7	/	3	2.333	1.222	1467
min 9th	15	/	7	2.143	1.100	1319
8th (Aug 7th)	49	/	25	1.960	0.971	1165
min 7th	9	/	5	1.800	0.848	1018
Maj 6th	5	/	3	1.667	0.737	884
Rough 5th (Dim 6th)	75	/	49	1.531	0.614	737
Aug 4th (Tritone)	7	/	5	1.400	0.485	583
dim 4th	9	/	7	1.286	0.363	435
min 3rd	25	/	21	1.190	0.252	302
Great limma	27	/	25	1.080	0.111	133
Unison	1	/	1	1.000	0.000	0

3, 7

Symmetry is based on interval of the twelfth.

Most Western intervals and chords are omitted.

**3, 7**
Interval name	Numerator	/	Denominator	Decimal	Diapasons	Cents
Perfect 12th	3	/	1	3.000	1.585	1902
Aug 11th	40,353,607	/	14,348,907	2.812	1.492	1790
?	6,561	/	2,401	2.733	1.450	1740
?	16,807	/	6,561	2.562	1.357	1628
?	14,348,907	/	5,764,801	2.489	1.316	1579
?	7	/	3	2.333	1.222	1467
?	282,475,249	/	129,140,163	2.187	1.129	1355
min 9th	729	/	343	2.125	1.088	1305
8th	117,649	/	59,049	1.992	0.995	1193
?	1,594,323	/	823,543	1.936	0.953	1144
?	49	/	27	1.815	0.860	1032
?	3,486,784,401	/	1,977,326,743	1.763	0.818	982
?	1,977,326,743	/	1,162,261,467	1.701	0.767	920
?	81	/	49	1.653	0.725	870
?	823,543	/	531,441	1.550	0.632	758
5th	177,147	/	117,649	1.506	0.590	709
Tritone	343	/	243	1.412	0.497	597
?	387,420,489	/	282,475,249	1.372	0.456	547
?	9	/	7	1.286	0.363	435
?	5,764,801	/	4,782,969	1.205	0.269	323
?	19,683	/	16,807	1.171	0.228	273
?	2,401	/	2,187	1.098	0.135	162
?	43,046,721	/	40,353,607	1.067	0.093	112
Unison	1	/	1	1.000	0.000	0

5, 7

Symmetry is based on interval of the seventeenth.

Most Western intervals and chords are omitted.

**5, 7**
Interval name	Numerator	/	Denominator	Decimal	Diapasons	Cents
?	5	/	1	5.000	2.322	2786
?	78,125	/	16,807	4.648	2.217	2660
?	1,220,703,125	/	282,475,249	4.321	2.112	2534
?	40,353,607	/	9,765,625	4.132	2.047	2456
?	2,401	/	625	3.842	1.942	2330
min 7th	25	/	7	3.571	1.837	2204
?	390,625	/	117,649	3.320	1.731	2078
?	6,103,515,625	/	1,977,326,743	3.087	1.626	1951
?	5,764,801	/	1,953,125	2.952	1.561	1874
?	343	/	125	2.744	1.456	1748
?	125	/	49	2.551	1.351	1621
min 10th	1,953,125	/	823,543	2.372	1.246	1495
min 9th	823,543	/	390,625	2.108	1.076	1291
?	49	/	25	1.960	0.971	1165
?	625	/	343	1.822	0.866	1039
Maj 6th	9,765,625	/	5,764,801	1.694	0.760	913
?	1,977,326,743	/	1,220,703,125	1.620	0.696	835
5th	117,649	/	78,125	1.506	0.591	709
?	7	/	5	1.400	0.485	583
?	3,125	/	2,401	1.302	0.380	456
?	48,828,125	/	40,353,607	1.210	0.275	330
?	282,475,249	/	244,140,625	1.157	0.210	252
?	16,807	/	15,625	1.076	0.105	126
Unison	1	/	1	1.000	0.000	0

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Last revised: 16 August 2008
visitors since 30 May 1999