Fudging

Fudging, or virtual tempering, is the use of one just intonation interval to approximate another.

Fudgers

Below are listed fudging intervals (or fudgers) which are all ratios between two 15-limit tonality diamond intervals (listed in the fifth column) which come within a comma of less than eight cents (listed in the fourth column) of a 15-limit tonality diamond interval (listed in the third column). If the comma is less than 1, the fudger is flat of the interval it approximates; if greater than 1, sharp of it.

Up-fudge

fudger	cents	approximately	comma(s)	(example) intervals	name
242/225	126.098	15/14, 14/13	3388/3375, 1573/1575	[15/11 22/15]
27/25	133.238	14/13, 13/12	351/350, 324/325	[10/9 6/5]	large limma
121/112	133.810	14/13, 13/12	1573/1568, 363/364	[14/11 11/8]
225/208	136.010	14/13, 13/12	225/224, 675/676	[16/15 15/13]
49/45	147.428	12/11	539/540	[15/14 7/6]	swetismic neutral second
35/32	155.140	12/11	385/384	[16/15 7/6]	septimal neutral second
54/49	168.213	11/10	540/539	[7/6 9/7]	Zalzal's mujannab
121/105	245.541	15/13	1573/1575	[14/11 22/15]
140/121	252.504	15/13	364/363	[11/10 14/11]
81/70	252.680	15/13	351/350	[10/9 9/7]
75/64	274.582	7/6	225/224	[16/15 5/4]
33/28	284.447	13/11	363/364	[12/11 9/7]
32/27	294.135	13/11	352/351	[9/8 4/3]
128/105	342.905	11/9	384/385	[7/4 32/15]
60/49	350.617	11/9	540/539	[7/6 10/7]
49/40	351.338	11/9	441/440	[8/7 7/5]
100/81	364.807	16/13	325/324	[9/5 20/9]
121/98	364.984	16/13	1573/1568	[14/11 11/7]
80/63	413.578	14/11	440/441	[9/8 10/7]
32/25	427.373	9/7	224/225	[5/4 8/5]
35/27	449.275	13/10	350/351	[6/5 14/9]
100/77	452.484	13/10	1000/1001	[11/10 10/7]
49/36	533.742	15/11	539/540	[8/7 14/9]
308/225	543.606	15/11, 11/8	3388/3375, 224/225	[15/14 22/15]
48/35	546.815	11/8	384/385	[7/6 8/5]
112/81	561.006	18/13	728/729	[9/8 14/9]
168/121	568.145	18/13	364/363	[11/7 24/11]
25/18	568.717	18/13	325/324	[6/5 5/3]
36/25	631.283	13/9	324/325	[10/9 8/5]
121/84	631.855	13/9	363/364	[12/11 11/7]
81/56	638.994	13/9	729/728	[14/9 9/4]
35/24	653.185	16/11	385/384	[16/15 14/9]
143/98	654.194	16/11	1573/1568	[14/13 11/7]
225/154	656.394	16/11, 22/15	225/224, 3375/3388	[22/15 15/7]
77/50	747.516	20/13	1001/1000	[10/7 11/5]
54/35	750.725	20/13	351/350	[10/9 12/7]
25/16	772.627	14/9	225/224	[16/15 5/3]
63/40	786.422	11/7	441/440	[10/9 7/4]
196/121	835.016	13/8	1568/1573	[11/7 28/11]
81/50	835.193	13/8	324/325	[10/9 9/5]
80/49	848.662	18/11	440/441	[7/5 16/7]
49/30	849.383	18/11	539/540	[15/14 7/4]
105/64	857.095	18/11	385/384	[16/15 7/4]
27/16	905.865	22/13	351/352	[16/15 9/5]
56/33	915.553	22/13	364/363	[15/14 20/11]
128/75	925.418	12/7	224/225	[5/4 32/15]
140/81	947.320	26/15	350/351	[9/7 20/9]
121/70	947.496	26/15	363/364	[14/11 11/5]
210/121	954.459	26/15	1575/1573	[22/15 28/11]
49/27	1031.787	20/11	539/540	[9/7 7/3]
64/35	1044.86	11/6	384/385	[7/6 32/15]
90/49	1052.572	11/6	540/539	[7/6 15/7]
416/225	1063.99	24/13, 13/7	676/675, 224/225	[15/13 32/15]
224/121	1066.19	24/13, 13/7	364/363, 1568/1573	[11/8 28/11]
50/27	1066.762	24/13, 13/7	325/324, 350/351	[6/5 20/9]
117/110	106.806	16/15	351/352	[10/9 13/11]
180/169	109.168	16/15	675/676	[13/12 15/13]
77/72	116.234	16/15, 15/14	385/384, 539/540	[12/11 7/6]	undecimal secor
28/25	196.198	9/8	224/225	[15/14 6/5]
224/195	240.030	15/13	224/225	[15/14 16/13]
52/45	250.304	15/13	676/675	[9/8 13/10]
77/60	431.875	9/7	539/540	[15/14 11/8]
110/81	529.812	15/11	242/243	[18/11 20/9]
160/117	541.876	15/11	352/351	[9/8 20/13]
117/80	658.124	22/15	351/352	[10/9 13/8]
81/55	670.188	22/15	243/242	[10/9 18/11]
120/77	768.125	14/9	540/539	[11/10 12/7]
45/26	949.696	26/15	675/676	[16/15 24/13]
195/112	959.970	26/15	225/224	[16/15 13/7]
25/14	1003.802	16/9	225/224	[6/5 15/7]
144/77	1083.766	28/15, 15/8	540/539, 384/385	[7/6 24/11]
169/90	1090.832	15/8	676/675	[15/13 13/6]
220/117	1093.194	15/8	352/351	[13/11 20/9]

Neutral/ambiguous

fudger	cents	approximately	comma(s)	(example) intervals	names
130/121	124.205	15/14, 14/13	364/363, 845/847	[11/10 13/11]
88/81	143.498	13/12, 12/11	352/351, 242/243	[9/8 11/9]	undecimal subtone
99/91	145.874	13/12, 12/11	1188/1183, 363/364	[13/11 9/7]
256/225	223.463	8/7	224/225	[15/8 32/15]
225/196	238.886	8/7	225/224	[28/15 15/7]
200/169	291.572	13/11	2200/2197	[13/10 20/13]
77/65	293.302	13/11	847/845	[13/11 7/5]
108/91	296.511	13/11	1188/1183	[13/12 9/7]
27/22	354.547	11/9, 16/13	243/242, 351/352	[10/9 15/11]
225/182	367.184	16/13	225/224	[26/15 15/7]
286/225	415.308	14/11	1573/1575	[15/13 22/15]
225/176	425.219	14/11	225/224	[16/15 15/11]
135/104	451.651	13/10	675/676	[16/15 18/13]
220/169	456.576	13/10	2200/2197	[13/11 20/13]
176/135	459.139	13/10	352/351	[9/8 22/15]
224/165	529.239	15/11	224/225	[15/14 16/11]
143/105	534.751	15/11	1573/1575	[14/13 22/15]
91/66	556.081	11/8, 18/13	364/363, 1183/1188	[11/7 13/6]
135/98	554.527	11/8	540/539	[14/9 15/7]
104/75	565.945	18/13	676/675	[15/13 8/5]
45/32	590.224	7/5	225/224	[16/15 3/2]
64/45	609.776	10/7	224/225	[9/8 8/5]
75/52	634.055	675/676	13/9	[16/15 20/13]
132/91	643.919	13/9, 16/11	1188/1183, 363/364	[13/12 11/7]
196/135	645.473	16/11	539/540	[15/14 14/9]
210/143	665.249	22/15	1575/1573	[22/15 28/13]
165/112	670.761	22/15	225/224	[16/15 11/7]
135/88	740.861	20/13	351/352	[16/15 18/11]
169/110	743.424	20/13	2197/2200	[20/13 26/11]
208/135	748.349	20/13	676/675	[9/8 26/15]
352/225	774.781	11/7	224/225	[15/11 32/15]
225/143	784.692	11/7	1575/1573	[22/15 30/13]
364/225	832.816	13/8	224/225	[15/14 26/15]
44/27	845.453	13/8, 18/11	352/351, 242/243	[12/11 16/9]
91/54	903.489	22/13	1183/1188	[9/7 13/6]
130/77	906.698	22/13	845/847	[14/13 20/11]
169/100	908.428	22/13	2197/2200	[20/13 13/5]
392/225	961.114	7/4	224/225	[15/14 28/15]
225/128	976.537	7/4	225/224	[16/15 15/8]
81/44	1056.502	11/6, 24/13	243/242, 351/352	[11/9 9/4]
225/121	1073.902	13/7, 28/15	1575/1573, 3375/3388	[22/15 30/11]
121/65	1075.795	13/7, 28/15	847/845, 363/364	[13/11 11/5]

Down-fudge

fudger	cents	approximately	commas	(example) intervals
128/117	155.562	12/11	352/351	[9/8 16/13]
169/154	160.911	11/10	845/847	[14/13 13/11]
100/91	163.274	11/10	1000/1001	[13/10 10/7]
182/165	169.767	11/10	364/363	[15/14 13/11]
72/65	177.069	10/9	324/325	[13/12 6/5]
195/176	177.478	10/9	351/352	[16/15 13/11]
49/44	186.334	10/9	441/440	[8/7 14/11]
39/35	187.343	10/9	351/350	[14/13 6/5]
135/121	189.543	10/9	243/242	[11/9 15/11]
121/108	196.771	9/8	242/243	[12/11 11/9]
55/49	199.980	9/8	440/441	[14/11 10/7]
91/81	201.534	9/8	728/729	[9/7 13/9]
169/150	206.473	9/8	676/675	[15/13 13/10]
44/39	208.835	9/8	352/351	[13/12 11/9]
154/135	227.965	8/7	539/540	[15/14 11/9]
63/55	235.104	8/7	441/440	[10/9 14/11]
55/48	235.677	8/7	385/384	[16/15 11/9]
121/104	262.108	7/6	363/364	[13/11 11/8]
64/55	262.368	7/6	384/385	[5/4 16/11]
90/77	270.080	7/6	540/539	[11/10 9/7]
117/100	271.810	7/6	351/350	[10/9 13/10]
198/169	274.173	7/6	1188/1183	[13/11 18/13]
140/117	310.702	6/5	350/351	[9/7 20/13]
77/64	320.144	6/5	385/384	[8/7 11/8]
65/54	320.976	6/5	325/324	[6/5 13/9]
135/112	323.353	6/5	225/224	[16/15 9/7]
39/32	342.483	11/9	351/352	[16/15 13/10]
56/45	378.602	5/4	224/225	[15/14 4/3]
81/65	380.979	5/4	324/325	[10/9 18/13]
96/77	381.811	5/4	384/385	[7/6 16/11]
169/135	388.877	5/4	676/675	[15/13 13/9]
33/26	412.745	14/11	363/364	[13/12 11/8]
143/112	423.020	14/11	1573/1568	[14/13 11/8]
216/169	424.810	14/11	1188/1183	[13/12 18/13]
169/132	427.782	9/7	1183/1188	[22/13 13/6]
50/39	430.145	9/7	350/351	[6/5 20/13]
104/81	432.708	9/7	728/729	[9/8 13/9]
165/128	439.587	9/7	385/384	[16/15 11/8]
156/121	439.847	9/7	364/363	[22/13 24/11]
117/88	493.120	4/3	351/352	[11/9 13/8]
121/91	493.282	4/3	363/364	[13/11 11/7]
225/169	495.482	4/3	675/676	[26/15 30/13]
162/121	505.184	4/3	243/242	[11/9 18/11]
75/56	505.757	4/3	225/224	[16/15 10/7]
196/143	545.806	11/8	1568/1573	[11/7 28/13]
169/121	578.419	7/5	845/847	[22/13 26/11]
88/63	578.582	7/5	440/441	[9/8 11/7]
200/143	580.782	7/5	1000/1001	[11/10 20/13]
108/77	585.721	7/5	540/539	[7/6 18/11]
77/54	614.279	10/7	539/540	[12/11 14/9]
143/100	619.218	10/7	1001/1000	[20/13 11/5]
63/44	621.418	10/7	441/440	[8/7 18/11]
242/169	621.581	10/7	847/845	[13/11 22/13]
72/49	666.258	22/15	540/539	[7/6 12/7]
112/75	694.243	3/2	224/225	[15/14 8/5]
121/81	694.816	3/2	242/243	[18/11 22/9]
338/225	704.518	3/2	676/675	[15/13 26/15]
182/121	706.718	3/2	364/363	[11/7 26/11]
176/117	706.880	3/2	352/351	[9/8 22/13]
121/78	760.153	14/9	363/364	[12/11 22/13]
256/165	760.413	14/9	384/385	[11/8 32/15]
81/52	767.292	14/9	729/728	[13/9 9/4]
39/25	769.855	14/9	351/350	[10/9 26/15]
264/169	772.218	14/9	1188/1183	[13/12 22/13]
169/108	775.190	11/7	1183/1188	[18/13 13/6]
224/143	776.980	11/7	1568/1573	[11/8 28/13]
52/33	787.255	11/7	364/363	[11/10 26/15]
270/169	811.123	8/5	675/676	[13/9 30/13]
77/48	818.189	8/5	385/384	[12/11 7/4]
130/81	819.021	8/5	325/324	[18/13 20/9]
45/28	821.398	8/5	225/224	[16/15 12/7]
64/39	857.517	18/11	352/351	[13/12 16/9]
224/135	876.647	5/3	224/225	[15/14 16/9]
108/65	879.024	5/3	324/325	[13/12 9/5]
128/77	879.856	5/3	384/385	[11/8 16/7]
117/70	889.298	5/3	351/350	[14/13 9/5]
169/99	925.827	12/7	1183/1188	[18/13 26/11]
200/117	928.190	12/7	350/351	[13/10 20/9]
77/45	929.920	12/7	539/540	[15/14 11/6]
55/32	937.632	12/7	385/384	[16/15 11/6]
208/121	937.892	12/7	364/363	[11/8 26/11]
96/55	964.323	7/4	384/385	[11/9 32/15]
110/63	964.896	7/4	440/441	[14/11 20/9]
135/77	972.035	7/4	540/539	[11/9 15/7]
39/22	991.165	16/9	351/352	[11/9 13/6]
300/169	993.527	16/9	675/676	[13/10 30/13]
162/91	998.466	16/9	729/728	[13/9 18/7]
98/55	1000.02	16/9	441/440	[10/7 28/11]
216/121	1003.229	16/9	243/242	[11/9 24/11]
242/135	1010.457	9/5	242/243	[15/11 22/9]
70/39	1012.657	9/5	350/351	[6/5 28/13]
88/49	1013.666	9/5	440/441	[14/11 16/7]
352/195	1022.522	9/5	352/351	[13/11 32/15]
65/36	1022.931	9/5	325/324	[6/5 13/6]
165/91	1030.233	20/11	363/364	[13/11 15/7]
91/50	1036.726	20/11	1001/1000	[10/7 13/5]
308/169	1039.089	20/11	847/845	[13/11 28/13]
117/64	1044.438	11/6	351/352	[16/13 9/4]
182/99	1054.126	11/6, 24/13	364/363, 1183/1188	[9/7 26/11]

Limit-raising and limit-lowering fudgers

A limit-raising fudger is a p prime limit interval which approximates to a q prime limit consonance, with p<q. An example is 100/77, which is 1001/1000 (1.7 cents) flat of 13/10, and which arises in 11-limit scales as the interval between 11/10 and 10/7, giving 13-limit harmony "for free", so to speak.

A limit-lowering fudger is an interval such as the marvelous fourth, 75/56, which is 225/224 (7.7 cents) sharp of 4/3, and which arises very often in 7-limit JI scales as the interval between 16/15 and 10/7, 7/5 and 15/8, 8/5 and 15/7, and 28/15 and 5/2, giving a 3-limit interval approximated in the 7-limit.

Tempering fudging commas

By tempering out the fudging commas, the error in the fudging may be distributed evenly. Since 225/224, 385/384 and 540/539 occur so commonly as fudging commas, marvel tempering in particular is often an excellent means to introduce smooth 7- and 11-limit harmonies into 5- or 7- limit scales. Adding 441/440 to that list results in miracle tempering, another excellent smoothing option.

However, indiscriminate tempering can lead to problems. For example, 112/75, 121/81, 338/225, 182/121 and 176/117 are all fudged fifths, with commas 676/675, 364/363, 352/351, 243/242 and 225/224. Tempering them all out leads to 34d tempering, a rather crude system by comparison.