Fudging

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

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.

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.

Fudging Intervals

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]