Pontiac

Pontiac is a 7-limit (and higher) temperament of the schismatic family. It is an extension of helmholtz temperament beyond the 5-limit but with the same simple chain-of-fifths structure (so that standard notation may be used). As in helmholtz temperament, 5/4 is mapped to the diminished fourth (e.g. C-F♭), and the new mapping specific to pontiac is that 7/4 is mapped to the quintuple augmented third (e.g. C-Exx#). This makes pontiac a ragismic temperament.

Immediate 11-limit extensions include helenoid (53 & 65), mapping 11/8 to -30 fifths, ponta (53 & 171), mapping 11/8 to -83 fifths, and pontic (118 & 171), mapping 11/8 to +88 fifths.

Pontiac was named by Gene Ward Smith in 2004.

Interval chain

#	Cents*	Approximate Ratios
		7-limit	17-limit Extension
			Helenoid	Ponta	Pontic
0	0.00	1/1
1	701.76	3/2
2	203.51	9/8
3	905.27	27/16	22/13
4	407.03	81/64
5	1108.78	243/128, 256/135
6	610.54	64/45
7	112.30	16/15
8	814.05	8/5
9	315.81	6/5
10	1017.57	9/5
11	519.32	27/20
12	21.08	81/80
13	722.84	243/160
14	224.59	256/225
15	926.35	128/75
16	428.11	32/25
17	1129.86	48/25
18	631.62	36/25
19	133.38	27/25
20	835.13	81/50
21	336.89	175/144	17/14	17/14	17/14
22	1038.65	175/96	20/11
23	540.40	175/128	15/11
24	42.16	128/125
25	743.92	192/125	20/13	20/13	20/13
26	245.67	144/125	15/13	15/13	15/13
27	947.43	140/81
28	449.19	35/27			22/17
29	1150.94	35/18
30	652.70	35/24	16/11
31	154.46	35/32	12/11
32	856.21	105/64	18/11
33	357.97	315/256	16/13	16/13	16/13
34	1059.73	448/243	24/13	24/13	24/13
35	561.48	112/81	18/13	18/13	18/13
36	63.24	28/27
37	765.00	14/9
38	266.75	7/6
39	968.51	7/4
40	470.27	21/16
41	1172.02	63/32
42	673.78	189/128
43	175.54	448/405
44	877.29	224/135
45	379.05	56/45
46	1080.81	28/15
47	582.56	7/5
48	84.32	21/20
49	786.08	63/40			11/7
50	287.83	189/160		13/11
51	989.59	567/320
52	491.35	896/675
53	1193.10	448/225
54	694.86	112/75
55	196.62	28/25
56	898.37	42/25
57	400.13	63/50
58	1101.89	189/100	17/9	17/9	17/9
59	603.64	567/400	17/12	17/12	17/12
60	105.40	1225/1152	17/16	17/16	17/16
61	807.16	1225/768
62	308.91	448/375
63	1010.67	224/125
64	512.43	168/125
65	14.18	126/125
66	715.94	189/125
67	217.70	245/216	17/15	17/15	17/15
68	919.45	245/144	17/10	17/10	17/10
69	421.21	245/192	14/11
70	1122.97	245/128
71	624.72	735/512, 896/625
72	126.48	672/625	14/13	14/13	14/13
73	828.24	392/243
74	329.99	98/81
75	1031.75	49/27		20/11
76	533.51	49/36		15/11
77	35.26	49/48
78	737.02	49/32
79	238.78	147/128
80	940.53	441/256
81	442.29	1323/1024, 1568/1215
82	1144.05	784/405
83	645.80	196/135		16/11
84	147.56	49/45		12/11
85	849.32	49/30		18/11
86	351.07	49/40			11/9
87	1052.83	147/80			11/6
88	554.59	441/320			11/8
89	56.34	1323/1280
90	758.10	3136/2025	17/11
91	259.86	784/675
92	961.61	392/225
93	463.37	98/75	17/13	17/13	17/13
94	1165.13	49/25
95	666.88	147/100			22/15
96	168.64	441/400			11/10
97	870.40	1323/800
98	372.15	3969/3200
99	1073.91	6272/3375
100	575.67	1568/1125
101	77.42	392/375
102	779.18	196/125
103	280.94	147/125
104	982.69	441/250
105	484.45	1323/1000
106	1186.21	1715/864
107	687.96	1715/1152
108	189.72	1715/1536
109	891.48	1715/1024, 3136/1875
110	393.23	784/625
111	1094.99	1176/625
112	596.75	343/243
113	98.50	343/324
114	800.26	343/216
115	302.02	343/288
116	1003.77	343/192
117	505.53	343/256
118	7.29	1029/1024
119	709.04	3087/2048, 4704/3125, 5488/3645
120	210.80	1372/1215
121	912.56	686/405			22/13
122	414.31	343/270		14/11
123	1116.07	343/180
124	617.83	343/240
125	119.58	343/320

* in 7-limit POTE tuning

Notation

Using pontiac can be a challenge because it defies the tradition of tertian harmony in circle-of-fifths notation. The just major triad on C is C-Fb-G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C-vE-G.

However, that which is considered sufficient to notate garibaldi may not be sufficient for pontiac when it comes to septimal and undecimal harmony, as 7/4 is a triple-up major sixth (C-^³A), which is still a lot of stacks of bending. The interval is often notated as a down-minor seventh such as in FJS and HEJI. Combination of these reasons suggests that another set of accidentals to represent 64/63, the septimal comma, or 5120/5103, the amount by which the septimal comma exceeds the syntonic comma, may be desired. Ponta, one notable extension to the 11-limit, identifies the undecimal quartertone of 33/32 by a stack of two septimal commas, and can benefit considerably from this new set of accidentals.

Tuning spectra

Helenoid

Gencom: [2 4/3; 352/351 385/384 561/560 625/624 729/728]

Gencom mapping: [⟨1 2 -1 19 -9 -10 29], ⟨0 -1 8 -39 30 33 -60]]

Eigenmonzo (Unchanged-interval)	Generator (¢)	Comments
11/10	701.5907
15/11	701.6066
11/8	701.6227
12/11	701.6335
11/9	701.6435
16/15	701.6759
14/11	701.7030	11-odd-limit minimax
22/17	701.7071
5/4	701.7108
17/14	701.7205
6/5	701.7379	5-odd-limit minimax
17/15	701.7416
18/17	701.7422
20/17	701.7447
24/17	701.7458
17/16	701.7493
15/14	701.7512
9/7	701.7544
7/5	701.7556	7-odd-limit minimax
10/9	701.7596	9-odd-limit minimax
7/6	701.7598
8/7	701.7648
17/13	701.7680	17-odd-limit minimax
14/13	701.7819	13 and 15-odd-limit minimax
16/13	701.8022
13/12	701.8067
18/13	701.8109
13/10	701.8314
15/13	701.8362
4/3	701.9550
13/11	703.5968

Ponta

Gencom: [2 4/3; 375/374 540/539 625/624 729/728 2200/2197]

Gencom mapping: [⟨1 2 -1 19 -31 -10 29], ⟨0 -1 8 -39 83 33 -60]]

Eigenmonzo (Unchanged-interval)	Generator (¢)	Comments
16/15	701.6759
5/4	701.7108
17/14	701.7205
6/5	701.7379	5-odd-limit minimax
17/15	701.7416
18/17	701.7422
20/17	701.7447
24/17	701.7458
17/16	701.7493
15/14	701.7512
9/7	701.7544
7/5	701.7556	7-odd-limit minimax
10/9	701.7596	9-odd-limit minimax
7/6	701.7598
8/7	701.7648
17/13	701.7680
22/17	701.7737	17-odd-limit minimax
14/13	701.7819
14/11	701.7829	11, 13 and 15-odd-limit minimax
13/11	701.7842
11/8	701.7914
12/11	701.7933
11/9	701.7952
11/10	701.7999
15/11	701.8020
16/13	701.8022
13/12	701.8067
18/13	701.8109
13/10	701.8314
15/13	701.8362
4/3	701.9550

Pontic

Gencom: [2 4/3; 441/440 595/594 625/624 729/728 2880/2873]

Gencom mapping: [⟨1 2 -1 19 40 -10 29], ⟨0 -1 8 -39 -88 33 -60]]

Eigenmonzo (Unchanged-interval)	Generator (¢)	Comments
22/17	701.6558
16/15	701.6759
14/11	701.6835
5/4	701.7108
11/9	701.7140
15/11	701.7163
12/11	701.7168
11/10	701.7188
11/8	701.7195	11-odd-limit minimax
17/14	701.7205
6/5	701.7379	5-odd-limit minimax
17/15	701.7416
13/11	701.7421	13, 15 and 17-odd-limit minimax
18/17	701.7422
20/17	701.7447
24/17	701.7458
17/16	701.7493
15/14	701.7512
9/7	701.7544
7/5	701.7556	7-odd-limit minimax
10/9	701.7596	9-odd-limit minimax
7/6	701.7598
8/7	701.7648
17/13	701.7680
14/13	701.7819
16/13	701.8022
13/12	701.8067
18/13	701.8109
13/10	701.8314
15/13	701.8362
4/3	701.9550