Algebraic tunings

Figures & data

Figure 1. Arithmetic sequences in the golden scales, with notes at $x=\mathrm{a\varphi }+b$ for integers $(a,b)$. The sequences are straight line segments in the $(a,b)$ plane, but curved in these coordinates. See section 5.

Figure 2. Sum-product phenomenon for the $S_N$ scale (upper), $S_{\text{β}}$ scale (middle) and Bohlen 833 scale (lower). The number of distinct sums, positive differences, and products are shown in blue, orange and green, respectively, plotted against the number of notes used. See section 7.

Table 1. Pisot units of degree 2 and 3, of magnitude less than 3, and not powers of smaller Pisot units.

Value	Minimal polynomial	$d_{\text{β}}\left(1\right)$	Name (Symbol)
1.32472	$x^3-x-1$	0.10001	Plastic(p)
1.46557	$x^3-x^2-1$	0.101	Supergolden
1.61803	$x^2-x-1$	0.11	Golden(ϕ)
1.83929	$x^3-x^2-x-1$	0.111	Tribonacci
2.20557	$x^3-2x^2-1$	0.201
2.24698	$x^3-2x^2-x+1$	$0.2\overline{01}$	Cyclotomic-7
2.41421	$x^2-2x-1$	0.21	Silver(s)
2.54682	$x^3-2x^2-x-1$	0.211
2.76929	$x^3-3x^2+x-1$	0.2201
2.83118	$x^3-2x^2-2x-1$	0.221
2.87939	$x^3-3x^2+1$	$0.22\overline{1}$	Cyclotomic-9

Note: The β-expansion of one (defined below) is $d_{\text{β}}\left(1\right)$ where the over bar denotes a periodic sequence of symbols. “Symbol” gives the notation in this paper; notation varies in the literature. Cyclotomic Pisot numbers (which also include ϕ and s) are discussed in Dettmann and Frankel (1993) and Bell and Hare (2005).

Table 2. Notes of the golden scale $S_N$.

Name	Frequency ratio	β-expansion	Lattice	Exact	Norm
α	1.00000	1	(0,1)	1	1
$\alpha \mathrm{♯}$	1.05573	1.000001	(−8, 14)	$2\sqrt{5}/{\varphi }^3$	20
$\text{β}\mathrm{♭}$	1.09017	1.00001	(5, −7)		−11
β	1.14590	1.0001	(−3, 6)	$3/{\varphi }^2$	9
γ	1.23607	1.001	(2, −2)	$2/\varphi$	−4
$\gamma \mathrm{♯}$	1.29180	1.001001	(−6, 11)		19
$\delta \mathrm{♭}$	1.32624	1.00101	(7, −10)		−19
δ	1.38197	1.01	(−1, 3)	$\sqrt{5}/\varphi$	5
$ϵ\mathrm{♭}$	1.47214	1.01001	(4, −5)		−11
ϵ	1.52786	1.0101	(−4, 8)	$4/{\varphi }^2$	16

Note: The scale $S_{\text{β}}$ consists of these apart from $\alpha \mathrm{♯}$ and $\gamma \mathrm{♯}$. The β-expansion and norm are to base ϕ. The lattice column gives $(a,b)$ for the expression $\mathrm{a\varphi }+b$. Transposing by a golden octave (factor of ϕ) flips the sign of the norms.

Table 3. Frequencies in the golden scales.

	α	$\alpha \mathrm{♯}$	$\text{β}\mathrm{♭}$	β	γ	$\gamma \mathrm{♯}$	$\delta \mathrm{♭}$	δ	$ϵ\mathrm{♭}$	ϵ
−4								6.854	7.301	7.578
−3	8.025	8.472	8.748	9.196	9.919	10.37	10.64	11.09	11.81	12.26
−2	12.98	13.71	14.16	14.88	16.05	16.78	17.22	17.94	19.12	19.84
−1	21.01	22.18	22.90	24.07	25.97	27.14	27.86	29.03	30.93	32.10
0	33.99	35.89	37.06	38.95	42.02	43.91	45.08	46.98	50.04	51.94
1	55.00	58.07	59.96	63.03	67.99	71.05	72.95	76.01	80.97	84.04
2	89.00	93.96	97.02	102	110	115	118	123	131	136
3	144	152	157	165	178	186	191	199	212	220
4	233	246	254	267	288	301	309	322	343	356
5	377	398	411	432	466	487	500	521	555	576
6	610	644	665	699	754	788	809	843	898	932
7	987	1042	1076	1131	1220	1275	1309	1364	1453	1508
8	1597	1686	1741	1830	1974	2063	2118	2207	2351	2440
9	2584	2728	2817	2961	3194	3338	3427	3571	3804	3948
10	4181	4414	4558	4791	5168	5401	5545	5778	6155	6388
11	6765	7142	7375	7752	8362	8739	8972	9349	9959	10336
12	10946	11556	11933	12543	13530	14140	14517

Note: All frequencies over 100Hz are within 0.05 of integers.

Table 4. Keyboard mapping of $S_{\text{β}}$, over the full MIDI range.

	α	$\text{β}\mathrm{♭}$	β	γ	$\delta \mathrm{♭}$	δ	$ϵ\mathrm{♭}$	ϵ
−4						C-1	D♭-1	D-1
−3	D♯-1	E-1	E♯-1	F♯-1	G-1	G♯-1	A-1	A♯-1
−2	B-1	C0	C♯0	D0	E♭0	E0	F0	F♯0
−1	G0	A♭0	A0	B♭0	C♭1	C1	D♭1	D1
0	D♯1	E1	E♯1	F♯1	G1	G♯1	A1	A♯1
1	B1	C2	C♯2	D2	E♭2	E2	F2	F♯2
2	G2	A♭2	A2	B♭2	C♭3	C3	D♭3	D3
3	D♯3	E3	E♯3	F♯3	G3	G♯3	A3	A♯3
4	B3	C4	C♯4	D4	E♭4	E4	F4	F♯4
5	G4	A♭4	A4	B♭4	C♭5	C5	D♭5	D5
6	D♯5	E5	E♯5	F♯5	G5	G♯5	A5	A♯5
7	B5	C6	C♯6	D6	E♭6	E6	F6	F♯6
8	G6	A♭6	A6	B♭6	C♭7	C7	D♭7	D7
9	D♯7	E7	E♯7	F♯7	G7	G♯7	A7	A♯7
10	B7	C8	C♯8	D8	E♭8	E8	F8	F♯8
11	G8	A♭8	A8	B♭8	C♭9	C9	D♭9	D9
12	D♯9	E9	E♯9	F♯9	G9

Note: The enharmonic equivalents are chosen so that the pitch classes $\{\alpha ,\text{β},\gamma ,\delta ,ϵ\}$ correspond to a minor scale on D♯, B or G.

Table 5. Intervals in the golden scales, in cents.

	α	$\alpha \mathrm{♯}$	$\text{β}\mathrm{♭}$	β	γ	$\gamma \mathrm{♯}$	$\delta \mathrm{♭}$	δ	$ϵ\mathrm{♭}$	ϵ
$\alpha \mathrm{♯}$	94
$\text{β}\mathrm{♭}$	149	56
β	236	142	86
γ	367	273	217	131
$\gamma \mathrm{♯}$	443	349	294	207	76
$\delta \mathrm{♭}$	489	395	339	253	122	46
δ	560	466	411	324	193	117	71
$ϵ\mathrm{♭}$	669	576	520	434	303	226	181	109
ϵ	734	640	584	498	367	291	245	174	64
α	833	739	684	597	466	390	344	273	164	99
$\alpha \mathrm{♯}$	927	833	778	691	560	484	438	367	257	193
$\text{β}\mathrm{♭}$	983	889	833	747	616	539	494	422	313	249
β	1069	975	919	833	702	626	580	509	399	335
γ	1200	1106	1051	964	833	757	711	640	531	466
$\gamma \mathrm{♯}$	1276	1182	1127	1041	909	833	788	716	607	543
$\delta \mathrm{♭}$	1322	1228	1172	1086	955	879	833	762	652	588
δ	1393	1299	1244	1157	1026	950	904	833	724	659
$ϵ\mathrm{♭}$	1503	1409	1353	1267	1136	1059	1014	943	833	769
ϵ	1567	1473	1417	1331	1200	1124	1078	1007	897	833
α	1666	1572	1517	1430	1299	1223	1177	1106	997	932
$\alpha \mathrm{♯}$	1760	1666	1611	1524	1393	1317	1271	1200	1091	1026
$\text{β}\mathrm{♭}$	1816	1722	1666	1580	1449	1372	1327	1256	1146	1082
β	1902	1808	1752	1666	1535	1459	1413	1342	1232	1168
γ	2033	1939	1884	1797	1666	1590	1544	1473	1364	1299
$\gamma \mathrm{♯}$	2109	2016	1960	1874	1743	1666	1621	1549	1440	1376
$\delta \mathrm{♭}$	2155	2061	2006	1919	1788	1712	1666	1595	1485	1421
δ	2226	2132	2077	1990	1859	1783	1737	1666	1557	1492
$ϵ\mathrm{♭}$	2336	2242	2186	2100	1969	1892	1847	1776	1666	1602
ϵ	2400	2306	2251	2164	2033	1957	1911	1840	1731	1666
α	2499	2405	2350	2263	2132	2056	2010	1939	1830	1765
$\alpha \mathrm{♯}$		2499	2444	2357	2226	2150	2104	2033	1924	1859
$\text{β}\mathrm{♭}$			2499	2413	2282	2205	2160	2089	1979	1915
β				2499	2368	2292	2246	2175	2066	2001
γ					2499	2423	2377	2306	2197	2132
$\gamma \mathrm{♯}$						2499	2454	2382	2273	2209
$\delta \mathrm{♭}$							2499	2428	2319	2254
δ								2499	2390	2326
$ϵ\mathrm{♭}$									2499	2435
ϵ										2499

Dettmann, C. P., and N. E. Frankel. 1993. “Structure Factor of Deterministic Fractals with Rotations.” Fractals 1 (2): 253–261. https://doi.org/10.1142/S0218348X93000265.

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Bell, Jason P., and Kevin G. Hare. 2005. “A Classification of (Some) Pisot-Cyclotomic Numbers.” Journal of Number Theory 115 (2): 215–229. https://doi.org/10.1016/j.jnt.2004.11.009.

 Web of Science ®Google Scholar