User:Aura/On 159edo Music Theory (Part 1)

Of all the multiples of 53edo, 159edo is the lowest multiple that is noteworthy for being accurate in the 2.3.5.11.17 subgroup while having structural compromises in the 7.13.19.23.29 subgroup. Despite the number of pitches in this tuning system making it perhaps best fit for digital instruments of various kinds in actual performance, it is nevertheless also useful as an interval classification scheme.

Intervals and Notation

This page or section deals with proposed concepts. The terminology and concepts used in it are developed by one person or a small group and may lack widespread adoption.

159edo contains all the intervals of 53edo and can be thought of as having three fields of 53edo each separated by a third of 53edo's step, however, the additional harmonies need naming, especially for those who seek to use 159edo as a means of extending the theory and praxis of Medieval, Neo-Medieval, and traditional Western Classical music.

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159edo also has its own variation on the dinner party rules— represented here by the Harmonic Compatibility Rating and Melodic Compatibility Rating columns in the following chart, where 10 is a full-blown friend relative to the root and −10 if a full-blown enemy relative to the root. Note that the Harmonic Compatibility and Melodic Compatibility ratings are based on octave-equivalence, and that some of the ratings are still speculative.

159edo Interval Names and Compatibility Ratings

Step	Cents	Interval and Note names			Compatibility rating
		SKULO-based interval names	Pythagorean-commatic-based interval names	SRS notation	Harmonic	Melodic
0	0	P1	Perfect Unison	D	10	10
1	7.5471698	R1	Wide Unison	D/	0	0
2	15.0943396	rK1	Narrow Superunison	D↑\	-10	-10
3	22.6415094	K1	Lesser Superunison	D↑	-10	-3
4	30.1886792	S1, kU1	Greater Superunison, Narrow Inframinor Second	Edb<, Dt<↓	-10	3
5	37.7358491	um2, RkU1	Inframinor Second, Wide Superunison	Edb>, Dt>↓	-9	10
6	45.2830189	kkm2, Rum2, rU1	Wide Inframinor Second, Narrow Ultraunison	Eb↓↓, Dt<\	-9	10
7	52.8301887	U1, rKum2	Ultraunison, Narrow Subminor Second	Dt<, Edb<↑	-9	10
8	60.3773585	sm2, Kum2, uA1	Lesser Subminor Second, Wide Ultraunison, Infra-Augmented Unison	Dt>, Eb↓\	-8	10
9	67.9245283	km2, RuA1, kkA1	Greater Subminor Second, Diptolemaic Augmented Unison	Eb↓, D#↓↓	-8	9
10	75.4716981	Rkm2, rKuA1	Wide Subminor Second, Lesser Sub-Augmented Unison	Eb↓/, Dt<↑	-7	9
11	83.0188679	rm2, KuA1	Narrow Minor Second, Greater Sub-Augmented Unison	Eb\, Dt>↑	-7	9
12	90.5660377	m2, kA1	Pythagorean Minor Second, Ptolemaic Augmented Unison	Eb, D#↓	-6	10
13	98.1132075	Rm2, RkA1	Artomean Minor Second, Artomean Augmented Unison	Eb/, D#↓/	-6	10
14	105.6603774	rKm2, rA1	Tendomean Minor Second, Tendomean Augmented Unison	D#\, Eb↑\	-5	10
15	113.2075472	Km2, A1	Ptolemaic Minor Second, Pythagorean Augmented Unison	D#, Eb↑	-5	10
16	120.7547170	RKm2, kn2, RA1	Wide Minor Second, Artoretromean Augmented Unison	Ed<↓, Eb↑/, D#/	-5	9
17	128.3018868	kN2, rKA1	Lesser Supraminor Second, Tendoretromean Augmented Unison	Ed>↓, D#↑\	-6	8
18	135.8490566	KKm2, rn2, KA1	Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Unison	Ed<\, Eb↑↑, D#↑	-7	6
19	143.3962264	n2, SA1	Artoneutral Second, Lesser Super-Augmented Unison	Ed<, Dt#<↓	-8	5
20	150.9433962	N2, RkUA1	Tendoneutral Second, Greater Super-Augmented Unison	Ed>, Dt#>↓	-7	6
21	158.4905660	kkM2, RN2, rUA1	Lesser Submajor Second, Retrodiptolemaic Augmented Unison	Ed>/, E↓↓, Dt#>↓/, D#↑↑	-6	8
22	166.0377358	Kn2, UA1	Greater Submajor Second, Ultra-Augmented Unison	Ed<↑, Dt#<, Fb↓/	-5	9
23	173.5849057	rkM2, KN2	Narrow Major Second	Ed>↑, E↓\, Dt#>, Fb\	-4	10
24	181.1320755	kM2	Ptolemaic Major Second	E↓, Fb	-3	10
25	188.6792458	RkM2	Artomean Major Second	E↓/, Fb/	-3	10
26	196.2264151	rM2	Tendomean Major Second	E\, Fb↑\	-2	10
27	203.7735849	M2	Pythagorean Major Second	E, Fb↑	-2	10
28	211.3207547	RM2	Wide Major Second	E/, Fd<↓	-1	10
29	218.8679245	rKM2	Narrow Supermajor Second	E↑\, Fd>↓	-1	10
30	226.4150943	KM2	Lesser Supermajor Second	E↑, Fd<\, Fb↑↑, Dx	-1	9
31	233.9622642	SM2, kUM2	Greater Supermajor Second, Narrow Inframinor Third	Fd<, Et<↓, E↑/	0	9
32	241.5094340	um3, RkUM2	Inframinor Third, Wide Supermajor Second	Fd>, Et>↓	-1	8
33	249.0566038	kkm3, KKM2, Rum3, rUM2	Wide Inframinor Third, Narrow Ultramajor Second, Semifourth	Fd>/, Et<\, F↓↓, E↑↑	0	8
34	256.6037736	UM2, rKum3	Ultramajor Second, Narrow Subminor Third	Et<, Fd<↑	-1	7
35	264.1509434	sm3, Kum3	Lesser Subminor Third, Wide Ultramajor Second	Et>, Fd>↑, F↓\	0	7
36	271.6981132	km3	Greater Subminor Third	F↓, Et>/, E#↓↓, Gbb	-1	7
37	279.2452830	Rkm3	Wide Subminor Third	F↓/, Et<↑	-1	8
38	286.7924528	rm3	Narrow Minor Third	F\, Et>↑	0	8
39	294.3396226	m3	Pythagorean Minor Third	F	-1	9
40	301.8867925	Rm3	Artomean Minor Third	F/	1	9
41	309.4339622	rKm3	Tendomean Minor Third	F↑\	4	10
42	316.9811321	Km3	Ptolemaic Minor Third	F↑, E#	7	10
43	324.5283019	RKm3, kn3	Wide Minor Third	Ft<↓, F↑/, Gdb<	4	9
44	332.0754717	kN3, ud4	Lesser Supraminor Third, Infra-Diminished Fourth	Ft>↓, Gdb>	1	9
45	339.6226415	KKm3, rn3, Rud4	Greater Supraminor Third, Retrodiptolemaic Diminished Fourth	Ft<\, F↑↑, Gdb<↑\, Gb↓↓	-1	8
46	347.1698113	n3, rKud4	Artoneutral Third, Lesser Sub-Diminished Fourth	Ft<, Gdb<↑	0	7
47	354.7169811	N3, sd4, Kud4	Tendoneutral Third, Greater Sub-Diminished Fourth	Ft>, Gdb>↑	-1	7
48	362.2641509	kkM3, RN3, kd4	Lesser Submajor Third, Retroptolemaic Diminished Fourth	Ft>/, F#↓↓, Gb↓	0	8
49	369.8113208	Kn3, Rkd4	Greater Submajor Third, Artoretromean Diminished Fourth	Ft<↑, Gb↓/	-1	9
50	377.3584906	rkM3, KN3, rd4	Narrow Major Third, Tendoretromean Diminished Fourth	Ft>↑, F#↓\, Gb\	3	9
51	384.9056604	kM3, d4	Ptolemaic Major Third, Pythagorean Diminished Fourth	Gb, F#↓	8	10
52	392.4528302	RkM3, Rd4	Artomean Major Third, Artomean Diminished Fourth	Gb/, F#↓/	4	10
53	400	rM3, rKd4	Tendomean Major Third, Tendomean Diminished Fourth	F#\, Gb↑\	1	9
54	407.5471698	M3, Kd4	Pythagorean Major Third, Ptolemaic Diminished Fourth	F#, Gb↑	-1	9
55	415.0943396	RM3, kUd4	Wide Major Third, Lesser Super-Diminished Fourth	F#/, Gd<↓, Gb↑/	0	8
56	422.6415094	rKM3, RkUd4	Narrow Supermajor Third, Greater Super-Diminished Fourth	F#↑\, Gd>↓	-1	7
57	430.1886792	KM3, rUd4, KKd4	Lesser Supermajor Third, Diptolemaic Diminished Fourth	F#↑, Gd<\, Gb↑↑	-1	6
58	437.7358491	SM3, kUM3, rm4, Ud4	Greater Supermajor Third, Ultra-Diminished Fourth	Gd<, F#↑/	0	5
59	445.2830189	m4, RkUM3	Paraminor Fourth, Wide Supermajor Third	Gd>, Ft#>↓	-1	3
60	452.8301887	Rm4, KKM3, rUM3	Wide Paraminor Fourth, Narrow Ultramajor Third	Gd>/, F#↑↑, G↓↓	-2	1
61	460.3773585	UM3, rKm4	Ultramajor Third, Narrow Grave Fourth	Gd<↑, Ft#<	-4	-2
62	467.9245283	s4, Km4	Lesser Grave Fourth, Wide Ultramajor Third	Gd>↑, G↓\	-7	-4
63	475.4716981	k4	Greater Grave Fourth	G↓, Abb	-6	-5
64	483.0188679	Rk4	Wide Grave Fourth	G↓/	-4	0
65	490.5660377	r4	Narrow Fourth	G\	1	5
66	498.1132075	P4	Perfect Fourth	G	9	10
67	505.6603774	R4	Wide Fourth	G/	1	8
68	513.2075472	rK4	Narrow Acute Fourth	G↑\	-3	6
69	520.7547170	K4	Lesser Acute Fourth	G↑	-5	5
70	528.3018868	S4, kM4	Greater Acute Fourth	Gt<↓, G↑/, Adb<	-3	5
71	535.8490566	RkM4, ud5	Wide Acute Fourth, Infra-Diminished Fifth	Gt>↓, Adb>	-2	5
72	543.3962264	rM4, Rud5	Narrow Paramajor Fourth, Retrodiptolemaic Diminished Fifth	Gt<\, G↑↑, Ab↓↓	-1	6
73	550.9433962	M4, rKud5	Paramajor Fourth, Lesser Sub-Diminished Fifth	Gt<, Adb<↑	0	7
74	558.4905660	RM4, uA4, Kud5	Infra-Augmented Fourth, Greater Sub-Diminished Fifth	Gt>, Adb>↑	-2	5
75	566.0377358	kkA4, RuA4, kd5	Diptolemaic Augmented Fourth, Retroptolemaic Diminished Fifth	Gt>/, G#↓↓, Ab↓	-3	4
76	573.5849057	rKuA4, Rkd5	Lesser Sub-Augmented Fourth, Artoretromean Diminished Fifth	Gt<↑, Ab↓/	-2	4
77	581.1320755	KuA4, rd5	Greater Sub-Augmented Fourth, Tendoretromean Diminished Fifth	Gt>↑, Ab\	0	5
78	588.6792458	kA4, d5	Ptolemaic Augmented Fourth, Pythagorean Diminished Fifth	Ab, G#↓	-5	6
79	596.2264151	RkA4, Rd5	Artomean Augmented Fourth, Artomean Diminished Fifth	G#↓/, Ab/	-9	7
80	603.7735849	rKd5, rA4	Tendomean Diminished Fifth, Tendomean Augmented Fourth	Ab↑\, G#\	-9	7
81	611.3207547	Kd5, A4	Ptolemaic Diminished Fifth, Pythagorean Augmented Fourth	Ab↑, G#	-5	6
82	618.8679245	kUd5, RA4	Lesser Super-Diminished Fifth, Artoretromean Augmented Fourth	Ad<↓, G#/	0	5
83	626.4150943	RkUd5, rKA4	Greater Super-Diminished Fifth, Tendoretromean Augmented Fourth	Ad>↓, G#↑\	-2	4
84	633.9622642	KKd5, rUDd5, KA4	Diptolemaic Diminished Fifth, Retroptolemaic Augmented Fourth	Ad<\, Ab↑↑, G#↑	-3	4
85	641.5094340	rm5, Ud5, kUA4	Ultra-Diminished Fifth, Lesser Super-Augmented Fourth	Ad<, Gt#<↓	-2	5
86	649.0566038	m5, RkUA4	Paraminor Fifth, Greater Super-Augmented Fourth	Ad>, Gt#>↓	0	7
87	656.6037736	Rm5, rUA4	Wide Paraminor Fifth, Retrodiptolemaic Augmented Fourth	Ad>/, G#↑, Ab↑↑	-1	6
88	664.1509434	rKm5, UA4	Narrow Grave Fifth, Ultra-Augmented Fourth	Ad<↑, Gt#<	-2	5
89	671.6981132	s5, Km5	Lesser Grave Fifth	Ad>↑, A↓\, Gt#>	-3	5
90	679.2452830	k5	Greater Grave Fifth	A↓	-5	5
91	686.7924528	Rk5	Wide Grave Fifth	A↓/	-3	6
92	694.3396226	r5	Narrow Fifth	A\	1	8
93	701.8867925	P5	Perfect Fifth	A	9	10
94	709.4339622	R5	Wide Fifth	A/	1	5
95	716.9811321	rK5	Narrow Acute Fifth	A↑\	-4	0
96	724.5283019	K5	Lesser Acute Fifth	A↑, Gx	-6	-5
97	732.0754717	S5, kM5	Greater Acute Fifth, Narrow Inframinor Sixth	At<↓, A↑/	-7	-4
98	739.6226415	um6, RkM5	Inframinor Sixth, Wide Acute Fifth	At>↓, Bdb>	-4	-2
99	747.1698113	Rm4, KKM3, rUM3	Narrow Paramajor Fifth, Wide Inframinor Sixth	At<\, Bb↓↓, A↑↑	-2	1
100	754.7169811	M5, rKum6	Paramajor Fifth, Narrow Subminor Sixth	At<, Bdb<↑	-1	3
101	762.2641509	sm6, Kum6, RM5, uA5	Lesser Subminor Sixth, Infra-Augmented Fifth	At>, Bb↓\	0	5
102	769.8113208	km6, RuA5, kkA5	Greater Subminor Sixth, Diptolemaic Augmented Fifth	Bb↓, At>/, A#↓↓	-1	6
103	777.3584906	Rkm6, rKuA5	Wide Subminor Sixth, Lesser Sub-Augmented Fifth	Bb↓/, At<↑	-1	7
104	784.9056604	rm6, KuA5	Narrow Minor Sixth, Greater Sub-Augmented Fifth	Bb\, At>↑, A#↓\	0	8
105	792.4528302	m6, kA5	Pythagorean Minor Sixth, Ptolemaic Augmented Fifth	Bb, A#↓	-1	9
106	800	Rm6, RkA5	Artomean Minor Sixth, Artomean Augmented Fifth	Bb/, A#↓/	1	9
107	807.5471698	rKm6, rA5	Tendomean Minor Sixth, Tendomean Augmented Fifth	A#\, Bb↑\	4	10
108	815.0943396	Km6, A5	Ptolemaic Minor Sixth, Pythagorean Augmented Fifth	A#, Bb↑	8	10
109	822.6415094	RKm6, kn6, RA5	Wide Minor Sixth, Artoretromean Augmented Fifth	Bd<↓, Bb↑/, A#/	3	9
110	830.1886792	kN6, rKA5	Lesser Supraminor Sixth, Tendoretromean Augmented Fifth	Bd>↓, A#↑\	-1	9
111	837.7358491	KKm6, rn6, KA5	Greater Supraminor Sixth, Retroptolemaic Augmented Fifth	Bd<\, Bb↑↑, A#↑	0	8
112	845.2830189	n6, SA5, kUA5	Artoneutral Sixth, Lesser Super-Augmented Fifth	Bd<, At#<↓	-1	7
113	852.8301887	N6, RkUA5	Tendoneutral Sixth, Greater Super-Augmented Fifth	Bd>, At#>↓	0	7
114	860.3773585	kkM6, RN6, rUA5	Lesser Submajor Sixth, Retrodiptolemaic Augmented Fifth	Bd>/, B↓↓, At#>↓/, A#↑↑	-1	8
115	867.9245283	Kn6, UA5	Greater Submajor Sixth, Ultra-Augmented Fifth	Bd<↑, At#<	1	9
116	875.4716981	rkM6, KN6	Narrow Major Sixth	Bd>↑, B↓\, At#>	4	9
117	883.0188679	kM6	Ptolemaic Major Sixth	B↓, Cb	7	10
118	890.5660377	RkM6	Artomean Major Sixth	B↓/	4	10
119	898.1132075	rM6	Tendomean Major Sixth	B\	1	9
120	905.6603774	M6	Pythagorean Major Sixth	B	-1	9
121	913.2075472	RM6	Wide Major Sixth	B/, Cd<↓	0	8
122	920.7547170	rKM6	Narrow Supermajor Sixth	B↑\, Cd>↓	-1	8
123	928.3018868	KM6	Lesser Supermajor Sixth	B↑, Cd<\, Cb↑↑, Ax	-1	7
124	935.8490566	SM6, kUM6	Greater Supermajor Sixth, Narrow Inframinor Seventh	Cd<, Bt<↓, B↑/	0	7
125	943.3962264	um7, RkUM6	Inframinor Seventh, Wide Supermajor Sixth	Cd>, Bt>↓	-1	7
126	950.9433962	KKM6, kkm7, rUM6, Rum7	Narrow Ultramajor Sixth, Wide Inframinor Seventh, Semitwelfth	Bt<\, Cd>/, B↑↑, C↓↓	0	8
127	958.4905660	UM6, rKum7	Ultramajor Sixth, Narrow Subminor Seventh	Bt<, Cd<↑	-1	8
128	966.0377358	sm7, Kum7	Lesser Subminor Seventh, Wide Ultramajor Sixth	Bt>, Cd>↑, C↓\	0	9
129	973.5849057	km7	Greater Subminor Seventh	C↓, Bt>/, B#↓↓, Dbb	-1	9
130	981.1320755	Rkm7	Wide Subminor Seventh	C↓/, Bt<↑	-1	10
131	988.6792458	rm7	Narrow Minor Seventh	C\, Bt>↑	-1	10
132	996.2264151	m7	Pythagorean Minor Seventh	C, B#↓	-2	10
133	1003.7735849	Rm7	Artomean Minor Seventh	C/, B#↓/	-2	10
134	1011.3207547	rKm7	Tendomean Minor Seventh	C↑\, B#\	-3	10
135	1018.8679245	kM2	Ptolemaic Minor Seventh	C↑, B#	-3	10
136	1026.4150943	RKm7, kn7	Wide Minor Seventh	Ct<↓, C↑/, Ddb<, B#/	-4	10
137	1033.9622642	kN7, ud8	Lesser Supraminor Seventh, Infra-Diminished Octave	Ct>↓, Ddb>, B#↑\	-5	9
138	1041.5094340	KKm7, rn7, Rud8	Greater Supraminor Seventh, Retrodiptolemaic Diminished Octave	Ct<\, C↑↑, Ddb<↑\, Db↓↓	-6	8
139	1049.0566038	n7, rKud8	Artoneutral Seventh, Lesser Sub-Diminished Octave	Ct<, Ddb<↑	-7	6
140	1056.6037736	N7, sd8	Tendoneutral Seventh, Greater Sub-Diminished Octave	Ct>, Ddb>↑	-8	5
141	1064.1509434	kkM7, RN7, kd8	Lesser Submajor Seventh, Diptolemaic Major Seventh, Retroptolemaic Diminished Octave	Ct>/, C#↓↓, Db↓	-7	6
142	1071.6981132	Kn7, Rkd8	Greater Submajor Seventh, Artoretromean Diminished Octave	Ct<↑, Db↓/	-6	8
143	1079.2452830	rkM7, KN7, rd8	Narrow Major Seventh, Tendoretromean Diminished Octave	Ct>↑, C#↓\, Db\	-5	9
144	1086.7924528	kM7, d8	Ptolemaic Major Seventh, Pythagorean Diminished Octave	Db, C#↓	-5	10
145	1094.3396226	RkM7, Rd8	Artomean Major Seventh, Artomean Diminished Octave	Db/, C#↓/	-5	10
146	1101.8867925	rM7, rKd8	Tendomean Major Seventh, Tendomean Diminished Octave	C#\, Db↑\	-6	10
147	1109.4339622	M7, Kd8	Pythagorean Major Seventh, Ptolemaic Diminished Octave	C#, Db↑	-6	10
148	1116.9811321	RM7, kUd8	Wide Major Seventh, Lesser Super-Diminished Octave	C#/, Dd<↓	-7	9
149	1124.5283019	rKM7, RkUd8	Narrow Supermajor Seventh, Greater Super-Diminished Octave	C#↑\, Dd>↓	-7	9
150	1132.0754717	km2, RuA1, kkA1	Lesser Supermajor Seventh, Diptolemaic Diminished Octave	C#↑, Db↑↑	-8	9
151	1139.6226415	SM7, kUM7, Ud8	Greater Supermajor Seventh, Narrow Infraoctave, Ultra-Diminished Octave	Dd<, C#↑/	-8	10
152	1147.1698113	u8, RkUM7	Infraoctave, Wide Supermajor Seventh	Dd>, Ct#>↓	-9	10
153	1154.7169811	KKM7, rUM7, Ru8	Narrow Ultramajor Seventh, Wide Infraoctave	C#↑↑, Dd>/	-9	10
154	1162.2641509	UM7, rKu8	Ultramajor Seventh, Narrow Suboctave	Ct#<, Dd<↑	-9	10
155	1169.8113208	s8, Ku8	Lesser Suboctave, Wide Ultramajor Seventh	Ct#>, Dd>↑	-10	3
156	1177.3584906	k8	Greater Suboctave	D↓	-10	-3
157	1184.9056604	Rk8	Wide Suboctave	D↓/	-10	-10
158	1192.4528302	r8	Narrow Octave	D\	0	0
159	1200	P8	Perfect Octave	D	10	10

Interval Naming Terminology

The table above contains terminology that needs to be defined, and so it will be defined in this section, for cross-references, see List of interval regions and Adaptive diatonic interval names #Interval regions.

Narrow and Wide - These terms both refer to intervals that are respectively narrower or wider than the expected range of an interval class by a single EDO-step

Greater and Lesser - When there are two distinct intervals within the expected range of a given interval class, for example, two supermajor seconds, these terms will be used to distinguish the smaller and larger intervals respectively

Paramajor - Where there are multiple distinct intervals within the semiaugmented category for fourths and fifths, this term refers to the intervals just shy of half a chroma sharp of the perfect forms of these intervals; this term comes from adding the para- prefix (meaning alongside or resembling here) to Ivan Wyschnegradsky's own usage of the term major fourth in order to disambiguate it from other uses of the same term, while generalizing it to refer to fifths as well as fourths.

Paraminor - Where there are multiple distinct intervals within the semidiminished category for fourths and fifths, this term refers to the intervals just shy of half a chroma flat of the perfect forms of these intervals; this term comes from adding the para- prefix (meaning alongside or resembling here) to Ivan Wyschnegradsky's own usage of the term minor fifth in order to disambiguate it from other uses of the same term, while generalizing it to refer to fourths as well as fifths.

Infra-Augmented - When there are multiple distinct intervals within the semiaugmented category for unisons, fourths, fifths, and octaves, this term refers to the intervals just shy of half a chroma flat of the augmented forms of these intervals

Ultra-Augmented - When there are multiple distinct intervals within the sesquiaugmented category for unisons, fourths, fifths, and octaves, this term refers to the intervals just shy of half a chroma sharp of the augmented forms of these intervals

Infra-Diminished - When there are multiple distinct intervals within the sesquidiminished category for unisons, fourths, fifths, and octaves, this term refers to the intervals just shy of half a chroma flat of the diminished forms of these intervals

Ultra-Diminished - When there are multiple distinct intervals within the semidiminished category for unisons, fourths, fifths, and octaves, this term refers to the intervals just shy of half a chroma sharp of the diminished forms of these intervals

Artomean and Tendomean - When there are multiple distinct intervals situated between the Pythagorean and Ptolemaic versions of both major and minor intervals, these terms refer respectively to the narrower and wider intervals within those spaces

Usage and Functions of Specific Interval Ranges

This page or section deals with proposed concepts. The terminology and concepts used in it are developed by one person or a small group and may lack widespread adoption.

In addition to knowing the intervals in 159edo, it also pays to be able to get a general idea as to which intervals can be used for what purposes, though, for Medieval and Neo-Medieval music and their extensions, we have to begin by assuming the starting pitch class is the final, while for later Western Classical music and its extensions, we have to begin by assuming the starting pitch class is the tonic. From there, distance from the final or tonic in terms of both logarithmic distance as well as by approximated Just Intonation relations both govern the usage of 159edo's various intervals.

Au

Despite being largely derived from traditional schools of thought in the area of functional harmony, the terminology for these functions, along with the distinction between different variations of a given function, is largely unique to me at this point, though I do hope it gets adopted more in the future. I should mention that one reason for my differences in terminology in the area of functional harmony is that I have this idea that music can be hung from the high sounds downwards just as much as it can be built from the low sounds upwards, and the consideration of this angle requires the reconception of the nature of certain functions.

Of all the intervals in 159edo, 5\159 is the first interval to be larger than what is called the fission boundary as per Dave Smey's video on the topic. Though there is a caveat which shall be discussed momentarily, the basic gist is that the fission boundary is where going back and forth between notes on either end of a given interval no longer sounds like a simple vibrato, but more like a dirty trill of sorts- basically a cross between a trill and a vibrato. The fission boundary further serves as the line separating melodic notes that can only be simple ornaments or quick passing tones from main melodic intervals, and since 5\159 is larger than this boundary it is the smallest interval that can serve as a main melodic interval.

The next landmark interval is 8\159, as this is the first interval to be larger than what Aura calls the gradient threshold, which is where going back and forth between notes on either end of a given interval no longer sounds like a dirty trill, but rather a clean trill. The gradient threshold doubles as the point beyond which microtonal intervals can begin to serve as proper leading-tones. This brings up the aforementioned caveat about Dave Smey's video on the fission boundary- specifically, it should be noted that Dave Smey fails to distinguish between the fission boundary and the gradient threshold, and assumes the fission boundary to be at about 60 cents on average, rather than at about 35 cents, where it actually is.

Finally, 33\159 is the first interval to be larger than what is called the trill threshold as per Dave Smey's video on the topic. This point is where going back and forth between notes on either end of a given interval no longer sounds like any kind of trill, and instead sounds like an arpeggio fragment. The trill threshold doubles as the boundary between intervals that are classified as steps, and those that are classified as leaps. As a consequence of this, the trill threshold marks the boundary where intervals cease to cause crowding in chords.

Because both the size of 159edo intervals and their representation of Just intonation affect the functions of intervals, a list of JI intervals will be included in the chart below. Note that 159edo can only represent 5-limit intervals on every third step as it inherits 53edo's 5-limit representation, and that prime limits higher than 17 will not be shown on the chart due to inconsistencies.

159edo Interval Functions and JI Representation

Step	Function			JI Representation
	Broad-Level Function	Medium-Level Function	Narrow-Level Function	5-Limit	7-Limit	11-Limit	13-Limit	17-Limit
0	Tonic			1/1
1	Supercommatic			N/A	225/224	243/242	351/350, 352/351	256/255
2	Supercommatic			N/A	?	100/99, 121/120	144/143	120/119
3	Superdietic			81/80	?	?	78/77	85/84
4	Superdietic			N/A	64/63	55/54, 56/55	?	52/51
5	Supergradient			N/A	?	45/44	?	51/50
6	Supergradient			?	?	?	40/39	192/187
7	Supergradient			N/A	?	33/32	?	34/33
8	Supersanguinant			N/A	28/27	512/495	?	88/85
9	Supersanguinant			25/24	?	?	26/25, 27/26	?
10	Supersanguinant			N/A	?	?	?	160/153
11	Supervicinant	Supercollocant	Proximosupercollocant	N/A	21/20	22/21	?	?
12	Supervicinant	Supercollocant	Proximosupercollocant	256/243, 135/128	?	?	?	?
13	Supervicinant	Supercollocant	Proximosupercollocant	N/A	?	128/121	55/52	18/17
14	Supervicinant	Supercollocant	Proximosupercollocant	N/A	?	?	?	17/16
15	Supervicinant	Supercollocant	Distosupercollocant	16/15	?	?	?	?
16	Supervicinant	Supercollocant	Distosupercollocant	N/A	15/14	275/256	?	?
17	Supervicinant	Supercollocant	Distosupercollocant	N/A	?	?	14/13	55/51, 128/119
18	Supervicinant	Superabrogant		27/25	?	?	13/12	?
19	Supervicinant	Superabrogant		N/A	?	88/81	?	?
20	Supervicinant	Superabrogant		N/A	?	12/11	?	?
21	Supervicinant	Superabrogant		?	?	?	128/117	561/512, 1024/935
22	Supervicinant	Superaequient		N/A	?	11/10	?	?
23	Supervicinant	Superaequient		N/A	567/512	243/220	?	425/384
24	Supervicinant	Superaequient		10/9	?	256/231	?	?
25	Supervicinant	Supertonic		N/A	?	?	143/128	512/459
26	Supervicinant	Supertonic		N/A	28/25	121/108	?	?
27	Supervicinant	Supertonic	Bidominant	9/8	?	?	?	?
28	Supervicinant	Supertonic		N/A	640/567	?	44/39	289/256
29	Contravaricant	Proximocontravaricant		N/A	?	25/22	?	17/15
30	Contravaricant	Proximocontravaricant		256/225	?	154/135	?	?
31	Contravaricant	Proximocontravaricant		N/A	8/7	55/48	?	?
32	Contravaricant	Mesocontravaricant		N/A	?	1024/891	?	?
33	Contravaricant	Mesocontravaricant		?	?	?	15/13	?
34	Contravaricant	Mesocontravaricant		N/A	?	297/256	?	?
35	Contravaricant	Distocontravaricant		N/A	7/6	64/55	?	?
36	Contravaricant	Distocontravaricant		75/64	?	?	117/100	?
37	Contravaricant	Distocontravaricant		N/A	?	?	?	20/17
38	Mediant	Proximomediant		N/A	?	33/28	13/11	85/72
39	Mediant	Proximomediant	Triservient	32/27	?	?	?	?
40	Mediant	Proximomediant		N/A	25/21	144/121	?	?
41	Mediant	Proximomediant		N/A	?	?	512/429	153/128
42	Mediant	Mesoproximomediant		6/5	?	77/64	?	?
43	Mediant	Mesoproximomediant		N/A	135/112	?	?	512/425
44	Mediant	Mesoproximomediant		N/A	?	40/33, 121/100	?	144/119, 165/136
45	Mediant	Mesomediant		?	?	?	39/32	17/14
46	Mediant	Mesomediant		N/A	?	11/9	?	?
47	Mediant	Mesomediant		N/A	?	27/22	?	?
48	Mediant	Mesomediant		?	?	?	16/13	21/17
49	Mediant	Mesodistomediant		N/A	?	?	26/21	68/55
50	Mediant	Mesodistomediant		N/A	56/45	1024/825	?	?
51	Mediant	Mesodistomediant		5/4	?	96/77	?	?
52	Mediant	Distomediant		N/A	?	?	?	64/51
53	Mediant	Distomediant		N/A	63/50	121/96	?	34/27
54	Mediant	Distomediant	Quadridominant	81/64	?	?	?	?
55	Mediant	Distomediant		N/A	?	14/11	33/26	108/85
56	Superobstant	Proximosuperobstant		N/A	?	?	143/112	51/40
57	Superobstant	Proximosuperobstant		32/25	?	?	50/39	?
58	Superobstant	Proximosuperobstant		N/A	9/7	165/128	?	?
59	Superobstant	Distosuperobstant		N/A	?	128/99	?	22/17
60	Superobstant	Distosuperobstant		?	?	?	13/10	?
61	Superobstant	Distosuperobstant		N/A	?	176/135	?	?
62	Superobstant	Distosuperobstant		N/A	21/16	55/42, 72/55	?	17/13
63	Pseudoservient			320/243, 675/512	?	?	?	?
64	Pseudoservient			N/A	?	33/25	?	45/34
65	Pseudoservient			N/A	?	968/729	?	85/64
66	Servient			4/3	?	?	?	?
67	Geminoservient			N/A	75/56	162/121	?	?
68	Geminoservient			N/A	?	121/90	?	?
69	Geminoservient			27/20	?	?	104/77	?
70	Geminoservient			N/A	256/189	110/81	?	?
71	Subient			N/A	?	15/11	?	?
72	Subient			?	?	?	160/117	256/187
73	Subient			N/A	?	11/8	?	?
74	Pseudosubient			N/A	112/81	?	?	?
75	Pseudosubient			25/18	?	?	18/13	?
76	Protosycophant			N/A	?	?	?	357/256
77	Protosycophant			N/A	7/5	?	?	?
78	Antitonic			45/32, 1024/729	?	?	?	?
79	Antitonic			N/A	?	?	?	24/17
80	Antitonic			N/A	?	?	?	17/12
81	Antitonic			64/45, 729/512	?	?	?	?
82	Prototyrant			N/A	10/7	?	?	?
83	Prototyrant			N/A	?	?	?	512/357
84	Pseudoimponent			36/25	?	?	13/9	?
85	Pseudoimponent			N/A	81/56	?	?	?
86	Imponent			N/A	?	16/11	?	?
87	Imponent			?	?	?	117/80	187/128
88	Imponent			N/A	?	22/15	?	?
89	Geminodominant			N/A	189/128	81/55	?	?
90	Geminodominant			40/27	?	?	77/52	?
91	Geminodominant			N/A	?	180/121	?	?
92	Geminodominant			N/A	112/75	121/81	?	?
93	Dominant			3/2	?	?	?	?
94	Pseudodominant			N/A	?	729/484	?	128/85
95	Pseudoservient			N/A	?	50/33	?	68/45
96	Pseudoservient			243/160, 1024/675	?	?	?	?
97	Subobstant	Distosubobstant		N/A	32/21	84/55, 55/36	?	26/17
98	Subobstant	Distosubobstant		N/A	?	135/88	?	?
99	Subobstant	Distosubobstant		?	?	?	20/13	?
100	Subobstant	Distosubobstant		N/A	?	99/64	?	17/11
101	Subobstant	Proximosubobstant		N/A	14/9	256/165	?	?
102	Subobstant	Proximosubobstant		25/16	?	?	39/25	?
103	Subobstant	Proximosubobstant		N/A	?	?	224/143	80/51
104	Contramediant	Distocontramediant		N/A	?	11/7	52/33	85/54
105	Contramediant	Distocontramediant	Quadriservient	128/81	?	?	?	?
106	Contramediant	Distocontramediant		N/A	100/63	192/121	?	27/17
107	Contramediant	Distocontramediant		N/A	?	?	?	51/32
108	Contramediant	Mesodistocontramediant		8/5	?	77/48	?	?
109	Contramediant	Mesodistocontramediant		N/A	45/28	825/512	?	?
110	Contramediant	Mesodistocontramediant		N/A	?	?	21/13	55/34
111	Contramediant	Mesocontramediant		?	?	?	13/8	34/21
112	Contramediant	Mesocontramediant		N/A	?	44/27	?	?
113	Contramediant	Mesocontramediant		N/A	?	18/11	?	?
114	Contramediant	Mesocontramediant		?	?	?	64/39	28/17
115	Contramediant	Mesoproximocontramediant		N/A	?	33/20, 200/121	?	119/72, 272/165
116	Contramediant	Mesoproximocontramediant		N/A	224/135	?	?	425/256
117	Contramediant	Mesoproximocontramediant		5/3	?	128/77	?	?
118	Contramediant	Proximocontramediant		N/A	?	?	429/256	256/153
119	Contramediant	Proximocontramediant		N/A	42/25	121/72	?	?
120	Contramediant	Proximocontramediant	Tridominant	27/16	?	?	?	?
121	Contramediant	Proximocontramediant		N/A	?	56/33	22/13	144/85
122	Varicant	Distovaricant		N/A	?	?	?	17/10
123	Varicant	Distovaricant		128/75	?	?	200/117	?
124	Varicant	Distovaricant		N/A	12/7	55/32	?	?
125	Varicant	Mesovaricant		N/A	?	512/297	?	?
126	Varicant	Mesovaricant		?	?	?	26/15	?
127	Varicant	Mesovaricant		N/A	?	891/512	?	?
128	Varicant	Proximovaricant		N/A	7/4	96/55	?	?
129	Varicant	Proximovaricant		225/128	?	135/77	?	?
130	Varicant	Proximovaricant		N/A	?	44/25	?	30/17
131	Subvicinant	Subtonic		N/A	567/320	?	39/22	512/289
132	Subvicinant	Subtonic	Biservient	16/9	?	?	?	?
133	Subvicinant	Subtonic		N/A	25/14	216/121	?	?
134	Subvicinant	Subtonic		N/A	?	?	256/143	459/256
135	Subvicinant	Subaequient		9/5	?	231/128	?	?
136	Subvicinant	Subaequient		N/A	1024/567	440/243	?	768/425
137	Subvicinant	Subaequient		N/A	?	20/11	?	?
138	Subvicinant	Subabrogant		?	?	?	117/64	1024/561, 935/512
139	Subvicinant	Subabrogant		N/A	?	11/6	?	?
140	Subvicinant	Subabrogant		N/A	?	81/44	?	?
141	Subvicinant	Subabrogant		50/27	?	?	24/13	?
142	Subvicinant	Subcollocant	Distosubcollocant	N/A	?	?	13/7	102/55, 119/64
143	Subvicinant	Subcollocant	Distosubcollocant	N/A	28/15	512/275	?	?
144	Subvicinant	Subcollocant	Distosubcollocant	15/8	?	?	?	?
145	Subvicinant	Subcollocant	Proximosubcollocant	N/A	?	?	?	32/17
146	Subvicinant	Subcollocant	Proximosubcollocant	N/A	?	121/64	104/55	17/9
147	Subvicinant	Subcollocant	Proximosubcollocant	243/128, 256/135	?	?	?	?
148	Subvicinant	Subcollocant	Proximosubcollocant	N/A	40/21	21/11	?	?
149	Subsanguinant			N/A	?	?	?	153/80
150	Subsanguinant			48/25	?	?	25/13, 52/27	?
151	Subsanguinant			N/A	27/14	495/256	?	85/44
152	Subgradient			N/A	?	64/33	?	33/17
153	Subgradient			?	?	?	39/20	187/96
154	Subgradient			N/A	?	88/45	?	100/51
155	Subdietic			N/A	63/32	108/55, 55/28	?	51/26
156	Superdietic			160/81	?	?	77/39	168/85
157	Supercommatic			N/A	?	99/50, 240/121	143/72	119/60
158	Supercommatic			N/A	448/225	484/243	700/351, 351/176	255/128
159	Tonic			2/1