Angel
Lua error in Module:MOS at line 46: attempt to index local 'equave' (a nil value).Angel is a name* proposed by Mason Green for the temperament that tempers out 81:80 (thus, it is a meantone system), and has a period that is a flattened 3:2, four of which make a pentave (5:1), while its generator is an octave. This temperament is very closely related to quarter-comma meantone (which was the standard for most Western classical music); the key difference is that meantone has a period of an octave and a fifth as a generator, whereas the roles of the fifth and octave are reversed in angel. (Due to the period being a fifth, setting the generator to an octave is equivalent to using a perfect fourth or whole tone as the generator instead).
If the pentaves are required to be exactly 5:1, then the fifths will be exactly the same size as the fifths of quarter-comma meantone (namely, the fourth root of five), but the octaves will be slightly flat (by less than half a cent). On the other hand, if the octaves are made perfect (making this a 31edo temperament), the pentaves will be slightly sharp. Both options are perceptually very close to one another.
More specifically, the term angel may refer to various MOSes and MODMOSes that are derived from this temperament. There are MOSes with 3, 4, 7, and 11 notes per period; these have 5, 7, 12, and 19 notes per octave and so may be considered the angel equivalents of the pentatonic, diatonic, chromatic, and enharmonic scales respectively.
Although angel scales are not octave-repeating, the fact that the generator is an octave makes them far less xenharmonic than one might think. You don't even have to train yourself to hear pentaves as equivalent, since the octave can still be thought of as a "pseudo-equivalency" due to its being the generator.
In particular, the angel MOS with 11 notes per period has long chains of ten octaves, which spans nearly the entire range of human hearing. Many if not most common-practice pieces can be easily translated into this scale, since the deviation from a purely octave-repeating system only becomes apparent for melodies and harmonies spanning several octaves. Compound intervals (spanning more than an octave) are sometimes perceived as more or less consonant than their simple counterparts; this is especially true for high-limit intervals like 11:8 (which is more consonant in compound form). Thus it may actually be beneficial to use a system that doesn't exactly repeat at the octave.
Straight-fretted angel guitars would be a possibility; such guitars would have unequally spaced frets and would need to be tuned in all-fifths, since the period is a fifth.
The generator range is 171.4 to 240 cents, placing it near the diatonic major second, usually representing a major second of some type. The dark (chroma-negative) generator is, however, its fifth complement (480 to 514.3 cents).
In the Angel scale, each tone has a 3/2 perfect fifth above it. The scale has two major chords and two minor chords.
Basic angel is in 7edf, which is a very good fifth-based equal tuning similar to 12edo.
Notation
There are 3 main ways to notate the angel scale. One method uses a simple sesquitave (fifth) repeating notation consisting of 4 naturals (eg. Do Re Mi Fa, Sol La Si Do). Given that 1-5/4-5/3 is fifth-equivalent to a tone cluster of 1-10/9-5/4, it may be more convenient to notate diatonic scales as repeating at the double or triple sesquitave (major ninth or thirteenth), however it does make navigating the genchain harder. This way, 5/3 is its own pitch class, distinct from 10/9. Notating this way produces a major ninth which is the Aeolian mode of Napoli[6L 2s] or a major thirteenth which is the Dorian mode of Bijou[9L 3s]. Since there are exactly 8 naturals in double sesquitave notation and 12 in triple sesquitave notation, letters A-H (FGABHCDEF) or dozenal digits (0123456789XE0 or D1234567FGACD with flats written C molle) may be used.
| Notation | Supersoft | Soft | Semisoft | Basic | Semihard | Hard | Superhard | ||
|---|---|---|---|---|---|---|---|---|---|
| Angel | Napoli | Bijou | ~15edf | ~11edf | ~18edf | ~7edf | ~17edf | ~10edf | ~13edf |
| Do#, Sol# | F# | 0#, D# | 1\15
46.153… |
1\11
63.157… |
2\18
77.419… |
1\7
100 |
3\17
124.137… |
2\10
141.176… |
3\13
163.63 |
| Reb, Lab | Gb | 1b, 1c | 3\15
138.461… |
2\11
126.315… |
3\18
116.129… |
2\17
82.758… |
1\10
70.588… |
1\13
54.54 | |
| Re, La | G | 1 | 4\15
184.615… |
3\11
189.473… |
5\18
193.548… |
2\7
200 |
5\17
206.896… |
3\10
211.764… |
4\13
218.18 |
| Re#, La# | G# | 1# | 5\15
230.769… |
4\11
252.631… |
7\18
270.967… |
3\7
300 |
8\17
331.034… |
5\10
352.941… |
7\13
381.81 |
| Mib, Sib | Ab | 2b, 2c | 7\15
323.076… |
5\11
315.789… |
8\18
309.677… |
7\17
289.655… |
4\10
282.352… |
5\13
272.72 | |
| Mi, Si | A | 2 | 8\15
369.230… |
6\11
378.947… |
10\18
387.096… |
4\7
400 |
10\17
413.793… |
6\10
423.529… |
8\13
436.36 |
| Mi#, Si# | A# | 2# | 9\15
415.384… |
7\11
442.105… |
12\18
464.516… |
5\7
500 |
13\17
537.931… |
8\10
564.705… |
11\13
600 |
| Fab, Dob | Bbb | 3bb, 3cc | 10\15
461.538… |
11\18
425.806… |
4\7
400 |
9\17
372.413… |
5\10
352.941… |
6\13
327.27 | |
| Fa, Do | Bb | 3b, 3c | 11\15
507.692… |
8\11
505.263… |
13\18
503.225… |
5\7
500 |
12\17
496.551… |
7\10
494.117… |
9\13
490.90 |
| Fa#, Do# | B | 3 | 12\15
553.846… |
9\11
568.421… |
15\18
580.645… |
6\7
600 |
15\17
620.689… |
9\10
635.294… |
12\13
654.54 |
| Fax, Dox | B# | 3# | 13\15
600 |
10\11
631.578… |
17\18
658.064… |
7\7
700 |
18\17
744.827… |
11\10
776.470… |
15\13
818.18 |
| Dob, Solb | Hb | 4b, 4c | 14\15
646.153… |
16\18
619.354… |
6\7
600 |
14\17
579.310… |
8\10
564.705… |
10\13
545.45 | |
| Do, Sol | H | 4 | 15\15
692.307… |
11\11
694.736… |
18\18
696.774… |
7\7
700 |
17\17
703.448… |
10\10
705.882… |
13\13
709.09 |
| Do#, Sol# | Η# | 4# | 16\15
738.461… |
12\11
757.894… |
20\18
774.193… |
8\8
800 |
20\17
827.586… |
12\10
847.058… |
16\13
872.72 |
| Reb, Lab | Cb | 5b, 5c | 18\15
830.769… |
13\11
821.052… |
21\18
812.903… |
19\17
786.206… |
11\10
776.470… |
14\13
763.63 | |
| Re, La | C | 5 | 19\18
876.923… |
14\11
884.210… |
23\18
890.322… |
9\5
900 |
22\17
910.344… |
13\10
917.647… |
17\13
927.27 |
| Re#, La# | C# | 5# | 20\15
923.076… |
15\11
947.368… |
25\18
967.741… |
10\7
1000 |
25\17
1034.482… |
15\10
1058.823… |
20\13
1090.90 |
| Mib, Sib | Db | 6b, 6c | 22\15
1015.384… |
16\11
1010.526… |
26\18
1006.451… |
24\17
993.103… |
14\10
988.235… |
18\13
981.81 | |
| Mi, Si | D | 6 | 23\15
1061.538… |
17\11
1073.684… |
28\18
1083.870… |
11\7
1100 |
27\17
1117.241… |
16\10
1129.411… |
21\9
1145.45 |
| Mi#, Si# | D# | 6# | 24\15
1107.692… |
18\11
1136.842… |
30\18
1161.290… |
12\7
1200 |
30\17
1241.379… |
18\10
1270.588… |
24\13
1309.09 |
| Fab, Dob | Ebb | 7bb, 7cc | 25\15
1153.846… |
29\18
1122.580… |
11\7
1100 |
26\17
1075.862… |
15\10
1058.823… |
19\13
1036.36 | |
| Fa, Do | Eb | 7b, 7c | 26\15
1200 |
19\11
1200 |
31\18
1200 |
12\7
1200 |
29\17
1200 |
17\10
1200 |
22\13
1200 |
| Fa#, Do# | E | 7 | 27\15
1246.153… |
20\11
1263.157… |
33\18
1277.419… |
13\7
1300 |
32\17
1324.137… |
19\10
1341.176… |
25\13
1363.63 |
| Fax, Dox | E# | 7# | 28\15
1292.307… |
21\11
1326.315… |
35\18
1354.838… |
14\7
1400 |
35\17
1448.275… |
21\10
1482.352… |
28\13
1527.27 |
| Dob, Solb | Fb | 8b, Fc | 29\15
1338.461… |
34\18
1316.129… |
13\7
1300 |
31\17
1282.758… |
18\10
1270.588… |
23\18
1254.54 | |
| Do, Sol | F | 8, F | 30\15
1384.615… |
22\11
1389.473… |
36\18
1393.548… |
14\7
1400 |
34\17
1406.896… |
20\10
1411.764… |
26\9
1418.18 |
| Do#, Sol# | F# | 8#, F# | 31\15
1430.769… |
23\11
1452.631… |
38\18
1470.967… |
15\7
1500 |
37\17
1531.034… |
22\10
1552.941… |
29\13
1581.81 |
| Reb, Lab | Gb | 9b, Gc | 33\15
1523.076… |
24\11
1515.789… |
39\18
1509.677… |
36\17
1489.655… |
21\10
1482.352… |
27\13
1472.72 | |
| Re, La | G | 9, G | 34\15
1569.230… |
25\11
1578.947… |
41\18
1587.096… |
16\7
1600 |
39\17
1613.793… |
23\10
1623.529… |
30\13
1636.36 |
| Re#, La# | G# | 9#, G# | 35\15
1615.384… |
26\11
1642.105… |
43\18
1664.516… |
17\7
1700 |
42\17
1737.931… |
25\10
1764.705… |
33\13
1800 |
| Mib, Sib | Ab | Xb, Ac | 37\15
1707.692… |
27\11
1705.263… |
44\18
1703.225… |
41\17
1696.551… |
24\10
1694.117… |
31\13
1690.90 | |
| Mi, Si | A | X, A | 38\15
1753.846… |
28\11
1768.421… |
46\18
1780.645… |
18\7
1800 |
44\17
1820.689… |
26\10
1835.294… |
34\13
1854.54 |
| Mi#, Si# | A# | X#, A# | 39\15
1800 |
29\11
1831.578… |
48\18
1858.064… |
19\7
1900 |
47\17
1944.827… |
28\10
1976.470… |
37\13
2018.18 |
| Fab, Dob | Bbb | Ebb, Ccc | 40\15
1846.153… |
47\18
1819.354… |
18\7
1800 |
43\17
1779.310… |
25\10
1764.705… |
32\13
1745.4̄5̄ | |
| Fa, Do | Bb | Eb, Cc | 41\15
1892.307… |
30\11
1894.736… |
49\18
1896.774… |
19\7
1900 |
46\17
1903.448… |
27\10
1905.882… |
35\13
1909.09 |
| Fa#, Do# | B | E, C | 42\15
1938.461… |
31\11
1957.894… |
51\18
1974.193… |
20\7
2000 |
49\17
2027.586… |
29\10
1976.470… |
38\13
2072.72 |
| Fax, Dox | B# | Ex, Cx | 43\15
1984.615… |
32\11
2021.052… |
53\18
2051.612… |
21\7
2100 |
52\17
2151.724… |
31\10
2188.235… |
41\13
2236.36 |
| Dob, Solb | Hb | 0b, Dc | 44\15
2030.769… |
52\18
2012.903… |
20\7
2000 |
48\17
1986.206… |
28\10
1967.470… |
36\13
1963.63 | |
| Do, Sol | H | 0, D | 45\15
2076.923… |
33\11
2084.210… |
54\18
2090.322… |
21\7
2100 |
51\17
2110.344… |
30\10
2117.647… |
39\13
2127.27 |
| Notation | Supersoft | Soft | Semisoft | Basic | Semihard | Hard | Superhard | ||
|---|---|---|---|---|---|---|---|---|---|
| Angel | Napoli | Bijou | ~15edf | ~11edf | ~18edf | ~7edf | ~17edf | ~10edf | ~13edf |
| Do#, Sol# | F# | 0#, D# | 1\15
46.6 |
1\11
63.63 |
2\18
77.7 |
1\7
100 |
3\17
123.529… |
2\10
140 |
3\13
161.538… |
| Reb, Lab | Gb | 1b, 1c | 3\15
140 |
2\11
127.27 |
3\18
116.6 |
2\17
82.352… |
1\10
70 |
1\13
53.846… | |
| Re, La | G | 1 | 4\15
186.6 |
3\11
190.90 |
5\18
194.4 |
2\7
200 |
5\17
205.882… |
3\10
210 |
4\13
215.384… |
| Re#, La# | G# | 1# | 5\15
233.3 |
4\11
254.54 |
7\18
272.2̄ |
3\7
300 |
8\17
329.411… |
5\10
350 |
7\13
376.923… |
| Mib, Sib | Ab | 2b, 2c | 7\15
326.6 |
5\11
318.18 |
8\18
311.1̄ |
7\17
288.235… |
4\10
280 |
5\13
269.230… | |
| Mi, Si | A | 2 | 8\15
373.3 |
6\11
381.81 |
10\18
388.8 |
4\7
400 |
10\17
411.764… |
6\10
420 |
8\13
430.769… |
| Mi#, Si# | A# | 2# | 9\15
420 |
7\11
445.45 |
12\18
466.6 |
5\7
500 |
13\17
535.294… |
8\10
560 |
11\13
592.307… |
| Fab, Dob | Bbb | 3bb, 3cc | 10\15
466.6 |
11\18
427.7 |
4\7
400 |
9\17
370.588… |
5\10
350 |
6\13
323.076.… | |
| Fa, Do | Bb | 3b, 3c | 11\15
513.3 |
8\11
509.09 |
13\18
505.5 |
5\7
500 |
12\17
494.117… |
7\10
490 |
9\13
484.615… |
| Fa#, Do# | B | 3 | 12\15
560 |
9\11
572.72 |
15\18
583.3 |
6\7
600 |
15\17
617.647… |
9\10
630 |
12\13
646.153… |
| Fax, Dox | B# | 3# | 13\15
606.6 |
10\11
636.36 |
17\18
661.6 |
7\7
700 |
18\17
741.176… |
11\10
770 |
15\13
807.692… |
| Dob, Solb | Hb | 4b, 4c | 14\15
653.3 |
16\18
622.2 |
6\7
600 |
14\17
576.470… |
8\10
560 |
10\13
538.461… | |
| Do, Sol | H | 4 | 700 | ||||||
| Do#, Sol# | Η# | 4# | 16\15
746.6 |
12\11
763.63 |
20\18
777.7 |
8\7
800 |
20\17
823.529… |
12\10
840 |
16\13
861.538… |
| Reb, Lab | Cb | 5b, 5c | 18\15
840 |
13\11
827.27 |
21\18
816.6 |
19\17
782.352… |
11\10
770 |
14\13
753.846… | |
| Re, La | C | 5 | 19\15
886.6 |
14\11
890.90 |
23\18
894.4 |
9\7
900 |
22\17
905.882… |
13\10
910 |
17\13
915.384… |
| Re#, La# | C# | 5# | 20\15
933.3 |
15\11
954.54 |
25\18
972.2̄ |
10\7
1000 |
25\17
1029.411… |
15\10
1050 |
20\13
1076.923… |
| Mib, Sib | Db | 6b, 6c | 22\15
1026.6 |
16\11
1018.18 |
26\18
1011.1 |
24\17
988.235… |
14\10
980 |
18\13
969.230… | |
| Mi, Si | D | 6 | 23\15
1073.3 |
17\11
1081.81 |
28\18
1088.8 |
11\7
1100 |
27\17
1111.764… |
16\10
1120 |
21\13
1130.769… |
| Mi#, Si# | D# | 6# | 24\15
1120 |
18\11
1145.45 |
30\18
1166.6 |
12\7
1200 |
30\17
1235.294… |
18\10
1260 |
24\13
1292.307… |
| Fab, Dob | Ebb | 7bb, 7cc | 25\15
1166.6 |
29\18
1127.7 |
11\7
1100 |
26\17
1070.588… |
15\10
1050 |
19\13
1023.076… | |
| Fa, Do | Eb | 7b, 7c | 26\15
1213.3 |
19\11
1209.09 |
31\18
1205.5 |
12\7
1200 |
29\17
1194.117… |
17\10
1190 |
22\13
1184.615… |
| Fa#, Do# | E | 7 | 27\15
1260 |
20\11
1272.72 |
33\18
1283.3 |
13\7
1300 |
32\17
1317.647… |
19\10
1330 |
25\13
1346.153… |
| Fax, Dox | E# | 7# | 28\15
1306.6 |
21\11
1336.36 |
35\18
1361.1 |
14\7
1400 |
35\17
1441.176… |
21\10
1470 |
28\13
1507.692… |
| Dob, Solb | Fb | 8b, Fc | 29\15
1333.3 |
34\18
1322.2 |
13\7
1300 |
31\17
1276.470… |
18\10
1260 |
23\13
1238.461… | |
| Do, Sol | F | 8, F | 1400 | ||||||
| Do#, Sol# | F# | 8#, F# | 31\15
1446.6 |
23\11
1463.63 |
38\18
1477.7̄ |
15\7
1500 |
37\17
1523.529… |
22\10
1540 |
29\13
1561.538… |
| Reb, Lab | Gb | 9b, Gc | 33\15
1540 |
24\11
1527.27 |
39\18
1516.6 |
36\17
1482.352… |
21\10
1470 |
27\13
1453.846… | |
| Re, La | G | 9, G | 34\15
1586.6 |
25\11
1590.90 |
41\18
1594.4 |
16\7
1600 |
39\17
1605.882… |
23\10
1610 |
30\13
1615.384… |
| Re#, La# | G# | 9#, G# | 35\15
1633.3 |
26\11
1654.54 |
43\18
1672.2 |
17\7
1700 |
42\17
1729.411… |
25\10
1750 |
33\13
1776.923… |
| Mib, Sib | Ab | Xb, Ac | 37\15
1726.6 |
27\11
1718.18 |
44\18
1711.1 |
41\17
1688.235… |
24\10
1680 |
31\13
1669.230… | |
| Mi, Si | A | X, A | 38\15
1773.3 |
28\11
1781.81 |
46\18
1788.8 |
18\7
1800 |
44\17
1811.764… |
26\10
1820 |
34\13
1830.769… |
| Mi#, Si# | A# | X#, A# | 39\15
1820 |
29\11
1845.45 |
48\18
1866.6 |
19\7
1900 |
47\17
1935.294… |
28\10
1960 |
37\13
1992.307… |
| Fab, Dob | Bbb | Ebb, Ccc | 40\15
1866.6 |
47\18
1827.7 |
18\7
1800 |
43\17
1770.588… |
25\10
1750 |
32\13
1723.076… | |
| Fa, Do | Bb | Eb, Cc | 41\15
1913.3 |
30\11
1909.09 |
49\18
1905.5 |
19\7
1900 |
46\17
1894.117… |
27\10
1890 |
35\13
1884.615… |
| Fa#, Do# | B | E, C | 42\15
1960 |
31\11
1972.72 |
51\18
1983.3 |
20\7
2000 |
49\17
2017.647… |
29\10
2030 |
38\13
2046.153… |
| Fax, Dox | B# | Ex, Cx | 43\15
2006.6 |
32\11
2036.36 |
53\18
2061.1 |
21\7
2100 |
52\17
2141.176… |
31\10
2170 |
41\13
2207.692… |
| Dob, Solb | Hb | 0b, Dc | 44\15
2053.3 |
52\18
2022.2 |
20\7
2000 |
48\17
1976.470… |
28\10
1960 |
36\13
1938.615… | |
| Do, Sol | H | 0, D | 2100 | ||||||
Intervals
| Generators | Sesquitave notation | Interval category name | Generators | Notation of 3/2 inverse | Interval category name |
|---|---|---|---|---|---|
| The 4-note MOS has the following intervals (from some root): | |||||
| 0 | Do, Sol | perfect unison | 0 | Do, Sol | sesquitave (just fifth) |
| 1 | Fa, Do | perfect fourth | -1 | Re, La | perfect second |
| 2 | Mib, Sib | minor third | -2 | Mi, Si | major third |
| 3 | Reb, Lab | diminished second | -3 | Fa#, Do# | augmented fourth |
| The chromatic 7-note MOS also has the following intervals (from some root): | |||||
| 4 | Dob, Solb | diminished sesquitave | -4 | Do#, Sol# | augmented unison (chroma) |
| 5 | Fab, Dob | diminished fourth | -5 | Re#, La# | augmented second |
| 6 | Mibb, Sibb | diminished third | -6 | Mi#, Si# | augmented third |
Genchain
The generator chain for this scale is as follows:
| Mibb
Sibb |
Fab
Dob |
Dob
Solb |
Reb
Lab |
Mib
Sib |
Fa
Do |
Do
Sol |
Re
La |
Mi
Si |
Fa#
Do# |
Do#
Sol# |
Re#
La# |
Mi#
Si# |
| d3 | d4 | d6 | d2 | m3 | P4 | P1 | P2 | M3 | A4 | A1 | A2 | A3 |
Modes
The mode names are based on the species of fifth:
| Mode | Scale | UDP | Interval type | ||
|---|---|---|---|---|---|
| name | pattern | notation | 2nd | 3rd | 4th |
| Lydian | LLLs | 3|0 | P | M | A |
| Major | LLsL | 2|1 | P | M | P |
| Minor | LLsL | 1|2 | P | m | P |
| Phrygian | LsLL | 0|3 | d | m | P |
Temperaments
The most basic rank-2 temperament interpretation of diatonic is Napoli. The name "Napoli" comes from the “Neapolitan” sixth triad spelled root-(p-2g)-(2p-3g) (p = 3/2, g = the whole tone) which serves as its minor triad approximating 5:6:8 in pental interpretations or 18:21:28 in septimal ones. Basic ~7edf fits both interpretations.
Napoli-Meantone
Subgroup: 3/2.6/5.8/5
POL2 generator: ~9/8 = 192.6406
Mapping: [⟨1 1 2], ⟨0 -2 -3]]
Napoli-Superpyth
Subgroup: 3/2.7/6.14/9
POL2 generator: ~8/7 = 218.6371
Mapping: [⟨1 1 2], ⟨0 -2 -3]]
Scale tree
The spectrum looks like this:
| Generator
(bright) |
Cents | L | s | L/s | Comments | |||
|---|---|---|---|---|---|---|---|---|
| Normalised | ed7\12 | |||||||
| 1\4 | 171.428… | 175 | 1 | 1 | 1.000 | Equalised | ||
| 6\23 | 180 | 182.608… | 6 | 5 | 1.200 | |||
| 11\42 | 180.821… | 183.3 | 11 | 9 | 1.222 | |||
| 5\19 | 181.81 | 184.210… | 5 | 4 | 1.250 | |||
| 14\53 | 182.608… | 184.905… | 14 | 11 | 1.273 | |||
| 9\34 | 183.050… | 185.294… | 9 | 7 | 1.286 | |||
| 4\15 | 184.615… | 186.6̄ | 4 | 3 | 1.333 | |||
| 11\41 | 185.915… | 187.804… | 11 | 8 | 1.375 | |||
| 7\26 | 186.6 | 188.461… | 7 | 5 | 1.400 | |||
| 10\37 | 187.5 | 189.189 | 10 | 7 | 1.429 | |||
| 13\48 | 187.951… | 189.583 | 13 | 9 | 1.444 | |||
| 16\59 | 188.235… | 189.830… | 16 | 11 | 1.4545 | |||
| 3\11 | 189.473… | 190.90 | 3 | 2 | 1.500 | Napoli-Meantone starts here | ||
| 17\62 | 190.654… | 191.935… | 17 | 11 | 1.5455 | |||
| 14\51 | 190.90 | 192.156… | 14 | 9 | 1.556 | |||
| 11\40 | 191.304… | 192.5 | 11 | 7 | 1.571 | |||
| 8\29 | 192 | 193.103… | 8 | 5 | 1.600 | |||
| 5\18 | 193.548… | 194.4 | 5 | 3 | 1.667 | |||
| 12\43 | 194.594 | 195.348… | 12 | 7 | 1.714 | |||
| 7\25 | 195.348… | 196 | 7 | 4 | 1.750 | |||
| 9\32 | 196.36 | 196.875 | 9 | 5 | 1.800 | |||
| 11\39 | 197.014… | 197.435… | 11 | 6 | 1.833 | |||
| 13\46 | 197.468… | 197.826… | 13 | 7 | 1.857 | |||
| 15\53 | 197.802… | 198.113… | 15 | 8 | 1.875 | |||
| 17\60 | 198.058… | 198.3̄ | 17 | 9 | 1.889 | |||
| 19\67 | 198.260… | 198.507… | 19 | 10 | 1.900 | |||
| 21\74 | 198.425… | 198.648 | 21 | 11 | 1.909 | |||
| 23\81 | 198.561… | 198.765… | 23 | 12 | 1.917 | |||
| 25\88 | 198.675… | 198.863 | 25 | 13 | 1.923 | |||
| 27\95 | 198.773… | 198.947… | 27 | 14 | 1.929 | |||
| 29\102 | 198.857… | 199.019… | 29 | 15 | 1.933 | |||
| 31\109 | 198.930… | 199.082… | 31 | 16 | 1.9375 | |||
| 33\116 | 198.994… | 199.137… | 33 | 17 | 1.941 | |||
| 35\123 | 199.052… | 199.186… | 35 | 18 | 1.944 | |||
| 2\7 | 200 | 200 | 2 | 1 | 2.000 | Napoli-Meantone ends, Napoli-Pythagorean begins | ||
| 19\66 | 201.769… | 201.51 | 19 | 9 | 2.111 | |||
| 17\59 | 201.980… | 201.694… | 17 | 8 | 2.125 | |||
| 15\52 | 202.247… | 201.923… | 15 | 7 | 2.143 | |||
| 13\45 | 202.597… | 202.2 | 13 | 6 | 2.167 | |||
| 11\38 | 203.076… | 202.631… | 11 | 5 | 2.200 | |||
| 9\31 | 203.773… | 203.225… | 9 | 4 | 2.250 | |||
| 7\24 | 204.878… | 204.16 | 7 | 3 | 2.333 | |||
| 12\41 | 205.714… | 204.878… | 12 | 5 | 2.400 | |||
| 5\17 | 206.896… | 205.882… | 5 | 2 | 2.500 | Napoli-Neogothic heartland is from here… | ||
| 18\61 | 207.692… | 206.557… | 18 | 7 | 2.571 | |||
| 13\44 | 208 | 206.81 | 13 | 5 | 2.600 | |||
| 8\27 | 208.695… | 207.407 | 8 | 3 | 2.667 | …to here | ||
| 11\37 | 209.523… | 208.108 | 11 | 4 | 2.750 | |||
| 14\47 | 210 | 208.510… | 14 | 5 | 2.800 | |||
| 17\57 | 210.309… | 208.771… | 17 | 6 | 2.833 | |||
| 20\67 | 210.526… | 208.955… | 20 | 7 | 2.857 | |||
| 23\77 | 210.687… | 209.09 | 23 | 8 | 2.875 | |||
| 3\10 | 211.764… | 210 | 3 | 1 | 3.000 | Napoli-Pythagorean ends, Napoli-Superpyth begins | ||
| 22\73 | 212.903… | 210.958… | 22 | 7 | 3.143 | |||
| 19\63 | 213.084… | 211.1 | 19 | 6 | 3.167 | |||
| 16\53 | 213.3 | 211.320… | 16 | 5 | 3.200 | |||
| 13\43 | 213.698… | 211.627… | 13 | 4 | 3.250 | |||
| 10\33 | 214.285… | 212.12 | 10 | 3 | 3.333 | |||
| 7\23 | 215.384… | 213.043… | 7 | 2 | 3.500 | |||
| 11\36 | 216.393… | 213.3 | 11 | 3 | 3.667 | |||
| 15\49 | 216.867… | 214.285… | 15 | 4 | 3.750 | |||
| 4\13 | 218.18 | 215.385… | 4 | 1 | 4.000 | |||
| 13\42 | 219.718… | 216.6 | 13 | 3 | 4.333 | |||
| 9\29 | 220.408… | 217.241… | 9 | 2 | 4.500 | |||
| 14\45 | 221.052… | 217.7 | 14 | 3 | 4.667 | |||
| 5\16 | 222.2 | 218.75 | 5 | 1 | 5.000 | Napoli-Superpyth ends | ||
| 16\51 | 223.255… | 219.607… | 16 | 3 | 5.333 | |||
| 11\35 | 223.728… | 220 | 11 | 2 | 5.500 | |||
| 17\54 | 224.175… | 220.370 | 17 | 3 | 5.667 | |||
| 6\19 | 225 | 221.052… | 6 | 1 | 6.000 | |||
| 1\3 | 240 | 233.3 | 1 | 0 | → inf | Paucitonic | ||
* Because this temperament almost seems too good to be true.