Angel: Difference between revisions
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'''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). | {{Infobox MOS | ||
| Name = Angel | |||
| Equave = 3/2 | |||
| nLargeSteps = 3 | |||
| nSmallSteps = 1 | |||
| Equalized = 2 | |||
| Paucitonic = 1 | |||
| Pattern = LLLs | |||
}}'''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|31edo]] temperament), the pentaves will be slightly sharp. Both options are perceptually very close to one another. | 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|31edo]] temperament), the pentaves will be slightly sharp. Both options are perceptually very close to one another. | ||
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Straight-fretted angel guitars would be a possibility; such guitars would have unequally spaced frets and would need to be tuned in [https://en.wikipedia.org/wiki/All_fifths_tuning all-fifths], since the period is a fifth. | Straight-fretted angel guitars would be a possibility; such guitars would have unequally spaced frets and would need to be tuned in [https://en.wikipedia.org/wiki/All_fifths_tuning all-fifths], since the period is a fifth. | ||
The generator range is 171.4 to 240 cents, placing it near the [[9/8|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 [[Generator|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. | |||
{| class="wikitable" | |||
|+ | |||
Cents | |||
! colspan="3" |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 | |||
46.1538… | |||
|1\11 | |||
63.1578… | |||
|2\18 | |||
77.4193… | |||
| rowspan="2" |1\7 | |||
100 | |||
|3\17 | |||
124.1379… | |||
|2\10 | |||
141.1764… | |||
|3\13 | |||
163.{{Overline|63}} | |||
|- | |||
|Reb, Lab | |||
|Gb | |||
|1b, 1c | |||
|3\15 | |||
138.4615… | |||
|2\11 | |||
126.3157… | |||
|3\18 | |||
116.1290… | |||
|2\17 | |||
82.7586… | |||
|1\10 | |||
70.5882… | |||
|1\13 | |||
54.{{Overline|54}} | |||
|- | |||
|'''Re, La''' | |||
|'''G''' | |||
|'''1''' | |||
|'''4\15''' | |||
'''184.6153…''' | |||
|'''3\11''' | |||
'''189.4736…''' | |||
|'''5\18''' | |||
'''193.5483…''' | |||
|'''2\7''' | |||
'''200''' | |||
|'''5\17''' | |||
'''206.8965…''' | |||
|'''3\10''' | |||
'''211.7647…''' | |||
|'''4\13''' | |||
'''218.{{Overline|18}}''' | |||
|- | |||
|Re#, La# | |||
|G# | |||
|1# | |||
|5\15 | |||
230.7692… | |||
|4\11 | |||
252.6315… | |||
|7\18 | |||
270.9677… | |||
| rowspan="2" |3\7 | |||
300 | |||
|8\17 | |||
331.0344… | |||
|5\10 | |||
352.9411… | |||
|7\13 | |||
381.{{Overline|81}} | |||
|- | |||
|Mib, Sib | |||
|Ab | |||
|2b, 2c | |||
|7\15 | |||
323.0769… | |||
|5\11 | |||
315.7894… | |||
|8\18 | |||
309.6774… | |||
|7\17 | |||
289.6551… | |||
|4\10 | |||
282.3529… | |||
|5\13 | |||
272.{{Overline|72}} | |||
|- | |||
|Mi, Si | |||
|A | |||
|2 | |||
|8\15 | |||
369.2307… | |||
|6\11 | |||
378.9473… | |||
|10\18 | |||
387.0967… | |||
|4\7 | |||
400 | |||
|10\17 | |||
413.7931… | |||
|6\10 | |||
423.5294… | |||
|8\13 | |||
436.{{Overline|36}} | |||
|- | |||
|Mi#, Si# | |||
|A# | |||
|2# | |||
|9\15 | |||
415.3846… | |||
| rowspan="2" |7\11 | |||
442.1052… | |||
|12\18 | |||
464.5161… | |||
|5\7 | |||
500 | |||
|13\17 | |||
537.9310… | |||
|8\10 | |||
564.7058… | |||
|11\13 | |||
600 | |||
|- | |||
|Fab, Dob | |||
|Bbb | |||
|3bb, 3cc | |||
|10\15 | |||
461.5384… | |||
|11\18 | |||
425.8064… | |||
|4\7 | |||
400 | |||
|9\17 | |||
372.4137… | |||
|5\10 | |||
352.9411… | |||
|6\13 | |||
327.{{Overline|27}} | |||
|- | |||
|'''Fa, Do''' | |||
|'''Bb''' | |||
|'''3b, 3c''' | |||
|'''11\15''' | |||
'''507.6923…''' | |||
|'''8\11''' | |||
'''505.2631…''' | |||
|'''13\18''' | |||
'''503.2258…''' | |||
|'''5\7''' | |||
'''500''' | |||
|'''12\17''' | |||
'''496.5517…''' | |||
|'''7\10''' | |||
'''494.1176…''' | |||
|'''9\13''' | |||
'''490.{{Overline|90}}''' | |||
|- | |||
|Fa#, Do# | |||
|B | |||
|3 | |||
|12\15 | |||
553.8461… | |||
|9\11 | |||
568.4210… | |||
|15\18 | |||
580.6451… | |||
|6\7 | |||
600 | |||
|15\17 | |||
620.6896… | |||
|9\10 | |||
635.2941… | |||
|12\13 | |||
654.{{Overline|54}} | |||
|- | |||
|Fax, Dox | |||
|B# | |||
|3# | |||
|13\15 | |||
600 | |||
| rowspan="2" |10\11 | |||
631.5789… | |||
|17\18 | |||
658.0645… | |||
|7\7 | |||
700 | |||
|18\17 | |||
744.8275… | |||
|11\10 | |||
776.4705… | |||
|15\13 | |||
818.{{Overline|18}} | |||
|- | |||
|Dob, Solb | |||
|Hb | |||
|4b, 4c | |||
|14\15 | |||
646.1538… | |||
|16\18 | |||
619.3548… | |||
|6\7 | |||
600 | |||
|14\17 | |||
579.3103… | |||
|8\10 | |||
564.7058… | |||
|10\13 | |||
545.{{Overline|45}} | |||
|- | |||
!Do, Sol | |||
!H | |||
!4 | |||
!'''15\15''' | |||
'''692.3076…''' | |||
!'''11\11''' | |||
'''694.7368…''' | |||
!'''18\18''' | |||
'''696.7741…''' | |||
!'''7\7''' | |||
'''700''' | |||
!'''17\17''' | |||
'''703.4482…''' | |||
!'''10\10''' | |||
'''705.8823…''' | |||
!'''13\13''' | |||
'''709.'''{{Overline|09}} | |||
|- | |||
|Do#, Sol# | |||
|Η# | |||
|4# | |||
|16\15 | |||
738.4615… | |||
|12\11 | |||
757.8947… | |||
|20\18 | |||
774.1935… | |||
| rowspan="2" |8\8 | |||
800 | |||
|20\17 | |||
827.5862… | |||
|12\10 | |||
847.0588… | |||
|16\13 | |||
872.{{Overline|72}} | |||
|- | |||
|Reb, Lab | |||
|Cb | |||
|5b, 5c | |||
|18\15 | |||
830.7692… | |||
|13\11 | |||
821.0526… | |||
|21\18 | |||
812.9032… | |||
|19\17 | |||
786.2068… | |||
|11\10 | |||
776.4705… | |||
|14\13 | |||
763.{{Overline|63}} | |||
|- | |||
|'''Re, La''' | |||
|'''C''' | |||
|'''5''' | |||
|'''19\18''' | |||
'''876.9230…''' | |||
|'''14\11''' | |||
'''884.2105…''' | |||
|'''23\18''' | |||
'''890.3225…''' | |||
|'''9\5''' | |||
'''900''' | |||
|'''22\17''' | |||
'''910.3448…''' | |||
|'''13\10''' | |||
'''917.6470…''' | |||
|'''17\13''' | |||
'''927.{{Overline|27}}''' | |||
|- | |||
|Re#, La# | |||
|C# | |||
|5# | |||
|20\15 | |||
923.0769… | |||
|15\11 | |||
947.3684… | |||
|25\18 | |||
967.7419… | |||
| rowspan="2" |10\7 | |||
1000 | |||
|25\17 | |||
1034.4828… | |||
|15\10 | |||
1058.8235… | |||
|20\13 | |||
1090.{{Overline|90}} | |||
|- | |||
|Mib, Sib | |||
|Db | |||
|6b, 6c | |||
|22\15 | |||
1015.3846… | |||
|16\11 | |||
1010.5263… | |||
|26\18 | |||
1006.4516… | |||
|24\17 | |||
993.1034… | |||
|14\10 | |||
988.2353… | |||
|18\13 | |||
981.{{Overline|81}} | |||
|- | |||
|Mi, Si | |||
|D | |||
|6 | |||
|23\15 | |||
1061.5385… | |||
|17\11 | |||
1073.6842… | |||
|28\18 | |||
1083.8709… | |||
|11\7 | |||
1100 | |||
|27\17 | |||
1117.2414… | |||
|16\10 | |||
1129.4117… | |||
|21\9 | |||
1145.{{Overline|45}} | |||
|- | |||
|Mi#, Si# | |||
|D# | |||
|6# | |||
|24\15 | |||
1107.6923… | |||
| rowspan="2" |18\11 | |||
1136.8421… | |||
|30\18 | |||
1161.2903… | |||
|12\7 | |||
1200 | |||
|30\17 | |||
1241.3793… | |||
|18\10 | |||
1270.5882… | |||
|24\13 | |||
1309.{{Overline|09}} | |||
|- | |||
|Fab, Dob | |||
|Ebb | |||
|7bb, 7cc | |||
|25\15 | |||
1153.8461… | |||
|29\18 | |||
1122.5806… | |||
|11\7 | |||
1100 | |||
|26\17 | |||
1075.8621… | |||
|15\10 | |||
1058.8235… | |||
|19\13 | |||
1036.{{Overline|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 | |||
1246.1538… | |||
|20\11 | |||
1263.1578… | |||
|33\18 | |||
1277.4193… | |||
|13\7 | |||
1300 | |||
|32\17 | |||
1324.1379… | |||
|19\10 | |||
1341.1764… | |||
|25\13 | |||
1363.{{Overline|63}} | |||
|- | |||
|Fax, Dox | |||
|E# | |||
|7# | |||
|28\15 | |||
1292.3076… | |||
| rowspan="2" |21\11 | |||
1326.3157… | |||
|35\18 | |||
1354.8387… | |||
|14\7 | |||
1400 | |||
|35\17 | |||
1448.2759… | |||
|21\10 | |||
1482.3529… | |||
|28\13 | |||
1527.{{Overline|27}} | |||
|- | |||
|Dob, Solb | |||
|Fb | |||
|8b, Fc | |||
|29\15 | |||
1338.4615… | |||
|34\18 | |||
1316.1290… | |||
|13\7 | |||
1300 | |||
|31\17 | |||
1282.7586… | |||
|18\10 | |||
1270.5882… | |||
|23\18 | |||
1254.{{Overline|54}} | |||
|- | |||
!Do, Sol | |||
!F | |||
!8, F | |||
!30\15 | |||
1384.6153… | |||
!22\11 | |||
1389.4736… | |||
!36\18 | |||
1393.5484… | |||
!14\7 | |||
1400 | |||
!34\17 | |||
1406.8965… | |||
!20\10 | |||
1411.7647… | |||
!26\9 | |||
1418.{{Overline|18}} | |||
|- | |||
|Do#, Sol# | |||
|F# | |||
|8#, F# | |||
|31\15 | |||
1430.7692… | |||
|23\11 | |||
1452.6315… | |||
|38\18 | |||
1470.9677… | |||
| rowspan="2" |15\7 | |||
1500 | |||
|37\17 | |||
1531.0345… | |||
|22\10 | |||
1552.9411… | |||
|29\13 | |||
1581.{{Overline|81}} | |||
|- | |||
|Reb, Lab | |||
|Gb | |||
|9b, Gc | |||
|33\15 | |||
1523.0769… | |||
|24\11 | |||
1515.7894… | |||
|39\18 | |||
1509.6774… | |||
|36\17 | |||
1489.6551… | |||
|21\10 | |||
1482.3529… | |||
|27\13 | |||
1472.{{Overline|72}} | |||
|- | |||
|'''Re, La''' | |||
|'''G''' | |||
|'''9, G''' | |||
|'''34\15''' | |||
'''1569.2307…''' | |||
|'''25\11''' | |||
'''1578.9473…''' | |||
|'''41\18''' | |||
'''1587.0967…''' | |||
|'''16\7''' | |||
'''1600''' | |||
|'''39\17''' | |||
'''1613.7931…''' | |||
|'''23\10''' | |||
'''1623.5294…''' | |||
|'''30\13''' | |||
'''1636.{{Overline|36}}''' | |||
|- | |||
|Re#, La# | |||
|G# | |||
|9#, G# | |||
|35\15 | |||
1615.3846… | |||
|26\11 | |||
1642.1053… | |||
|43\18 | |||
1664.5161… | |||
| rowspan="2" |17\7 | |||
1700 | |||
|42\17 | |||
1737.9310… | |||
|25\10 | |||
1764.7058… | |||
|33\13 | |||
1800 | |||
|- | |||
|Mib, Sib | |||
|Ab | |||
|Xb, Ac | |||
|37\15 | |||
1707.6923… | |||
|27\11 | |||
1705.2631… | |||
|44\18 | |||
1703.2258… | |||
|41\17 | |||
1696.5517… | |||
|24\10 | |||
1694.1176… | |||
|31\13 | |||
1690.{{Overline|90}} | |||
|- | |||
|Mi, Si | |||
|A | |||
|X, A | |||
|38\15 | |||
1753.8461… | |||
|28\11 | |||
1768.4210… | |||
|46\18 | |||
1780.6451… | |||
|18\7 | |||
1800 | |||
|44\17 | |||
1820.6896… | |||
|26\10 | |||
1835.2941… | |||
|34\13 | |||
1854.{{Overline|54}} | |||
|- | |||
|Mi#, Si# | |||
|A# | |||
|X#, A# | |||
|39\15 | |||
1800 | |||
| rowspan="2" |29\11 | |||
1831.5789… | |||
|48\18 | |||
1858.0645… | |||
|19\7 | |||
1900 | |||
|47\17 | |||
1944.8276… | |||
|28\10 | |||
1976.4705… | |||
|37\13 | |||
2018.{{Overline|18}} | |||
|- | |||
|Fab, Dob | |||
|Bbb | |||
|Ebb, Ccc | |||
|40\15 | |||
1846.1538… | |||
|47\18 | |||
1819.3548… | |||
|18\7 | |||
1800 | |||
|43\17 | |||
1779.3103… | |||
|25\10 | |||
1764.7058… | |||
|32\13 | |||
1745.4̄5̄ | |||
|- | |||
|'''Fa, Do''' | |||
|'''Bb''' | |||
|Eb, Cc | |||
|'''41\15''' | |||
'''1892.3076…''' | |||
|'''30\11''' | |||
'''1894.7368…''' | |||
|'''49\18''' | |||
'''1896.7741…''' | |||
|'''19\7''' | |||
'''1900''' | |||
|'''46\17''' | |||
'''1903.4482…''' | |||
|'''27\10''' | |||
'''1905.8823…''' | |||
|'''35\13''' | |||
'''1909.{{Overline|09}}''' | |||
|- | |||
|Fa#, Do# | |||
|B | |||
|E, C | |||
|42\15 | |||
1938.4615… | |||
|31\11 | |||
1957.8947… | |||
|51\18 | |||
1974.1935… | |||
|20\7 | |||
2000 | |||
|49\17 | |||
2027.5862… | |||
|29\10 | |||
1976.4705… | |||
|38\13 | |||
2072.{{Overline|72}} | |||
|- | |||
|Fax, Dox | |||
|B# | |||
|Ex, Cx | |||
|43\15 | |||
1984.6153… | |||
| rowspan="2" |32\11 | |||
2021.0526… | |||
|53\18 | |||
2051.6129… | |||
|21\7 | |||
2100 | |||
|52\17 | |||
2151.7241… | |||
|31\10 | |||
2188.2353… | |||
|41\13 | |||
2236.{{Overline|36}} | |||
|- | |||
|Dob, Solb | |||
|Hb | |||
|0b, Dc | |||
|44\15 | |||
2030.7692… | |||
|52\18 | |||
2012.9032… | |||
|20\7 | |||
2000 | |||
|48\17 | |||
1986.2068… | |||
|28\10 | |||
1967.4705… | |||
|36\13 | |||
1963.{{Overline|63}} | |||
|- | |||
!Do, Sol | |||
!H | |||
!0, D | |||
!45\15 | |||
2076.9230… | |||
!33\11 | |||
2084.2105… | |||
!54\18 | |||
2090.3225… | |||
!21\7 | |||
2100 | |||
!51\17 | |||
2110.3448… | |||
!30\10 | |||
2117.6470… | |||
!39\13 | |||
2127.{{Overline|27}} | |||
|} | |||
{| class="wikitable" | |||
|+Relative cents | |||
! colspan="3" |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.{{Overline|6}}'' | |||
|1\11 | |||
''63.{{Overline|63}}'' | |||
|2\18 | |||
''77.{{Overline|7}}'' | |||
| rowspan="2" |1\7 | |||
''100'' | |||
|3\17 | |||
''123.5294…'' | |||
|2\10 | |||
''140'' | |||
|3\13 | |||
''161.5385…'' | |||
|- | |||
|Reb, Lab | |||
|Gb | |||
|1b, 1c | |||
|3\15 | |||
''140'' | |||
|2\11 | |||
''127.{{Overline|27}}'' | |||
|3\18 | |||
''116.{{Overline|6}}'' | |||
|2\17 | |||
''82.3529…'' | |||
|1\10 | |||
''70'' | |||
|1\13 | |||
''53.8461…'' | |||
|- | |||
|'''Re, La''' | |||
|'''G''' | |||
|'''1''' | |||
|'''4\15''' | |||
'''''186.{{Overline|6}}''''' | |||
|'''3\11''' | |||
'''''190.{{Overline|90}}''''' | |||
|'''5\18''' | |||
'''''194.{{Overline|4}}''''' | |||
|'''2\7''' | |||
'''''200''''' | |||
|'''5\17''' | |||
'''''205.8823…''''' | |||
|'''3\10''' | |||
'''''210''''' | |||
|'''4\13''' | |||
'''''215.3846…''''' | |||
|- | |||
|Re#, La# | |||
|G# | |||
|1# | |||
|5\15 | |||
''233.{{Overline|3}}'' | |||
|4\11 | |||
''254.{{Overline|54}}'' | |||
|7\18 | |||
''272.2̄'' | |||
| rowspan="2" |3\7 | |||
''300'' | |||
|8\17 | |||
''329.4117…'' | |||
|5\10 | |||
''350'' | |||
|7\13 | |||
''376.9230…'' | |||
|- | |||
|Mib, Sib | |||
|Ab | |||
|2b, 2c | |||
|7\15 | |||
''326.{{Overline|6}}'' | |||
|5\11 | |||
''318.{{Overline|18}}'' | |||
|8\18 | |||
''311.1̄'' | |||
|7\17 | |||
''288.2353…'' | |||
|4\10 | |||
''280'' | |||
|5\13 | |||
''269.2307…'' | |||
|- | |||
|Mi, Si | |||
|A | |||
|2 | |||
|8\15 | |||
''373.{{Overline|3}}'' | |||
|6\11 | |||
''381.{{Overline|81}}'' | |||
|10\18 | |||
''388.{{Overline|8}}'' | |||
|4\7 | |||
''400'' | |||
|10\17 | |||
''411.7647…'' | |||
|6\10 | |||
''420'' | |||
|8\13 | |||
''430.7692…'' | |||
|- | |||
|Mi#, Si# | |||
|A# | |||
|2# | |||
|9\15 | |||
''420'' | |||
| rowspan="2" |7\11 | |||
''445.{{Overline|45}}'' | |||
|12\18 | |||
''466.{{Overline|6}}'' | |||
|5\7 | |||
''500'' | |||
|13\17 | |||
''535.2941…'' | |||
|8\10 | |||
''560'' | |||
|11\13 | |||
''592.3076…'' | |||
|- | |||
|Fab, Dob | |||
|Bbb | |||
|3bb, 3cc | |||
|10\15 | |||
''466.{{Overline|6}}'' | |||
|11\18 | |||
''427.{{Overline|7}}'' | |||
|4\7 | |||
''400'' | |||
|9\17 | |||
''370.5882…'' | |||
|5\10 | |||
''350'' | |||
|6\13 | |||
''323.0769.…'' | |||
|- | |||
|'''Fa, Do''' | |||
|'''Bb''' | |||
|'''3b, 3c''' | |||
|'''11\15''' | |||
'''''513.{{Overline|3}}''''' | |||
|'''8\11''' | |||
'''''509.{{Overline|09}}''''' | |||
|'''13\18''' | |||
'''''505.{{Overline|5}}''''' | |||
|'''5\7''' | |||
'''''500''''' | |||
|'''12\17''' | |||
'''''494.1176…''''' | |||
|'''7\10''' | |||
'''''490''''' | |||
|'''9\13''' | |||
'''''484.6153…''''' | |||
|- | |||
|Fa#, Do# | |||
|B | |||
|3 | |||
|12\15 | |||
''560'' | |||
|9\11 | |||
''572.{{Overline|72}}'' | |||
|15\18 | |||
''583. {{Overline|3}}'' | |||
|6\7 | |||
''600'' | |||
|15\17 | |||
''617.6470…'' | |||
|9\10 | |||
''630'' | |||
|12\13 | |||
''646.1538…'' | |||
|- | |||
|Fax, Dox | |||
|B# | |||
|3# | |||
|13\15 | |||
''606.{{Overline|6}}'' | |||
| rowspan="2" |10\11 | |||
''636.{{Overline|36}}'' | |||
|17\18 | |||
''661. {{Overline|6}}'' | |||
|7\7 | |||
''700'' | |||
|18\17 | |||
''741.1764…'' | |||
|11\10 | |||
''770'' | |||
|15\13 | |||
''807.6923…'' | |||
|- | |||
|Dob, Solb | |||
|Hb | |||
|4b, 4c | |||
|14\15 | |||
''653.{{Overline|3}”'' | |||
|16\18 | |||
''622.{{Overline|2}}'' | |||
|6\7 | |||
''600'' | |||
|14\17 | |||
''576.4705…'' | |||
|8\10 | |||
''560'' | |||
|10\13 | |||
''538.4615…'' | |||
|- | |||
!Do, Sol | |||
!H | |||
!4 | |||
! colspan="7" |''700'' | |||
|- | |||
|Do#, Sol# | |||
|Η# | |||
|4# | |||
|16\15 | |||
''746.{{Overline|6}}'' | |||
|12\11 | |||
''763.{{Overline|63}}'' | |||
|20\18 | |||
''777.{{Overline|7}}'' | |||
| rowspan="2" |8\7 | |||
''800'' | |||
|20\17 | |||
''823.5294…'' | |||
|12\10 | |||
''840'' | |||
|16\13 | |||
''861.5385…'' | |||
|- | |||
|Reb, Lab | |||
|Cb | |||
|5b, 5c | |||
|18\15 | |||
''840'' | |||
|13\11 | |||
''827.{{Overline|27}}'' | |||
|21\18 | |||
''816.{{Overline|6}}'' | |||
|19\17 | |||
''782.3529…'' | |||
|11\10 | |||
''770'' | |||
|14\13 | |||
''753.8461…'' | |||
|- | |||
|'''Re, La''' | |||
|'''C''' | |||
|'''5''' | |||
|'''19\15''' | |||
'''''886.{{Overline|6}}''''' | |||
|'''14\11''' | |||
'''''890.{{Overline|90}}''''' | |||
|'''23\18''' | |||
'''''894. {{Overline|4}}''''' | |||
|'''9\7''' | |||
'''''900''''' | |||
|'''22\17''' | |||
'''''905.8823…''''' | |||
|'''13\10''' | |||
'''''910''''' | |||
|'''17\13''' | |||
'''''915.3846…''''' | |||
|- | |||
|Re#, La# | |||
|C# | |||
|5# | |||
|20\15 | |||
''933.{{Overline|3}}'' | |||
|15\11 | |||
''954.{{Overline|54}}'' | |||
|25\18 | |||
''972.2̄'' | |||
| rowspan="2" |10\7 | |||
''1000'' | |||
|25\17 | |||
''1029.4117…'' | |||
|15\10 | |||
''1050'' | |||
|20\13 | |||
''1076.9230…'' | |||
|- | |||
|Mib, Sib | |||
|Db | |||
|6b, 6c | |||
|22\15 | |||
''1026.{{Overline|6}}'' | |||
|16\11 | |||
''1018.{{Overline|18}}'' | |||
|26\18 | |||
''1011.{{Overline|1}}'' | |||
|24\17 | |||
''988.2353…'' | |||
|14\10 | |||
''980'' | |||
|18\13 | |||
''969.2307…'' | |||
|- | |||
|Mi, Si | |||
|D | |||
|6 | |||
|23\15 | |||
''1073.{{Overline|3}}'' | |||
|17\11 | |||
''1081.{{Overline|81}}'' | |||
|28\18 | |||
''1088.{{Overline|8}}'' | |||
|11\7 | |||
''1100'' | |||
|27\17 | |||
''1111.7647…'' | |||
|16\10 | |||
''1120'' | |||
|21\13 | |||
''1130.7692…'' | |||
|- | |||
|Mi#, Si# | |||
|D# | |||
|6# | |||
|24\15 | |||
''1120'' | |||
| rowspan="2" |18\11 | |||
''1145.{{Overline|45}}'' | |||
|30\18 | |||
''1166.{{Overline|6}}'' | |||
|12\7 | |||
''1200'' | |||
|30\17 | |||
''1235.2941…'' | |||
|18\10 | |||
''1260'' | |||
|24\13 | |||
''1292.3076…'' | |||
|- | |||
|Fab, Dob | |||
|Ebb | |||
|7bb, 7cc | |||
|25\15 | |||
''1166.{{Overline|6}}'' | |||
|29\18 | |||
''1127. {{Overline|7}}'' | |||
|11\7 | |||
''1100'' | |||
|26\17 | |||
''1070.5882…'' | |||
|15\10 | |||
''1050'' | |||
|19\13 | |||
''1023.0769…'' | |||
|- | |||
|'''Fa, Do''' | |||
|'''Eb''' | |||
|'''7b, 7c''' | |||
|'''26\15''' | |||
'''''1213.{{Overline|3}}''''' | |||
|'''19\11''' | |||
'''''1209.{{Overline|09}}''''' | |||
|'''31\18''' | |||
'''''1205.{{Overline|5}}''''' | |||
|'''12\7''' | |||
'''''1200''''' | |||
|'''29\17''' | |||
'''''1194.1176…''''' | |||
|'''17\10''' | |||
'''''1190''''' | |||
|'''22\13''' | |||
'''''1184.6153…''''' | |||
|- | |||
|Fa#, Do# | |||
|E | |||
|7 | |||
|27\15 | |||
''1260'' | |||
|20\11 | |||
''1272.{{Overline|72}}'' | |||
|33\18 | |||
''1283.{{Overline|3}}'' | |||
|13\7 | |||
''1300'' | |||
|32\17 | |||
''1317.6470…'' | |||
|19\10 | |||
''1330'' | |||
|25\13 | |||
''1346.1538…'' | |||
|- | |||
|Fax, Dox | |||
|E# | |||
|7# | |||
|28\15 | |||
''1306.{{Overline|6}}'' | |||
| rowspan="2" |21\11 | |||
''1336.{{Overline|36}}'' | |||
|35\18 | |||
''1361.{{Overline|1}}'' | |||
|14\7 | |||
''1400'' | |||
|35\17 | |||
''1441.1764…'' | |||
|21\10 | |||
''1470'' | |||
|28\13 | |||
''1507.6923…'' | |||
|- | |||
|Dob, Solb | |||
|Fb | |||
|8b, Fc | |||
|29\15 | |||
''1333.{{Overline|3}}'' | |||
|34\18 | |||
''1322.{{Overline|2}}'' | |||
|13\7 | |||
''1300'' | |||
|31\17 | |||
''1276.4705…'' | |||
|18\10 | |||
''1260'' | |||
|23\13 | |||
''1238.4615…'' | |||
|- | |||
!Do, Sol | |||
!F | |||
!8, F | |||
! colspan="7" |''1400'' | |||
|- | |||
|Do#, Sol# | |||
|F# | |||
|8#, F# | |||
|31\15 | |||
''1446.{{Overline|6}}'' | |||
|23\11 | |||
''1463.{{Overline|63}}'' | |||
|38\18 | |||
''1477.7̄'' | |||
| rowspan="2" |15\7 | |||
''1500'' | |||
|37\17 | |||
''1523.5294…'' | |||
|22\10 | |||
''1540'' | |||
|29\13 | |||
''1561.5385…'' | |||
|- | |||
|Reb, Lab | |||
|Gb | |||
|9b, Gc | |||
|33\15 | |||
''1540'' | |||
|24\11 | |||
''1527.{{Overline|27}}'' | |||
|39\18 | |||
''1516.{{Overline|6}}'' | |||
|36\17 | |||
''1482.3529…'' | |||
|21\10 | |||
''1470'' | |||
|27\13 | |||
''1453.8461…'' | |||
|- | |||
|'''Re, La''' | |||
|'''G''' | |||
|'''9, G''' | |||
|'''34\15''' | |||
'''''1586.{{Overline|6}}''''' | |||
|'''25\11''' | |||
'''''1590.{{Overline|90}}''''' | |||
|'''41\18''' | |||
'''''1594.{{Overline|4}}''''' | |||
|'''16\7''' | |||
'''''1600''''' | |||
|'''39\17''' | |||
'''''1605.8823…''''' | |||
|'''23\10''' | |||
'''''1610''''' | |||
|'''30\13''' | |||
'''''1615.3846…''''' | |||
|- | |||
|Re#, La# | |||
|G# | |||
|9#, G# | |||
|35\15 | |||
''1633.{{Overline|3}}'' | |||
|26\11 | |||
''1654.{{Overline|54}}'' | |||
|43\18 | |||
''1672.{{Overline|2}}'' | |||
| rowspan="2" |17\7 | |||
''1700'' | |||
|42\17 | |||
''1729.4117…'' | |||
|25\10 | |||
''1750'' | |||
|33\13 | |||
''1776.9230…'' | |||
|- | |||
|Mib, Sib | |||
|Ab | |||
|Xb, Ac | |||
|37\15 | |||
''1726.{{Overline|6}}'' | |||
|27\11 | |||
''1718.{{Overline|18}}'' | |||
|44\18 | |||
''1711.{{Overline|1}}'' | |||
|41\17 | |||
''1688.2353…'' | |||
|24\10 | |||
''1680'' | |||
|31\13 | |||
''1669.2307…'' | |||
|- | |||
|Mi, Si | |||
|A | |||
|X, A | |||
|38\15 | |||
''1773.{{Overline|3}}'' | |||
|28\11 | |||
''1781.{{Overline|81}}'' | |||
|46\18 | |||
''1788.{{Overline|8}}'' | |||
|18\7 | |||
''1800'' | |||
|44\17 | |||
''1811.7647…'' | |||
|26\10 | |||
''1820'' | |||
|34\13 | |||
''1830.7692…'' | |||
|- | |||
|Mi#, Si# | |||
|A# | |||
|X#, A# | |||
|39\15 | |||
''1820'' | |||
| rowspan="2" |29\11 | |||
''1845.{{Overline|45}}'' | |||
|48\18 | |||
''1866.{{Overline|6}}'' | |||
|19\7 | |||
''1900'' | |||
|47\17 | |||
''1935.2941…'' | |||
|28\10 | |||
''1960'' | |||
|37\13 | |||
''1992.3076…'' | |||
|- | |||
|Fab, Dob | |||
|Bbb | |||
|Ebb, Ccc | |||
|40\15 | |||
''1866.{{Overline|6}}'' | |||
|47\18 | |||
''1827. {{Overline|7}}'' | |||
|18\7 | |||
''1800'' | |||
|43\17 | |||
''1770.5882…'' | |||
|25\10 | |||
''1750'' | |||
|32\13 | |||
''1723.0769…'' | |||
|- | |||
|'''Fa, Do''' | |||
|'''Bb''' | |||
|Eb, Cc | |||
|'''41\15''' | |||
'''''1913.{{Overline|3}}''''' | |||
|'''30\11''' | |||
'''''1909.{{Overline|09}}''''' | |||
|'''49\18''' | |||
'''''1905.{{Overline|5}}''''' | |||
|'''19\7''' | |||
'''''1900''''' | |||
|'''46\17''' | |||
'''''1894.1176…''''' | |||
|'''27\10''' | |||
'''''1890''''' | |||
|'''35\13''' | |||
'''''1884.6153…''''' | |||
|- | |||
|Fa#, Do# | |||
|B | |||
|E, C | |||
|42\15 | |||
''1960'' | |||
|31\11 | |||
''1972.{{Overline|72}}'' | |||
|51\18 | |||
''1983.{{Overline|3}}'' | |||
|20\7 | |||
''2000'' | |||
|49\17 | |||
''2017.6470…'' | |||
|29\10 | |||
''2030'' | |||
|38\13 | |||
''2046.15385…'' | |||
|- | |||
|Fax, Dox | |||
|B# | |||
|Ex, Cx | |||
|43\15 | |||
''2006.{{Overline|6}}'' | |||
| rowspan="2" |32\11 | |||
''2036.3636'' | |||
|53\18 | |||
''2061.{{Overline|1}}'' | |||
|21\7 | |||
''2100'' | |||
|52\17 | |||
''2141.1764…'' | |||
|31\10 | |||
''2170'' | |||
|41\13 | |||
''2207.6923…'' | |||
|- | |||
|Dob, Solb | |||
|Hb | |||
|0b, Dc | |||
|44\15 | |||
''2053.{{Overline|3}}'' | |||
|52\18 | |||
''2022.{{Overline|2}}'' | |||
|20\7 | |||
''2000'' | |||
|48\17 | |||
''1976.4705…'' | |||
|28\10 | |||
''1960'' | |||
|36\13 | |||
1938.6153… | |||
|- | |||
!Do, Sol | |||
!H | |||
!0, D | |||
! colspan="7" |2100 | |||
|} | |||
==Intervals== | |||
{| class="wikitable" | |||
!Generators | |||
!Sesquitave notation | |||
!Interval category name | |||
!Generators | |||
!Notation of 3/2 inverse | |||
!Interval category name | |||
|- | |||
| colspan="6" |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 | |||
|- | |||
| colspan="6" |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: | |||
{| class="wikitable" | |||
|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: | |||
{| class="wikitable" | |||
!Mode | |||
!Scale | |||
![[Modal UDP Notation|UDP]] | |||
! colspan="3" |Interval type | |||
|- | |||
!name | |||
!pattern | |||
!notation | |||
!2nd | |||
!3rd | |||
!4th | |||
|- | |||
|Lydian | |||
|LLLs | |||
|<nowiki>3|0</nowiki> | |||
|P | |||
|M | |||
|A | |||
|- | |||
|Major | |||
|LLsL | |||
|<nowiki>2|1</nowiki> | |||
|P | |||
|M | |||
|P | |||
|- | |||
|Minor | |||
|LLsL | |||
|<nowiki>1|2</nowiki> | |||
|P | |||
|m | |||
|P | |||
|- | |||
|Phrygian | |||
|LsLL | |||
|<nowiki>0|3</nowiki> | |||
|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 <code>root-(p-2g)-(2p-3g)</code> (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 | |||
[[Comma]] list: [[81/80]] | |||
[[POL2]] generator: ~9/8 = [[Tel:192.6406|192.6406]] | |||
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -2 -3}}] | |||
[[Vals]]: {{val list|~(7edf, 11edf, 18edf)}} | |||
==='''Napoli-Superpyth'''=== | |||
[[Subgroup]]: 3/2.7/6.14/9 | |||
[[Comma]] list: [[64/63]] | |||
[[POL2]] generator: ~8/7 = [[Tel:218.6371|218.6371]] | |||
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -2 -3}}] | |||
[[Vals]]: {{val list|~(7edf, 10edf, 13edf, 16edf)}} | |||
====Scale tree==== | |||
The spectrum looks like this: | |||
{| class="wikitable" | |||
! colspan="3" rowspan="2" |Generator | |||
(bright) | |||
! colspan="2" |Cents | |||
! rowspan="2" |L | |||
! rowspan="2" |s | |||
! rowspan="2" |L/s | |||
! rowspan="2" |Comments | |||
|- | |||
!<u>Normalised</u> | |||
!''ed7\12'' | |||
|- | |||
|1\4 | |||
| | |||
| | |||
|<u>171.428…</u> | |||
|''175'' | |||
|1 | |||
|1 | |||
|1.000 | |||
|Equalised | |||
|- | |||
|6\23 | |||
| | |||
| | |||
|<u>180</u> | |||
|''182.608…'' | |||
|6 | |||
|5 | |||
|1.200 | |||
| | |||
|- | |||
| | |||
|11\42 | |||
| | |||
|<u>180.821…</u> | |||
|''183.{{Overline|3}}'' | |||
|11 | |||
|9 | |||
|1.222 | |||
| | |||
|- | |||
|5\19 | |||
| | |||
| | |||
|<u>181.{{Overline|81}}</u> | |||
|''184.210…'' | |||
|5 | |||
|4 | |||
|1.250 | |||
| | |||
|- | |||
| | |||
|14\53 | |||
| | |||
|<u>182.608…</u> | |||
|''184.905…'' | |||
|14 | |||
|11 | |||
|1.273 | |||
| | |||
|- | |||
| | |||
|9\34 | |||
| | |||
|<u>183.050…</u> | |||
|''185.294…'' | |||
|9 | |||
|7 | |||
|1.286 | |||
| | |||
|- | |||
|4\15 | |||
| | |||
| | |||
|<u>184.615…</u> | |||
|''186.6̄'' | |||
|4 | |||
|3 | |||
|1.333 | |||
| | |||
|- | |||
| | |||
|11\41 | |||
| | |||
|<u>185.915…</u> | |||
|''187.804…'' | |||
|11 | |||
|8 | |||
|1.375 | |||
| | |||
|- | |||
| | |||
|7\26 | |||
| | |||
|<u>186.{{Overline|6}}</u> | |||
|''188.461…'' | |||
|7 | |||
|5 | |||
|1.400 | |||
| | |||
|- | |||
| | |||
|10\37 | |||
| | |||
|<u>187.5</u> | |||
|''189.{{Overline|189}}'' | |||
|10 | |||
|7 | |||
|1.429 | |||
| | |||
|- | |||
| | |||
|13\48 | |||
| | |||
|<u>187.951…</u> | |||
|''189.58{{Overline|3}}'' | |||
|13 | |||
|9 | |||
|1.444 | |||
| | |||
|- | |||
| | |||
|16\59 | |||
| | |||
|<u>188.235…</u> | |||
|''189.830…'' | |||
|16 | |||
|11 | |||
|1.4545 | |||
| | |||
|- | |||
|3\11 | |||
| | |||
| | |||
|<u>189.473…</u> | |||
|''190.{{Overline|90}}'' | |||
|3 | |||
|2 | |||
|1.500 | |||
|Napoli-Meantone starts here | |||
|- | |||
| | |||
|17\62 | |||
| | |||
|<u>190.654…</u> | |||
|''191.935…'' | |||
|17 | |||
|11 | |||
|1.5455 | |||
| | |||
|- | |||
| | |||
|14\51 | |||
| | |||
|<u>190.{{Overline|90}}</u> | |||
|''192.156…'' | |||
|14 | |||
|9 | |||
|1.556 | |||
| | |||
|- | |||
| | |||
|11\40 | |||
| | |||
|<u>191.304…</u> | |||
|''192.5'' | |||
|11 | |||
|7 | |||
|1.571 | |||
| | |||
|- | |||
| | |||
|8\29 | |||
| | |||
|<u>192</u> | |||
|''193.103…'' | |||
|8 | |||
|5 | |||
|1.600 | |||
| | |||
|- | |||
| | |||
|5\18 | |||
| | |||
|<u>193.548…</u> | |||
|''194.{{Overline|4}}'' | |||
|5 | |||
|3 | |||
|1.667 | |||
| | |||
|- | |||
| | |||
| | |||
|12\43 | |||
|<u>194.{{Overline|594}}</u> | |||
|''195.348…'' | |||
|12 | |||
|7 | |||
|1.714 | |||
| | |||
|- | |||
| | |||
|7\25 | |||
| | |||
|<u>195.348…</u> | |||
|''196'' | |||
|7 | |||
|4 | |||
|1.750 | |||
| | |||
|- | |||
| | |||
|9\32 | |||
| | |||
|<u>196.{{Overline|36}}</u> | |||
|''196.875'' | |||
|9 | |||
|5 | |||
|1.800 | |||
| | |||
|- | |||
| | |||
|11\39 | |||
| | |||
|<u>197.014…</u> | |||
|''197.435…'' | |||
|11 | |||
|6 | |||
|1.833 | |||
| | |||
|- | |||
| | |||
|13\46 | |||
| | |||
|<u>197.468…</u> | |||
|''197.826…'' | |||
|13 | |||
|7 | |||
|1.857 | |||
| | |||
|- | |||
| | |||
|15\53 | |||
| | |||
|<u>197.802…</u> | |||
|''198.113…'' | |||
|15 | |||
|8 | |||
|1.875 | |||
| | |||
|- | |||
| | |||
|17\60 | |||
| | |||
|<u>198.058…</u> | |||
|''198.3̄'' | |||
|17 | |||
|9 | |||
|1.889 | |||
| | |||
|- | |||
| | |||
|19\67 | |||
| | |||
|<u>198.260…</u> | |||
|''198.507…'' | |||
|19 | |||
|10 | |||
|1.900 | |||
| | |||
|- | |||
| | |||
|21\74 | |||
| | |||
|<u>198.425…</u> | |||
|''198.{{Overline|648}}'' | |||
|21 | |||
|11 | |||
|1.909 | |||
| | |||
|- | |||
| | |||
|23\81 | |||
| | |||
|<u>198.561…</u> | |||
|''198.765…'' | |||
|23 | |||
|12 | |||
|1.917 | |||
| | |||
|- | |||
| | |||
|25\88 | |||
| | |||
|<u>198.675…</u> | |||
|''198.8{{Overline|63}}'' | |||
|25 | |||
|13 | |||
|1.923 | |||
| | |||
|- | |||
| | |||
|27\95 | |||
| | |||
|<u>198.773…</u> | |||
|''198.947…'' | |||
|27 | |||
|14 | |||
|1.929 | |||
| | |||
|- | |||
| | |||
|29\102 | |||
| | |||
|<u>198.857…</u> | |||
|''199.019…'' | |||
|29 | |||
|15 | |||
|1.933 | |||
| | |||
|- | |||
| | |||
|31\109 | |||
| | |||
|<u>198.930…</u> | |||
|''199.082…'' | |||
|31 | |||
|16 | |||
|1.9375 | |||
| | |||
|- | |||
| | |||
|33\116 | |||
| | |||
|<u>198.994…</u> | |||
|''199.137…'' | |||
|33 | |||
|17 | |||
|1.941 | |||
| | |||
|- | |||
| | |||
|35\123 | |||
| | |||
|<u>199.052…</u> | |||
|''199.186…'' | |||
|35 | |||
|18 | |||
|1.944 | |||
| | |||
|- | |||
|2\7 | |||
| | |||
| | |||
|<u>200</u> | |||
|''200'' | |||
|2 | |||
|1 | |||
|2.000 | |||
|Napoli-Meantone ends, Napoli-Pythagorean begins | |||
|- | |||
| | |||
|19\66 | |||
| | |||
|<u>201.769…</u> | |||
|''201.{{Overline|51}}'' | |||
|19 | |||
|9 | |||
|2.111 | |||
| | |||
|- | |||
| | |||
|17\59 | |||
| | |||
|<u>201.980…</u> | |||
|''201.694…'' | |||
|17 | |||
|8 | |||
|2.125 | |||
| | |||
|- | |||
| | |||
|15\52 | |||
| | |||
|<u>202.247…</u> | |||
|''201.923…'' | |||
|15 | |||
|7 | |||
|2.143 | |||
| | |||
|- | |||
| | |||
|13\45 | |||
| | |||
|<u>202.597…</u> | |||
|''202.{{Overline|2}}'' | |||
|13 | |||
|6 | |||
|2.167 | |||
| | |||
|- | |||
| | |||
|11\38 | |||
| | |||
|<u>203.076…</u> | |||
|''202.631…'' | |||
|11 | |||
|5 | |||
|2.200 | |||
| | |||
|- | |||
| | |||
|9\31 | |||
| | |||
|<u>203.773…</u> | |||
|''203.225…'' | |||
|9 | |||
|4 | |||
|2.250 | |||
| | |||
|- | |||
| | |||
|7\24 | |||
| | |||
|<u>204.878…</u> | |||
|''204.1{{Overline|6}}'' | |||
|7 | |||
|3 | |||
|2.333 | |||
| | |||
|- | |||
| | |||
| | |||
|12\41 | |||
|<u>205.714…</u> | |||
|''204.878…'' | |||
|12 | |||
|5 | |||
|2.400 | |||
| | |||
|- | |||
| | |||
|5\17 | |||
| | |||
|<u>206.896…</u> | |||
|''205.882…'' | |||
|5 | |||
|2 | |||
|2.500 | |||
|Napoli-Neogothic heartland is from here… | |||
|- | |||
| | |||
| | |||
|18\61 | |||
|<u>207.692…</u> | |||
|''206.557…'' | |||
|18 | |||
|7 | |||
|2.571 | |||
| | |||
|- | |||
| | |||
| | |||
|13\44 | |||
|<u>208</u> | |||
|''206.{{Overline|81}}'' | |||
|13 | |||
|5 | |||
|2.600 | |||
| | |||
|- | |||
| | |||
|8\27 | |||
| | |||
|<u>208.695…</u> | |||
|''207.{{Overline|407}}'' | |||
|8 | |||
|3 | |||
|2.667 | |||
|…to here | |||
|- | |||
| | |||
|11\37 | |||
| | |||
|<u>209.523…</u> | |||
|''208.{{Overline|108}}'' | |||
|11 | |||
|4 | |||
|2.750 | |||
| | |||
|- | |||
| | |||
|14\47 | |||
| | |||
|<u>210</u> | |||
|''208.510…'' | |||
|14 | |||
|5 | |||
|2.800 | |||
| | |||
|- | |||
| | |||
|17\57 | |||
| | |||
|<u>210.309…</u> | |||
|''208.771…'' | |||
|17 | |||
|6 | |||
|2.833 | |||
| | |||
|- | |||
| | |||
|20\67 | |||
| | |||
|<u>210.526…</u> | |||
|''208.955…'' | |||
|20 | |||
|7 | |||
|2.857 | |||
| | |||
|- | |||
| | |||
|23\77 | |||
| | |||
|<u>210.687…</u> | |||
|''209.{{Overline|09}}'' | |||
|23 | |||
|8 | |||
|2.875 | |||
| | |||
|- | |||
|3\10 | |||
| | |||
| | |||
|<u>211.764…</u> | |||
|''210'' | |||
|3 | |||
|1 | |||
|3.000 | |||
|Napoli-Pythagorean ends, Napoli-Superpyth begins | |||
|- | |||
| | |||
|22\73 | |||
| | |||
|<u>212.903…</u> | |||
|''210.958…'' | |||
|22 | |||
|7 | |||
|3.143 | |||
| | |||
|- | |||
| | |||
|19\63 | |||
| | |||
|<u>213.084…</u> | |||
|''211.{{Overline|1}}'' | |||
|19 | |||
|6 | |||
|3.167 | |||
| | |||
|- | |||
| | |||
|16\53 | |||
| | |||
|<u>213.{{Overline|3}}</u> | |||
|''211.320…'' | |||
|16 | |||
|5 | |||
|3.200 | |||
| | |||
|- | |||
| | |||
|13\43 | |||
| | |||
|<u>213.698…</u> | |||
|''211.627…'' | |||
|13 | |||
|4 | |||
|3.250 | |||
| | |||
|- | |||
| | |||
|10\33 | |||
| | |||
|<u>214.285…</u> | |||
|''212.{{Overline|12}}'' | |||
|10 | |||
|3 | |||
|3.333 | |||
| | |||
|- | |||
| | |||
|7\23 | |||
| | |||
|<u>215.384…</u> | |||
|''213.043…'' | |||
|7 | |||
|2 | |||
|3.500 | |||
| | |||
|- | |||
| | |||
|11\36 | |||
| | |||
|<u>216.393…</u> | |||
|''213.{{Overline|3}}'' | |||
|11 | |||
|3 | |||
|3.667 | |||
| | |||
|- | |||
| | |||
|15\49 | |||
| | |||
|<u>216.867…</u> | |||
|''214.285…'' | |||
|15 | |||
|4 | |||
|3.750 | |||
| | |||
|- | |||
|4\13 | |||
| | |||
| | |||
|<u>218.{{Overline|18}}</u> | |||
|''215.385…'' | |||
|4 | |||
|1 | |||
|4.000 | |||
| | |||
|- | |||
| | |||
|13\42 | |||
| | |||
|<u>219.718…</u> | |||
|''216.{{Overline|6}}'' | |||
|13 | |||
|3 | |||
|4.333 | |||
| | |||
|- | |||
| | |||
|9\29 | |||
| | |||
|<u>220.408…</u> | |||
|''217.241…'' | |||
|9 | |||
|2 | |||
|4.500 | |||
| | |||
|- | |||
| | |||
|14\45 | |||
| | |||
|<u>221.052…</u> | |||
|''217.{{Overline|7}}'' | |||
|14 | |||
|3 | |||
|4.667 | |||
| | |||
|- | |||
|5\16 | |||
| | |||
| | |||
|<u>222.{{Overline|2}}</u> | |||
|''218.75'' | |||
|5 | |||
|1 | |||
|5.000 | |||
|Napoli-Superpyth ends | |||
|- | |||
| | |||
|16\51 | |||
| | |||
|<u>223.255…</u> | |||
|''219.607…'' | |||
|16 | |||
|3 | |||
|5.333 | |||
| | |||
|- | |||
| | |||
|11\35 | |||
| | |||
|<u>223.728…</u> | |||
|''220'' | |||
|11 | |||
|2 | |||
|5.500 | |||
| | |||
|- | |||
| | |||
|17\54 | |||
| | |||
|<u>224.175…</u> | |||
|''220.{{Overline|370}}'' | |||
|17 | |||
|3 | |||
|5.667 | |||
| | |||
|- | |||
|6\19 | |||
| | |||
| | |||
|<u>225</u> | |||
|''221.052…'' | |||
|6 | |||
|1 | |||
|6.000 | |||
| | |||
|- | |||
|1\3 | |||
| | |||
| | |||
|<u>240</u> | |||
|''233.{{Overline|3}}'' | |||
|1 | |||
|0 | |||
|→ inf | |||
|Paucitonic | |||
|} | |||
<nowiki />* Because this temperament almost seems too good to be true. | <nowiki />* Because this temperament almost seems too good to be true. | ||
[[Category:31edo]] | [[Category:31edo]] | ||
[[Category:meantone]] | [[Category:meantone]] | ||
[[Category:pentave]] | [[Category:pentave]] |