1318:
190:
130:
1200:. Beyond this point, the regular major and minor thirds approximate simple ratios of numbers with prime factors 2-3-7, such as the 9/7 or septimal major third (435.084 cents) and 7/6 or septimal minor third (266.871 cents). At the same time, the regular tones more and more closely approximate a large 8/7 tone (231.174 cents), and regular minor sevenths the "harmonic seventh" at the simple ratio of 7/4 (968.826 cents). This septimal range extends out to around 711.11 cents or
1362:
33:
1303:
generated tonal intervals, and (b) the corresponding partials of a similarly-generated pseudo-Harmonic timbre. Hence, the relationship between the syntonic temperament and its note-aligned timbres can be seen as a generalization of the special relationship between Just
Intonation and the Harmonic Series.
985:. For any tuning, the chromatic semitone is the space between a flat note and its natural, or a natural note and its sharp; between a white key and either the black key above it (if tuned as a sharp) or the black key below it (if tuned as a flat); in most tunings, the two intervals are different. The
647:
These two extremes are not included as "regular" diatonic tunings, because to be "regular" the pattern of five large and two small steps has to be preserved; everything in between is regular, however small the semitones are without vanishing completely, or however large they become while still being
1369:
The legend of Figure 2 (on the right side of the figure) shows a stack of P5s centered on D. Each resulting note represents an interval in the syntonic temperament with D as the tonic. The body of the figure shows how the widths (from D) of these intervals change as the width of the P5 is changed
1313:
If one considers the syntonic temperament's tuning continuum as a string, and individual tunings as beads on that string, then one can view much of the traditional microtonal literature as being focused on the differences among the beads, whereas the syntonic temperament can be viewed as being
1485:
is based on the syntonic temperament. Guido 2.0 seeks to achieve a 10x increase in the efficiency of music education by exposing the invariant properties of music's syntonic temperament (octave invariance, transpositional invariance, tuning invariance, and fingering invariance) with geometric
1302:
This combination is necessary and sufficient to define a set of relationships among tonal intervals that is invariant across the syntonic temperament's tuning range. Hence, it also defines an invariant mapping -- all across the tuning continuum -- between (a) the notes at these (pseudo-Just)
240:
with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's. In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size (TTTS or a permutation of that) which makes it a
1192:
If the fifth is tuned slightly sharp of just, between 702.4 and 705.9 cents, the result is very sharp major thirds with ratios near 14/11 (417.508 cents) and very flat minor thirds around 13/11 (289.210 cents). These tunings are known as "parapythagorean" tunings.
153:
209:
197:
1341:
As shown in the figure at right, the tonally valid tuning range of the syntonic temperament includes a number of historically important tunings, such as the currently popular 12-tone equal division of the octave (12-edo tuning, also known as
1333:
for the syntonic temperament. The fingering-pattern of any given musical structure is the same in any tuning on the syntonic temperament's tuning continuum. The combination of an isomorphic keyboard and continuously variable tuning supports
600:
177:
165:
1438:
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402:
1408:
1393:
1456:
1378:
1181:- the amount by which a chain of eight fifths reduced to an octave is sharper than the just minor sixth 8/5. So for instance, a 1/8 schisma temperament will achieve a pure 8/5 in an ascending chain of eight fifths.
1233:
The "ultraseptimal" or "ultrapythagorean" range encompasses the sharpmost extreme, between 711.11 cents as seen in 27-tone equal temperament all the way to the upper bound of the regular diatonic at 720 cents or
1423:
1310:, a novel expansion of the framework of tonality, which includes timbre effects such as primeness, conicality, and richness, and tonal effects such as polyphonic tuning bends and dynamic tuning progressions.
1033:. For any tuning, the diatonic semitone is the relative pitch difference on a standard keyboard between two white keys that have no black key between them. The pattern of chromatic and diatonic semitones is
137:
1329:. Because the syntonic temperament and the Wicki-Hayden note-layout are generated using the same generator and period, they are isomorphic with each other; hence, the Wicki-Hayden note-layout is an
1071:
However, his "range of recognizability" is more restrictive than "regular diatonic tuning". For instance, he requires the diatonic semitone to be at least 25 cents in size. See for a summary.
265:-s are semitones which are half, or approximately half the size of the tone. But in the more general regular diatonic tunings, the two steps can be of any relation within the range between
1215:(685.71 cents) and the range of historical meantones beginning around 19 tone equal temperament (694.74 cents). Here, the diatonic semitones approach the size of the whole tone.
783:, ascending fourths, reduced to the octave), but the tempered fifth is the more usual choice, and in any case, because fifths and fourths are octave complements, rising by
1565:
725:
and so on; in all those examples the result is "reduced to the octave" (lowered by an octave whenever a note in the sequence exceeds an octave above the starting tone).
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300:(fifth, p5, between 685.71 ¢ and 720 ¢). Note that regular diatonic tunings are not limited to the notes of any particular diatonic scale used to describe them.
1317:
1858:
663:
The notes are connected together through a chain of six fifths reduced to the octave, or equivalently, through ascending fifths and descending fourths (e.g.
775:, i.e. regular temperaments with two generators: the octave and the tempered fifth. One can use the tempered fourth as an alternative generator (e.g. as
413:
322:
1298:
the "tuning range" of P5 temperings in which the generated minor second is neither larger than the generated major second, nor smaller than the unison.
1790:
50:
1923:
1446:(53-edo), a stack of 53 such intervals - each just 3/44 of a cent short of a pure fifth - makes 31 octaves, producing 53-edo tuning. S/T = 4/9.
1515:
1041:
system is the limit as the chromatic semitone tends to zero, and the five tone system in the limit as the diatonic semitone tends to zero.
1933:
1211:
The "inframeantone" or "flattone" range is the flatmost extreme, where the fifth is between the lower bound for the regular diatonic of
97:
1295:(i.e., in which the syntonic comma is tempered to zero, making the generated major third as wide as two generated major seconds); and
1242:
Diatonic scales constructed in equal temperaments can have fifths either wider or narrower than a just 3/2. Here are a few examples:
1548:
116:
69:
1896:
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54:
1416:(31-edo), a stack of 31 such intervals would show equal gaps between each such interval, producing 31-edo tuning. S/T = 3/5.
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2081:
83:
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1783:
2019:
1386:, the intervals converge on just 7 widths (assuming octave equivalence of 0 and 1200 cents), producing 7-edo. S/T = 0.
65:
1863:
129:
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relationships are preserved. For instance, in all regular diatonic tunings, just as for the
Pythagorean diatonic:
43:
1848:
1201:
1197:
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At 705.882 cents, with fifths tempered in the wide direction by 3.929 cents, the result is the diatonic scale in
1120:
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the continuum of tunings in which the tempered perfect fifth (P5) is the generator and the octave is the period;
2132:
1326:
1235:
1212:
2268:
193:
T and S in various equal temperaments (*5-tone and 7-tone are the limits of and not regular diatonic tunings)
1928:
1853:
1776:
1365:
Figure 2: Change in widths of intervals of the syntonic temperament across its tuning continuum (tonic is D)
909:. Three notes spaced by a chromatic and diatonic semitone make a whole tone between the first and the last:
1886:
629:. As the semitones get larger, eventually the steps are all the same size, and the result is in seven tone
2124:
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When the fifths are slightly flatter than in just intonation, then we are in the region of the historical
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A chain of three tones spaced in equal-sized fifths (reduced to the octave) generates a whole tone (e.g.
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Maintaining an invariant mapping between notes and partials, across the entire tuning range, enables
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1401:(19-edo), the gaps between these 19 intervals are all equal, producing 19-edo tuning. S/T = 2/3.
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1972:
1911:
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798:) and the chain of fifths can be continued in either direction to obtain a twelve tone system
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1980:
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595:{\displaystyle \ {\mathsf {p5}}={\tfrac {\ 1\ }{2}}\left(\ T+1200{\text{¢}}\right)=3\ T+s\ }
1675:
Proceedings of the Annual
Meeting of the South Central Chapter of the College Music Society
1149:
comma meantone - achieves pure major sevenths of almost exactly 15/8; fifth is 697.67 cents
956:, as the space between the first and the last; it is the change of pitch needed to raise a
2238:
1878:
1218:
The range between 690.91 cents (the fifth of 33-tone equal temperament, which reporesents
1080:
1722:
1599:"Isomorphic Controllers and Dynamic Tuning: Invariant Fingering over a Tuning Continuum"
1916:
1891:
1838:
1799:
1431:(12-edo), the sharp notes and flat notes are equal, producing 12-edo tuning. S/T = 1/2.
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1953:
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1001:
above, is the change in pitch of a sequence of six notes spaced by fifths, e.g. from
927:
691:
A sequence of six tones spaced in fourths generates a semitone in the same way (e.g.
225:
1624:
1579:
17:
2175:
2015:
1818:
1136:; achieves major thirds extremely close to 5/4 (387.1 cents); fifth is 696.77 cents
297:
161:, and also (both written the same as 12-tone in Easley Blackwood notation) 17-tone
1598:
1238:. As one tends towards 5 equal, the diatonic semitones become smaller and smaller.
1162:
comma meantone - achieves a rational diatonic tritone 45/32; fifth is 698.18 cents
764:. However, these strange scales are only mentioned here to dismiss them; they not
932:, usually prefixed by the name of the tuning system that generates it, such as a
2263:
2258:
2248:
1843:
1823:
1035: c d c d d c d c d c d d
32:
1566:"The Structure of Recognizable Diatonic Tunings by Easley Blackwood - a review"
1321:
Figure 1: The syntonic temperament's tuning continuum, from (Milne et al. 2007)
1228:-comma meantone) and 685.71 cents has been called the "deeptone" range by some.
752:
scales with two large steps and five small steps, and eventually, when all the
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2222:
1615:
229:
1671:"Dynamic Tonality: Extending the Framework of Tonality into the 21st Century"
1639:
1509:
488:{\displaystyle \ T={\tfrac {\ 1\ }{5}}\left(\ 1200{\text{¢}}-2\ s\ \right)\ }
397:{\displaystyle \ s={\tfrac {\ 1\ }{2}}\left(\ 1200{\text{¢}}-5\ T\ \right)\ }
236:" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the
2243:
2197:
1514:(MS thesis). Program in Media Arts & Sciences. Machover, Tod (advisor).
233:
1166:
When the fifths are exactly 3/2, or around 702 cents, the result is the
952:
A chain of eight notes spaced in fifths generates a chromatic semitone,
2217:
2167:
1703:
Milne, A., Sethares, W.A., Tiedje, S., Prechtl, A., and
Plamondon, J.,
1178:
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1901:
1828:
237:
1049:
The regular diatonic tunings include all linear temperaments within
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2202:
2190:
1768:
1464:, the pitches converge on just 5 widths, producing 5-edo. S/T = 1.
1360:
1316:
188:
1117:-Comma meantone; achieves pure minor thirds of almost exactly 6/5
713:
A chain of four tones spaced in fourths generates a minor third (
2207:
1325:
The notes of the syntonic temperament are best played using the
1177:, where the temperament is measured in terms of a fraction of a
1772:
1091:
12-Tone equal temperament, practically indistinguishable from
26:
1358:), non-equal (Pythagorean, meantone), circulating, and Just.
1478:
investigates the musical theory of the syntonic temperament.
1354:. Tunings in the syntonic temperament can be equal (12-edo,
874:, etc., another moment of symmetry with two interval sizes.
651:"Regular" here is understood in the sense of a mapping from
1597:
Milne, Andrew; Sethares, William; Plamondon, James (2007).
926:
The small difference in pitch between the two is called a
1751:
710:) generates a major third, consisting of two whole tones.
1705:"Spectral Tools for Dynamic Tonality and Audio Morphing"
1486:
invariance. Guido 2.0 is the Music
Education aspect of
1173:
For fifths slightly narrower than 3/2, the result is a
1061:
the fifth tempered to between 4/7 and 3/5 of an octave;
525:
427:
336:
507:
416:
325:
1638:
Milne, Andrew; Sethares, William; Plamondon, James.
1469:
Research projects regarding the syntonic temperament
1370:
across the syntonic temperament's tuning continuum.
2231:
2148:
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2064:
1971:
1962:
1877:
1806:
57:. Unsourced material may be challenged and removed.
883:, there are actually two: the chromatic semitone,
594:
487:
396:
702:A sequence of five fifths spaced in fifths (e.g.
311:, and the fifth (p5), given one of the values:
1669:Plamondon, J., Milne, A., and Sethares, W.A.,
1541:The Structure of Recognizable Diatonic Tunings
1511:Dynamic intonation for synthesizer performance
1152:55 tone equal temperament—Equivalent to
1139:43 tone equal temperament—Equivalent to
1067:the major second larger than the minor second.
1055:The Structure of Recognizable Diatonic Tunings
1037:or some mixed-around version of it. Here, the
787:produces the same result as rising by fifths.
1784:
1686:Milne, A., Sethares, W.A. and Plamondon, J.,
1314:focused on the commonality along the string.
1268:, and 43 have fifths narrower than a just 3/2
303:One may determine the corresponding cents of
8:
1859:List of intervals in 5-limit just intonation
744:further, so that it becomes larger than the
1968:
1791:
1777:
1769:
1257:, and 27 have fifths wider than a just 3/2
1064:the major and minor seconds both positive;
1614:
1523:
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524:
512:
511:
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457:
426:
415:
366:
335:
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117:Learn how and when to remove this message
245:with the tempered fifth as a generator.
128:
1500:
790:All regular diatonic tunings are also
771:All regular diatonic tunings are also
740:and continues to increase the size of
516:
513:
1516:Massachusetts Institute of Technology
1083:, which distribute or temper out the
877:Instead of there being one semitone,
7:
1688:Tuning Continua and Keyboard Layouts
1665:
1663:
1508:Denckla, Benjamin Frederick (1997).
1075:Significant regions within the range
648:strictly smaller than a whole tone.
55:adding citations to reliable sources
760:, so a division of the octave into
1053:"Range of Recognizability" in his
25:
1564:Serafini, Carlo (9 August 2015).
1185:achieves a good approximation to
1934:Ptolemy's intense diatonic scale
1692:Journal of Mathematics and Music
31:
1539:Blackwood, Easley (July 2014).
1273:Syntonic temperament and timbre
607:When the (diatonic) semitones,
42:needs additional citations for
1543:. Princeton University Press.
1281:describes the combination of
1207:That leaves the two extremes:
1:
1580:"1-6 Syntonic Comma Meantone"
1897:Harry Partch's 43-tone scale
1344:12-tone “equal temperament”
728:If one breaks the rule for
2307:
1864:List of meantone intervals
1483:Guido 2.0 research project
1057:for diatonic tunings with
1854:List of musical intervals
1849:Consonance and dissonance
1616:10.1162/comj.2007.31.4.15
1202:27 tone equal temperament
1198:17 tone equal temperament
1183:53 tone equal temperament
1121:31 tone equal temperament
1105:19 tone equal temperament
282:at the high extreme) and
66:"Regular diatonic tuning"
1481:The music theory of the
1327:Wicki-Hayden note layout
1260:12 (and its multiples),
1236:5 tone equal temperament
1213:7 tone equal temperament
1045:Range of recognizability
964:tone; for instance from
756:-s vanish the result is
611:, are reduced to zero (
222:regular diatonic tuning
185:regular diatonic scales
1709:Computer Music Journal
1603:Computer Music Journal
1366:
1322:
1187:Schismatic temperament
1175:Schismatic temperament
838:♯, where the interval
596:
489:
398:
217:
186:
2121:Temperament ordinaire
1474:The research program
1364:
1320:
1123:—Equivalent to
1107:—Equivalent to
792:generated collections
736:must be smaller than
597:
490:
399:
261:-s are tones and the
192:
132:
1924:List of compositions
1452:at P5 = 720.0 cents
1374:At P5 ≈ 685.7 cents
1338:as described above.
1291:that start with the
1279:syntonic temperament
1204:, or a bit further.
903:is another name for
505:
414:
323:
296:at the low extreme)
257:described here, the
147:Maneri-Sims notation
51:improve this article
18:Syntonic temperament
2291:Linear temperaments
1647:The Open University
1331:isomorphic keyboard
796:moments of symmetry
773:linear temperaments
653:Pythagorean diatone
499:perfect fifth
2158:Chinese musicology
1944:Scale of harmonics
1939:Pythagorean tuning
1887:Euler–Fokker genus
1723:"The Tone Diamond"
1367:
1352:Pythagorean tuning
1323:
1051:Easley Blackwood's
768:diatonic tunings.
655:such that all the
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542:
485:
444:
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353:
243:Linear temperament
218:
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2278:
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1170:diatonic tuning.
938:Pythagorean comma
631:equal temperament
627:equal temperament
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253:For the ordinary
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16:(Redirected from
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2117:Well temperament
2103:Regular diatonic
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1949:Tonality diamond
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1750:. Archived from
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1727:Dynamic Tonality
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1336:Dynamic tonality
1308:Dynamic tonality
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1087:. They include:
1081:meantone tunings
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618:) the octave is
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2271:(Bohlen–Pierce)
2239:833 cents scale
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1958:
1879:Just intonation
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1800:Musical tunings
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1721:Milne, Andrew.
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1289:Comma sequences
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1134:-Comma meantone
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940:(23.5 ¢), or a
911:
910:
905:
904:
899:
898:
893:
892:
884:
879:
878:
869:
868:
863:
862:
857:
856:
854:is the same as
849:
848:
841:
840:
833:
832:
825:
824:
817:
816:
809:
808:
802:F C G D A E B F
801:
800:
785:perfect fourths
778:
777:
757:
753:
745:
741:
737:
733:
716:
715:
705:
704:
694:
693:
683:
682:
672:
666:
665:
635:
633:
625:or a five tone
620:
619:
612:
608:
548:
544:
527:
503:
502:
450:
446:
429:
412:
411:
359:
355:
338:
321:
320:
308:
304:
290:
283:
273:
266:
262:
258:
255:diatonic scales
251:
228:consisting of "
210:
208:
207:
206:
198:
196:
195:
194:
178:
176:
175:
174:
166:
164:
163:
162:
154:
152:
151:
150:
138:
136:
135:
134:
123:
112:
106:
103:
60:
58:
48:
36:
23:
22:
15:
12:
11:
5:
2304:
2302:
2294:
2293:
2283:
2282:
2276:
2275:
2273:
2272:
2266:
2261:
2256:
2251:
2246:
2241:
2235:
2233:
2229:
2228:
2226:
2225:
2220:
2215:
2205:
2200:
2195:
2194:
2193:
2188:
2183:
2178:
2170:
2165:
2160:
2154:
2152:
2146:
2145:
2142:
2141:
2115:
2113:
2109:
2108:
2106:
2105:
2100:
2095:
2090:
2085:
2070:
2068:
2062:
2061:
2059:
2058:
2053:
2048:
2043:
2038:
2033:
2028:
2023:
2013:
2008:
2003:
1998:
1993:
1988:
1983:
1977:
1975:
1966:
1960:
1959:
1957:
1956:
1951:
1946:
1941:
1936:
1931:
1926:
1921:
1920:
1919:
1914:
1904:
1899:
1894:
1892:Harmonic scale
1889:
1883:
1881:
1875:
1874:
1872:
1871:
1866:
1861:
1856:
1851:
1846:
1841:
1839:Interval ratio
1836:
1831:
1826:
1821:
1816:
1810:
1808:
1804:
1803:
1798:
1796:
1795:
1788:
1781:
1773:
1765:
1764:
1748:"Musica Facta"
1739:
1713:
1696:
1694:, Spring 2008.
1679:
1659:
1640:"The X-System"
1630:
1589:
1571:
1556:
1549:
1531:
1499:
1498:
1496:
1493:
1492:
1491:
1479:
1470:
1467:
1466:
1465:
1450:
1447:
1434:At P5 ≈ 701.9
1432:
1419:At P5 = 700.0
1417:
1404:At P5 ≈ 696.8
1402:
1389:At P5 ≈ 694.7
1387:
1300:
1299:
1296:
1293:syntonic comma
1286:
1274:
1271:
1270:
1269:
1258:
1240:
1239:
1231:
1230:
1229:
1164:
1163:
1150:
1137:
1118:
1102:
1085:syntonic comma
1076:
1073:
1069:
1068:
1065:
1062:
1046:
1043:
934:syntonic comma
723:
722:
711:
700:
689:
678:
603:
602:
588:
585:
582:
576:
573:
569:
560:
557:
554:
547:
540:
533:
523:
518:
515:
500:
496:
495:
480:
473:
467:
464:
456:
449:
442:
435:
425:
422:
409:
405:
404:
389:
382:
376:
373:
365:
358:
351:
344:
334:
331:
318:
250:
247:
125:
124:
39:
37:
30:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2303:
2292:
2289:
2288:
2286:
2270:
2267:
2265:
2262:
2260:
2257:
2255:
2252:
2250:
2247:
2245:
2242:
2240:
2237:
2236:
2234:
2230:
2224:
2221:
2219:
2216:
2213:
2212:Carnatic raga
2209:
2206:
2204:
2201:
2199:
2196:
2192:
2189:
2187:
2184:
2182:
2181:Turkish makam
2179:
2177:
2174:
2173:
2171:
2169:
2166:
2164:
2161:
2159:
2156:
2155:
2153:
2147:
2138:
2134:
2130:
2126:
2122:
2118:
2114:
2110:
2104:
2101:
2099:
2096:
2094:
2091:
2089:
2086:
2083:
2079:
2078:quarter-comma
2075:
2072:
2071:
2069:
2067:
2063:
2057:
2054:
2052:
2049:
2047:
2044:
2042:
2039:
2037:
2034:
2032:
2029:
2027:
2024:
2021:
2017:
2014:
2012:
2009:
2007:
2004:
2002:
1999:
1997:
1994:
1992:
1989:
1987:
1984:
1982:
1979:
1978:
1976:
1974:
1970:
1967:
1965:
1961:
1955:
1954:Tonality flux
1952:
1950:
1947:
1945:
1942:
1940:
1937:
1935:
1932:
1930:
1927:
1925:
1922:
1918:
1915:
1913:
1910:
1909:
1908:
1905:
1903:
1900:
1898:
1895:
1893:
1890:
1888:
1885:
1884:
1882:
1880:
1876:
1870:
1867:
1865:
1862:
1860:
1857:
1855:
1852:
1850:
1847:
1845:
1842:
1840:
1837:
1835:
1832:
1830:
1827:
1825:
1822:
1820:
1817:
1815:
1812:
1811:
1809:
1805:
1801:
1794:
1789:
1787:
1782:
1780:
1775:
1774:
1771:
1754:on 2014-05-17
1753:
1749:
1743:
1740:
1728:
1724:
1717:
1714:
1710:
1706:
1700:
1697:
1693:
1689:
1683:
1680:
1676:
1672:
1666:
1664:
1660:
1648:
1641:
1634:
1631:
1626:
1622:
1617:
1612:
1608:
1604:
1600:
1593:
1590:
1585:
1581:
1575:
1572:
1567:
1560:
1557:
1552:
1550:9780691610887
1546:
1542:
1535:
1532:
1526:
1525:10.1.1.929.58
1521:
1517:
1513:
1512:
1504:
1501:
1494:
1489:
1484:
1480:
1477:
1473:
1472:
1468:
1460:
1451:
1448:
1442:
1433:
1427:
1418:
1412:
1403:
1397:
1388:
1382:
1373:
1372:
1371:
1363:
1359:
1357:
1353:
1350:tunings, and
1349:
1345:
1339:
1337:
1332:
1328:
1319:
1315:
1311:
1309:
1304:
1297:
1294:
1290:
1287:
1284:
1283:
1282:
1280:
1272:
1267:
1263:
1259:
1256:
1252:
1248:
1245:
1244:
1243:
1237:
1232:
1217:
1216:
1214:
1210:
1209:
1208:
1205:
1203:
1199:
1194:
1190:
1188:
1184:
1180:
1176:
1171:
1169:
1151:
1138:
1135:
1122:
1119:
1106:
1103:
1090:
1089:
1088:
1086:
1082:
1074:
1072:
1066:
1063:
1060:
1059:
1058:
1056:
1052:
1044:
1042:
1040:
1032:
1024:
1016:
1008:
988:
984:
971:
963:
959:
950:
948:
939:
936:(21.5 ¢), or
935:
931:
930:
922:
918:
914:
890:
875:
873:
861:
853:
845:
837:
829:
821:
813:
805:
797:
794:(also called
793:
788:
786:
782:
779:B E A D G C F
774:
769:
767:
763:
751:
731:
726:
720:
712:
709:
701:
698:
690:
687:
679:
675:
670:
667:F C G D A E B
662:
661:
660:
658:
654:
649:
642:
638:
632:
628:
615:
586:
583:
580:
574:
571:
567:
558:
555:
552:
545:
538:
531:
521:
501:
498:
497:
478:
471:
465:
462:
454:
447:
440:
433:
423:
420:
410:
407:
406:
387:
380:
374:
371:
363:
356:
349:
342:
332:
329:
319:
316:
315:
312:
301:
299:
293:
286:
280:
276:
269:
256:
248:
246:
244:
239:
235:
231:
227:
226:musical scale
223:
213:
201:
191:
181:
169:
157:
148:
141:
131:
121:
118:
110:
99:
96:
92:
89:
85:
82:
78:
75:
71:
68: –
67:
63:
62:Find sources:
56:
52:
46:
45:
40:This article
38:
34:
29:
28:
19:
2269:Lambda scale
2176:Arabic maqam
2133:Werckmeister
2102:
1964:Temperaments
1756:. Retrieved
1752:the original
1742:
1730:. Retrieved
1726:
1716:
1708:
1699:
1691:
1682:
1674:
1650:. Retrieved
1646:
1633:
1609:(4): 15–32.
1606:
1602:
1592:
1583:
1574:
1559:
1540:
1534:
1510:
1503:
1488:Musica Facta
1487:
1476:Musica Facta
1475:
1368:
1340:
1335:
1324:
1312:
1305:
1301:
1278:
1276:
1241:
1206:
1195:
1191:
1172:
1165:
1078:
1070:
1054:
1048:
1038:
1026:
1018:
1010:
1002:
986:
978:
965:
961:
957:
951:
928:
920:
916:
912:
888:
876:
867:
855:
847:
839:
831:
823:
815:
807:
799:
795:
789:
776:
770:
765:
749:
729:
727:
724:
714:
703:
692:
681:
664:
650:
640:
636:
613:
606:
302:
291:
284:
278:
274:
267:
252:
221:
219:
173:and 19-tone
113:
104:
94:
87:
80:
73:
61:
49:Please help
44:verification
41:
2264:Delta scale
2259:Gamma scale
2249:Alpha scale
2151:non-Western
2149:Traditional
1844:Pitch class
1824:Millioctave
1807:Measurement
1711:, in press.
1168:Pythagorean
1039:seven equal
748:, one gets
695:E A D G C F
232:" (T) and "
145:, 72-tone (
107:August 2018
2254:Beta scale
2232:Non-octave
2223:Tetrachord
2125:Kirnberger
2088:Schismatic
1758:2015-09-19
989:semitone,
960:tone to a
891:semitone,
887:, and the
643:= 171.43 ¢
270:= 171.43 ¢
77:newspapers
2244:A12 scale
2198:Octoechos
2163:Shí-èr-lǜ
2112:Irregular
1929:Otonality
1869:Microtone
1520:CiteSeerX
1277:The term
995:, called
750:irregular
730:"regular"
706:C G D A E
621:T T T T T
463:−
408:full tone
372:−
234:semitones
2285:Category
2129:Vallotti
2082:septimal
2074:Meantone
1834:Interval
1732:28 March
1652:28 March
1625:27906745
1490:(above).
1348:meantone
987:diatonic
974:♭
942:53
889:diatonic
864:♭
762:tritones
657:interval
317:semitone
249:Overview
133:12-tone
2218:Slendro
2168:Dastgah
2093:Miracle
2056:96-tone
2051:72-tone
2046:58-tone
2041:53-tone
2036:41-tone
2031:34-tone
2026:31-tone
2016:24-tone
2011:23-tone
2006:22-tone
2001:19-tone
1996:17-tone
1991:15-tone
1986:12-tone
1917:7-limit
1912:5-limit
1677:(2009).
1449:etc....
1346:), the
1223:⁄
1179:schisma
1157:⁄
1144:⁄
1129:⁄
1112:⁄
1096:⁄
766:regular
717:A D G C
674:C major
616:= 240 ¢
287:= 240 ¢
224:is any
211:Play 31
199:Play 53
91:scholar
2186:Mugham
2172:Maqam
2066:Linear
2020:pieces
1981:6-tone
1902:Hexany
1829:Savart
1623:
1547:
1522:
1356:31-edo
590:
578:
563:¢
550:
535:
529:
509:
483:
475:
469:
459:¢
452:
437:
431:
418:
392:
384:
378:
368:¢
361:
346:
340:
327:
238:octave
93:
86:
79:
72:
64:
2203:Pelog
2191:Muqam
2137:Young
2098:Magic
1973:Equal
1907:Limit
1814:Pitch
1673:, in
1643:(PDF)
1621:S2CID
1495:Notes
962:major
958:minor
929:comma
732:that
684:C G D
289:(for
272:(for
230:tones
98:JSTOR
84:books
2208:Raga
1819:Cent
1734:2017
1654:2017
1545:ISBN
1458:Play
1440:Play
1425:Play
1410:Play
1395:Play
1380:Play
846:♯ -
559:1200
455:1200
364:1200
205:and
179:Play
167:Play
155:Play
139:Play
70:news
1611:doi
1025:to
1017:or
1009:to
977:to
945:TET
917:d c
913:c d
758:s s
671:in
294:= 0
53:by
2287::
2135:,
2131:,
2127:,
2080:,
1725:.
1707:,
1690:,
1662:^
1645:.
1619:.
1607:31
1605:.
1601:.
1582:.
1518:.
1266:31
1264:,
1262:19
1255:22
1253:,
1251:17
1249:,
1247:15
1189:.
1098:11
915:=
897:;
830:♯
822:♯
814:♯
806:♯
699:).
688:).
677:).
645:).
639:=
307:,
277:=
220:A
149:)
2214:)
2210:(
2139:)
2123:(
2119:/
2084:)
2076:(
2022:)
2018:(
1792:e
1785:t
1778:v
1761:.
1736:.
1656:.
1627:.
1613::
1586:.
1568:.
1553:.
1528:.
1225:2
1221:1
1159:6
1155:1
1146:5
1142:1
1131:4
1127:1
1114:3
1110:1
1094:1
1029:C
1021:B
1013:F
1005:E
998:S
992:D
981:E
968:E
954:c
924:.
921:T
919:=
906:S
900:D
894:D
885:c
880:S
870:B
858:B
850:G
842:F
834:A
826:D
818:G
810:C
754:T
746:T
742:s
738:T
734:s
721:)
641:T
637:s
634:(
623:,
614:T
609:s
587:s
584:+
581:T
575:3
572:=
568:)
556:+
553:T
546:(
539:2
532:1
522:=
517:5
514:p
479:)
472:s
466:2
448:(
441:5
434:1
424:=
421:T
388:)
381:T
375:5
357:(
350:2
343:1
333:=
330:s
309:T
305:s
292:s
285:T
279:T
275:s
268:T
263:s
259:T
120:)
114:(
109:)
105:(
95:·
88:·
81:·
74:·
47:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
