Knowledge (XXG)

Tetrachord

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segment of the chromatic scale, or, in a more historically oriented context, to refer to the six chromatic notes used to fill the interval of a perfect fourth, usually found in descending bass lines. It may also be used to describes sets of fewer than four notes, when used in scale-like fashion to span the interval of a perfect fourth.
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He continues with four other possible genres "dividing the tone in quarters, eighths, thirds, half thirds, quarter thirds, and combining them in diverse manners". Later, he presents possible positions of the frets on the lute, producing ten intervals dividing the interval of a fourth between the
1910:
Theorists of the later 20th century often use the term "tetrachord" to describe any four-note set when analysing music of a variety of styles and historical periods. The expression "chromatic tetrachord" may be used in two different senses: to describe the special case consisting of a four-note
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The number of strings on the classical lyre varied at different epochs, and possibly in different localities – four, seven and ten having been favorite numbers. Larger scales are constructed from conjunct or disjunct tetrachords. Conjunct tetrachords share a note, while disjunct tetrachords are
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are said to also appear particularly in Indian music. In this case, the lower "tetrachord" totals six semitones (a tritone). The following elements produce 36 combinations when joined by halfstep. These 36 combinations together with the 36 combinations described above produce the so-called "72
1948:
Tetrachords based upon equal-tempered tuning were also used to approximate common heptatonic scales in use in Indian, Hungarian, Arabian and Greek musics. Western theorists of the 19th and 20th centuries, convinced that any scale should consist of two tetrachords and a tone, described various
1902:–F, etc.). The 19th-century theorists of ancient Greek music believed that this had also been the case in Antiquity, and imagined that there had existed Dorian, Phrygian or Lydian tetrachords. This misconception was denounced in Otto Gombosi's thesis (1939). 204:
A chromatic tetrachord has a characteristic interval that is greater than about half the total interval of the tetrachord, yet not as great as four-fifths of the interval (between about 249 and 398 cents). Classically, the characteristic interval is a
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All these scales are formed by two complete disjunct tetrachords: contrarily to Greek and Medieval theory, the tetrachords change here from scale to scale (i.e., the C major tetrachord would be C–D–E–F, the D major one
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combinations supposed to correspond to a variety of exotic scales. For instance, the following diatonic intervals of one, two or three semitones, always totaling five semitones, produce 36 combinations when joined by
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The descending tetrachord from tonic to dominant, typically in minor (e.g. A–G–F–E in A minor), had been used since the Renaissance to denote a lamentation. Well-known cases include the ostinato bass of Dido's aria
2482:). It should be mentioned that Al-Farabi's, among other Islamic treatises, also contained additional division schemes as well as providing a gloss of the Greek system as Aristoxenian doctrines were often included. 1484: 145:
as the basic unit for determining tuning, where ancient Greeks used the tetrachord. Ancient Greek theorists recognized that the octave is a fundamental interval but saw it as built from two tetrachords and a
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in an octave, this would create a 25-tone scale. A more inclusive description (where Ottoman, Persian and Arabic overlap), of the scale divisions is that of 24 quarter tones (see also
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An enharmonic tetrachord has a characteristic interval that is greater than about four-fifths the total tetrachord interval. Classically, the characteristic interval is a
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texture, but in contrast to the lament, here the tetrachordal fragment – ascending or descending and possibly reordered – is set in the upper voice in the manner of a
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The Greeks analyzed genera using various terms, including diatonic, enharmonic, and chromatic. Scales are constructed from conjunct or disjunct tetrachords.
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21, no. 1 (Spring 1999): 43–73; Tuire Kuusi, "Subset-Class Relation, Common Pitches, and Common Interval Structure Guiding Estimations of Similarity",
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A diatonic tetrachord has a characteristic interval that is less than or equal to half the total interval of the tetrachord (or approximately 249 
1729:. Given the following vocabulary of tetrachords (the digits give the number of semitones in consecutive intervals of the tetrachord, adding to five): 1620: 571:
In all cases, the extreme notes of the tetrachords, E – B, and A – E, remain fixed, while the notes in between are different depending on the genus.
3394: 282:, meaning "compressed"). This is the case for the chromatic and enharmonic tetrachords, but not the diatonic (meaning "stretched out") tetrachord. 2490:
The tetrachord, a fundamentally incomplete fragment, is the basis of two compositional forms constructed upon repetition of that fragment: the
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of 9/8 (a Pythagorean major second). Alternating conjunct and disjunct tetrachords form a scale that repeats in octaves (as in the familiar
3530: 2784: 1520: 1311: 3487: 1064: 3349: 3304: 3201: 3090: 3082: 2858: 814: 3077:, pp. 1, 18, 68, 70, 73, 87, 88, 21, 119, 123, 124, 125, 138, 143, 171, 174, and 223. New Haven and London: Yale University Press. 2579:
texture, wherein the bass descends through some tetrachord, possibly that of the previous paragraph, but usually one suggesting a
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The fourth genre, corresponding to the Greek chromatic, has a tone and a half, a semitone and a semitone, as G–A
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or the kithara, with the implicit understanding that the four strings produced adjacent (i.e., conjunct) notes.
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The first genre, corresponding to the Greek diatonic, is composed of a tone, a tone and a semitone, as G–A–B–C.
1532: 3100: 2614:. Because of the extreme brevity of the theme and number of repetitions required, and free of the binding of 2530:("complaint", "lamentation"), has been used until today. A variant form, the full chromatic descent (e.g. A–G 3540: 3450: 3380: 3018: 2903: 2631: 164: 2884: 3467: 2636: 175:) of tetrachords. These genera are characterized by the largest of the three intervals of the tetrachord: 3292:
Divisions of the Tetrachord: A Prolegomenon [introduction] to the Construction of Musical Scales
2919: 2889: 2515: 2384: 2374: 1937: 1847: 1693: 1552: 188:). This characteristic interval is usually slightly smaller (approximately 200 cents), becoming a 3502: 2930: 269: 3472: 3411: 2736: 3477: 3189: 3135:, 2 vols., translated by John Fenstermaker (Paris: Alphonse Leduc, 1962): 2:35. ASIN: B0006CNH8E. 2772: 580: 356:
As the three genera simply represent ranges of possible intervals within the tetrachord, various
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Reynold Simpson, "New Sketches, Old Fragments, and Schoenberg's Third String Quartet, Op. 30",
2763:, 2 vols., translated by John Fenstermaker. Paris: Alphonse Leduc, 1962, 2:35. ASIN: B0006CNH8E. 224: 1686: 1676: 1661: 1646: 1608: 1593: 1578: 1563: 3438: 3345: 3300: 3197: 3086: 3078: 2854: 2801: 2780: 2615: 2603: 2572: 1722: 1635: 2118:
The second genre is composed of a tone, three quarter tones and three quarter tones, as G–A–B
3497: 2990: 2970: 2925: 2846: 2656: 2544: 2022: 1726: 1627: 1336:, created in such a manner from the diatonic genus), but this was not the only arrangement. 55: 555: 539: 495: 479: 463: 411: 395: 250: 3492: 3013: 2914: 2748: 2619: 2611: 2413: 2121: 1940:, the term may have the special sense of any consecutive four notes of a twelve-tone row. 563: 547: 531: 487: 471: 419: 403: 258: 192:. Classically, the diatonic tetrachord consists of two intervals of a tone and one of a 3372: 3286: 3031: 2994: 2980: 2966: 2934: 2898: 2651: 2556: 1928: 1333: 285:
Whatever the tuning of the tetrachord, its four degrees are named, in ascending order,
122: 108: 59: 47: 39: 209:(approximately 300 cents), and the two smaller intervals are equal semitones, e.g. A–G 3524: 3423: 3362: 3327: 2956: 2948: 2826: 2677: 2503: 2479: 1632: 1282: 1274: 1266: 1258: 1035: 1027: 1019: 1011: 783: 775: 767: 759: 218: 199: 179: 159: 67: 2129:
The third genre has a tone and a quarter, three quarter tones and a semitone, as G–A
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Joanny Grosset, "Inde. Histoire de la musique depuis l'origine jusqu'à nos jours",
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Brent Auerbach, "Tiered Polyphony and Its Determinative Role in the Piano Music of
2952: 2894: 2475: 2428: 1556: 242: 189: 185: 147: 63: 31: 2917:'s 'Construction of Twelve Verticals': An Aspect of Harmony in the Serial Music", 1548: 1080:
Here is a representative Pythagorean tuning of the enharmonic genus attributed to
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The second note of the tetrachord (in descending order) is the first of the scale.
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The third note of the tetrachord (in descending order) is the first of the scale.
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Two Greek tetrachords in the enharmonic genus, forming an enharmonic Dorian scale
3407: 3403: 3280:] (reprint) (in French). Translated by Rodolphe d'Erlanger. Paris: Geuthner. 3255:, 2 vols., translated by John Fenstermaker (Paris: Alphonse Leduc, 1962): 2:110. 3070: 2907:
38, no. 1 (Spring 1994): 21–41; Ian Quinn, "Listening to Similarity Relations",
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If one considers that the interval of a fourth between the strings of the lute (
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music divides the interval of a fourth differently than the Greek. For example,
1919: 238: 206: 3242:, 2 vols., translated by John Fenstermaker. Paris: Alphonse Leduc, 1937): 1:14. 2983:: Tonality, Motivic Sets and the Referential Pitch-Class Specific Collection", 3457: 3196:, translated by Laurie Schwartz. (Portland, Oregon: Amadeus Press, 1996): 19. 1950: 1777: 826: 58:. In traditional music theory, a tetrachord always spanned the interval of a 3462: 3337: 3269: 2607: 2576: 2108: 268:
When the composite of the two smaller intervals is less than the remaining (
17: 3509: 3433: 2032: 1639: 1081: 193: 71: 2941:, With Notes on Narrative, Symmetry, Quantitative Flux and Heraclitus", 2474:) corresponds to a tetrachord, and that there are two tetrachords and a 1776:
the following scales could be derived by joining two tetrachords with a
1614:(b–a–g–f). This tetrachord spans a tritone instead of a perfect fourth. 1108:                  2523: 371:
The first note of the tetrachord is also the first note of the scale.
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Benedict Taylor, "Modal Four-Note Pitch Collections in the Music of
2587:, is repeated again and again over the length of the composition. 1626: 1547: 305:
and, for the second tetrachord in the construction of the system,
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are fixed, and a perfect fourth apart, while the position of the
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Ellen Rosand, "The Descending Tetrachord: An Emblem of Lament",
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4, no. 2 (Spring–Summer 1966): 14–22; Annie K. Yih, "Analysing
134:, originally in reference to harp-like instruments such as the 2471: 3093:(pbk). Allen Forte (1985). "Pitch-Class Set Analysis Today". 3240:
Cours complet d'improvisation a l'orgue: Exercices preparées
1292: 1045: 793: 3344:. Longman Music Series. New York and London: Longman Inc.. 3146:
Encyclopédie de la musique et Dictionnaire du Conservatoire
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17, In Celebration of Arnold Schoenberg (1) (1992): 85–101.
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Mathiesen, Thomas J. (2001). "Greece §I: Ancient". In
2526:'s Mass in C minor, KV 427, etc. This tetrachord, known as 1932:
or simply a "4-element set" – a set of any four pitches or
74:, not necessarily related to a particular tuning system. 66:)—but in modern use it means any four-note segment of a 861:              2590:
Another musical form, of the same time period, is the
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39, no. 2 (Summer 2001): 108–158; Joseph N. Straus, "
2779:, 2 vols. (New York: Carl Fischer, 1941), 1:112–114. 2111:
describes four genres of the division of the fourth:
94:—"string" or "note"). In ancient Greek music theory, 2853:(Boston: Northeastern University Press, 1993): 120. 2851:
The "Style hongrois" in the Music of Western Europe
3051:Nors S. Josephson, "On Some Apparent Sketches for 2933:, "An Imagined Drama of Competitive Opposition in 27:Series of four notes separated by three intervals 3106:Conference 1984): 29–58, citations on 48–51, 53. 2618:to tetrachord in the lament, the breadth of the 3358:The New Grove Dictionary of Music and Musicians 3323:The New Grove Dictionary of Music and Musicians 2687:The New Grove Dictionary of Music and Musicians 2555:There exists a short, free musical form of the 272:) interval, the three-note group is called the 54:) is a series of four notes separated by three 3099:4, nos. 1 & 2 (March–July: Special Issue: 3034:, "Beethoven's Interrupted Tetrachord and the 3388: 2777:The Schillinger System of Musical Composition 8: 2583:. This tetrachord, treated as a very short 1559:(also known as the upper minor tetrachord): 3299:. Hanover, New Hampshire: Frog Peak Music. 583:of the diatonic and chromatic tetrachords: 3395: 3381: 3373: 2947:28, no. 2/3 (July–October 2009): 373–422; 62:, a 4:3 frequency proportion (approx. 498 3181: 3169: 3157: 3148:, vol. 1, Paris, Delagrave, 1914, p. 325. 2965:11, no. 2 (Spring–Summer 1973): 204–209; 2872:Tonarten und Stimmungen der Antiken Musik 2800:(Pacific, Missouri: Mel Bay, 2010): 111. 2602:(OE spur). It is also a set of harmonic 3213: 2724: 2712: 2700: 2038: 1955: 1782: 1731: 1341: 1304:Enharmonic tetrachord pythagorean tuning 3253:Cours complet d'improvisation a l'orgue 3133:Cours Complet d'Improvisation a l'Orgue 3003:2, no. 1 (Autumn–Winter 1963): 102–113. 2761:Cours Complet d'Improvisation a l'Orgue 2668: 1057:Chromatic tetrachord pythagorean tuning 3186:Das Arabische Tonsystem im Mittelalter 1696:: i–v6–iv6–V in C minor (bassline: c–b 824: 805:Diatonic tetrachord pythagorean tuning 609:          603:          2959:: String Quartet (1967): Two Views", 2874:, Kopenhagen, Ejnar Munksgaard, 1939. 2819:"Double harmonic scale and its modes" 2571:. It is typically a set of harmonic 1535:divisions by Chalmers after Hofmann. 86:(from Greek—"four of something") and 7: 2973:: A Few Observations of His Music", 2542:–F–E in A minor), has been known as 2518:'s Mass in B minor, BWV 232, or the 241:, and the two smaller intervals are 3184:, pp. 165–179; Liberty Manik, 1725:tuning were used to explain common 1382:Eratosthenes’ chromatic tetrachord 2929:25, no. 1 (September 2007): 1–11; 1926:to mean what he elsewhere calls a 100:greater and lesser perfect systems 25: 3188:(Leiden, E. J. Brill, 1969): 42; 2017:Indian-specific tetrachord system 3355:Roeder, John. 2001. "Set (ii)". 2989:19, no. 2 (July 2000): 203–229; 2893:32, no. 1 (Spring 2010): 44–59; 2412: 2383: 2373: 2131: 2120: 1709: 1685: 1675: 1660: 1645: 1638:creates a descending tetrachord 1615: 1607: 1592: 1577: 1562: 1517: 1480: 1443: 1406: 1369: 1308: 1061: 825:Problems playing this file? See 809: 562: 554: 546: 538: 530: 494: 486: 478: 470: 462: 418: 410: 402: 394: 257: 249: 116:); the notes between these were 3320:Anonymous. 2001. "Tetrachord". 3022:52, no. 2 (Fall 2008): 273–320. 2997:'s Composition for Orchestra", 1531:This is a partial table of the 3229:65, no. 3 (1979): 346–59. 2622:in litany is usually notable. 1345:Didymos’ chromatic tetrachord 1: 3285:Chalmers, John H. Jr (1993). 3075:The Structure of Atonal Music 2798:Advanced Lead Guitar Concepts 510:Diatonic: C–B | A–G–F–E | D–C 437:Diatonic: D–C–B | A–G–F–E | D 141:Modern music theory uses the 3361:, second edition, edited by 3326:, second edition, edited by 3058:Archiv für Musikwissenschaft 2901:and Intervallic Weighting", 1456:Ptolemy’s intense chromatic 2031:Tetrachords separated by a 1922:occasionally uses the term 1880:1 2 2 : 2 : 1 3 1 1869:2 1 2 : 2 : 2 2 1 1858:2 2 1 : 2 : 1 2 2 1844:1 3 1 : 2 : 1 3 1 1833:2 1 2 : 2 : 1 3 1 1822:2 2 1 : 2 : 1 3 1 1811:2 1 2 : 2 : 1 2 2 1800:2 2 1 : 2 : 2 2 1 374:Diatonic: E–D–C–B | A–G–F–E 167:theory distinguishes three 98:signified a segment of the 3557: 3531:Ancient Greek music theory 3053:Sibelius's Eighth Symphony 2027:Hindustani classical music 2020: 157: 154:Ancient Greek music theory 3419: 3000:Perspectives of New Music 2976:Perspectives of New Music 2962:Perspectives of New Music 2910:Perspectives of New Music 1896:–G, the C minor one C–D–E 1419:Ptolemy’s soft chromatic 1288: 1089: 1041: 840: 789: 588: 579:Here are the traditional 126: 112: 43: 3289:; Carter Scholz (eds.). 3274:Kitābu l-mūsīqī al-kabīr 3061:61, no. 1 (2004): 54–67. 3019:Journal of Music Theory 2904:Journal of Music Theory 2870:Otto Johannes Gombosi, 2796:Joshua Craig Podolsky, 2632:All-interval tetrachord 2500:When I am laid in earth 1721:Tetrachords based upon 3251:Marcel Dupré, (1962). 3194:The Music of the Arabs 2737:"Phrygian Progression" 2637:Diatonic and chromatic 1877:Upper minor + harmonic 1718: 1624: 1297: 1050: 798: 229: 130:). It literally means 51: 3227:The Musical Quarterly 3101:King's College London 2920:Music Theory Spectrum 2890:Music Theory Spectrum 2887:'s American Period", 2516:Johann Sebastian Bach 1906:20th-century analysis 1848:Double harmonic scale 1786:Component tetrachords 1694:Phrygian half cadence 1630: 1553:Descending tetrachord 1551: 1493:Archytas’ enharmonic 1296: 1049: 797: 227: 3369:. London: Macmillan. 3334:. London: Macmillan. 2897:and Jack Douthett, " 2741:Classical Music Blog 513:Chromatic: C–B | A–G 82:The name comes from 3342:Basic Atonal Theory 3117:Theory and Practice 2486:Compositional forms 1855:Major + upper minor 1841:Harmonic + harmonic 1808:Minor + upper minor 581:Pythagorean tunings 575:Pythagorean tunings 165:Ancient Greek music 3190:Habib Hassan Touma 2773:Joseph Schillinger 2747:2011-10-06 at the 2620:harmonic excursion 2545:Passus duriusculus 2045:Upper tetrachords 1962:Upper tetrachords 1944:Non-Western scales 1938:twelve-tone theory 1719: 1625: 1298: 1051: 799: 230: 3518: 3517: 3488:"Ode-to-Napoleon" 3160:, pp. 56–57. 3042:5 (1991): 77–100. 2931:Joshua B. Mailman 2806:978-0-7866-8236-2 2616:chord progression 2468: 2467: 2097: 2096: 2042:Lower tetrachords 2036:karnatic modes". 2014: 2013: 1959:Lower tetrachords 1887: 1886: 1883:Neapolitan minor 1774: 1773: 1727:heptatonic scales 1723:equal temperament 1715: 1621: 1529: 1528: 1523: 1486: 1449: 1412: 1375: 1325: 1324: 1314: 1078: 1077: 1067: 836: 835: 815: 16:(Redirected from 3548: 3397: 3390: 3383: 3374: 3310: 3281: 3278:La musique arabe 3256: 3249: 3243: 3236: 3230: 3223: 3217: 3211: 3205: 3179: 3173: 3167: 3161: 3155: 3149: 3142: 3136: 3129: 3120: 3113: 3107: 3068: 3062: 3049: 3043: 3036:Seventh Symphony 3029: 3023: 3010: 3004: 2926:Music Perception 2881: 2875: 2868: 2862: 2847:Jonathan Bellman 2844: 2838: 2837: 2835: 2834: 2825:. Archived from 2823:docs.solfege.org 2815: 2809: 2794: 2788: 2770: 2764: 2757: 2751: 2734: 2728: 2722: 2716: 2710: 2704: 2698: 2692: 2691: 2673: 2657:Tetratonic scale 2541: 2540: 2535: 2534: 2494:and the litany. 2417: 2416: 2407: 2406: 2399: 2398: 2388: 2387: 2378: 2377: 2368: 2367: 2360: 2359: 2344: 2342: 2341: 2338: 2335: 2327: 2325: 2324: 2321: 2318: 2310: 2308: 2307: 2304: 2301: 2293: 2291: 2290: 2287: 2284: 2276: 2274: 2273: 2270: 2267: 2259: 2257: 2256: 2253: 2250: 2242: 2240: 2239: 2236: 2233: 2225: 2223: 2222: 2219: 2216: 2208: 2206: 2205: 2202: 2199: 2191: 2189: 2188: 2185: 2182: 2174: 2172: 2171: 2168: 2165: 2154: 2153: 2145: 2144: 2136: 2135: 2125: 2124: 2039: 1956: 1901: 1900: 1895: 1894: 1830:Minor + harmonic 1819:Major + harmonic 1792:Resulting scale 1783: 1738:Halfstep String 1732: 1717: 1716: 1707: 1706: 1701: 1700: 1690: 1689: 1680: 1679: 1672: 1671: 1665: 1664: 1657: 1656: 1650: 1649: 1623: 1622: 1612: 1611: 1604: 1603: 1597: 1596: 1589: 1588: 1582: 1581: 1574: 1573: 1567: 1566: 1525: 1524: 1488: 1487: 1451: 1450: 1414: 1413: 1377: 1376: 1342: 1330:disjunctive tone 1316: 1315: 1295: 1249: 1247: 1246: 1243: 1240: 1229: 1227: 1226: 1223: 1220: 1209: 1207: 1206: 1203: 1200: 1187: 1185: 1184: 1181: 1178: 1168: 1166: 1165: 1162: 1159: 1149: 1147: 1146: 1143: 1140: 1130: 1128: 1127: 1124: 1121: 1087: 1086: 1069: 1068: 1048: 1002: 1000: 999: 996: 993: 982: 980: 979: 976: 973: 962: 960: 959: 956: 953: 940: 938: 937: 934: 931: 921: 919: 918: 915: 912: 902: 900: 899: 896: 893: 883: 881: 880: 877: 874: 849:    838: 837: 817: 816: 796: 750: 748: 747: 744: 741: 730: 728: 727: 724: 721: 710: 708: 707: 704: 701: 688: 686: 685: 682: 679: 669: 667: 666: 663: 660: 650: 648: 647: 644: 641: 631: 629: 628: 625: 622: 597:    586: 585: 567: 566: 559: 558: 551: 550: 543: 542: 535: 534: 524: 523: 518: 517: 499: 498: 491: 490: 483: 482: 475: 474: 467: 466: 457: 456: 451: 450: 445: 444: 423: 422: 415: 414: 407: 406: 399: 398: 388: 387: 382: 381: 264: 262: 261: 254: 253: 214: 213: 129: 128: 115: 114: 45: 21: 3556: 3555: 3551: 3550: 3549: 3547: 3546: 3545: 3536:Music of Greece 3521: 3520: 3519: 3514: 3415: 3401: 3317: 3315:Further reading 3307: 3284: 3268: 3260: 3259: 3250: 3246: 3237: 3233: 3224: 3220: 3212: 3208: 3180: 3176: 3168: 3164: 3156: 3152: 3143: 3139: 3130: 3123: 3114: 3110: 3069: 3065: 3050: 3046: 3030: 3026: 3014:Johannes Brahms 3011: 3007: 2939:Scrivo in Vento 2899:Vector Products 2882: 2878: 2869: 2865: 2845: 2841: 2832: 2830: 2817: 2816: 2812: 2795: 2791: 2771: 2767: 2758: 2754: 2749:Wayback Machine 2735: 2731: 2723: 2719: 2711: 2707: 2699: 2695: 2675: 2674: 2670: 2665: 2628: 2612:chorale prelude 2548:in the Baroque 2538: 2537: 2532: 2531: 2508:Dido and Aeneas 2488: 2411: 2404: 2403: 2396: 2395: 2382: 2372: 2365: 2364: 2357: 2356: 2339: 2336: 2333: 2332: 2330: 2322: 2319: 2316: 2315: 2313: 2305: 2302: 2299: 2298: 2296: 2288: 2285: 2282: 2281: 2279: 2271: 2268: 2265: 2264: 2262: 2254: 2251: 2248: 2247: 2245: 2237: 2234: 2231: 2230: 2228: 2220: 2217: 2214: 2213: 2211: 2203: 2200: 2197: 2196: 2194: 2186: 2183: 2180: 2179: 2177: 2169: 2166: 2163: 2162: 2160: 2142: 2141: 2130: 2119: 2102: 2029: 2019: 1946: 1917: 1908: 1898: 1897: 1892: 1891: 1850:or Gypsy major 1836:Harmonic minor 1825:Harmonic major 1803:Diatonic major 1789:Halfstep string 1710: 1704: 1703: 1698: 1697: 1691: 1684: 1681: 1674: 1669: 1668: 1666: 1659: 1654: 1653: 1651: 1644: 1616: 1613: 1606: 1601: 1600: 1598: 1591: 1586: 1585: 1583: 1576: 1571: 1570: 1568: 1561: 1546: 1541: 1533:superparticular 1518: 1481: 1444: 1407: 1370: 1328:separated by a 1321: 1320: 1319: 1318: 1317: 1309: 1306: 1299: 1293: 1244: 1241: 1238: 1237: 1235: 1224: 1221: 1218: 1217: 1215: 1204: 1201: 1198: 1197: 1195: 1182: 1179: 1176: 1175: 1173: 1163: 1160: 1157: 1156: 1154: 1144: 1141: 1138: 1137: 1135: 1125: 1122: 1119: 1118: 1116: 1074: 1073: 1072: 1071: 1070: 1062: 1059: 1052: 1046: 997: 994: 991: 990: 988: 977: 974: 971: 970: 968: 957: 954: 951: 950: 948: 935: 932: 929: 928: 926: 916: 913: 910: 909: 907: 897: 894: 891: 890: 888: 878: 875: 872: 871: 869: 855:    832: 831: 823: 821: 820: 819: 818: 810: 807: 800: 794: 745: 742: 739: 738: 736: 725: 722: 719: 718: 716: 705: 702: 699: 698: 696: 683: 680: 677: 676: 674: 664: 661: 658: 657: 655: 645: 642: 639: 638: 636: 626: 623: 620: 619: 617: 577: 561: 553: 545: 537: 529: 521: 520: 515: 514: 493: 485: 477: 469: 461: 454: 453: 448: 447: 442: 441: 417: 409: 401: 393: 392:Enharmonic: E–D 385: 384: 379: 378: 353:, are movable. 256: 248: 246: 211: 210: 196:, e.g. A–G–F–E. 162: 156: 121: 107: 80: 28: 23: 22: 15: 12: 11: 5: 3554: 3552: 3544: 3543: 3541:Musical scales 3538: 3533: 3523: 3522: 3516: 3515: 3513: 3512: 3507: 3506: 3505: 3500: 3495: 3490: 3485: 3480: 3475: 3470: 3460: 3455: 3454: 3453: 3443: 3442: 3441: 3431: 3426: 3420: 3417: 3416: 3404:Pitch segments 3402: 3400: 3399: 3392: 3385: 3377: 3371: 3370: 3353: 3335: 3316: 3313: 3312: 3311: 3305: 3295:. foreword by 3287:Larry Polansky 3282: 3258: 3257: 3244: 3238:Marcel Dupré, 3231: 3218: 3206: 3182:Al-Farabi 2001 3174: 3170:Al-Farabi 2001 3162: 3158:Al-Farabi 2001 3150: 3137: 3131:Marcel Dupré, 3121: 3108: 3104:Music Analysis 3096:Music Analysis 3063: 3044: 3032:Robert Gauldin 3024: 3005: 2995:Godfrey Winham 2986:Music Analysis 2967:Milton Babbitt 2944:Music Analysis 2876: 2863: 2839: 2810: 2789: 2785:978-0306775215 2765: 2759:Marcel Dupré, 2752: 2729: 2717: 2715:, p. 103. 2705: 2693: 2667: 2666: 2664: 2661: 2660: 2659: 2654: 2649: 2644: 2639: 2634: 2627: 2624: 2487: 2484: 2466: 2465: 2462: 2459: 2456: 2453: 2450: 2447: 2444: 2441: 2438: 2435: 2432: 2425: 2424: 2421: 2418: 2408: 2400: 2392: 2389: 2379: 2369: 2361: 2353: 2350: 2346: 2345: 2328: 2311: 2294: 2277: 2260: 2243: 2226: 2209: 2192: 2175: 2158: 2148: 2147: 2138: 2127: 2116: 2101: 2098: 2095: 2094: 2091: 2087: 2086: 2083: 2079: 2078: 2075: 2071: 2070: 2067: 2063: 2062: 2059: 2055: 2054: 2051: 2047: 2046: 2043: 2018: 2015: 2012: 2011: 2008: 2004: 2003: 2000: 1996: 1995: 1992: 1988: 1987: 1984: 1980: 1979: 1976: 1972: 1971: 1968: 1964: 1963: 1960: 1945: 1942: 1916: 1913: 1907: 1904: 1885: 1884: 1881: 1878: 1874: 1873: 1872:Melodic minor 1870: 1867: 1863: 1862: 1861:Melodic major 1859: 1856: 1852: 1851: 1845: 1842: 1838: 1837: 1834: 1831: 1827: 1826: 1823: 1820: 1816: 1815: 1814:Natural minor 1812: 1809: 1805: 1804: 1801: 1798: 1794: 1793: 1790: 1787: 1772: 1771: 1768: 1764: 1763: 1760: 1756: 1755: 1752: 1748: 1747: 1744: 1740: 1739: 1736: 1683: 1673: 1658: 1643: 1605: 1590: 1575: 1560: 1555:in the modern 1545: 1542: 1540: 1537: 1527: 1526: 1515: 1512: 1509: 1506: 1503: 1500: 1497: 1494: 1490: 1489: 1478: 1475: 1472: 1469: 1466: 1463: 1460: 1457: 1453: 1452: 1441: 1438: 1435: 1432: 1429: 1426: 1423: 1420: 1416: 1415: 1404: 1401: 1398: 1395: 1392: 1389: 1386: 1383: 1379: 1378: 1367: 1364: 1361: 1358: 1355: 1352: 1349: 1346: 1334:diatonic scale 1323: 1322: 1307: 1302: 1301: 1300: 1291: 1290: 1289: 1286: 1285: 1279: 1277: 1271: 1269: 1263: 1261: 1254: 1253: 1250: 1233: 1230: 1213: 1210: 1193: 1189: 1188: 1171: 1169: 1152: 1150: 1133: 1131: 1113: 1112: 1109: 1106: 1103: 1101: 1098: 1096: 1092: 1091: 1076: 1075: 1060: 1055: 1054: 1053: 1044: 1043: 1042: 1039: 1038: 1032: 1030: 1024: 1022: 1016: 1014: 1007: 1006: 1003: 986: 983: 966: 963: 946: 942: 941: 924: 922: 905: 903: 886: 884: 866: 865: 862: 859: 856: 853: 850: 847: 843: 842: 834: 833: 822: 808: 803: 802: 801: 792: 791: 790: 787: 786: 780: 778: 772: 770: 764: 762: 755: 754: 751: 734: 731: 714: 711: 694: 690: 689: 672: 670: 653: 651: 634: 632: 614: 613: 610: 607: 604: 601: 598: 595: 591: 590: 576: 573: 569: 568: 526: 511: 508: 505: 501: 500: 458: 438: 434: 433: 430: 429:Phrygian scale 426: 425: 390: 377:Chromatic: E–D 375: 372: 369: 266: 265: 222: 221: 216: 202: 197: 182: 158:Main article: 155: 152: 79: 76: 60:perfect fourth 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 3553: 3542: 3539: 3537: 3534: 3532: 3529: 3528: 3526: 3511: 3508: 3504: 3501: 3499: 3496: 3494: 3491: 3489: 3486: 3484: 3481: 3479: 3476: 3474: 3471: 3469: 3466: 3465: 3464: 3461: 3459: 3456: 3452: 3449: 3448: 3447: 3444: 3440: 3437: 3436: 3435: 3432: 3430: 3427: 3425: 3422: 3421: 3418: 3413: 3409: 3405: 3398: 3393: 3391: 3386: 3384: 3379: 3378: 3375: 3368: 3364: 3363:Stanley Sadie 3360: 3359: 3354: 3351: 3350:0-582-28117-2 3347: 3343: 3339: 3336: 3333: 3329: 3328:Stanley Sadie 3325: 3324: 3319: 3318: 3314: 3308: 3306:0-945996-04-7 3302: 3298: 3294: 3293: 3288: 3283: 3279: 3275: 3271: 3267: 3266: 3265: 3264: 3254: 3248: 3245: 3241: 3235: 3232: 3228: 3222: 3219: 3216:, p. 20. 3215: 3214:Chalmers 1993 3210: 3207: 3203: 3202:0-931340-88-8 3199: 3195: 3191: 3187: 3183: 3178: 3175: 3172:, p. 58. 3171: 3166: 3163: 3159: 3154: 3151: 3147: 3141: 3138: 3134: 3128: 3126: 3122: 3118: 3112: 3109: 3105: 3102: 3098: 3097: 3092: 3091:0-300-02120-8 3088: 3084: 3083:0-300-01610-7 3080: 3076: 3072: 3067: 3064: 3060: 3059: 3054: 3048: 3045: 3041: 3037: 3033: 3028: 3025: 3021: 3020: 3015: 3009: 3006: 3002: 3001: 2996: 2992: 2991:J. K. Randall 2988: 2987: 2982: 2978: 2977: 2972: 2971:Edgard Varèse 2968: 2964: 2963: 2958: 2957:Martin Boykan 2954: 2950: 2949:John Harbison 2946: 2945: 2940: 2936: 2932: 2928: 2927: 2922: 2921: 2916: 2912: 2911: 2906: 2905: 2900: 2896: 2892: 2891: 2886: 2880: 2877: 2873: 2867: 2864: 2860: 2859:1-55553-169-5 2856: 2852: 2848: 2843: 2840: 2829:on 2015-06-18 2828: 2824: 2820: 2814: 2811: 2807: 2803: 2799: 2793: 2790: 2786: 2782: 2778: 2774: 2769: 2766: 2762: 2756: 2753: 2750: 2746: 2742: 2738: 2733: 2730: 2727:, p. 11. 2726: 2725:Chalmers 1993 2721: 2718: 2714: 2713:Chalmers 1993 2709: 2706: 2702: 2701:Chalmers 1993 2697: 2694: 2689: 2688: 2683: 2679: 2672: 2669: 2662: 2658: 2655: 2653: 2650: 2648: 2645: 2643: 2640: 2638: 2635: 2633: 2630: 2629: 2625: 2623: 2621: 2617: 2613: 2609: 2605: 2601: 2597: 2593: 2588: 2586: 2582: 2578: 2574: 2570: 2566: 2562: 2558: 2553: 2551: 2547: 2546: 2529: 2525: 2521: 2517: 2513: 2509: 2505: 2504:Henry Purcell 2501: 2495: 2493: 2485: 2483: 2481: 2480:Arabian maqam 2477: 2473: 2463: 2460: 2457: 2454: 2451: 2448: 2445: 2442: 2439: 2436: 2433: 2430: 2427: 2426: 2422: 2419: 2415: 2409: 2401: 2393: 2390: 2386: 2380: 2376: 2370: 2362: 2354: 2351: 2348: 2347: 2329: 2312: 2295: 2278: 2261: 2244: 2227: 2210: 2193: 2176: 2159: 2156: 2155: 2152: 2139: 2134: 2128: 2123: 2117: 2114: 2113: 2112: 2110: 2106: 2099: 2092: 2089: 2088: 2084: 2081: 2080: 2076: 2073: 2072: 2068: 2065: 2064: 2060: 2057: 2056: 2052: 2049: 2048: 2044: 2041: 2040: 2037: 2034: 2028: 2024: 2023:Carnatic rāga 2016: 2009: 2006: 2005: 2001: 1998: 1997: 1993: 1990: 1989: 1985: 1982: 1981: 1977: 1974: 1973: 1969: 1966: 1965: 1961: 1958: 1957: 1954: 1952: 1943: 1941: 1939: 1935: 1934:pitch classes 1931: 1930: 1925: 1921: 1914: 1912: 1905: 1903: 1882: 1879: 1876: 1875: 1871: 1868: 1866:Minor + major 1865: 1864: 1860: 1857: 1854: 1853: 1849: 1846: 1843: 1840: 1839: 1835: 1832: 1829: 1828: 1824: 1821: 1818: 1817: 1813: 1810: 1807: 1806: 1802: 1799: 1797:Major + major 1796: 1795: 1791: 1788: 1785: 1784: 1781: 1780:(2) between: 1779: 1769: 1766: 1765: 1761: 1758: 1757: 1753: 1750: 1749: 1745: 1742: 1741: 1737: 1734: 1733: 1730: 1728: 1724: 1695: 1688: 1678: 1663: 1648: 1641: 1637: 1634: 1629: 1610: 1595: 1580: 1565: 1558: 1554: 1550: 1543: 1538: 1536: 1534: 1516: 1513: 1510: 1507: 1504: 1501: 1498: 1495: 1492: 1491: 1479: 1476: 1473: 1470: 1467: 1464: 1461: 1458: 1455: 1454: 1442: 1439: 1436: 1433: 1430: 1427: 1424: 1421: 1418: 1417: 1405: 1402: 1399: 1396: 1393: 1390: 1387: 1384: 1381: 1380: 1368: 1365: 1362: 1359: 1356: 1353: 1350: 1347: 1344: 1343: 1340: 1337: 1335: 1331: 1305: 1287: 1284: 1280: 1278: 1276: 1272: 1270: 1268: 1264: 1262: 1260: 1256: 1255: 1251: 1234: 1231: 1214: 1211: 1194: 1191: 1190: 1172: 1170: 1153: 1151: 1134: 1132: 1115: 1114: 1110: 1107: 1104: 1102: 1099: 1097: 1094: 1093: 1088: 1085: 1083: 1058: 1040: 1037: 1033: 1031: 1029: 1025: 1023: 1021: 1017: 1015: 1013: 1009: 1008: 1004: 987: 984: 967: 964: 947: 944: 943: 925: 923: 906: 904: 887: 885: 868: 867: 863: 860: 857: 854: 851: 848: 845: 844: 839: 830: 828: 806: 788: 785: 781: 779: 777: 773: 771: 769: 765: 763: 761: 757: 756: 752: 735: 732: 715: 712: 695: 692: 691: 673: 671: 654: 652: 635: 633: 616: 615: 611: 608: 605: 602: 599: 596: 593: 592: 587: 584: 582: 574: 572: 565: 557: 549: 541: 533: 528:Enharmonic: C 527: 512: 509: 506: 503: 502: 497: 489: 481: 473: 465: 460:Enharmonic: D 459: 439: 436: 435: 431: 428: 427: 421: 413: 405: 397: 391: 376: 373: 370: 367: 366: 365: 363: 359: 354: 352: 348: 344: 340: 336: 332: 328: 324: 320: 316: 312: 308: 304: 300: 296: 292: 288: 283: 281: 277: 276: 271: 260: 252: 244: 243:quarter tones 240: 236: 232: 231: 226: 220: 217: 208: 203: 201: 198: 195: 191: 187: 183: 181: 178: 177: 176: 174: 170: 166: 161: 160:Genus (music) 153: 151: 149: 144: 139: 137: 133: 124: 119: 110: 105: 101: 97: 93: 89: 85: 77: 75: 73: 69: 65: 61: 57: 53: 49: 41: 37: 33: 19: 3468:All-trichord 3451:All-interval 3445: 3367:John Tyrrell 3356: 3341: 3332:John Tyrrell 3321: 3297:Lou Harrison 3291: 3277: 3273: 3262: 3261: 3252: 3247: 3239: 3234: 3226: 3221: 3209: 3193: 3185: 3177: 3165: 3153: 3145: 3140: 3132: 3116: 3111: 3103: 3094: 3074: 3066: 3056: 3047: 3039: 3027: 3017: 3008: 2998: 2984: 2974: 2960: 2953:Eleanor Cory 2942: 2938: 2924: 2918: 2908: 2902: 2895:Steven Block 2888: 2879: 2871: 2866: 2850: 2842: 2831:. Retrieved 2827:the original 2822: 2813: 2797: 2792: 2776: 2768: 2760: 2755: 2740: 2732: 2720: 2708: 2703:, p. 8. 2696: 2686: 2671: 2599: 2595: 2591: 2589: 2564: 2560: 2557:Romantic Era 2554: 2550:Figurenlehre 2549: 2543: 2527: 2519: 2511: 2507: 2499: 2496: 2489: 2469: 2149: 2103: 2030: 1947: 1933: 1927: 1923: 1918: 1915:Atonal usage 1909: 1888: 1775: 1720: 1544:Romantic era 1530: 1338: 1329: 1326: 1079: 578: 570: 504:Lydian scale 440:Chromatic: D 368:Dorian scale 361: 357: 355: 350: 346: 342: 338: 334: 330: 326: 322: 318: 314: 310: 306: 302: 298: 294: 290: 286: 284: 279: 273: 267: 172: 168: 163: 140: 132:four strings 131: 117: 103: 95: 91: 90:(from Greek 87: 83: 81: 52:tetrachordum 35: 32:music theory 29: 3408:cardinality 3071:Allen Forte 2682:Tyrrell, J. 2647:Lament bass 2585:ground bass 2349:Note name: 1920:Allen Forte 1767:Upper Minor 1636:progression 1090:Enharmonic 446:–C–B | A–G 270:incomposite 239:major third 207:minor third 171:(singular: 102:bounded by 44:τετράχορδoν 18:Tetrachords 3525:Categories 3503:Schoenberg 3458:Pentachord 3446:Tetrachord 3338:Rahn, John 2915:Stravinsky 2833:2015-04-12 2663:References 2608:homophonic 2604:variations 2598:(Fr.), or 2581:minor mode 2577:homophonic 2573:variations 2565:complainte 2520:Qui tollis 2512:Crucifixus 2476:major tone 2021:See also: 1951:whole step 1924:tetrachord 1778:whole step 1735:Tetrachord 1539:Variations 841:Chromatic 827:media help 383:–C–B | A–G 329:, and the 219:Enharmonic 190:whole tone 148:whole tone 127:κινούμενοι 96:tetrachord 36:tetrachord 3510:Aggregate 3493:Petrushka 3473:Chromatic 3463:Hexachord 3272:(2001) . 3270:Al-Farabi 2678:Sadie, S. 2567:(Fr.) or 2561:complaint 2559:, called 2492:complaint 2151:strings: 2109:Al-Farabi 1557:B Locrian 1100:parhypate 852:parhypate 600:parhypate 589:Diatonic 339:parhypate 299:hypermese 291:parhypate 200:Chromatic 104:immovable 56:intervals 3478:Diatonic 3439:Viennese 3434:Trichord 3340:. 1980. 3085:(cloth) 3073:(1973). 3040:Intégral 2745:Archived 2684:(eds.). 2626:See also 2539:♯ 2533:♯ 2405:♭ 2397:♭ 2366:♯ 2358:♯ 2143:♯ 2033:halfstep 1899:♭ 1893:♯ 1759:Harmonic 1705:♭ 1699:♭ 1670:♭ 1655:♭ 1640:bassline 1633:Phrygian 1602:♭ 1587:♭ 1572:♭ 1105:lichanos 1082:Archytas 858:lichanos 606:lichanos 536:–B | A–G 522:♭ 519:–F–E | D 516:♭ 476:–B | A–G 455:♭ 452:–F–E | D 449:♭ 443:♭ 408:–B │ A–G 386:♭ 380:♭ 351:paranete 343:lichanos 331:paramese 315:paranete 307:paramese 295:lichanos 212:♭ 194:semitone 180:Diatonic 72:tone row 3263:Sources 2981:Debussy 2600:lytanie 2596:litanie 2528:lamento 2343:⁠ 2331:⁠ 2326:⁠ 2314:⁠ 2309:⁠ 2297:⁠ 2292:⁠ 2280:⁠ 2275:⁠ 2263:⁠ 2258:⁠ 2246:⁠ 2241:⁠ 2229:⁠ 2224:⁠ 2212:⁠ 2207:⁠ 2195:⁠ 2190:⁠ 2178:⁠ 2173:⁠ 2161:⁠ 2157:Ratio: 2105:Persian 2100:Persian 1511:(28:27) 1505:(36:35) 1474:(22:21) 1468:(12:11) 1437:(28:27) 1431:(15:14) 1400:(20:19) 1394:(19:18) 1363:(16:15) 1357:(25:24) 1248:⁠ 1236:⁠ 1228:⁠ 1216:⁠ 1208:⁠ 1196:⁠ 1186:⁠ 1174:⁠ 1167:⁠ 1155:⁠ 1148:⁠ 1136:⁠ 1129:⁠ 1117:⁠ 1001:⁠ 989:⁠ 981:⁠ 969:⁠ 961:⁠ 949:⁠ 939:⁠ 927:⁠ 920:⁠ 908:⁠ 901:⁠ 889:⁠ 882:⁠ 870:⁠ 749:⁠ 737:⁠ 729:⁠ 717:⁠ 709:⁠ 697:⁠ 687:⁠ 675:⁠ 668:⁠ 656:⁠ 649:⁠ 637:⁠ 630:⁠ 618:⁠ 301:), and 245:, e.g. 118:movable 113:ἑστῶτες 106:notes ( 92:chordon 78:History 3498:Sacher 3483:Mystic 3348:  3303:  3200:  3089:  3081:  2935:Carter 2885:Dvořák 2857:  2804:  2783:  2652:Tetrad 2592:litany 2569:lament 2524:Mozart 2510:, the 2093:1 1 3 2085:1 2 2 2077:2 1 2 2069:1 3 1 2061:2 2 1 2053:3 1 1 2010:1 1 3 2002:1 2 2 1994:2 1 2 1986:1 3 1 1978:2 2 1 1970:3 1 1 1929:tetrad 1770:1 2 2 1762:1 3 1 1754:2 1 2 1746:2 2 1 1095:hypate 846:hypate 594:hypate 552:–E | D 492:–E | D 362:chroai 358:shades 323:hypate 321:. The 317:, and 287:hypate 280:pyknós 278:(from 275:pyknón 235:ditone 169:genera 143:octave 3424:Monad 3276:[ 2429:Cents 2146:–B–C. 2137:–B–C. 2090:1 2 3 2082:2 1 3 2074:1 3 2 2066:2 2 2 2058:3 1 2 2050:3 2 1 2007:1 1 3 1999:1 2 2 1991:2 1 2 1983:1 3 1 1975:2 2 1 1967:3 1 1 1936:. 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Index

Tetrachords
music theory
Greek
Latin
intervals
perfect fourth
cents
scale
tone row
greater and lesser perfect systems
Greek
Greek
lyre
octave
whole tone
Genus (music)
Ancient Greek music
Diatonic
cents
whole tone
semitone
Chromatic
minor third
Enharmonic

ditone
major third
quarter tones
double flat
half flat

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