Chapter II. GREEK TUNINGS Greek music theory is highly complex and difficult, with its al- phabetical notation, the dependence of musical rhythm upon poetic meter, and all the rest of it. Our confusion is not lessened by the fact that scholars quarrel about the exact interpretation of the modal scales and that a pitifully scant remnant of the music itself is available for study today. Fortunately it is possible to understand the essentials of Greek tuning theories without enter- ing into the other and more controversial aspects of Greek mu- sical science. Moreover, it is advisable that the Greek tuning lore be presented in some detail in order that the attitude of many sixteenth and seventeenth century theorists may be clarified. The foundation of the Greek scale was the tetrachord, a de- scending series of four notes in the compass of the modern per- fect fourth. Most typical was the Dorian tetrachord, with two tones and then a semitone, as A G F E or E D C B. Two or more tetrachords could be combined by conjunction, as the above tetra- chords would be with E a common note. Or they might be com- bined by disjunction, as the above tetrachords would be in reverse order, with a whole tone between B and A. Tetrachords combined alternately by conjunction and by disjunction correspond to our natural heptatonic scale. The Greeks had three genera— diatonic, chromatic, and en- harmonic. A diatonic tetrachord contained two tones and a semi- tone, variously arranged, the Dorian tetrachord having the order shown above, as A G F E. In the chromatic tetrachord the second string (as G) was lowered until the two lower intervals in the tetrachord were equal. Thus A G" F E represents the process of formation better than the more commonly shown A F* F E. In the enharmonic tetrachord the second string was lowered still further until it was in unison with the third string; the third string was then tuned half way between the second and fourth strings. In notes the enharmonic tetrachord would be A G"" F E or A F F E. Thus in the chromatic tetrachord there were the consecu- tive semitones that we associate with the modern chromatic genus; but the enharmonic tetrachord contained real quarter tones, whereas our enharmonically equivalent notes, as F*5 and E, differ by a comma, 1/9 tone, or at most by a diesis, 1/5 tone. TUNING AND TEMPERAMENT Claudius Ptolemy has presented the most complete list of tunings advocated by various theorists, including himself. * These (with one exception to be discussed later) were shown by the ratios of the three consecutive intervals that constituted the tetrachord, and also by string- lengths for the octave lying be- tween 120 and 60, using sexagesimal fractions where necessary. The octave is the Dorian octave, as from E to E, with B-A the disjunctive tone, always with 9:8 ratio. Ptolemy's tables are given here (Tables 1-21) with comments following. The frac- tions have been changed into decimal notation. Greek Enharmonic Tunings Table 1. Archytas' Enharmonic Lengths 60.00 75.00 77.14 80.00 90.00 112.50 115.71 120.00 Names ECCBA F F E Ratios 5/4 36/35 28/27 9/8 5/4 36/35 28/27 Cents 1200 814 765 702 498 112 63 Table 2. Aristoxenus' Enharmonic Lengths 60.00 76.00 78.00 80.00 90.00 114.00 117.00 120.00 b b Names E C C B A F F E Parts 16 2 2 10 24 3 3 Cents 1200 791 746 702 498 89 44 0 Table 3. Eratosthenes' Enharmonic Lengths 60.00 75.00 77.50 80.00 90.00 112.50 116.25 120.00 b b Names ECCBA F F E Ratios 5/4 24/23 46/45 9/8 5/4 24/23 46/45 Cents 1200 814 740 702 498 112 38 0 *Claudii Ptolemaei Harmonicorum libri tres. Latin translation by John Wallis (London, 1699). 16 GREEK TUNINGS Greek Chromatic Tunings Table 4. Archytas' Chromatic Lengths 60.00 71.11 77.14 80.00 90.00 106.67 115.71 120.00 Names E Db C B A Gb F E Ratios 32/27 243/224 28/27 9/8 32/27 243/224 28/27 Cents 1200 906 765 702 498 204 63 0 Table 5. Aristoxenus' Chromatic Malakon Lengths 60.00 74.67 77.33 80.00 90.00 112.00 116.00 120.00 Db C B A Gb F E 23" 23 10 22 4 4 821 761 702 498 119 59 0 Table 6. Aristoxenus' Chromatic Hemiolion Lengths 60.00 74.00 77.00 80.00 90.00 111.00 115.50 120.00 Names E Db C B A Gb F E Parts 14 3 3 10 21 42 4* Cents 1200 837 768 702 498 135 66 0 Table 7. Aristoxenus' Chromatic Tonikon Lengths 60.00 72.00 76.00 80.00 90.00 108.00 114.00 120.00 Names E Db C B A Gb F E Parts 12 4 4 10 18 6 6 Cents 1200 884 791 702 498 182 89 0 Names E Parts 14 3 Cents 1200 17 TUNING AND TEMPERAMENT Table 8. Eratosthenes' Chromatic Lengths 60.00 72.00 76.00 80.00 90.00 108.00 114.00 120.00 Names E Db C B A Gb F E Ratios 6/5 19/18 20/19 9/8 6/5 19/18 20/19 Cents 1200 884 791 702 498 182 89 0 Table 9. Didymus' Chromatic Lengths 60.00 72.00 75.00 80.00 90.00 108.00 112.50 120.00 Names E Db C B A Gb F E Ratios 6/5 25/24 16/15 9/8 6/5 25/24 16/15 Cents 1200 884 814 702 498 182 112 0 Table 10. Ptolemy's Chromatic Malakon Lengths 60.00 72.00 77.14 80.00 90.00 108.00 115.71 120.00 Names E Db C B A Gb F E Ratios 6/5 15/14 28/27 9/8 6/5 15/14 28/27 Cents 1200 884 765 702 498 182 63 0 Table 11. Ptolemy's Chromatic Syntonon Lengths 60.00 70.00 76.36 80.00 90.00 105.00 114.55 120.00 Names E Db C B A Gb F E Ratios 7/6 12/11 22/21 9/8 7/6 12/11 22/21 Cents 1200 933 783 702 498 231 81 0 18 GREEK TUNINGS Greek Diatonic Tunings Table 12. Archytas' Diatonic Lengths 60.00 67.50 77.14 80.00 90.00 101.25 115.71 120.00 Names EDCBAG F E Ratios 9/8 8/7 28/27 9/8 9/8 8/7 28/27 Cents 1200 996 765 702 498 294 63 0 Table 13. Aristoxenus' Diatonic Malakon Lengths 60.00 70.00 76.00 80.00 90.00 105.00 114.00 120.00 Names EDCBAG F E Parts 10 6 4 10 15 9 6 Cents 1200 933 791 702 498 231 89 0 Table 14. Aristoxenus' Diatonic Syntonon Lengths 60.00 68.00 76.00 80.00 90.00 102.00 114.00 120.00 Names EDCBAG F E Parts 8 8 4 10 12 12 6 Cents 1200 983 791 702 498 281 89 0 Table 15. Eratosthenes' Diatonic Lengths 60.00 67.50 75.94 80.00 90.00 101.25 113.91 120.00 Names EDCBAG F E Ratios 9/8 9/8 256/243 9/8 9/8 9/8 256/243 Cents 1200 996 792 702 498 294 90 0 19 TUNING AND TEMPERAMENT Table 16. Didymus' Diatonic Lengths 60.00 67.50 75.00 80.00 90.00 101.25 112.50 120.00 Names EDCBAG F E Ratios 9/8 10/9 16/15 9/8 9/8 10/9 16/15 Cents 1200 996 814 702 498 294 112 0 Table 17. Ptolemy's Diatonic Malakon Lengths 60.00 68.57 76.19 80.00 90.00 102.86 114.27 120.00 Names EDCBAG F E Ratios 8/7 10/9 21/20 9/8 8/7 10/9 21/20 Cents 1200 969 787 702 498 265 85 0 Table 18. Ptolemy's Diatonic Toniaion Lengths 60.00 67.30 77.14 80.00 90.00 101.25 115.71 120.00 Names EDCBA G F E Ratios 9/8 8/7 28/27 9/8 9/8 8/7 28/27 Cents 1200 996 765 702 498 294 63 0 Table 19. Ptolemy's Diatonic Ditoniaion Lengths 60.00 67.50 75.94 80.00 90.00 101.25 113.91 120.00 Names EDCBA G F E Ratios 9/8 9/8 256/243 9/8 9/8 9/8 256/243 Cents 1200 996 792 702 498 294 90 0 Table 20. Ptolemy's Diatonic Syntonon Lengths 60.00 66.67 75.00 80.00 90.00 100.00 112.50 120.00 Names EDCBAG F E Ratios 10/9 9/8 16/15 9/8 10/9 9/8 16/15 Cents 1200 1018 814 702 498 316 112 0 20 GREEK TUNINGS Table 21. Ptolemy's Diatonic Hemiolon Lengths 60.00 66.67 73.33 80.00 90.00 100.00 110.00 120.00 Names EDCBA G F E Ratios 10/9 11/10 12/11 9/8 10/9 11/10 12/11 Cents 1200 1018 853 702 498 316 151 0 Only two of these seventeen or eighteen independent tunings have had any great influence upon modern music theory— the third and fourth of Ptolemy's diatonic scales, commonly called the "ditonic" and the "syntonic." The former is the same as Era- tosthenes' diatonic, and is the old Pythagorean tuning. It gains its name from the fact that its major third (ditone) consists of a pair of equal tones. The latter, the "tightly stretched" in con- trast to the "soft" (malakon), is what we know as just intonation. Didymus' diatonic contains the same intervals as Ptolemy's syn- tonic diatonic, but with the minor tone (10:9) below the major tone (9:8) instead of the reverse. Didymus' arrangement is the more logical for constructing amonochord; Ptolemy's in terms of the harmonic series. The theorists of the sixteenth and seventeenth centuries, eager to bolster their ideas with classical prototypes, pointed out that the just tuning was that of Didymus and Ptolemy. But they ignored the other diatonic tunings of Ptolemy. They liked to point out further that in three of the enharmonic tunings the pure major third (5:4) appears, and in four of the chromatic tunings the pure minor third (6:5). But only Didymus used en- harmonic and chromatic tunings that really resembled just into- nation. His chromatic is tuned precisely as E, C*, C, etc., would be in just intonation, using the chromatic semitone, 25:24, which appears in no other tuning. In his enharmonic, not only does the major third have the ratio 5:4, but the small intervals are "equal" quarter tones, resulting from an arithmetical division of the 16:15 semitone.* The other nine enharmonic and chromatic tun- ings depart more or less from Didymus' standard. 21 TUNING AND TEMPERAMENT Let us examine more of the peculiarities of these Greek tun- ings. Archytas has used the same ratio (28:27) for the lowest interval in each genus, thus having an interval (63 cents) that is much larger than most of the semitones and smaller than the quarter tones. The ditonic semitone, 256:243, is about the same size as Ptolemy's "soft" semitone, 21:20, being a comma smaller than the syntonic semitone, 16:15. The tones range from mini- mum, 11:10, through minor, 10:9, and major, 9:8, to maximum, 8:7. Archytas' minor third, 32:27, is a comma larger than the syntonic third, 6:5, and more than a comma smaller than Ptol- emy's minor third, 7:6. Eratosthenes' major third, 19:15, is about the same size as the Pythagorean ditone, 81:64, and is about a ditonic comma larger than the syntonic third, 5:4. Ever since his own age a great controversy has raged about the teachings of Aristoxenus. Instead of using ratios, he divided the tetrachord into 30 parts, of which, in his diatonic syntonon, each tone has 12 parts, each semitone 6. Thus, if we are to take him at his word, Aristoxenus was here describing equal tem- perament. The sixteenth and seventeenth century theorists were of the opinion that such was his intention, the advocates of equal temperament opposing the name of Aristoxenus to that of Ptolemy. Ptolemy himself did not so understand Aristoxenus' doctrines. With a fundamental of 120 units, the perfect fourth above has 90 units. Thus, as shown in the tables, Ptolemy subtracted Aris- toxenus' "parts" from 120. His enharmonic then agrees with that of Eratosthenes, and his chromatic tonikon with the latter 's chromatic. But Aristoxenus' diatonic syntonon does not then quite agree with the Pythagorean (ditonic) diatonic, although the latter is the only Greek tuning that contains two equal tones. His diatonic malakon, as Ptolemy has shown it, is unlike any of the other tunings; whereas in its succession of intervals— large, medium, small — it resembles Ptolemy's diatonic malakon or chromatic syntonon. So it seems quite likely that Aristoxenus did not intend to ex- press any new tunings by his adding together of parts of a tone, but simply to indicate in a general way the impression that cur- rent tunings made upon the ear. But his vagueness has made possible all sorts of wild speculations. It is even possible, by 22 GREEK TUNINGS an improper manipulation of the figures, to argue that Aris- toxenuswas a proponent of just intonation. Take his enharmonic: 24 + 3 + 3. Add these numbers to 90 in reverse order as before, getting 90 93 96 120. Then consider these numbers to be fre- quencies rather than string- lengths. The result is practically thesameasDidymus': 5/4 x 32/31 x 31/30. Or take Aristoxenus' diatonic syntonon: 12 + 12 + 6. Treat it as we have just treated his enharmonic, getting 90 96 108 120. If these are then taken as frequencies, we have Ptolemy's syntonic, 10/9 x 9/8 x 16/15. The paramount principle in Ptolemy's tunings was the use of superparticular proportion, a ratio in which the antecedent ex- ceeds the consequent by unity. (The Latin prefix "sesqui" is conveniently used to describe these ratios, e.g., "sesquiquarta," meaning 5/4.) Ptolemy used 5/4, 6/5, 7/6, 8/7, etc. Seven of the eight tunings that bear his own name are constructed entirely of superparticular proportions, the eighth being the ditonic, or Pythagorean. Seven tunings that he has ascribed to other writers also use these ratios exclusively, including all of Didymus' tun- ings, Archytas' enharmonic and diatonic, and Eratosthenes' chro- matic (Aristoxenus' chromatic tonikon). In just intonation the ratios are, of course, superparticular, and this feature only would have appealed to Ptolemy and his contemporaries. For, despite the many apparently just intervals used in the given tun- ings, Ptolemy recognized no consonances other than those of the Pythagorean tuning— fourth, fifth, octave, eleventh, twelfth, and fifteenth. It is easy to obtain, by algebra, all the possible divisions of the tetrachord built up entirely by superparticular proportions. (A theory for the superparticular division of tones is shown in connection with Colonna, in Chapter VIL) Eliminating those in which one interval is considerably smaller than the smallest enharmonic quarter tone (46:45), we find that, collectively, the Greeks had not omitted many possibilities. Other enharmonic tunings similar to Ptolemy's would be 5/4 x 22/21 x 56/55 and 5/4 x 26/25 x 40/39. Chromatic tunings would include 6/5 x 13/12 x 40/39; 7/6 x 9/8 x 64/63; 7/6 x 10/9 x 36/35; and 7/6 x 15/14 x 16/15. Two others are difficult to classify: 8/7 x 13/12 x 14/13 might best be considered a chromatic tuning, something 23 TUNING AND TEMPERAMENT like 14 + 8 + 8 in Aristoxenus' parts. And 8/7 x 8/7 x 49/48 is undoubtedly a variant of the ditonic tuning, but with a quarter tone instead of a semitone at the bottom, perhaps 14 + 14 + 2. In later chapters we shall see many echoes of Greek tuning methods, not only in such well-known systems as the Pythagorean and the just, but also in the modified systems, such as Ganassi's, and in irregular systems, such as Dowland's. Unusual super- particular intervals are used by Colonna in the poorest tuning system shown in this book, and also by Awraamoff , whose system is even worse. 24