source file: m1420.txt Date: Mon, 18 May 1998 18:19:37 -0400 Subject: RE: tetrachords From: "Paul H. Erlich" There seem to be two schools of thought about the melodic structure of scales; one is based on tetrachords, the other on "maximal evenness". Let's flesh out what they mean for some common scales. First, let's consider the modes of the major scale. All of them are maximally even in 12-, 19- or 26-tone equal temperament; that is, there is no way to more closely approximate a 7-tone equal-tempered scale in these tunings than the major scale and its modes. Now look at the modes themselves: (C D E F) (G A B C) (D E F G) (A B C D) (E F G A) (B C D E) F (G A B (C) D E F) (G A B (C) D E F) G or G (A B C (D) E F G) A (B C D (E) F G A) or (A B C (D) E F G) A (B C D (E) F G A) B Each octave species contains two identical tetrachords, shown in parentheses. In the first three modes, the tetrachords are disjunct, and in the last four, they are conjunct. So in the first three modes, any melody played in one tetrachord can be repeated an approximate 3:2 away, and in the last four, any melody played in one tetrachord can be played an approximate 4:3 away. Why is this important? Well, one theory is that the minor third was the first melodic interval ever used, and the most primitive melodies had only two notes a minor third apart. See for yourself how easy it is to sing just about any text in such a melody, and how common it is in popular sing-song. Later, a third note was added, a whole-tone away so that the outer two notes formed a perfect fourth, the 4:3 ratio. This interval is a simple enough ratio so that even without harmony, had some sort of pretonal strength to it. This three-note scale, or "trichord" was then repeated either starting on the last note of the previous trichord, or a whole-tone higher such that the last note of the second trichord was an octave, or 2:1 ratio, away from the first note of the first trichord. Thus were formed all the modes of the pentatonic scale: (C D F) (G A C) (D F G) (A C D) F (G A (C) D F) (G A (C) D F) G or G (A C (D) F G) (A C (D) F G) A (These are maximally even pentatonic scales in 7- or 12-tone equal temperaments). Then the minor thirds were filled in. In the West, the minor thirds were divided into a whole tone (already familiar) and a half tone (new). Ordering these two intervals the same way in both trichords leads to two identical tetrachords, thus the modes of the major scale above. In the Arabic world, the minor thirds were often divided instead into two equal parts, 3/4-tones which were not familiar from the trichords. This led to the following scales (+ means quarter-tone sharp, - means quarter-tone flat): (C D E- F) (G A B- C) (D E- F G) (A B- C D) F (G A B- (C) D E- F) (G A B- (C) D E- F) G or G (A B- C (D) E- F G) (A B- C (D) E- F G) A Also appearing was a scale, Mohajira, which put the 3/4-tones at opposite ends of the tetrachords and the whole-tone in the middle: (E- F G A-) (B- C D E-) (This is a maximally even heptatonic scale in 17-, 24-, or 31-tone equal temperament). Many scales in the East may have been formed by halving the whole tones, rather than the minor thirds, in each trichord. Though a clear violation of the maximal evenness principle, some of the resulting scales are found outside the West: (C C# D F) (G G# A C) (D F F# G) (A C C# D) F (G G# A (C) C# D F) (G G# A (C) C# D F) G G (A C C# (D) F F# G) (A C C# (D) F F# G) A Also appearing was a scale (Major Gypsy or Persian) which put the half-tones at opposite ends of the tetrachords and the minor third in the middle: (C# D F F#) (G# A C C#) In India, musicians may have had a very high sensitivity to intonation, which forced them to distinguish two types of whole tones. Tuning the original minor third to 6:5, the whole tone completing the 4:3 would have to be 10:9, while the one that, along with the two 4:3s, completed the 2:1, would have to be a 9:8. They appear to have reckoned the former as 3 srutis, the latter as 4 srutis, and the minor third as 6 srutis, making a 22-sruti octave. There seems to have been an early attempt at dividing the minor third into two equal parts, resulting in a scale known as Gandhaara grama, which did not survive the ancient period and whose true nature is shrouded in mystery. More successful were the Western-type approach of dividing the minor third into a half-tone (2 srutis) and a whole-tone (4 srutis). When the two minor thirds were divided the same way, the resulting "diatonic" scales could have the following sequences of intervals: (3 2 4) (3 2 4) 4 (3 4 2) (3 4 2) 4 (4 2 3) (4 2 3) 4 (4 3 2) (4 3 2) 4 (3 2 4) 4 (3 2 4) (3 4 2) 4 (3 4 2) (4 2 3) 4 (4 2 3) (4 3 2) 4 (4 3 2) 4 (3 2 4) (3 2 4) 4 (3 4 2) (3 4 2) 4 (4 2 3) (4 2 3) 4 (4 3 2) (4 3 2) There are four distinct scale types here, each represented by three modes. The four scale types are the ones that appear to have been most important in a period in Indian music history, according to a paper by Lewis Rowell, John Clough, and others. They show that these scales can be singled out from all 7-out-of-22 scales by demanding two properties. One is that there be only one tritone (half-octave interval). The other is a sort of second-order maximal evenness involving the numbers 7, 12, and 22. I pointed out to John Clough that a different but equally simple definition of second-order maximal evenness allows one to obtain the four scale types without the additional tritone rule. However, I did not make a big deal out of it, since I believe the acoustical-tetrachordal derivation to be more relevant. In modern Indian scales the two minor thirds were, however, not always divided with the intervals in the same order. Also the Eastern-type approaches of dividing the whole-tone are used in India. Finally, even the scales above each have only three octave species with identical tetrachords. Probably, the majority of Indian scales cannot be considered to have two identical tetrachords. If you read my paper on decatonic scales in 22-tone equal temperament, you will see that one version of the scale has the property that all its octave species contain two identical pentachords, either conjunct or disjunct, while another version is maximally even. The pentachordal scale can be derived from the Indian scales above by treating the 22 srutis as equal and dividing all 4-sruti intervals in half. As far as melodic appeal goes, I think the tetrachord (trichord, pentachord) concept wins, as evidenced by comparing the two scales played melodically. However, the maximally even scale contains 8 consonant tetrads to the pentachordal scale's 6. Another case of melody and harmony being at odds for scale construction (something Ivor Darreg liked to point out).