Hello, William and all. Recently Marcel De Velde has introduced a Melisse series of 12-note scales in Pythagorean tuning, and my purpose here is to show how his method can be adapted to 24-ed2 in a series of Quasi-Melisse scales that provide one method for exploring 24 and some of its resources. Here I'm going to explain the Quasi-Melisse concept as it applies to 24-ed2 without getting into the specifics of the beautiful Melisse tunings in Pythagorean, which deserve attention in their own right. My purpose is to focus on a very creative strategy for exploring 24, either to get oriented to or to "rediscover" some of its possibilities. One very natural way for many people to view 24-ed2 is as a system with two 12-note chains of fifths, or also circles of fifths, in 12-ed2. There are actually three tuning parameters, so to speak: the 2/1 octave or period of the tuning; the 700-cent fifths generating each 12-note chain; and the 50-cent or quartertone spacing between the chains. From a comparative viewpoint, we could look at 24-ed2 of one of the many tunings which can be grouped into two 12-note chains or circles at some distance or spacing apart. The intervals of each 12-note chain are familiar to anyone acquainted with 12-ed2; but it's the intervals that result from mixing notes of the two chains or circles, including that intriguing 50-cent quartertone step itself that defines the space between them, that are of special interest in exploring 24. To use the Quasi-Melisse method, let's start with a 12-ed2 chain. Since the Dorian mode is one of my favorites, I'll make D the 1/1. D Eb E F F# G G# A Bb B C C# D 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 In this familiar 12-ed2 set, we have only a single adjacent step size between any two notes: the 100-cent semitone, of course. I've arbitrarily decided to call the note at a 600-cent tritone above D by the name G#, although Ab would have been just as correct. But the way I've spelled things, for "antiquarian" reasons so not important here (those used to a Pythagorean or meantone tuning in Eb-G# may understand), helps us in setting out on our Quasi-Melisse quest to know 24 better. That method, as applied here, is to start at the top of the chain of fifths as I've arbitrarily named the notes -- here with G# at 600 cents -- and replace a note in our usual 12-ed2 set with a note from its sister chain in 24-ed2. Here we'll do that by substituting a note 50 cents lower, giving this as our first Quasi-Melisse scale: D Eb E F F# G G^ A Bb B C C# D 0 100 200 300 400 500 550 700 800 900 1000 1100 1200 ! quasi-melisse24-1.scl ! Marcel De Velde's Melisse series adapted to 24-ed2 (#1) 12 ! 100.00000 200.00000 300.00000 400.00000 500.00000 550.00000 700.00000 800.00000 900.00000 1000.00000 1100.00000 2/1 We've only changed one note, introducing a 550-cent step, a "superfourth" or small tritone or diminished fifth very close to 11/8. This note lets us try a jazzy version of a Near Eastern maqam or modality called Maqam Nakriz: Nakriz-5 Nahawand-4 |------------------|-------------| D E F G^ A B C D 0 200 300 550 700 900 1000 1200 200 100 250 150 200 100 200 Maqam Nakriz has a lower fifth, or pentachord, also known as Nakriz, which from one view could be described as a lower tone around 9/8, D-E, plus a division of the fourth E-A or tetrachord called Hijaz, E-F-G^-A, here 100-250-150 cents. Then, on the 3/2, we have the same upper tetrachord as the familiar Dorian mode: 200-100-200 cents, a usual tone-semitone-tone. This version of Nakriz alters only one note of Dorian: the 4/3 step at a tempered 500 cents gets replaced with a "lightly augmented fourth" at 550 cents or a tempered 11/8. But we get lots of new steps and intervals: the Nakriz pentachord or Nakriz-5 for short (Eric Ederer is my source for this helpful notation) introduces both a neutral second step at 150 cents, and the fascinating "plus-tone" or very small minor third at 250 cents -- somewhere between the large 8/7 tone (231 cents) and the small 7/6 third (267 cents). It can make a great middle step for Nakriz and other maqamat including a Hijaz tetrachord (or a Nakriz pentachord with its lower tone followed by such a tetrachord), with 2-5-3 or 3-5-2 steps of 24 both beautiful tunings of Hijaz. So far, we've replaced only G#, the note at the top of the chain of fifths, with a note 50 cents lower from our second chain of fifths. Moving down the chain of our original 12-note set, we next replace C#, the major seventh at 1100 cents above D, with C^ at 1050 cents, a virtually just 11/6. D Eb E F F# G G^ A Bb B C C^ D 0 100 200 300 400 500 550 700 800 900 1000 1050 1200 ! quasi-melisse24-2.scl ! Marcel De Velde's Melisse series adapted to 24-ed2 (#2) 12 ! 100.00000 200.00000 300.00000 400.00000 500.00000 550.00000 700.00000 800.00000 900.00000 1000.00000 1050.00000 2/1 Now we have some new options, one of which is a variation on Nakriz called Basandidah, with the same lower Nakriz-5 genus, but an upper genus of Rast-4 or 200-150-150 cents. Nakriz-5 Rast-4 |------------------|-------------| D E F G^ A B C^ D 0 200 300 550 700 900 1050 1200 200 100 250 150 200 150 150 Note that we can still, with Quasi-Melisse scale #2, play Maqam Nakriz with C, as well as Basandidah with C^. Another choice now open to us is a colorful and expressive version of a very popular Arab modality: Maqam Nahawand. Often this is compared to a Western minor scale, but our subset of 24-ed2 includes a Nahawand with a difference: Nahawand-4 T Hijaz-4 |-------------|-----|--------------| D E F G A Bb C^ D 0 200 300 500 700 800 1050 1200 200 100 200 200 100 250 150 Nahawand has the same lower tetrachord (Nahawand-4) as Dorian, followed by a middle tone leading to the upper tetrachord on A: Hijaz, here 100-250-150 cents, as in our tuning of Nakriz. The minor sixth, neutral seventh step a near-just 11/6, and the 250-cent step between them together make this quite different than a Western modality or tonality such as Dorian or minor. Nahawand as a maqam is far more intricate, with various possible tetrachords, inflections, and modulations: but the immediate purpose here is just to give some glimpses of what is possible by combining notes from our two 12-note chains. Our next step down the chain of fifths after G# and C# is to replace F# (the 400-cent major third above D) with a note 50 cents lower: a neutral third at 350 cents, F^. We can think of this as representing either 11/9 (347 cents) or 27/22 (355 cents). D Eb E F F^ G G^ A Bb B C C^ D 0 100 200 300 350 500 550 700 800 900 1000 1050 1200 ! quasi-melisse24-3.scl ! Marcel De Velde's Melisse series adapted to 24-ed2 (#3) 12 ! 100.00000 200.00000 300.00000 350.00000 500.00000 550.00000 700.00000 800.00000 900.00000 1000.00000 1050.00000 2/1 One obvious option at this point is to introduce the premier maqam of the Arab or Turkish system, Rast. We have available the two most common forms of this maqam: an ascending form with the neutral seventh C^ that we used in Basandidah and Nahawand; and a descending form with the minor seventh C. Rast-4 T Rast-4 |-------------|-----|--------------| D E F^ G A B C^ D 0 200 350 500 700 900 1050 1200 200 150 150 200 200 150 150 Rast-4 T Nahawand-4 |-------------|-----|--------------| D E F^ G A B C D 0 200 350 500 700 900 1000 1200 200 150 150 200 200 100 200 There are lots more possibilities, but the availability of the most common forms of ascending and descending Rast is a main landmark of Quasi-Melisse in scale #3 of the series. Note that the upper Nahawand tetrachord of the descending form is identical to that of Dorian or Mixolydian, which may give this form of Rast a certain resemblance to these popular Western modes. Moving down the chain of fifths in our Quasi-Melisse process, from G#, C#, and F#, our next location for replacement is B, the 900-cent major sixth -- and, for the first time, a note of the D Dorian mode. While this means that our 12-note set will not include all the notes of Dorian on D, we may be able the mode elsewhere -- as well as knowing that it remains present on D in the full 24-ed2 set. The idea is to focus for the moment on other possibilities. D Eb E F F^ G G^ A Bb Bv C C^ D 0 100 200 300 350 500 550 700 800 850 1000 1050 1200 ! quasi-melisse24-4.scl ! Marcel De Velde's Melisse series adapted to 24-ed2 (#4) 12 ! 100.00000 200.00000 300.00000 350.00000 500.00000 550.00000 700.00000 800.00000 850.00000 1000.00000 1050.00000 2/1 Among many other things, two additional variations on Rast now become available to us. One of them is the classic medieval form of this maqam, with two conjunct tetrachords of Rast on 1/1 and 4/3, and a beautiful tetrachord on the 3/2 called Bayyati, 150-150-300 cents, which at least in a modern interpretation of this conjunct form of Rast often gets lots of emphasis. Rast-4 T Bayyati-4 |-------------|-----|--------------| D E F^ G A Bv C D 0 200 350 500 700 850 1000 1200 200 150 150 200 150 150 200 The other choice, not so prominent in Arab theory but important in some 20th-century Turkish theory, is what Jacques Dudon calls a "folk Rast" with neutral sixth and seventh: Rast-4 T `Iraq-4 |-------------|-----|--------------| D E F^ G A Bv C^ D 0 200 350 500 700 850 1050 1200 200 150 150 200 150 200 150 The upper tetrachord here is sometimes recognized in modern Arab theory, following medieval theory, as `Iraq: 150-200-150 cents. Jacques Dudon has rediscovered this type of tetrachord from his own musical and spiritual perspective, and called it Mohajira. Indeed, the `Iraq or Mohajira tetrachord is a good segue to our next scale, Quasi-Melisse #5, where, moving down the chain of fifths G#-C#-F#-B, we next replace E, the 200-cent major second above D near 9/8, with a note on the other chain of fifths 50 cents lower: Dv at 150 cents or a near-just 12/11. D Eb Ev F F^ G G^ A Bb Bv C C^ D 0 100 150 300 350 500 550 700 800 850 1000 1050 1200 ! quasi-melisse24-5.scl ! Marcel De Velde's Melissa series adapted to 24-ed2 (#5) 12 ! 100.00000 150.00000 300.00000 350.00000 500.00000 550.00000 700.00000 800.00000 850.00000 1000.00000 1050.00000 2/1 Among many other things, the scale supports two beautiful forms of Jacques Dudon's Mohajira tuning. The first has two symmetrical Mohajira tetrachords of 150-200-150 cents with a middle tone between them: Mohajira-4 T Mohajira-4 |-------------|-----|---------------| D Ev F^ G A Bv C^ D 0 150 350 500 700 850 1050 1200 150 200 150 200 150 200 150 The second form, which Jacques Dudon calls Ibina, has a lower Mohajira tetrachord and an upper tetrachord of Bayyati on the fifth step, like the upper Bayyati tetrachord of a conjunct Rast. Mohajira-4 T Bayyati-4 |-------------|-----|---------------| D Ev F^ G A Bv C D 0 150 350 500 700 850 1000 1200 150 200 150 200 150 150 200 The name Ibina actually refers to Dudon's specific JI tuning of this modality at 1/1-13/12-11/9-4/3-3/2-13/8-16/9-2/1, but I also use it as a convenient name for this modal pattern generally. At this point, we have reached the limit of how many notes we can alter and still maintain a perfect fifth above D at 700 cents. However, one more replacement will bring us to the point where we have six notes from each chain or circle of fifths. Having replaced G#-C#-F#-B-E, we therefore now substitute for A, the fifth at 700 cents above D, the note on the other chain which is 50 cents lower: Av at 650 cents, a near-just 16/11. D Eb Ev F F^ G G^ Av Bb Bv C C^ D 0 100 150 300 350 500 550 650 800 850 1000 1050 1200 ! quasi-melisse24-6.scl ! Marcel De Velde's Melisse series adapted to 24-ed2 (#6) 12 ! 100.00000 150.00000 300.00000 350.00000 500.00000 550.00000 650.00000 800.00000 850.00000 1000.00000 1050.00000 2/1 In classical Western theory, the idea of a mode without a 3/2 has often been problematic, with lots of debate in the 16th century about whether the B modes (often now known as Locrian) should count -- a mostly theoretical debate, since these modes were not then generally in use, although certain Gregorian chants centering on B might arguably be assigned to them. Modern Western forms such as jazz, however, take a more relaxed attitude about Locrian and similar forms. In Near Eastern music, however, some very prominent modes may have no 3/2 fifth, and one worth noting here is Maqam `Iraq. While lots of modes tend to divide into tetrachords and tone, or sometimes also pentachords, `Iraq is one of the main maqamat with a lower trichord or three-note genus called Sikah, which refers to the third step of Rast in its usual placement. Here, however, we will simply locate Maqam `Iraq (and its lower Sikah trichord) on our 1/1, D. Sikah-3 Bayyati-4 Rast-3 |---------|-------------|--------| D Ev F^ G Av Bv Cv D 0 150 350 500 650 850 1050 1200 150 200 150 150 200 200 150 Maqam `Iraq divides into a lower Sikah-3, a middle Bayyati-4, and an upper Rast-3 -- with the upper trichord often tending to reach beyond the octave to Ev above it, 150 cents higher, completing a Rast tetrchord with D-Ev as a leading tone. The maqam as a modal form gets more intricate, but even this simple "textbook" seven-note or heptatonic suggests some of the possibilities of modality without a 3/2 step. Given the symmetries of 24-ed2, these six Quasi-Melisse scales may suffice to suggest some of the possibilities opened by combining notes from the two 12-note chains. Exactly the same method can be used with other systems have two 12-note chains of fifths at some arbitrary distance apart -- not necessarily an ed2 system. It can also apply to systems with more than 12 notes in a chain -- for example, 17-note or 29-note chains for wide-fifth tunings. However, this survey of Quasi-Melisse has only shown a few possibilities of these scales focusing mostly on maqam and neutral intervals. There's lots more, which may invite more articles on Marcel De Velde's scale generation method as applied to 24-ed2. Most appreciatively, Margo Schulter mschulter@calweb.com September 26, 2013