Extension and Recombination of The Messiaen’s Limited Transpositions Modes with The Interval Cycles Technique
Author:Chenming Zhang
China,Henan Province +86 18838135530
Abstract:
The Interval Cycles technique as an important way of pitch structure after the 20th Century, presents the important development power recently. The Interval Cycles technique pitch structure resource includes a widely used, such as: chromatic, diatonic, eight sounds scale, pentatonic scale, six sound scale, diatonic scale,Olivier Messiaen limited trans postions modes, and many other pitch structure resources.From the perspective of the classification and parametric algorithm of interval cycles by Edward Gollin, an American Musical Theorist, in my this paper studies the properties of non-multiple Aggregate interval cycles including Messiaen’s limited transpostions modes patterns and their relations with limited transpostions modes patterns.
Tag:
Interval Cycles; Compound Interval Cycle; Multi-Aggregate Cycles;Messiaen’s Limited Transpostions Modes
In the past 100 years of music creation and research, mathematical and physical logic as a trend, but also a more rational way of thinking affects the composition and analysis of composers and music theorists. Among them, interval circulation has been widely used in the works of composers such as Bartok and Ives, as a kind of mathematical logic of post-modern music in terms of pitch structure. In theory, the classification of interval cycles and related operations are becoming more and more perfect with the efforts of contemporary theorists such as American theorist Edward Gollin.
In the following, I mainly uses the relevant features described by Edward Gollin for the non-multiple Aggregate interval cycle and the related operations he puts forward to make a new interpretation of the limited transpostions mode that is widely known by French composer Messiaen from the perspective of interval cycle , And expand the new limited transpostions mode and form an algorithm, and at the same time innovate and research on the method and logic of splitting the same limited shift mode into different non-multiple cluster intervals.
1. Isomorphic relationship between Non-multiple Aggregate Interval Cycles and Limited transpostion mode
In a recent study by the American theorist Edward Gollin, it divides Compound interval cycles into two types of multi-Aggregate Interval Cycles and non-multi-Aggregate Interval Cycles. Edward Gollin defines multi-Aggregate Interval Cycles, which refers to those “interval groups that mix cycles with multiple intervals,” The phenomenon of returning to the original cycle point many times during the cycle, and finally reaching the starting point with the original interval cycle group value. “For example, Ord.PCI ((2,2,1) and (1,2, 2) or (2,1,2) constituted by the cycle (where Ord.PCI stands for ordered pitch intervals, the numbers separated by commas in parentheses represent the composition value of the interval cycle). See Figure 1. In Figure 1 Square brackets, pointed brackets, and ellipses respectively circled three types of cyclic Aggregate represented by pitch-class, so they are all multi-aggregate cycles.
Figure 1 (2,2,1) and (1,2,2) or (2,1,2) multiple aggregate cycles
Then the Non-multiple Aggregate Cycles refers to the phenomenon that the return to the original cycle point can be completed with the original interval cycle group value only once. For example: Ord.PCI (2), a single pitch cycle with a whole tone as the cycle value, constitutes a diatonic scale, and all its constituent sounds are represented by digital scales. If it is “0 2 4 6 8 10 0”, its subsequent recycling will continue. “0 2 4 6 8 10 0”. Another example: Ord.PCI (2,1,1) is “0 2 3 4 6 7 8 10 11 0” and its subsequent cycle is also “0 2 3 4 6 7 8 10 11 0”
According to the author’s experiment, the single interval cycle conforms to the characteristics of non-multiple aggregate cycles, while the mixed interval cycle must meet the characteristics that the sum of its cycle composition values cannot be prime with each other, and at the same time the calculated total number of sounds cannot exceed 12 In order to meet the characteristics of non-multiple aggregate cycles. The author pointed out in “On the Theory of Edward Gollin’s Interval Cycle Parameter Algorithm” that “all non-multiple aggreate cycles also conform to the logic of compound splitting on the basis of a single interval cycle.” Then at the digital level with modulo 12 as the logic In the case of calculation, we only need to consider the single interval level is only 1 to 6, because its complement with 12 has the same characteristics as the original interval level. Then, as long as it is divisible by 12 in the interval levels of 1 to 6, the mixed intervals can be split based on it, thereby forming a non-multiple cluster loop. Then there is only 5 in it, and the interval level 1 cannot be further subdivided, so it cannot be split into a new compound interval cycle. Of course, if the total number of sounds in a mixed interval loop is greater than 12, then multiple cluster loops will definitely be generated in the case of forming repeated digital levels, because the repeated digital levels will disrupt the original after the loop returns to the starting point. Cyclic group value.
Then the scale composed of a compound interval cycle with such characteristics has 1, unable to exhaust 12 semitones, 2, the sum of its cycle group value and 12 is not a prime number relationship, that is to say, according to Green’s sum of cycle group value SUM and 12 The greatest common divisor GCD can be used to express the number of its mode shifts logically, it contains only limited mode shift possibilities. Then this coincides with the scale logic of the limited shift mode proposed by the French composer Messiaen In Yang Liqing’s book “Sincerity, Elegance, and Sincerity-Messiaen’s Musical Language”, it is pointed out that “Messiaen’s finite shift pattern is composed of an octave divided into several equal parts by the same interval, with multiple cycles. The cells with the same structure have a limited possibility of displacement. “Then, by verifying the author’s conclusion, the existing finite displacement model of Messiaen is consistent with the characteristics of non-multiple aggregate cycles.
2. The algorithm and extension of non-multiple Aggregate interval cycles and limited transposition mode
(1) The expression of the limited transposition mode converted into the interval cycle algorithm mode
The following shift modes in the limited shift mode are listed in the form starting with the C sound (0), and expressed as a digital sound level of 1-11 under the premise of modulo 12.
Mode 1 diatonic Ord.PCI (2) 0 2 4 6 8 10 0
Mode 2 Octave Scale Ord. PCI (1,2) 0 1 3 4 6 7 9 10 0
Mode 3 Ord.PCI (2,1,1) 0 2 3 4 6 7 8 10 11 0
Mode 4 Ord.PCI (1,1,3,1) 0 1 2 5 6 7 8 11 0
Mode 5 Ord.PCI (1,4,1) 0 1 5 6 7 11 0
Mode 6 Ord.PCI (2,2,1,1) 0 2 4 5 6 8 10 11 0
Mode 7 Ord.PCI (1,1,1,2,1) 0 1 2 3 5 6 7 8 9 11 0
So far, we have obtained seven new expressions that convert Messiaen’s finite shift mode to non-multi-cluster cyclic mode, so Messiaen’s seven finite shift modes can be studied with Green’s interval cyclic mode algorithm .
(2) The calculation of Edward Gollin’s maximum convention value for the number of times of Mesian’s finite shift modulation shift
Edward pointed out that, for example, a mixed interval loop with (4,5) as the group value, and at the same time its GCD (12,9) = 3, it can be found that the loop has only three different shift forms. This cycle constitutes an octave scale, that is, the magnitude of the number of times of finite shift modulation in Messiah.
It is verified by the author that the algorithm is also applicable to the calculation of all Meixian finite shift modes. For example, Ord. PCI cycle of the fourth finite shift modulation mode of Meixi’an is 1,1,3,1. Then SUM = 6, GCD = 6, so this finite shift modulation includes 6 finite shifts.
(3) Non-multiple cluster interval cycle perspective generates a new limited shift pattern
First of all, according to Edward’s theory of simplifying the mixing interval cycle, the sum of all the combined values of intervals in the finite shift mode is listed as SUM
Mode 1 diatonic Ord.PCI (2) SUM = 2
Mode 2 octave Ord.PCI (1,2) SUM = 3
Mode 3 Ord.PCI (2,1,1) SUM = 4
Mode 4 Ord.PCI (1,1,3,1) SUM = 6
Mode 5 Ord.PCI (1,4,1) SUM = 6
Mode 6 Ord.PCI (2,2,1,1) SUM = 6
Mode 7 Ord.PCI (1,1,1,2,1) SUM = 6
In this way, it can be found that if the above SUM value is Ord.PCI, a cycle is formed, that is, all the simplified cycles are single-tone cycles, and they cannot exhaust 12 semitones. At the same time, they are also non-prime. The author’s discussion of the characteristics of non-multiple aggregate cycles. In the case where Ord.PCI1, 5, 7, and 11 cannot be split, of course, Ord.PCI 2 cannot be split, so we can split Ord.PCI (3 ), Ord.PCI (4), Ord.PCI (6) These single intervals are new combinational logic to form a new limited shift mode.
Before the split, it is necessary to solve the problem of the order of occurrence of Ord.PCI in the loop formed in the Mesian finite shift modulation. According to Edward’s theory and experiment, Ord.PCI will not change in the interval cycle. The principle of the result. The author has verified that the theory is valid for only two different ordered intervals, and when splitting the ordered interval 6, there will be 1 + 2 + 3, which has 3 mutually different ordered intervals. In the case of elements, then the arrangement of these three mutually different elements in the circular space is only 2 groups as shown in Figure 2, the three groups of numbers divided by the three arrows to the left hemisphere are the same form 1 + 3 + 2 , 3 + 2 + 1,2 + 1 + 3; the three arrows to the right of the hemisphere divide another form: 1 + 2 + 3, 2 + 3 + 1, 3 + 1 + 2; There are only two sets of arrangements for the cycle of different ordered interval elements.
figure 2
At this point, we can start to generate a new limited shift pattern by way of interval rotation. After splitting Ord.PCI (3), Ord.PCI (4), Ord.PCI (6), these single intervals are new combinations. In logic, 3 can only be split into 1 + 2; therefore, only the interval number 4 and interval number 6 can be split, that is, 4 can be split into 1 + 3, 1 + 1 + 2; 6 can be split into 1 + 5, 1 + 4 + 1, 1 + 3 + 1 + 1, 1 + 3 + 2, 1 + 2 + 1 + 1 + 1; 1 + 2 + 2 + 1; 2 + 4; thus comparing the patterns of Messi, 5 modes can be deduced, namely 1 + 3; 1 + 5; 1 + 3 + 2; 1 + 2 + 3; 2 + 4; we take 1 + 5 as an example, then SUM6 is firstly non-coprime and less than or equal 6. Secondly, after calculating GCD = 6 L = 4, the mode includes 6 finite shifts, and the number of sounds is 4 as follows:
(1,5) -Cycle 0 1 6 7 0 So this is a new limited-shift mode with 6 shifts, and the number of sounds is 4
Such new limited shift modes also include:
Hexascale Ord.PCI (1,3) –Cycle SUM = 4 d = 4 L = 6 Four shifts, number 6;
Ord.PCI (1,2,3) -Cycle 0 1 3 6 7 9 0 SUM = 6 d = 6 L = 6; Six shifts, sound number 6,
Ord.PCI (1,3,2) -cycle 0 1 4 6 7 10 0 SUM = 6 d = 6 L = 6, six shifts, number of sounds 6,
Ord.PCI (1,5) -Cycle 0 1 6 7 0 SUM = 6 d = 6 L = 4, 6 shifts, 4 sounds.
At this point, another five finite shift patterns have been generated.
3. The possibility of splitting different non-multiple aggregate intervals with the same limited shift mode
When you look at the finite shift mode in the form of interval cycles, you will find that the same finite shift mode can be split into different non-multiple cluster interval cycles.
For example, with (4,5) -cycle, the cycle generates an octave scale,
I.e. 0 4 9 1 6 T 3 7 0
At the same time Ord.PCI (1, 2) can also produce a same octave scale,
That is 0 1 3 4 6 7 9 T 0, it is verified that the two are the same scale structure.
Then we can find that the SUM values 9 and 3 of these two cycles are complementary in the modulo 12 space, that is, they are the same as the greatest common divisor of 12. Therefore, according to the formula for calculating the length of the double-cycle mixed cycle number provided by Edward “L = 2 (12 / d); d = GCD (12, SUM); SUM = x + y (mod12)”, their phone number must be The same, that is, when the value of d is unchanged, L is equivalent.
Therefore, we can split all the limited shift patterns as long as the number of sounds does not change, just make sure that the high-level results of each loop in the new loop appear in the original limited shift pattern only once. . The author here lists six other finite shift modes with non-multiple cluster loop characteristics except for the first finite shift mode of Messiaen’s first limited shift mode, and all available mixed interval cycles of the five newly generated limited shift modes. Split results. Only the original arrangement after splitting is listed in the result, and its transposed form can be obtained by sequentially shifting the first ordered interval to the end. For example, (1,8) cycle includes (8,1). (2,1,5) cycle its transposition includes (1,5,2) (5,2,1).
The author here lists all the possible splitting modes of the finite shift mode with non-multi-cluster cyclic properties as shown in the following table, and lists the calculation procedures in the appendix.
Limited transposition mode splits non-multiple Aggregate interval cycles list
1. All the results of the second limited transposition mode (1,2) -cycle
（1,8） （4,5） （4,11） （7,2） （7,8） （10,5） （10,11）
2. The third limited transposition mode (2,1,1) -cycle
（2,1,5） （2,5,9） （2,5,1） (2,9,5) (2,9,9) (3,3,10) (3,3,2) (3,7,6) (3,7,10) (3,11,2) (3,11,6) (6,1,1) (6,1,9) (6,5,5) (6,5,9) (6,9,1) (6,9,5) (7,3,6) (7,3,10) (7,7,2) (7,7,6) (7,11,2) (7,11,10) (10,1,5) (10,1,9) (10,5,1) (10,5,5) (10,9,1) (10,9,9) (11,3,2) (11,3,6) (11,7,10) (11,7,2) (11,11,6) (11,11,10)
3. The fourth limited transposition mode (1,1,3,1) -cycle
（1,1,9,7） （1,4,3,10） （1,4,9,4） （1,7,3,7） （1,7,9,1） （1,10,3,4） （1,10,9,10） （2,3,2,11） （2,3,8,5） （2,5,4,7） （2,5,10,1） （2,9,2,5） （2,9,8,11） （2,11,4,1） （2,11,10,7） （5,2,1,10） （5,2,7,4） （5,3,5,5） （5,3,11,11） （5,8,1,4） (5,8,7,10) (5,9,5,11) (5,9,11,5) (7,1,3,7) (7,1,9,1) (7,4,3,4) (7,4,9,10) (7,7,3,1) （7,7,9,7） （7,10,3,10） (7,10,9,4) （8,3,2,5） （8,3,8,11） （8,5,4,1） （8,5,10,7） （8,9,2,11） （8,9,8,5） （8,11,4,7） （8,11,10,1） （11,2,1,4） （11,2,7,10） （11,3,5,11） （11,3,11,5） (11,8,1,10) (11,8,7,4) (11,9,5,5) (11,9,11,11)
4. The fifth Limited transposition mode(1,4,1) -cycle
（1,10,7） （5,2,11） （5,8,5） （7,4,7） （7,10,1） (11,2,5) (11,8,11)
5. The sixth Limited transposition mode (2,2,1,1) -cycle
（2,2,7,7）(2,3,5,8） （2,3,11,2） （2,8,1,7） （2,8,7,1） （2,9,5,2） （2,9,11,8） （4,1,3,10） （4,1,9,4） （4,4,3,7） (4,4,9,1) (4,7,3,4) (4,7,9,10) (4,10,3,1) (4,10,9,7) (5,3,2,8) (5,3,8,2) (5,5,4,4) (5,5,10,10) (5,9,2,2) (5,9,8,8) (5,11,4,10) (5,11,10,4) (8,3,5,2) (8,3,11,8) (8,8,1,1) (8,8,7,7) （8,9,5,8） （8,9,11,2） (8,2,1,7) (8,2,7,1) (10,1,3,4) (10,1,9,10) (10,4,3,1) (10,4,9,7) (10,7,3,10) (10,7,9,4) (10,10,3,7) (10,10,9,1) (11,3,2,2) (11,3,8,8) (11,5,4,10) (11,5,10,4) (11,9,2,8) (11,9,8,2) (11,11,4,4) (11,11,10,10)
6. The seventh Limited transposition mode (1,1,1,2,1) -cycle
（1,1,1,8,7） （1,1,3,4,9） (1,1,3,10,3) (1,1,7,2,7) (1,1,7,8,1) (1,1,9,4,3) (1,1,9,10,9) (1,2,2,3,10) (1,2,2,9,4) (1,2,5,3,7) (1,2,5,9,1) (1,2,8,3,4) (1,2,8,9,10) (1,2,11,3,1) (1,2,11,9,7) (1,4,3,1,9) (1,4,3,7,3) (1,4,4,5,4) (1,4,4,11,10) (1,4,9,1,3) (1,4,9,7,9) (1,4,10,5,10) (1,4,10,11,4) (1,7,1,2,7)(1,7,1,8,1) (1,7,3,4,3) (1,7,3,10,9) (1,7,7,2,1) (1,7,7,8,7) (1,7,9,4,9) (1,7,9,10,3) (1,8,2,3,4) (1,8,2,9,10) (1,8,5,3,1) （1,8,5,9,7）(1,8,8,3,10) (1,8,8,9,4) (1,8,11,3,7) (1,8,11,9,1) (1,10,3,1,3) （1,10,3,7,9） （1,10,4,5,10） （1,10,4,11,4） (1,10,9,1,9) (1,10,9,7,3) (1,10,10,5,4) (1,10,10,11,10) （2,1,2,2,11） （2,1,2,8,5） （2,1,4,4,7） （2,1,4,10,1）（2,1,8,2,5） （2,1,8,8,11） （2,1,10,4,1） （2,1,10,10,7） （2,3,2,2,9） （2,3,2,8,3） （2,3,4,4,5） （2,3,4,10,11） （2,3,8,2,3） （2,3,8,8,9） （2,3,10,4,11） （2,3,10,10,5） （2,5,2,2,7） （2,5,2,8,1） （2,5,4,4,3）（2,5,4,10,9） （2,5,8,2,1） (2,5,8,8,7) (2,5,10,4,9) (2,5,10,10,3) (2,7,2,2,5) (2,7,2,8,11) (2,7,4,4,1) (2,7,4,10,7) (2,7,8,2,11) (2,7,8,8,5) (2,7,10,4,7） （2,7,10,10,1）(2,9,2,2,3) (2,9,2,8,9) (2,9,4,4,11) (2,9,4,10,5) (2,9,8,2,9) （2,9,8,8,3） （2,9,10,4,5） （2,9,10,10,11） （2,11,2,2,1） （2,11,2,8,7） （2,11,4,4,9）（2,11,4,10,3） （2,11,8,2,7） （2,11,8,8,1） （2,11,10,4,3） （2,11,10,10,9） （3,2,2,1,10） （3,2,2,7,4） （3,2,3,5,5） （3,2,3,11,11） （3,2,8,1,4） （3,2,8,7,10） （3,2,9,5,11） （3,2,9,11,5） （3,4,1,3,7） （3,4,1,9,1） （3,4,4,3,4） （3,4,4,9,10） （3,4,7,3,1） （3,4,7,9,7） （3,4,10,3,10） （3,4,10,9,4） （3,5,3,2,5） （3,5,3,8,11）(3,5,5,4,1) (3,5,5,10,7) (3,5,9,2,11) (3,5,9,8,5) (3,5,11,4,7) (3,5,11,10,1) (3,8,2,1,4) (3,8,2,7,10) (3,8,3,5,11) (3,8,3,11,5) (3,8,8,1,10) （3,8,8,7,4）(3,8,9,5,5) (3,8,9,11,11) (3,10,1,3,1) (3,10,1,9,7) (3,10,4,3,10) (3,10,4,9,4) (3,10,7,3,7) (3,10,7,9,1) (3,10,10,3,4) (3,10,10,9,10) (3,11,3,2,11) (3,11,3,8,5) (3,11,5,4,7) (3,11,5,10,1) (3,11,9,2,5) (3,11,9,8,11) (3,11,11,4,1) (3,11,11,10,7) (5,2,1,1,9) (5,2,1,7,3) (5,2,2,5,4) (5,2,2,11,10) (5,2,7,1,3) (5,2,7,7,9) (5,2,8,5,10) (5,2,8,11,4) (5,3,1,4,5) (5,3,1,10,11) (5,3,5,2,3) （5,3,5,8,9）（5,3,7,4,11） （5,3,7,10,5） （5,3,11,2,9）(5,3,11,8,3) (5,4,4,1,4) (5,4,4,7,10) (5,4,5,5,11) (5,4,5,11,5) (5,4,10,1,10) (5,4,10,7,4) (5,4,11,5,5) (5,4,11,11,11) (5,8,1,1,3) (5,8,1,7,9) (5,8,2,5,10) (5,8,2,11,4) (5,8,7,1,9) (5,8,7,7,3) (5,8,8,5,4) （5,8,8,11,10） （5,9,1,4,11） （5,9,1,10,5）（5,9,5,2,9） (5,9,5,8,3) (5,9,7,4,5) (5,9,7,10,11) (5,9,11,2,3) (5,9,11,8,9) (5,10,4,1,10) (5,10,4,7,4) (5,10,5,5,5) （5,10,5,11,11） （5,10,10,1,4） （5,10,10,7,10） （5,10,11,5,11） （5,10,11,11,5） (7,1,1,2,7) (7,1,1,8,1) (7,1,3,4,3) (7,1,3,10,9) (7,1,7,2,1) (7,1,7,8,7) (7,1,9,4,9) (7,1,7,10,3) (7,2,2,3,4) (7,2,2,9,10) (7,2,5,3,1) (7,2,5,9,7) (7,2,8,3,10) (7,2,8,9,4) (7,2,11,3,7) (7,2,11,9,1) (7,4,3,1,3) (7,4,3,7,9) (7,4,4,5,10) (7,4,4,11,4) (7,4,9,1,9) (7,4,9,7,3) （7,4,10,5,4） （7,4,10,11,10） （7,7,1,2,1） （7,7,1,8,7） （7,7,3,4,9） （7,7,3,10,3） （7,7,7,2,7） (7,7,7,8,1) （7,7,9,4,3） （7,7,9,10,9） （7,8,2,3,10） （7,8,2,9,4） （7,8,5,3,7）（7,8,5,9,1）(7,8,8,3,4) (7,8,8,9,10) (7,8,11,3,1) (7,8,11,9,7) (7,10,3,1,9) (7,10,3,7,3) (7,10,4,5,4) (7,10,4,11,10) (7,10,9,1,3) (7,10,9,7,9) (7,10,10,5,10) (7,10,10,11,4) (8,1,2,2,5) (8,1,2,8,11) (8,1,4,4,1) (8,1,4,10,7) (8,1,8,2,11) (8,1,8,8,5) (8,1,10,4,7) (8,1,10,10,1) （8,3,2,2,3） （8,3,2,8,9） （8,3,4,4,11） （8,3,4,10,5） （8,3,8,2,9） （8,3,8,8,3） （8,3,10,4,5） （8,3,8,10,11） （8,5,2,2,1） (8,5,2,8,7) (8,5,4,4,9) (8,5,4,10,3) (8,5,8,2,7) (8,5,8,8,1) (8,5,10,4,3) （8,5,10,10,9）（8,7,2,2,11） （8,7,2,8,5） （8,7,4,4,7） （8,7,4,10,1）(8,7,8,2,5) (8,7,8,8,11) (8,7,10,4,1) (8,7,10,10,7) (8,9,2,2,9) (8,9,2,8,3) (8,9,4,4,5) (8,9,4,10,11) (8,9,8,2,3) (8,9,8,8,9) (8,9,10,4,11) （8,9,10,10,5） （8,11,2,2,7） （8,11,2,8,1） （8,11,4,4,3） （8,11,4,10,9） （8,11,8,2,1） （8,11,8,8,7） （8,11,10,4,9） （8,11,10,10,3） （9,2,2,1,4） （9,2,2,7,10） （9,2,3,5,11） （9,2,3,11,5） （9,2,8,1,10） （9,2,8,7,4）（9,2,9,5,5） (9,2,9,11,11) (9,4,1,3,1) (9,4,1,9,7) (9,4,4,3,10) (9,4,4,9,4) (9,4,7,3,7) (9,4,7,9,1) (9,4,10,3,4) (9,4,10,9,10) (9,5,3,2,11) (9,5,3,8,5) (9,5,5,4,7) (9,5,5,10,1) (9,5,9,2,5) (9,5,9,8,11) (9,5,11,4,1) (9,5,11,10,7) (9,8,2,1,10) (9,8,2,7,4) (9,8,3,5,5) (9,8,3,11,11) (9,8,8,1,4) (9,8,8,7,10) (9,8,9,5,11) (9,8,9,11,5) (9,10,1,3,7) (9,10,1,9,1) (9,10,4,3,4) (9,10,4,9,10) (9,10,7,3,1) (9,10,7,9,7) (9,10,10,3,10) (9,10,10,9,4) (9,11,3,2,5) (9,11,3,8,11) (9,11,5,4,1) (9,11,5,10,7) (9,11,7,2,1) (9,11,7,8,7) (9,11,9,4,9) (9,11,9,10,3) (11,2,1,1,3) (11,2,1,7,9) (11,2,2,5,10) (11,2,2,11,4) (11,2,7,1,9) (11,2,7,7,3) (11,2,8,5,4) (11,2,8,11,10) (11,3,1,4,11) (11,3,1,10,5) (11,3,5,2,9) (11,3,5,8,3) (11,3,7,4,5) (11,3,7,10,11) (11,3,11,2,3) (11,3,11,8,9) (11,4,4,1,10) (11,4,4,7,4) (11,4,5,5,5) (11,4,5,11,11)(11,4,10,1,4) (11,4,10,7,10) (11,4,11,5,11) (11,4,11,11,5) (11,8,1,1,9)(11,8,1,7,3) (11,8,2,5,4) (11,8,2,11,10) (11,8,7,1,3) (11,8,7,7,9) (11,8,8,5,10) (11,8,8,11,4) （11,9,1,4,5） （11,9，1,10,11） （11,9,5,2,3） (11,9,5,8,9) (11,9,7,4,11) (11,9,7,10,5) (11,9,11,2,9) (11,9,11,8,3) (11,10,4,1,4) (11,10,4,7,10) (11,10,5,5,11) (11,10,5,11,5) (11,10,10,1,10) (11,10,10,7,4) (11,10,11,5,5) (11,10,11,11,11)
7. The new mode one (1,5) -cycle
（7,11）
8. New mode two (2,4) -cycle
（8,10）
9. The new mode three (1,3) -cycle
（1,7） （5,3） （5,11） （9,7） （9,11）
10. The new mode four (1,2,3) -cycle
（1,8,9,） （3,4,11） （3,10，5） （7,2,9） （7,8,3） （9,4,5） （9,10,11）
11. The new mode five (1, 3, 2) -cycle
（1,9,8） （4,3,11） （4,9,5） （7,3,8） （7,9,2） （10,3,5） （10,9,11）
Conclusion
By interpreting the limited transposition mode from the perspective of non-multiple aggregate cycles in the interval cycle, it can be concluded that the scale composed of the limited transposition mode is non-multiple in the commonness between the non-multiple aggregate interval cycles and the Messiaen’s limited transposition mode, while non-multiple aggregate cycles also include all single interval cycles, but in the mixed interval cycle, it can only be achieved if the total value of the loop composition with the non-prime relationship of 12 is established. The mixed interval loop cannot exhaust 12 semitones, and many features are consistent with the limited transposition mode. At the same time, 5 new types of limited transposition modes are obtained during the process of splitting Ord.PCI (4) and Ord.PCI (6). Combination of patterns. In the experiment of splitting the same finite shift mode into different non-multiple aggregate intervals, a list of 11 limited transposition modes split into different non-multiple aggregate cycles was obtained, and the algorithm process was summarized. The data in this table can be applied to creation and analysis research. At the same time, the classification table has an interpretation of the application of The list pitch-class sets theory in the interval cycles. I will make another article for research and discussion.
Appendix
(1) The steps of splitting the limited transposition mode of the two-tone cycle
Let: SUM = x + 12n (n is a natural number), and the two-tone cycle is (a, b) -cycle.
a + b = SUM = x + 12n, then b = x + 12n-a
In order to ensure that the tone level generated by the interval cycle appears only once, a + b + a ≠ 12n, and b + a + b ≠ 12n,
Then a + b + a = a + x + 12n-a + a = x + 12n + a ≠ 12n ,; b + a + b = 2x + 24n-2a + a = 2x + 24n-a ≠ 12n
Then determine the prototype of the pattern to be split, and list the sound column, determine the SUM value x, bring it into the formula in the previous step, determine the value of a ≠, and sequentially bring the pitch difference between the pitch in the sound column and 0 , Find all possibilities. If (1,5) cycle mode expands to 0 1 6 7 0; SUM = 6, bring x + 12m + a = 6 + 12n + a ≠ 12n, then a ≠ 6, bring 2x + 24n-a = 12 + 24n-a ≠ 12n, then a ≠ 0, then a can only be equal to 1 or 7 relative to the mode, and calculate the corresponding b value, you can get the cycle (1,5); (7,11).
(2) Steps of splitting the limited transposition mode of the three-step cycle
The three-cycle cycle finite shift mode split requires first splitting the SUM into two-cycle loops. For the method, see the two-interval cycle limited shift mode split. Then, on the basis of splitting into two intervals (a, b) cycle, splitting b into c, d. You can get the three intervals cycle (a, c, d) -cycle, in order to ensure that the three intervals cycle The generated sound level appears only once, then a + c ≠ 12n, d + a ≠ 12n. A = SUM-cd, bring it in to get SUM-c-d + c ≠ 12n, then SUM-d ≠ 12n, SUM- c-d + d ≠ 12n, then SUM-c ≠ 12n, bring the difference in pitch level in the limited shift mode to be split in sequence, and get all the results.
For example, the mode five (1,4,1) cycle is expanded to 0 1 5 6 7 11 0, then according to the split method of the two-tone cycle can get a ≠ 6, and a ≠ 0. Bring the SUM value 6 into the three-tone interval The two range formulas of cyclic splitting can be obtained 6-d ≠ 12n; 6-c ≠ 12n; then d, c are not equal to 6. The first step is split into (1,5) (5,1) (7, 11) (11,7) The second step is split into, (1,4,1) (1,10,7) (5,2,11) (5,8,5) (7,4,7) ( 7,10,1) (11,2,5) (11,8,11)
(3) Steps of splitting the limited transposition mode of the four-interval loop
To split the limited shift mode of the four-pitch cycle, you must first split the limited shift mode into the limited-shift mode of the three-pitch cycle. Split (a, c, d) -cycle into (a, c, e, f) -cycle, then in order to ensure that the tone level produced by the four-cycle cycle only appears once, then c + e ≠ 12n; f + a ≠ 12n, SUM-e or f is not equal to 12n, and all interval differences are brought in sequence to find all possibilities. For example, if we take the mode four (1,1,3,1) -cycle as an example, we first split it into a cycle of three intervals, and we take one of them (5,2,11) as e ≠ 10 f ≠ 7 e Or f is not equal to 6, then bring in all the difference in sound levels in sequence to get (5,2,1,10) (5,2,7,4)
(4) The steps of splitting the five-cycle cyclic limited transposition mode
In order to split the limited shift mode of the five-cycle cycle, the limited shift mode needs to be split into the limited shift mode of the four-cycle cycle first. For the method, see Splitting the Limited Shift Mode of the Four-Pitch Cycle. Split (a, c, e, f) -cycle into (a, c, e, g, h) -cycle, then in order to ensure that the tone level produced by the four-interval cycle appears only once, then h + a ≠ 12n, g + e ≠ 12n, c + e + g ≠ 12n, h + a + c ≠ 12n, SUM-h or g ≠ 12n. The difference in pitch level brought into the sequence of sounds in sequence to find all possibilities. For example, take mode seven (1,1,1,2,1) -cycle as an example to split it into a four-tone cycle mode, we take one of them (11,4,5,10) as an example, then g is not equal to 7 or 3, h is not equal to 1 or 9, and g and h are not equal to 6, successively bring the difference in sound level, you can get (11,4,5,5,5) (11,4,5,11,11 ) Two possibilities.
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