Friday, 8 April 2016

Farey Sequence Tables for FM Synthesis


The choice of Carrier:Modulator (C:M) frequency ratio is an interesting area of research, because it is one of the fundamental aspects of FM programming. Although I mentioned the classic book by Chowning & Bristow in the recent Matrixsynth NYZ DRN4 DECLASSIFIED, space and time didn't allow me to mention another classic text by Barry Truax: Organizational Techniques for C:M Ratios in Frequency Modulation. If you're interested in FM programming, and you don't mind a tiny bit of maths via the Farey Sequence, I highly recommend it. If you want to dig deeper I've made some useful tables for FM synth programmers in this post. There's also a handful of youtube FM synthesis vids at the end.

Truax's paper covers a number of aspects related to choosing and organising C:M ratios. He firstly covers why the C:M ratio is important: "because it controls the set of partials (called sidebands) in the resultant spectrum." Following this is a discussion on the degree of harmonicity and inharmonicity of the C:M ratio and working with integer ratios, which allow for approximations of real numbers in any case. He notes that one of Chowning's favourite ratios 1:1.414 can be approximated by the integer ratio 5:7. The mystery number 1.414 is simply the square root of two.

Approximating irrational numbers with rational numbers is achieved by rationalisation, and 1.414 can be roughly approximated by 7/5. To convert this to a C:M form of 1:1.414 we can simply swap the numerator and denominator of the approximation and use this as the C:M ratio e.g. 5:7. This can be generally applied to any rational approximation. Another ratio favoured by Chowning is 1:1.618, see for example the video of Stria (1977) at the end of this post. The number 1.618 is called the golden mean, and is popular in many areas. It is also known as the most irrational number because it is the hardest to approximate with rational numbers. This American Mathematical Society page has useful approximations for this number :


From that article we could choose ratios with varying degrees of accuracy up to 34/21 which would give a required C:M of 21:34. If we wanted our ratio pitched a bit lower we could use 8/5 to give 5:8.

Later on in his paper, in the section "Organizing c:m Ratios According to Spectral Identity and Uniqueness" is where Truax presents the Farey Sequence as an organisational technique for C:M ratios. It's also described in his FM tutorial here, if you don't have access to CMJ or the book.

Arras (1980) is one of Truax's classic pieces using the Farey Sequence for C:M ratios :


You can also find the definition of the Farey Sequence online at Mathworld here. The complete Farey Sequence of order 9 generated with my little program in the Appendix below is :

{0,1},{1,9},{1,8},{1,7},{1,6},{1,5},{2,9},{1,4},{2,7},{1,3},{3,8},{2,5},{3,7},{4,9},{1,2},{5,9},{4,7},{3,5},{5,8},{2,3},{5,7},{3,4},{7,9},{4,5},{5,6},{6,7},{7,8},{8,9},{1,1}

and to use these integer number pairs in FM, the C:M ratio is substituted as {C,M}, so {1,9} would give a C:M ratio of 1:9. The complete sequence for FM is usually given without 0:1, and in Normal Form (NF) which gives a spectrally unique C:M ratio per sequence pair :

{1,9},{1,8},{1,7},{1,6},{1,5},{2,9},{1,4},{2,7},{1,3},{3,8},{2,5},{3,7},{4,9},{1,2}

Once the Farey Sequence is in NF, each ratio can then be used (via some simple maths) to obtain a family of ratios sharing the same spectral identity. Truax expands on this and explains the maths in the paper, as well as on the web page. Here is my pen and paper working for the first 8 C:M ratios of the 2:5 family:


Below is an Appendix of Farey Sequence tables useful for FM synth programmers / composers interested in application to C:M ratios. I picked orders 15 and 31, because this fits in well with Yamaha four and six/eight operator integer ratio ranges (i.e. using the coarse adjustment only, and fine set to zero). On Yamaha FM synths it is possible to get to some higher number integers using the frequency fine control, as well as some reasonable approximations beyond the exact range. I include order 9 for easy cross reference with Barry Truax's paper and FM Tutorial web page.

Here is a visual example of one of the ways I've used 3 different spectral families, 3:4 (from the 1:4 family), 3:5 (from the 2:5 family) and NF ratio 3:7 mapped to a parallel modulator algorithm :


The picture back at the top of this post shows one example of how I've used the parallel carrier algorithm. There we can see NF ratio 4:9 and its family member 5:9, as well as the NF ratio 2:9 which is from a different spectral family.

Additional Bonus FM Homework Items 

These are just a few picks of some good youtube vids related to FM synthesis. The two music pieces are single examples, and both composer's have done other FM works. 


John Chowning's Stria (1977), you can read a short interview by Curtis Roads which gives an overview of this piece here: http://www.o-art.org/history/LongDur/Chowning.html


Laurie Spiegel's Concerto Generator (1977) played on the Bell Labs Hal Alles synth. The internals of this synth are also presented in a handful of short papers in Foundations of Computer Music (see references above). For more FM music by Laurie Spiegel I highly recommend the album Unseen Worlds, with the sound sources done entirely on a Yamaha TX816 FM synth. Some info on this piece from Laurie's youtube page: 

"The interactive software I wrote and am playing in this video recycles my keyboard input into an accompaniment to my continued playing, which is why I called it a "concerto generator". I use part of one of the keyboards for control data entry, and the small switches upper right to access pre-entered numerical patterns. The sliders are mainly pre-Yamaha FM synthesis parameter controls, for the number of harmonics and amplitude and frequency of the FM modulator and carrier that constituted each musical voice." 


An Audio Engineering Society youtube video of John Chowning on the origins of FM. Radio Web MACBA also did a great interview with John Chowning available to download here: http://rwm.macba.cat/en/sonia/john-chowning-/capsula 


A really good Matrixsynth vid with Dave Bristow :"This is a fascinating and significant bit of synthesizer history starting with the Yamaha CS80 through the DX7 from the man that was actually there. Yamaha hired Dave Bristow to showcase the Yamaha CS80 followed by asking him to provide input on their first FM synth the GS1.  It was sold as a preset synth but in the video you will see a programmer for it.  Dave was the person that Yamaha had program the presets for it followed by the GS2, DX7 and more.  You'll see the prototype for what became the DX7. " 

References

Chowning, J & Bristow, D. 1986. FM Theory & Applications - By Musicians For Musicians. Tokyo: Yamaha.  

Barry Truax website http://www.sfu.ca/~truax/ 

Truax, B. 1982. Timbral Construction in Arras as a Stochastic Process, Computer Music Journal, 6, 3. http://www.sfu.ca/~truax/arras.html

Truax, B. 1978. Organizational Techniques for C:M Ratios in Frequency Modulation, Computer Music Journal, 1, 4. Note this paper is reprinted in Roads C. & Strawn J.  eds. Foundations of Computer Music,  MIT Press, 1985. A number of key FM papers are reproduced in this book, including Chowning's, which are useful reading in addition to Chowning & Bristow.

John Chowning's page at Stanford: https://ccrma.stanford.edu/people/john-chowning

Appendix: Farey Sequence Tables in Complete and Normal Form of Order 9, 15 and 31 arranged by Carrier and Modulator

The following tables were computed using a short program I wrote in Mathematica. I used the recent (well it was back in September 2009 when I first wrote the program) algorithm by Olivier Gerard on the Sequence Fanatics discussion list here :

http://list.seqfan.eu/pipermail/seqfan/2009-April/001413.html

which gives the raw FareySequence[order] function for computing the Farey Sequence in Mathematica. I wrote a little program which call's FareySequence[order] and formats the results to produce tables in various configurations. In particular arranging the complete and NF sequences along rows by carrier and modulator I've found very useful. I also left in the 1:1 ratio in the NF orderings for completeness (Truax mentions including 1:1 in his paper).

I've included my Mathematica code to compute these tables, which simply calls FareySequence[order] to obtain the raw Sequence data. Gerard's function is not listed in the code below, see link above for the Function declaration.

(* Farey Sequence Table Generator *)
(* Computes Farey Sequence Tables of a given order in Complete and Normal Form, arranged by Carrier and Modulator *)
(* D Burraston 2009, revised for web version March 2016 *)

order = 31; 
Print["ORDER : ",order];

(* computer raw sequence data *)
FareyNumerators[order];
FareyDenominators[order];
allfarey = FareySequence[order];

(* sort by carrier *)
sortedfarey = Table[Cases[Drop[allfarey,1],{n,_}],{n,0,order}];
 (* sort by modulator *)
sortedfarey2 = Table[Cases[Drop[allfarey,1],{_,n}],{n,0,order}];

(* convert Sequence to Normal Form *)
allfarey2 = Take[allfarey,1+IntegerPart[Length[allfarey]/2]];
allfareyfinal = Delete[allfarey2,1] ; 
allfareyfinal = Append[allfareyfinal,Last[allfarey]]; 

(* sort by carrier *)
sortedfarey3 = Table[Cases[allfareyfinal,{n,_}],{n,0,order}];
(* sort by modulator *)
sortedfarey4 = Table[Cases[allfareyfinal,{_,n}],{n,0,order}];

(* export sorted tables to data files *) 
Export["FAREY1.txt",sortedfarey, "Table"]
Export["FAREY2.txt",sortedfarey2, "Table"]
Export["FAREY3.txt",sortedfarey3, "Table"]
Export["FAREY4.txt",sortedfarey4, "Table"]

My tiny program could easily be improved in many ways, for example by adding some additional code to arrange the columns so that the numerator and denominators are spaced out numerically, which would give a matrix-like representation. Alternatively, you could load these tables into a spreadsheet and work with them from there.



--------ORDER 9--------


{1, 9} {1, 8} {1, 7} {1, 6} {1, 5} {1, 4} {1, 3} {1, 2} {1, 1}
{2, 9} {2, 7} {2, 5} {2, 3}
{3, 8} {3, 7} {3, 5} {3, 4}
{4, 9} {4, 7} {4, 5}
{5, 9} {5, 8} {5, 7} {5, 6}
{6, 7}
{7, 9} {7, 8}
{8, 9}
Order 9 C:M Ratios Sorted by Carrier


{1, 1}
{1, 2}
{1, 3} {2, 3}
{1, 4} {3, 4}
{1, 5} {2, 5} {3, 5} {4, 5}
{1, 6} {5, 6}
{1, 7} {2, 7} {3, 7} {4, 7} {5, 7} {6, 7}
{1, 8} {3, 8} {5, 8} {7, 8}
{1, 9} {2, 9} {4, 9} {5, 9} {7, 9} {8, 9}
Order 9 C:M Ratios Sorted by Modulator

{1, 9} {1, 8} {1, 7} {1, 6} {1, 5} {1, 4} {1, 3} {1, 2} {1, 1}
{2, 9} {2, 7} {2, 5}
{3, 8} {3, 7}
{4, 9}
Order 9 Normal Form C:M Ratios Sorted by Carrier


{1, 1}
{1, 2}
{1, 3}
{1, 4}
{1, 5} {2, 5}
{1, 6}
{1, 7} {2, 7} {3, 7}
{1, 8} {3, 8}
{1, 9} {2, 9} {4, 9}
Order 9 Normal Form C:M Ratios Sorted by Modulator





--------ORDER 15--------

{1, 15} {1, 14} {1, 13} {1, 12} {1, 11} {1, 10} {1, 9} {1, 8} {1, 7} {1, 6} {1, 5} {1, 4} {1, 3} {1, 2} {1, 1}
{2, 15} {2, 13} {2, 11} {2, 9} {2, 7} {2, 5} {2, 3}
{3, 14} {3, 13} {3, 11} {3, 10} {3, 8} {3, 7} {3, 5} {3, 4}
{4, 15} {4, 13} {4, 11} {4, 9} {4, 7} {4, 5}
{5, 14} {5, 13} {5, 12} {5, 11} {5, 9} {5, 8} {5, 7} {5, 6}
{6, 13} {6, 11} {6, 7}
{7, 15} {7, 13} {7, 12} {7, 11} {7, 10} {7, 9} {7, 8}
{8, 15} {8, 13} {8, 11} {8, 9}
{9, 14} {9, 13} {9, 11} {9, 10}
{10, 13} {10, 11}
{11, 15} {11, 14} {11, 13} {11, 12}
{12, 13}
{13, 15} {13, 14}
{14, 15}
Order 15 C:M Ratios Sorted by Carrier




{1, 1}
{1, 2}
{1, 3} {2, 3}
{1, 4} {3, 4}
{1, 5} {2, 5} {3, 5} {4, 5}
{1, 6} {5, 6}
{1, 7} {2, 7} {3, 7} {4, 7} {5, 7} {6, 7}
{1, 8} {3, 8} {5, 8} {7, 8}
{1, 9} {2, 9} {4, 9} {5, 9} {7, 9} {8, 9}
{1, 10} {3, 10} {7, 10} {9, 10}
{1, 11} {2, 11} {3, 11} {4, 11} {5, 11} {6, 11} {7, 11} {8, 11} {9, 11} {10, 11}
{1, 12} {5, 12} {7, 12} {11, 12}
{1, 13} {2, 13} {3, 13} {4, 13} {5, 13} {6, 13} {7, 13} {8, 13} {9, 13} {10, 13} {11, 13} {12, 13}
{1, 14} {3, 14} {5, 14} {9, 14} {11, 14} {13, 14}
{1, 15} {2, 15} {4, 15} {7, 15} {8, 15} {11, 15} {13, 15} {14, 15}
Order 15 C:M Ratios Sorted by Modulator




{1, 15} {1, 14} {1, 13} {1, 12} {1, 11} {1, 10} {1, 9} {1, 8} {1, 7} {1, 6} {1, 5} {1, 4} {1, 3} {1, 2} {1, 1}
{2, 15} {2, 13} {2, 11} {2, 9} {2, 7} {2, 5}
{3, 14} {3, 13} {3, 11} {3, 10} {3, 8} {3, 7}
{4, 15} {4, 13} {4, 11} {4, 9}
{5, 14} {5, 13} {5, 12} {5, 11}
{6, 13}
{7, 15}
Order 15 Normal Form C:M Ratios Sorted by Carrier




{1, 1}
{1, 2}
{1, 3}
{1, 4}
{1, 5} {2, 5}
{1, 6}
{1, 7} {2, 7} {3, 7}
{1, 8} {3, 8}
{1, 9} {2, 9} {4, 9}
{1, 10} {3, 10}
{1, 11} {2, 11} {3, 11} {4, 11} {5, 11}
{1, 12} {5, 12}
{1, 13} {2, 13} {3, 13} {4, 13} {5, 13} {6, 13}
{1, 14} {3, 14} {5, 14}
{1, 15} {2, 15} {4, 15} {7, 15}
Order 15 Normal Form C:M Ratios Sorted by Modulator





--------ORDER 31--------

{1, 31} {1, 30} {1, 29} {1, 28} {1, 27} {1, 26} {1, 25} {1, 24} {1, 23} {1, 22} {1, 21} {1, 20} {1, 19} {1, 18} {1, 17} {1, 16} {1, 15} {1, 14} {1, 13} {1, 12} {1, 11} {1, 10} {1, 9} {1, 8} {1, 7} {1, 6} {1, 5} {1, 4} {1, 3} {1, 2} {1, 1}
{2, 31} {2, 29} {2, 27} {2, 25} {2, 23} {2, 21} {2, 19} {2, 17} {2, 15} {2, 13} {2, 11} {2, 9} {2, 7} {2, 5} {2, 3}
{3, 31} {3, 29} {3, 28} {3, 26} {3, 25} {3, 23} {3, 22} {3, 20} {3, 19} {3, 17} {3, 16} {3, 14} {3, 13} {3, 11} {3, 10} {3, 8} {3, 7} {3, 5} {3, 4}
{4, 31} {4, 29} {4, 27} {4, 25} {4, 23} {4, 21} {4, 19} {4, 17} {4, 15} {4, 13} {4, 11} {4, 9} {4, 7} {4, 5}
{5, 31} {5, 29} {5, 28} {5, 27} {5, 26} {5, 24} {5, 23} {5, 22} {5, 21} {5, 19} {5, 18} {5, 17} {5, 16} {5, 14} {5, 13} {5, 12} {5, 11} {5, 9} {5, 8} {5, 7} {5, 6}
{6, 31} {6, 29} {6, 25} {6, 23} {6, 19} {6, 17} {6, 13} {6, 11} {6, 7}
{7, 31} {7, 30} {7, 29} {7, 27} {7, 26} {7, 25} {7, 24} {7, 23} {7, 22} {7, 20} {7, 19} {7, 18} {7, 17} {7, 16} {7, 15} {7, 13} {7, 12} {7, 11} {7, 10} {7, 9} {7, 8}
{8, 31} {8, 29} {8, 27} {8, 25} {8, 23} {8, 21} {8, 19} {8, 17} {8, 15} {8, 13} {8, 11} {8, 9}
{9, 31} {9, 29} {9, 28} {9, 26} {9, 25} {9, 23} {9, 22} {9, 20} {9, 19} {9, 17} {9, 16} {9, 14} {9, 13} {9, 11} {9, 10}
{10, 31} {10, 29} {10, 27} {10, 23} {10, 21} {10, 19} {10, 17} {10, 13} {10, 11}
{11, 31} {11, 30} {11, 29} {11, 28} {11, 27} {11, 26} {11, 25} {11, 24} {11, 23} {11, 21} {11, 20} {11, 19} {11, 18} {11, 17} {11, 16} {11, 15} {11, 14} {11, 13} {11, 12}
{12, 31} {12, 29} {12, 25} {12, 23} {12, 19} {12, 17} {12, 13}
{13, 31} {13, 30} {13, 29} {13, 28} {13, 27} {13, 25} {13, 24} {13, 23} {13, 22} {13, 21} {13, 20} {13, 19} {13, 18} {13, 17} {13, 16} {13, 15} {13, 14}
{14, 31} {14, 29} {14, 27} {14, 25} {14, 23} {14, 19} {14, 17} {14, 15}
{15, 31} {15, 29} {15, 28} {15, 26} {15, 23} {15, 22} {15, 19} {15, 17} {15, 16}
{16, 31} {16, 29} {16, 27} {16, 25} {16, 23} {16, 21} {16, 19} {16, 17}
{17, 31} {17, 30} {17, 29} {17, 28} {17, 27} {17, 26} {17, 25} {17, 24} {17, 23} {17, 22} {17, 21} {17, 20} {17, 19} {17, 18}
{18, 31} {18, 29} {18, 25} {18, 23} {18, 19}
{19, 31} {19, 30} {19, 29} {19, 28} {19, 27} {19, 26} {19, 25} {19, 24} {19, 23} {19, 22} {19, 21} {19, 20}
{20, 31} {20, 29} {20, 27} {20, 23} {20, 21}
{21, 31} {21, 29} {21, 26} {21, 25} {21, 23} {21, 22}
{22, 31} {22, 29} {22, 27} {22, 25} {22, 23}
{23, 31} {23, 30} {23, 29} {23, 28} {23, 27} {23, 26} {23, 25} {23, 24}
{24, 31} {24, 29} {24, 25}
{25, 31} {25, 29} {25, 28} {25, 27} {25, 26}
{26, 31} {26, 29} {26, 27}
{27, 31} {27, 29} {27, 28}
{28, 31} {28, 29}
{29, 31} {29, 30}
{30, 31}
Order 31 C:M Ratios Sorted by Carrier


{1, 1}
{1, 2}
{1, 3} {2, 3}
{1, 4} {3, 4}
{1, 5} {2, 5} {3, 5} {4, 5}
{1, 6} {5, 6}
{1, 7} {2, 7} {3, 7} {4, 7} {5, 7} {6, 7}
{1, 8} {3, 8} {5, 8} {7, 8}
{1, 9} {2, 9} {4, 9} {5, 9} {7, 9} {8, 9}
{1, 10} {3, 10} {7, 10} {9, 10}
{1, 11} {2, 11} {3, 11} {4, 11} {5, 11} {6, 11} {7, 11} {8, 11} {9, 11} {10, 11}
{1, 12} {5, 12} {7, 12} {11, 12}
{1, 13} {2, 13} {3, 13} {4, 13} {5, 13} {6, 13} {7, 13} {8, 13} {9, 13} {10, 13} {11, 13} {12, 13}
{1, 14} {3, 14} {5, 14} {9, 14} {11, 14} {13, 14}
{1, 15} {2, 15} {4, 15} {7, 15} {8, 15} {11, 15} {13, 15} {14, 15}
{1, 16} {3, 16} {5, 16} {7, 16} {9, 16} {11, 16} {13, 16} {15, 16}
{1, 17} {2, 17} {3, 17} {4, 17} {5, 17} {6, 17} {7, 17} {8, 17} {9, 17} {10, 17} {11, 17} {12, 17} {13, 17} {14, 17} {15, 17} {16, 17}
{1, 18} {5, 18} {7, 18} {11, 18} {13, 18} {17, 18}
{1, 19} {2, 19} {3, 19} {4, 19} {5, 19} {6, 19} {7, 19} {8, 19} {9, 19} {10, 19} {11, 19} {12, 19} {13, 19} {14, 19} {15, 19} {16, 19} {17, 19} {18, 19}
{1, 20} {3, 20} {7, 20} {9, 20} {11, 20} {13, 20} {17, 20} {19, 20}
{1, 21} {2, 21} {4, 21} {5, 21} {8, 21} {10, 21} {11, 21} {13, 21} {16, 21} {17, 21} {19, 21} {20, 21}
{1, 22} {3, 22} {5, 22} {7, 22} {9, 22} {13, 22} {15, 22} {17, 22} {19, 22} {21, 22}
{1, 23} {2, 23} {3, 23} {4, 23} {5, 23} {6, 23} {7, 23} {8, 23} {9, 23} {10, 23} {11, 23} {12, 23} {13, 23} {14, 23} {15, 23} {16, 23} {17, 23} {18, 23} {19, 23} {20, 23} {21, 23} {22, 23}
{1, 24} {5, 24} {7, 24} {11, 24} {13, 24} {17, 24} {19, 24} {23, 24}
{1, 25} {2, 25} {3, 25} {4, 25} {6, 25} {7, 25} {8, 25} {9, 25} {11, 25} {12, 25} {13, 25} {14, 25} {16, 25} {17, 25} {18, 25} {19, 25} {21, 25} {22, 25} {23, 25} {24, 25}
{1, 26} {3, 26} {5, 26} {7, 26} {9, 26} {11, 26} {15, 26} {17, 26} {19, 26} {21, 26} {23, 26} {25, 26}
{1, 27} {2, 27} {4, 27} {5, 27} {7, 27} {8, 27} {10, 27} {11, 27} {13, 27} {14, 27} {16, 27} {17, 27} {19, 27} {20, 27} {22, 27} {23, 27} {25, 27} {26, 27}
{1, 28} {3, 28} {5, 28} {9, 28} {11, 28} {13, 28} {15, 28} {17, 28} {19, 28} {23, 28} {25, 28} {27, 28}
{1, 29} {2, 29} {3, 29} {4, 29} {5, 29} {6, 29} {7, 29} {8, 29} {9, 29} {10, 29} {11, 29} {12, 29} {13, 29} {14, 29} {15, 29} {16, 29} {17, 29} {18, 29} {19, 29} {20, 29} {21, 29} {22, 29} {23, 29} {24, 29} {25, 29} {26, 29} {27, 29} {28, 29}
{1, 30} {7, 30} {11, 30} {13, 30} {17, 30} {19, 30} {23, 30} {29, 30}
{1, 31} {2, 31} {3, 31} {4, 31} {5, 31} {6, 31} {7, 31} {8, 31} {9, 31} {10, 31} {11, 31} {12, 31} {13, 31} {14, 31} {15, 31} {16, 31} {17, 31} {18, 31} {19, 31} {20, 31} {21, 31} {22, 31} {23, 31} {24, 31} {25, 31} {26, 31} {27, 31} {28, 31} {29, 31} {30, 31}
Order 31 C:M Ratios Sorted by Modulator


{1, 31} {1, 30} {1, 29} {1, 28} {1, 27} {1, 26} {1, 25} {1, 24} {1, 23} {1, 22} {1, 21} {1, 20} {1, 19} {1, 18} {1, 17} {1, 16} {1, 15} {1, 14} {1, 13} {1, 12} {1, 11} {1, 10} {1, 9} {1, 8} {1, 7} {1, 6} {1, 5} {1, 4} {1, 3} {1, 2} {1, 1}
{2, 31} {2, 29} {2, 27} {2, 25} {2, 23} {2, 21} {2, 19} {2, 17} {2, 15} {2, 13} {2, 11} {2, 9} {2, 7} {2, 5}
{3, 31} {3, 29} {3, 28} {3, 26} {3, 25} {3, 23} {3, 22} {3, 20} {3, 19} {3, 17} {3, 16} {3, 14} {3, 13} {3, 11} {3, 10} {3, 8} {3, 7}
{4, 31} {4, 29} {4, 27} {4, 25} {4, 23} {4, 21} {4, 19} {4, 17} {4, 15} {4, 13} {4, 11} {4, 9}
{5, 31} {5, 29} {5, 28} {5, 27} {5, 26} {5, 24} {5, 23} {5, 22} {5, 21} {5, 19} {5, 18} {5, 17} {5, 16} {5, 14} {5, 13} {5, 12} {5, 11}
{6, 31} {6, 29} {6, 25} {6, 23} {6, 19} {6, 17} {6, 13}
{7, 31} {7, 30} {7, 29} {7, 27} {7, 26} {7, 25} {7, 24} {7, 23} {7, 22} {7, 20} {7, 19} {7, 18} {7, 17} {7, 16} {7, 15}
{8, 31} {8, 29} {8, 27} {8, 25} {8, 23} {8, 21} {8, 19} {8, 17}
{9, 31} {9, 29} {9, 28} {9, 26} {9, 25} {9, 23} {9, 22} {9, 20} {9, 19}
{10, 31} {10, 29} {10, 27} {10, 23} {10, 21}
{11, 31} {11, 30} {11, 29} {11, 28} {11, 27} {11, 26} {11, 25} {11, 24} {11, 23}
{12, 31} {12, 29} {12, 25}
{13, 31} {13, 30} {13, 29} {13, 28} {13, 27}
{14, 31} {14, 29}
{15, 31}
Order 31 Normal Form C:M Ratios Sorted by Carrier



{1, 1}
{1, 2}
{1, 3}
{1, 4}
{1, 5} {2, 5}
{1, 6}
{1, 7} {2, 7} {3, 7}
{1, 8} {3, 8}
{1, 9} {2, 9} {4, 9}
{1, 10} {3, 10}
{1, 11} {2, 11} {3, 11} {4, 11} {5, 11}
{1, 12} {5, 12}
{1, 13} {2, 13} {3, 13} {4, 13} {5, 13} {6, 13}
{1, 14} {3, 14} {5, 14}
{1, 15} {2, 15} {4, 15} {7, 15}
{1, 16} {3, 16} {5, 16} {7, 16}
{1, 17} {2, 17} {3, 17} {4, 17} {5, 17} {6, 17} {7, 17} {8, 17}
{1, 18} {5, 18} {7, 18}
{1, 19} {2, 19} {3, 19} {4, 19} {5, 19} {6, 19} {7, 19} {8, 19} {9, 19}
{1, 20} {3, 20} {7, 20} {9, 20}
{1, 21} {2, 21} {4, 21} {5, 21} {8, 21} {10, 21}
{1, 22} {3, 22} {5, 22} {7, 22} {9, 22}
{1, 23} {2, 23} {3, 23} {4, 23} {5, 23} {6, 23} {7, 23} {8, 23} {9, 23} {10, 23} {11, 23}
{1, 24} {5, 24} {7, 24} {11, 24}
{1, 25} {2, 25} {3, 25} {4, 25} {6, 25} {7, 25} {8, 25} {9, 25} {11, 25} {12, 25}
{1, 26} {3, 26} {5, 26} {7, 26} {9, 26} {11, 26}
{1, 27} {2, 27} {4, 27} {5, 27} {7, 27} {8, 27} {10, 27} {11, 27} {13, 27}
{1, 28} {3, 28} {5, 28} {9, 28} {11, 28} {13, 28}
{1, 29} {2, 29} {3, 29} {4, 29} {5, 29} {6, 29} {7, 29} {8, 29} {9, 29} {10, 29} {11, 29} {12, 29} {13, 29} {14, 29}
{1, 30} {7, 30} {11, 30} {13, 30}
{1, 31} {2, 31} {3, 31} {4, 31} {5, 31} {6, 31} {7, 31} {8, 31} {9, 31} {10, 31} {11, 31} {12, 31} {13, 31} {14, 31} {15, 31}
Order 31 Normal Form C:M Ratios Sorted by Modulator