Fokker block: Difference between revisions

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From the values for T[i] for each of the four temperaments, we find that the generator range for pajara is -7 to 3, since we obtain the even numbers from -14 to 6. The others are magic from -9 to 12, orwell from -4 to 17 and porcupine from -8 to 13.

We can pass from a Fokker group basis to a chroma basis in various ways. One way begins by finding the [[Tenney-Euclidean_Tuning#~~The~~ Frobenius projection ~~map~~|Frobenius projection ~~map~~]] P_k corresponding to each temperament wedgie W_k, and from that the dual projection ~~map~~ Q_k. Q_k has the property that each chroma except c_k is an eigenvector with eigenvalue 1. Hence, the matrix product of the Q_i with i≠k has a single eigenvalue of 1, corresponding to c_k, which allows us to find c_k. From the Fokker group basis [pajara, magic, orwell, porccupine] we may find in this way the dual chroma basis [385/384, 176/175, 100/99, 225/224]. Taking the monzo matrix for 385/384, 175/176, 100/99, 225/224 and 36/35, inverting and transposing, we obtain [ {{val|12 19 28 34 42}, -{{val|3 5 7 9 10}, {{val|9 14 21 25 31}}, -{{val|7 11 16 20 24}}, {{val|22 35 51 62 76}} ]. From this and the previously obtained generator ranges, we find that

We can pass from a Fokker group basis to a chroma basis in various ways. One way begins by finding the [[Tenney-Euclidean_Tuning#Frobenius tuning and Frobenius projection matrix|Frobenius projection matrix]] P_k corresponding to each temperament wedgie W_k, and from that the dual projection matrix Q_k. Q_k has the property that each chroma except c_k is an eigenvector with eigenvalue 1. Hence, the matrix product of the Q_i with i≠k has a single eigenvalue of 1, corresponding to c_k, which allows us to find c_k. From the Fokker group basis [pajara, magic, orwell, porccupine] we may find in this way the dual chroma basis [385/384, 176/175, 100/99, 225/224]. Taking the monzo matrix for 385/384, 175/176, 100/99, 225/224 and 36/35, inverting and transposing, we obtain [ {{val|12 19 28 34 42}, -{{val|3 5 7 9 10}, {{val|9 14 21 25 31}}, -{{val|7 11 16 20 24}}, {{val|22 35 51 62 76}} ]. From this and the previously obtained generator ranges, we find that

<math>S[i] = (36/35)^i (385/384)^{\lfloor(12i+14)/22\rfloor} (175/176)^{\lfloor(-3i+9)/22\rfloor} (100/99)^{\lfloor(9i+4)/22\rfloor} (224/225)^{\lfloor(-7i+13)/22\rfloor}</math>

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 From the values for T[i] for each of the four temperaments, we find that the generator range for pajara is -7 to 3, since we obtain the even numbers from -14 to 6. The others are magic from -9 to 12, orwell from -4 to 17 and porcupine from -8 to 13.
-We can pass from a Fokker group basis to a chroma basis in various ways. One way begins by finding the [[Tenney-Euclidean_Tuning#The Frobenius projection map|Frobenius projection map]] P_k corresponding to each temperament wedgie W_k, and from that the dual projection map Q_k. Q_k has the property that each chroma except c_k is an eigenvector with eigenvalue 1. Hence, the matrix product of the Q_i with i≠k has a single eigenvalue of 1, corresponding to c_k, which allows us to find c_k. From the Fokker group basis [pajara, magic, orwell, porccupine] we may find in this way the dual chroma basis [385/384, 176/175, 100/99, 225/224]. Taking the monzo matrix for 385/384, 175/176, 100/99, 225/224 and 36/35, inverting and transposing, we obtain [ {{val|12 19 28 34 42}, -{{val|3 5 7 9 10}, {{val|9 14 21 25 31}}, -{{val|7 11 16 20 24}}, {{val|22 35 51 62 76}} ]. From this and the previously obtained generator ranges, we find that
+We can pass from a Fokker group basis to a chroma basis in various ways. One way begins by finding the [[Tenney-Euclidean_Tuning#Frobenius tuning and Frobenius projection matrix|Frobenius projection matrix]] P_k corresponding to each temperament wedgie W_k, and from that the dual projection matrix Q_k. Q_k has the property that each chroma except c_k is an eigenvector with eigenvalue 1. Hence, the matrix product of the Q_i with i≠k has a single eigenvalue of 1, corresponding to c_k, which allows us to find c_k. From the Fokker group basis [pajara, magic, orwell, porccupine] we may find in this way the dual chroma basis [385/384, 176/175, 100/99, 225/224]. Taking the monzo matrix for 385/384, 175/176, 100/99, 225/224 and 36/35, inverting and transposing, we obtain [ {{val|12 19 28 34 42}, -{{val|3 5 7 9 10}, {{val|9 14 21 25 31}}, -{{val|7 11 16 20 24}}, {{val|22 35 51 62 76}} ]. From this and the previously obtained generator ranges, we find that
 <math>S[i] = (36/35)^i (385/384)^{\lfloor(12i+14)/22\rfloor}  (175/176)^{\lfloor(-3i+9)/22\rfloor}  (100/99)^{\lfloor(9i+4)/22\rfloor}  (224/225)^{\lfloor(-7i+13)/22\rfloor}</math>

Fokker block: Difference between revisions

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