## ----include = FALSE---------------------------------------------------------- knitr::opts_chunk$set( collapse = TRUE, comment = "#>" ) ## ----setup-------------------------------------------------------------------- library(musicMCT) ## ----------------------------------------------------------------------------- blues <- c(0, 3, 5, 6, 7, 10) ## ----------------------------------------------------------------------------- double_harmonic <- realize_stepword(c(1, 3, 1, 2, 1, 3, 1)) blackkey_pentatonic <- maxeven(5, 12) all_interval_tetrachord <- sc(4, 29) print(double_harmonic) print(blackkey_pentatonic) print(all_interval_tetrachord) ## ----------------------------------------------------------------------------- blackkey_pentatonic[3] ## ----------------------------------------------------------------------------- c_major <- c(0, 2, 4, 5, 7, 9, 11) coord_to_edo(c_major) coord_to_edo(blues) ## ----------------------------------------------------------------------------- cis_diminished <- c(1, 0, -1) coord_from_edo(cis_diminished) ## ----------------------------------------------------------------------------- augmented_triad <- edoo(3) equiheptatonic_scale <- edoo(7) print(augmented_triad) print(equiheptatonic_scale) # edoo(n) does look like the origin after applying coord_to_edo(): coord_to_edo(equiheptatonic_scale) ## ----------------------------------------------------------------------------- g <- 2 i <- 4 sim(blues)[g+1, i] ## ----------------------------------------------------------------------------- tetrachord_arrangement <- getineqmat(4) print(tetrachord_arrangement) ## ----------------------------------------------------------------------------- dom7 <- c(0, 4, 7, 10) x <- coord_to_edo(dom7) hyperplane_v <- tetrachord_arrangement[7, ] print(hyperplane_v) # Let's leave off the last column: hyperplane_v <- hyperplane_v[-5] print(hyperplane_v) # Now we can take the dot product of v and x as in the example: hyperplane_v %*% x ## ----------------------------------------------------------------------------- signvector(dom7) ## ----------------------------------------------------------------------------- signvector(c_major)[23] ## ----------------------------------------------------------------------------- # P_theta: ineqsym(a=2, b=0, card=5) # P_psi: ineqsym(a=2, b=1, card=5) ## ----------------------------------------------------------------------------- # Using the transformation Theta: ineqsym(set=blackkey_pentatonic, a=2, b=0) # Using the transformation Psi: ineqsym(set=blackkey_pentatonic, a=2, b=1) ## ----------------------------------------------------------------------------- image_of_blues <- ineqsym(blues, 5, 0) print(image_of_blues) fortenum(image_of_blues) ## ----------------------------------------------------------------------------- spectrumcount(blackkey_pentatonic) spectrumcount(c_major) ## ----------------------------------------------------------------------------- spectrumcount(blackkey_pentatonic) spectrumcount(ineqsym(blackkey_pentatonic, 2, 0)) spectrumcount(blues) spectrumcount(image_of_blues) ## ----------------------------------------------------------------------------- melodic_minor <- c(0, 2, 3, 5, 7, 9, 11) melmin_image <- ineqsym(melodic_minor, 2, 0) spectrumcount(melodic_minor) spectrumcount(melmin_image) ## ----------------------------------------------------------------------------- k <- 2 rows_with_skips <- get_relevant_rows(k, tetrachord_arrangement) print(rows_with_skips) skip_subarrangement <- tetrachord_arrangement[rows_with_skips, ] print(skip_subarrangement) ## ----------------------------------------------------------------------------- ineqsym(card=5, a=1, b=0, involution=TRUE) ## ----------------------------------------------------------------------------- blues_involution <- ineqsym(blues, involution=TRUE) print(blues_involution) fortenum(blues_involution) normal_form(blues_involution, optic="optc") normal_form(image_of_blues, optic="optc") ## ----fig.width=5, fig.height=5, fig.fullwidth=TRUE---------------------------- tetrachord_F <- c(0, 2, 4, 7) tetrachord_G <- c(0, 1, 4, 8) brightnessgraph(tetrachord_F) ## ----fig.width=5, fig.height=5, fig.fullwidth=TRUE---------------------------- brightnessgraph(tetrachord_G) ## ----fig.width=5, fig.height=5, fig.fullwidth=TRUE---------------------------- brightnessgraph(tetrachord_F, show_sums=FALSE, show_pitches=FALSE) ## ----fig.width=5, fig.height=5, fig.fullwidth=TRUE---------------------------- brightnessgraph(tetrachord_G, show_sums=FALSE, show_pitches=FALSE) ## ----------------------------------------------------------------------------- vlsig(tetrachord_F, index=2) vlsig(tetrachord_G, index=1) ## ----------------------------------------------------------------------------- iswellformed(blackkey_pentatonic) iswellformed(ineqsym(set=blackkey_pentatonic, a=2, b=0)) iswellformed(ineqsym(set=blackkey_pentatonic, a=2, b=1)) ## ----------------------------------------------------------------------------- blackkey_pentatonic |> ineqsym(a=2, b=0) |> iswellformed() ## ----------------------------------------------------------------------------- just_diatonic <- j(dia) # Defined in footnote 7 of "Symmetries" iswellformed(just_diatonic) isgwf(just_diatonic) howfree(just_diatonic) # Thus it is pairwise well-formed. # Like all PWF scales, it is trivalent: spectrumcount(just_diatonic) ## ----fig.width=5, fig.height=5, fig.fullwidth=TRUE---------------------------- pentachord_F <- c(0, 1, 4, 6, 9) pentachord_G <- c(0, 2, 3, 5, 7) brightnessgraph(pentachord_F, show_sums=FALSE, show_pitches=FALSE) ## ----fig.width=5, fig.height=5, fig.fullwidth=TRUE---------------------------- brightnessgraph(pentachord_G, show_sums=FALSE, show_pitches=FALSE) ## ----fig.width=5, fig.height=5, fig.fullwidth=TRUE---------------------------- brightnessgraph(blues) ## ----fig.width=5, fig.height=5, fig.fullwidth=TRUE---------------------------- brightnessgraph(image_of_blues) ## ----fig.width=5, fig.height=5, fig.fullwidth=TRUE---------------------------- brightnessgraph(blues_involution) ## ----------------------------------------------------------------------------- pentatonic_palette <- scale_palette(blackkey_pentatonic) print(pentatonic_palette) apply(pentatonic_palette, 2, iswellformed) ## ----------------------------------------------------------------------------- apply(pentatonic_palette, 2, evenness) |> round(digits=5) ## ----------------------------------------------------------------------------- blues_palette <- scale_palette(blues, include_involution=FALSE) print(blues_palette) ## ----------------------------------------------------------------------------- apply(blues_palette, 2, normal_form, optic="optc") ## ----------------------------------------------------------------------------- more_blues <- scale_palette(blues, include_involution=TRUE) print(more_blues) ## ----------------------------------------------------------------------------- golomb_ruler <- c(0, 1, 4, 9, 11) golomb_scale <- convert(golomb_ruler, 15, 12) coord_to_edo(golomb_scale) |> round(digits=3) golomb_scale ineqsym(golomb_scale, a=4, b=3) |> round(digits=3) # This computation could also be done in 15-tone equal temperament, # working directly with golomb_ruler and using the "edo" parameter: coord_to_edo(golomb_ruler, edo=15) ineqsym(golomb_ruler, a=4, b=3, edo=15) ## ----------------------------------------------------------------------------- P <- ineqsym(card=5, a=4, b=3) print(P) eigen(P) ## ----------------------------------------------------------------------------- eigenvector_y <- c(0, -1, 1, 0, 0) eigenvector_z <- c(1, 0, 0, -1, 0) eigenscale <- coord_from_edo(eigenvector_y + eigenvector_z) eigenscale_involution <- coord_from_edo(-1 * (eigenvector_y + eigenvector_z)) print(eigenscale) print(eigenscale_involution) ineqsym(eigenscale, a=4, b=3) ## ----------------------------------------------------------------------------- eigenscale_involution saturate(-1, eigenscale) ## ----------------------------------------------------------------------------- dorian <- c(0, 2, 3, 5, 7, 9, 10) sigma_6_0 <- ineqsym(card=7, a=6, b=0) inversion <- -1 * sigma_6_0 inversion %*% coord_to_edo(dorian) |> coord_from_edo() |> t() |> as.numeric() ## ----------------------------------------------------------------------------- phrygian <- c(0, 1, 3, 5, 7, 8, 10) inversion %*% coord_to_edo(phrygian) |> coord_from_edo() |> t() |> as.numeric() ## ----fig.width=5, fig.height=5, fig.fullwidth=TRUE---------------------------- inverted_blues <- ineqsym(blues, a=5, b=0) |> saturate(r=-1) print(inverted_blues) clockface(blues) ## ----fig.width=5, fig.height=5, fig.fullwidth=TRUE---------------------------- clockface(inverted_blues) ## ----------------------------------------------------------------------------- tni(blues, 0) ## ----------------------------------------------------------------------------- forza <- c(0, 2, 3, 7) destino <- ineqsym(forza, a=1, b=2, involution=TRUE) ## ----------------------------------------------------------------------------- print(destino) tn(forza, 3) ## ----------------------------------------------------------------------------- ineqsym(forza, a=3, b=0, involution=FALSE) tni(forza, 3) ## ----------------------------------------------------------------------------- ait <- c(0, 1, 4, 6) scale_palette(ait, include_involution=TRUE) ## ----------------------------------------------------------------------------- scale_palette(forza, include_involution=TRUE) ## ----------------------------------------------------------------------------- schbeg <- sc(6, 44) scale_palette(schbeg) # Only 12 entries isym(schbeg) # Not inversionally symmetrical ineqsym(schbeg, a=5, b=0) # Is symmetrical under this transformation ## ----------------------------------------------------------------------------- set <- c(0, 1, 4, 8) fortenum(set) (set * 5) %% 12 |> fortenum() (set * 7) %% 12 |> fortenum() set_image <- ineqsym(set, a=3, b=0) print(set_image) fortenum(set_image) ## ----------------------------------------------------------------------------- just_chromatic_scale <- j(1, m2, 2, m3, 3, 4, jtt, 5, m6, 6, m7, 7) print(just_chromatic_scale) ## ----------------------------------------------------------------------------- adjusted_chromatic_scale <- ineqsym(just_chromatic_scale, a=5, b=0) print(adjusted_chromatic_scale) ## ----------------------------------------------------------------------------- primeform(just_chromatic_scale) primeform(adjusted_chromatic_scale) ## ----------------------------------------------------------------------------- spectrumcount(just_chromatic_scale) spectrumcount(adjusted_chromatic_scale)