Modes and Their Chords in an Equal Tempered Scale
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This GeoGebra example demonstrates the structure of seven musical modes (Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, Locrian) in an Equal Tempered Scale and their corresponding chord constructions. It uses GeoGebra commands including Sequence to generate mode intervals, Length to calculate scale sizes, Textfield and Input for user interaction, Sum for chord note summation, If for conditional logic, and Polygon for potential interval visualization. The core mathematical concepts include the frequency ratio of equal temperament (2^(n/12)), discrete sequence patterns of mode intervals, and the mathematical structure of chords. Suitable for high school and university students exploring the mathematics behind music, and for mathematics teachers demonstrating practical applications of sequences in music theory.
GeoGebra Commands & Parameters
Sequence(1, n, 1, 12) → noteListLength(noteList) → nTextfield(noteList) → InputRemoveUndefined(Zip[If[s ≟ 1, k / n], s, noteList, k, Sequence[0, n - 1]]) → fractionsSum(noteList) → modeSizeIf(Element[noteList, 101] ≟ 1, ℯ^((((2 * π) * ί) * 100 / n))) → A101If(Element[noteList, 102] ≟ 1, ℯ^((((2 * π) * ί) * 101 / n))) → A102If(Element[noteList, 103] ≟ 1, ℯ^((((2 * π) * ί) * 102 / n))) → A103If(Element[noteList, 104] ≟ 1, ℯ^((((2 * π) * ί) * 103 / n))) → A104If(Element[noteList, 105] ≟ 1, ℯ^((((2 * π) * ί) * 104 / n))) → A105If(Element[noteList, 106] ≟ 1, ℯ^((((2 * π) * ί) * 105 / n))) → A106If(Element[noteList, 107] ≟ 1, ℯ^((((2 * π) * ί) * 106 / n))) → A107If(Element[noteList, 108] ≟ 1, ℯ^((((2 * π) * ί) * 107 / n))) → A108If(Element[noteList, 109] ≟ 1, ℯ^((((2 * π) * ί) * 108 / n))) → A109If(Element[noteList, 110] ≟ 1, ℯ^((((2 * π) * ί) * 109 / n))) → A110If(Element[noteList, 111] ≟ 1, ℯ^((((2 * π) * ί) * 110 / n))) → A111If(Element[noteList, 112] ≟ 1, ℯ^((((2 * π) * ί) * 111 / n))) → A112If(Element[noteList, 113] ≟ 1, ℯ^((((2 * π) * ί) * 112 / n))) → A113If(Element[noteList, 114] ≟ 1, ℯ^((((2 * π) * ί) * 113 / n))) → A114If(Element[noteList, 115] ≟ 1, ℯ^((((2 * π) * ί) * 114 / n))) → A115If(Element[noteList, 116] ≟ 1, ℯ^((((2 * π) * ί) * 115 / n))) → A116If(Element[noteList, 117] ≟ 1, ℯ^((((2 * π) * ί) * 116 / n))) → A117If(Element[noteList, 118] ≟ 1, ℯ^((((2 * π) * ί) * 117 / n))) → A118If(Element[noteList, 119] ≟ 1, ℯ^((((2 * π) * ί) * 118 / n))) → A119If(Element[noteList, 120] ≟ 1, ℯ^((((2 * π) * ί) * 119 / n))) → A120If(Element[noteList, 121] ≟ 1, ℯ^((((2 * π) * ί) * 120 / n))) → A121If(Element[noteList, 122] ≟ 1, ℯ^((((2 * π) * ί) * 121 / n))) → A122If(Element[noteList, 123] ≟ 1, ℯ^((((2 * π) * ί) * 122 / n))) → A123If(Element[noteList, 124] ≟ 1, ℯ^((((2 * π) * ί) * 123 / n))) → A124If(Element[noteList, 125] ≟ 1, ℯ^((((2 * π) * ί) * 124 / n))) → A125If(Element[noteList, 126] ≟ 1, ℯ^((((2 * π) * ί) * 125 / n))) → A126If(Element[noteList, 127] ≟ 1, ℯ^((((2 * π) * ί) * 126 / n))) → A127If(Element[noteList, 128] ≟ 1, ℯ^((((2 * π) * ί) * 127 / n))) → A128If(Element[noteList, 129] ≟ 1, ℯ^((((2 * π) * ί) * 128 / n))) → A129If(Element[noteList, 130] ≟ 1, ℯ^((((2 * π) * ί) * 129 / n))) → A130If(Element[noteList, 131] ≟ 1, ℯ^((((2 * π) * ί) * 130 / n))) → A131If(Element[noteList, 132] ≟ 1, ℯ^((((2 * π) * ί) * 131 / n))) → A132If(Element[noteList, 133] ≟ 1, ℯ^((((2 * π) * ί) * 132 / n))) → A133If(Element[noteList, 134] ≟ 1, ℯ^((((2 * π) * ί) * 133 / n))) → A134If(Element[noteList, 135] ≟ 1, ℯ^((((2 * π) * ί) * 134 / n))) → A135If(Element[noteList, 136] ≟ 1, ℯ^((((2 * π) * ί) * 135 / n))) → A136If(Element[noteList, 137] ≟ 1, ℯ^((((2 * π) * ί) * 136 / n))) → A137If(Element[noteList, 138] ≟ 1, ℯ^((((2 * π) * ί) * 137 / n))) → A138If(Element[noteList, 139] ≟ 1, ℯ^((((2 * π) * ί) * 138 / n))) → A139If(Element[noteList, 140] ≟ 1, ℯ^((((2 * π) * ί) * 139 / n))) → A140If(Element[noteList, 141] ≟ 1, ℯ^((((2 * π) * ί) * 140 / n))) → A141If(Element[noteList, 142] ≟ 1, ℯ^((((2 * π) * ί) * 141 / n))) → A142If(Element[noteList, 143] ≟ 1, ℯ^((((2 * π) * ί) * 142 / n))) → A143If(Element[noteList, 144] ≟ 1, ℯ^((((2 * π) * ί) * 143 / n))) → A144If(Element[noteList, 145] ≟ 1, ℯ^((((2 * π) * ί) * 144 / n))) → A145If(Element[noteList, 146] ≟ 1, ℯ^((((2 * π) * ί) * 145 / n))) → A146If(Element[noteList, 147] ≟ 1, ℯ^((((2 * π) * ί) * 146 / n))) → A147If(Element[noteList, 148] ≟ 1, ℯ^((((2 * π) * ί) * 147 / n))) → A148If(Element[noteList, 149] ≟ 1, ℯ^((((2 * π) * ί) * 148 / n))) → A149If(Element[noteList, 150] ≟ 1, ℯ^((((2 * π) * ί) * 149 / n))) → A150If(Element[noteList, 151] ≟ 1, ℯ^((((2 * π) * ί) * 150 / n))) → A151If(Element[noteList, 152] ≟ 1, ℯ^((((2 * π) * ί) * 151 / n))) → A152If(Element[noteList, 153] ≟ 1, ℯ^((((2 * π) * ί) * 152 / n))) → A153If(Element[noteList, 154] ≟ 1, ℯ^((((2 * π) * ί) * 153 / n))) → A154If(Element[noteList, 155] ≟ 1, ℯ^((((2 * π) * ί) * 154 / n))) → A155If(Element[noteList, 156] ≟ 1, ℯ^((((2 * π) * ί) * 155 / n))) → A156If(Element[noteList, 157] ≟ 1, ℯ^((((2 * π) * ί) * 156 / n))) → A157If(Element[noteList, 158] ≟ 1, ℯ^((((2 * π) * ί) * 157 / n))) → A158If(Element[noteList, 159] ≟ 1, ℯ^((((2 * π) * ί) * 158 / n))) → A159If(Element[noteList, 160] ≟ 1, ℯ^((((2 * π) * ί) * 159 / n))) → A160If(Element[noteList, 161] ≟ 1, ℯ^((((2 * π) * ί) * 160 / n))) → A161If(Element[noteList, 162] ≟ 1, ℯ^((((2 * π) * ί) * 161 / n))) → A162If(Element[noteList, 163] ≟ 1, ℯ^((((2 * π) * ί) * 162 / n))) → A163If(Element[noteList, 164] ≟ 1, ℯ^((((2 * π) * ί) * 163 / n))) → A164If(Element[noteList, 165] ≟ 1, ℯ^((((2 * π) * ί) * 164 / n))) → A165If(Element[noteList, 166] ≟ 1, ℯ^((((2 * π) * ί) * 165 / n))) → A166If(Element[noteList, 167] ≟ 1, ℯ^((((2 * π) * ί) * 166 / n))) → A167If(Element[noteList, 168] ≟ 1, ℯ^((((2 * π) * ί) * 167 / n))) → A168If(Element[noteList, 169] ≟ 1, ℯ^((((2 * π) * ί) * 168 / n))) → A169If(Element[noteList, 170] ≟ 1, ℯ^((((2 * π) * ί) * 169 / n))) → A170If(Element[noteList, 171] ≟ 1, ℯ^((((2 * π) * ί) * 170 / n))) → A171If(Element[noteList, 172] ≟ 1, ℯ^((((2 * π) * ί) * 171 / n))) → A172If(Element[noteList, 173] ≟ 1, ℯ^((((2 * π) * ί) * 172 / n))) → A173If(Element[noteList, 174] ≟ 1, ℯ^((((2 * π) * ί) * 173 / n))) → A174If(Element[noteList, 175] ≟ 1, ℯ^((((2 * π) * ί) * 174 / n))) → A175If(Element[noteList, 176] ≟ 1, ℯ^((((2 * π) * ί) * 175 / n))) → A176If(Element[noteList, 177] ≟ 1, ℯ^((((2 * π) * ί) * 176 / n))) → A177If(Element[noteList, 178] ≟ 1, ℯ^((((2 * π) * ί) * 177 / n))) → A178If(Element[noteList, 179] ≟ 1, ℯ^((((2 * π) * ί) * 178 / n))) → A179If(Element[noteList, 180] ≟ 1, ℯ^((((2 * π) * ί) * 179 / n))) → A180If(Element[noteList, 181] ≟ 1, ℯ^((((2 * π) * ί) * 180 / n))) → A181If(Element[noteList, 182] ≟ 1, ℯ^((((2 * π) * ί) * 181 / n))) → A182If(Element[noteList, 183] ≟ 1, ℯ^((((2 * π) * ί) * 182 / n))) → A183If(Element[noteList, 184] ≟ 1, ℯ^((((2 * π) * ί) * 183 / n))) → A184If(Element[noteList, 185] ≟ 1, ℯ^((((2 * π) * ί) * 184 / n))) → A185If(Element[noteList, 186] ≟ 1, ℯ^((((2 * π) * ί) * 185 / n))) → A186If(Element[noteList, 187] ≟ 1, ℯ^((((2 * π) * ί) * 186 / n))) → A187If(Element[noteList, 188] ≟ 1, ℯ^((((2 * π) * ί) * 187 / n))) → A188If(Element[noteList, 189] ≟ 1, ℯ^((((2 * π) * ί) * 188 / n))) → A189If(Element[noteList, 190] ≟ 1, ℯ^((((2 * π) * ί) * 189 / n))) → A190If(Element[noteList, 191] ≟ 1, ℯ^((((2 * π) * ί) * 190 / n))) → A191If(Element[noteList, 192] ≟ 1, ℯ^((((2 * π) * ί) * 191 / n))) → A192If(Element[noteList, 193] ≟ 1, ℯ^((((2 * π) * ί) * 192 / n))) → A193If(Element[noteList, 194] ≟ 1, ℯ^((((2 * π) * ί) * 193 / n))) → A194If(Element[noteList, 195] ≟ 1, ℯ^((((2 * π) * ί) * 194 / n))) → A195If(Element[noteList, 196] ≟ 1, ℯ^((((2 * π) * ί) * 195 / n))) → A196If(Element[noteList, 197] ≟ 1, ℯ^((((2 * π) * ί) * 196 / n))) → A197If(Element[noteList, 198] ≟ 1, ℯ^((((2 * π) * ί) * 197 / n))) → A198If(Element[noteList, 199] ≟ 1, ℯ^((((2 * π) * ί) * 198 / n))) → A199If(Element[noteList, 200] ≟ 1, ℯ^((((2 * π) * ί) * 199 / n))) → A200If(Element[noteList, 201] ≟ 1, ℯ^((((2 * π) * ί) * 200 / n))) → A201If(Element[noteList, 202] ≟ 1, ℯ^((((2 * π) * ί) * 201 / n))) → A202If(Element[noteList, 203] ≟ 1, ℯ^((((2 * π) * ί) * 202 / n))) → A203If(Element[noteList, 204] ≟ 1, ℯ^((((2 * π) * ί) * 203 / n))) → A204If(Element[noteList, 205] ≟ 1, ℯ^((((2 * π) * ί) * 204 / n))) → A205If(Element[noteList, 1] ≟ 1, ℯ^((((2 * π) * ί) * 0 / n))) → A1If(Element[noteList, 2] ≟ 1, ℯ^((((2 * π) * ί) * 1 / n))) → A2If(Element[noteList, 3] ≟ 1, ℯ^((((2 * π) * ί) * 2 / n))) → A3If(Element[noteList, 4] ≟ 1, ℯ^((((2 * π) * ί) * 3 / n))) → A4If(Element[noteList, 5] ≟ 1, ℯ^((((2 * π) * ί) * 4 / n))) → A5If(Element[noteList, 6] ≟ 1, ℯ^((((2 * π) * ί) * 5 / n))) → A6If(Element[noteList, 7] ≟ 1, ℯ^((((2 * π) * ί) * 6 / n))) → A7If(Element[noteList, 8] ≟ 1, ℯ^((((2 * π) * ί) * 7 / n))) → A8If(Element[noteList, 9] ≟ 1, ℯ^((((2 * π) * ί) * 8 / n))) → A9If(Element[noteList, 10] ≟ 1, ℯ^((((2 * π) * ί) * 9 / n))) → A10If(Element[noteList, 11] ≟ 1, ℯ^((((2 * π) * ί) * 10 / n))) → A11If(Element[noteList, 12] ≟ 1, ℯ^((((2 * π) * ί) * 11 / n))) → A12RemoveUndefined(A1:A12) → modeNoteListPolygon({Element[modeNoteList, chordNote1], Element[modeNoteList, chordNote2], Element[modeNoteList, chordNote3], Element[modeNoteList, chordNote4]}) → chord
AI Drawing Prompt Reference
A graphic showing the structure of seven musical modes in an equal-tempered scale, including scale notes and chord notes for each mode. Modes are arranged in order: Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, Locrian, with different colors to distinguish modes and chords. The graphic also includes frequency ratio calculations and chord note summation to illustrate the mathematical structure. It should clearly show interval relationships between modes and chord construction.
The prompt above was compiled from the original drawing commands and is for reference only.