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isoharmonic chords


In just intonation, Isoharmonic chords are built by successive jumps up the harmonic series by some number of steps. Since the harmonic series is arranged such that each higher step is smaller than the one before it, all isoharmonic chords have this same shape -- with diminishing step size as one ascends. All isoharmonic chords are equal-hertz chords, meaning that the frequencies of the notes are in an arithmetic sequence with an equal difference in cycles per second between successive notes. However, not all equal-hertz chords are isoharmonic chords, since the ratios between the notes need not be integers. An isoharmonic "chord" may function more like a "scale" than a chord (depending on the composition of course), but we will use the word "chord" on this page for consistency.

class i

The simplest isoharmonic chords are built by stepping up the harmonic series by single steps (adjacent steps in the harmonic series). Take, for instance, 4:5:6:7, the harmonic seventh chord. We may call these class i isoharmonic chords. There is one class i series (the harmonic series), which looks like this:

harmonic
1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16
cents diff

1200

702

498

386

316

267

231

204

182

165

151

139

128

119

112


Some "scales" built this way: otones12-24, otones20-40...

class ii

The next simplest isoharmonic chords are built by stepping up the harmonic series by two (skipping every other harmonic). This gives us chords such as 3:5:7:9 (the primary tetrad in the Bohlen-Pierce tuning system) and 9:11:13:15. Note that if you start on an even number, your chord is equivalent to a class i harmonic chord: 4:6:8:10 = 2:3:4:5. Thus, there is one class ii series (the series of all odd harmonics):

harmonic
1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31
cents diff

1902

884

583

435

347

289

248

217

193

173

157

144

133

124

115


class iii

Class iii isoharmonic chords are less common and more complex sounding. They include chords such as 7:10:13:16 and 14:17:20:23. Note that if you start on a number divisible by three, you'll again get a chord reducible to class i (eg. 9:12:15 = 3:4:5). There are two series for class iii:

harmonic
1

4

7

10

13

16

19

22

25

28

31

34

37

40

43

46
cents diff

2400

969

617

454

359

298

254

221

196

176

160

146

135

125

117


harmonic
2

5

8

11

14

17

20

23

26

29

32

35

38

41

44

47
cents diff

1586

814

551

418

336

281

242

212

189

170

155

142

132

122

114


class iv

harmonic
1

5

9

13

17

21

25

29

33

37

41

45

49

53

57

61
cents diff

2786

1018

637

464

366

302

257

224

198

178

161

147

136

126

117


harmonic
3

7

11

15

19

23

27

31

35

39

43

47

51

55

59

63
cents diff

1467

782

537

409

331

278

239

210

187

169

154

141

131

122

114


class v

harmonic
1

6

11

16

21

26

31

36

41

46

51

56

61

66

71

76
cents diff

3102

1049

649

471

370

306

259

225

199

179

162

148

136

126

118


harmonic
2

7

12

17

22

27

32

37

42

47

52

57

62

67

72

77
cents diff

2169

933

603

446

355

294

251

219

195

175

159

146

134

125

116


harmonic
3

8

13

18

23

28

33

38

43

48

53

58

63

68

73

78
cents diff

1698

841

563

424

341

284

244

214

190

172

156

143

132

123

115


harmonic
4

9

14

19

24

29

34

39

44

49

54

59

64

69

74

79
cents diff

1404

765

529

404

328

275

238

209

186

168

153

141

130

121

113