If there is a group of notes in the form of a chord or scale and we want to make a homotonic functional analysis with respect to the groups of notes that precede and follow it, then, the first step is to analyze their fundamentals, that is to find the convergent chords which configure totally or partial the group of notes.

The most convenient way to do this is to analyse directly the intervalic relationship between notes. It will be sufficient to find two essential intervals of convergent chords: major 3rd. (or minor 6th.) and augmented 4th. (or diminished 5th.). Indeed, in many cases it will be sufficient with the major 3rd. interval class. Once these intervals are found it is possible to have an idea close to the real chord convergence and so of its fundamentals.

For example, if we have the chord E:Bb:C#:F#:A (Fig. 19 (a)): the major 3rds. (including enharmonic changes) are: A:C# and F#:A#(Bb). Indeed, having A and F# as a fundamentals, the chord is separated into two convergent chords. The fundamental figure will, then, be: A_F#º

If we take now the chord: C:G:C#:G# (Fig. 19 (b)): it could be thought that, as the chord contains two 5ths., the fundamentals would be C and C#. As it was stated on pages 15 and 16 the 5ths'potencial convergence (as the 8ths) is very weak and in fact, this example is a trick to hide the real convergences.¹ In this chord there is only a major 3rd. Ab(G#):C, the rest of the function is given by the tritone G:C#(Db) in its convergent direction towards Eb (because Eb is the 5th. of Ab and because the only other possible direction for the tritone (A) is against Ab). In this way, the two fundamentals are Ab and Eb –this latter one virtual– and the fundamental figure: Ab^Eb.

Only if the intervalic analysis is complicated or one is not sure of the results (or, why not, for curiosity) it is possible to look up the numerical tables (Table 3) at the end of this appendix where for every group of notes it will be easy to find their representative in the chord-mode class table and deduct their fundamentals.

Taking as an example the same chords as before (Fig 20):

First of all it is necessary to put the chord in scale form (into an octave according to the order of the chromatic scale, if there are any repeated notes, they are eliminated). It is possible to start with any note of the chord (here, we show the bass as first degree of the scale to be clearer). Then one looks at the number of semitones between the different degrees in the scale just formed; this gives us one digit numbers² which are considered as digits of one whole number (in this case 2313). Beside number 2313 in this appendix table numbers 61 and 2 are found. 61 indicates that the representative of the chord is num. 61 in the chord-mode class tables (from the five notes group) and 2 indicates that the 2nd. degree (or the 2nd. note) of the scale made (F#) is the fundamental bass³ of the chord –according to the representative structure–.

Comparing the representative fundamental bass in the chord-mode class tables (E) with the fundamental bass of the chord (F#) we obtain the transposition relation between the chord and the representative (major 2nd.). It is only then needed to apply the transposition (major 2nd.) to the representative fundamentals (G_Eº) to obtain the chord fundamentals: A_F#º.

The process for the other chord is the same: first set out the chord in scale form and looking at the intervals the number 161 is obtained. This number gives us numbers 26 and 4 in the table. The 4th. note of the scale (G#) is the fundamental bass while 26 chord-mode class fundamental bass is C. That is, transposing the representative fundamentals (C^Gº) up an augmented 5th. (up a minor 6th. or down a major 3rd.) the result is the chord fundamentals: Ab^Ebº.

Note: The fact that in these two examples the bass of the chords given are the same as the bass of the representatives is merely a coincidence.

(WEB Edition Addendum: You can find the numbers pressing Ctrl+F)

1 1 2

2 5 2

3 4 1

4 3 1

5 2 2

6 6 1

6 6 2

7 2 1

8 3 2

9 4 2

11 1 2

12 13 1

13 16 1

14 4 2

15 3 1

16 14 1

17 6 2

18 15 3

19 19 3

21 19 1

22 2 2

23 17 2

24 10 2

25 7 3

26 18 3

27 9 2

28 2 1

29 13 3

31 15 1

32 9 3

33 11 1

34 12 1

35 8 3

36 11 3

37 17 1

38 16 3

41 6 3

42 18 1

43 8 1

44 5 1

44 5 3

44 5 2

45 12 3

46 10 1

47 4 1

51 14 2

52 7 1

53 12 2

54 8 2

55 7 2

56 3 3

61 3 2

62 10 3

63 11 2

64 18 2

65 14 3

71 4 3

72 17 3

73 9 1

74 6 1

81 16 2

82 2 3

83 15 2

91 13 2

92 19 2

111 7 3

112 28 3

113 5 2

114 21 3

115 8 3

116 2 2

117 27 3

118 20 3

119 7 2

121 34 1

122 18 2

123 14 1

124 32 1

125 25 4

126 22 4

127 36 1

128 28 2

131 4 1

132 41 4

133 39 1

134 3 1

135 33 4

136 43 4

137 5 1

141 26 2

142 1 1

143 24 2

144 17 2

145 6 1

146 21 2

151 30 1

151 30 3

152 11 4

153 29 3

154 16 1

155 8 2

161 26 4

162 38 4

163 37 1

164 2 1

171 4 3

172 13 3

173 27 2

181 34 3

182 20 2

191 7 1

211 20 4

212 36 2

213 43 1

214 6 2

215 16 2

216 37 2

217 13 4

218 34 4

221 13 1

222 12 2

223 23 1

224 19 1

225 10 2

226 12 1

227 18 1

231 38 2

232 9 4

233 15 2

234 35 2

235 23 4

236 14 4

241 11 2

242 42 2

242 42 4

243 31 2

244 19 4

245 32 4

251 1 3

252 9 2

253 10 1

254 25 3

261 41 2

262 12 4

263 22 3

271 18 4

272 36 4

281 28 1

311 27 4

312 22 1

313 33 1

314 17 3

315 29 4

316 38 1

317 4 4

321 37 3

322 10 3

323 35 3

324 31 3

325 9 3

326 41 3

331 29 1

332 31 4

333 40 1

333 40 4

333 40 2

333 40 3

334 15 1

335 39 4

341 24 4

342 15 4

343 35 1

344 3 4

351 39 3

352 23 3

353 33 3

361 14 3

362 43 3

371 5 4

411 2 3

412 25 1

413 3 2

414 24 3

415 11 1

416 26 1

421 16 3

422 19 2

423 15 3

424 42 1

424 42 3

425 1 4

431 17 4

432 35 4

433 31 1

434 24 1

441 3 3

442 19 3

443 17 1

451 32 3

452 6 4

461 21 1

511 8 4

512 32 2

513 39 2

514 1 2

515 30 4

515 30 2

521 6 3

522 23 2

523 9 1

524 11 3

531 33 2

532 10 4

533 29 2

541 25 2

542 16 4

551 8 1

611 21 4

612 14 2

613 41 1

614 26 3

621 43 2

622 12 3

623 38 3

631 22 2

632 37 4

641 2 4

711 5 3

712 18 3

713 4 2

721 36 3

722 13 2

731 27 1

811 28 4

812 34 2

821 20 1

911 7 4

1111 33 3

1112 45 2

1113 18 3

1114 15 3

1115 35 3

1116 13 3

1117 14 3

1118 33 2

1121 20 1

1122 43 3

1123 53 3

1124 46 1

1125 51 3

1126 42 3

1127 45 1

1131 9 2

1132 36 5

1133 7 2

1134 8 2

1135 64 2

1136 18 2

1141 55 3

1142 3 2

1143 54 3

1144 40 3

1145 15 2

1151 58 4

1152 52 3

1153 16 3

1154 35 2

1161 2 2

1162 34 5

1163 13 2

1171 41 3

1172 14 2

1181 33 1

1211 41 5

1212 31 1

1213 59 1

1214 6 5

1215 44 5

1216 62 1

1217 20 5

1221 50 2

1222 23 4

1223 47 2

1224 38 2

1225 22 4

1226 43 2

1231 57 1

1232 24 5

1233 19 1

1234 30 1

1235 53 2

1241 49 5

1242 32 5

1243 60 1

1244 46 5

1251 5 4

1252 48 4

1253 51 2

1261 31 4

1262 42 2

1271 45 5

1311 2 4

1312 5 1

1313 4 1

1314 37 4

1315 17 1

1316 9 1

1321 65 4

1322 27 5

1323 56 4

1324 66 4

1325 36 4

1331 10 5

1332 61 5

1333 63 1

1334 7 1

1341 4 4

1342 29 4

1343 8 1

1351 59 4

1352 64 1

1361 18 1

1411 58 1

1412 49 2

1413 10 2

1414 1 3

1415 55 2

1421 1 1

1422 39 3

1423 12 1

1424 3 1

1431 37 2

1432 11 1

1433 54 2

1441 6 3

1442 40 2

1451 15 1

1511 55 5

1512 57 3

1513 65 1

1514 58 3

1521 17 4

1522 25 4

1523 52 2

1531 44 3

1532 16 2

1541 35 1

1611 9 4

1612 50 4

1613 2 1

1621 62 4

1622 34 4

1631 13 1

1711 20 3

1712 41 2

1721 14 1

1811 33 5

2111 14 4

2112 42 4

2113 64 3

2114 40 4

2115 16 4

2116 34 1

2117 41 4

2121 62 2

2122 22 5

2123 30 2

2124 60 2

2125 48 5

2126 31 5

2131 17 2

2132 66 5

2133 63 2

2134 29 5

2135 59 5

2141 1 4

2142 12 2

2143 11 2

2144 6 4

2151 65 2

2152 25 5

2153 44 4

2161 50 5

2162 62 5

2171 20 4

2211 34 2

2212 48 1

2213 29 1

2214 11 3

2215 25 1

2216 50 1

2221 25 2

2222 28 3

2223 26 2

2224 28 2

2225 23 3

2231 39 1

2232 21 2

2233 26 1

2234 47 1

2241 27 3

2242 28 1

2243 38 1

2251 23 2

2252 22 3

2261 43 1

2311 52 5

2312 32 2

2313 61 2

2314 39 5

2315 57 5

2321 12 4

2322 21 4

2323 21 1

2324 24 4

2331 56 2

2332 26 5

2333 19 5

2341 47 5

2342 30 5

2351 53 1

2411 3 4

2412 24 2

2413 27 2

2414 49 4

2421 66 2

2422 28 5

2423 32 4

2431 38 5

2432 60 5

2441 46 4

2511 36 2

2512 23 1

2513 5 3

2521 22 2

2522 48 3

2531 51 1

2611 43 5

2612 31 3

2621 42 1

2711 45 4

3111 13 4

3112 51 4

3113 8 3

3114 54 4

3115 52 4

3116 2 3

3121 44 1

3122 38 3

3123 19 2

3124 32 1

3125 5 5

3131 37 5

3132 56 5

3133 61 1

3134 4 5

3141 10 3

3142 39 4

3143 37 3

3151 57 4

3152 17 5

3161 9 5

3211 16 5

3212 60 3

3213 63 3

3214 12 3

3215 65 3

3221 11 4

3222 26 3

3223 21 3

3224 27 4

3231 61 3

3232 21 5

3233 56 3

3241 24 3

3242 66 3

3251 36 3

3311 54 5

3312 19 3

3313 56 1

3314 10 4

3321 63 4

3322 26 4

3323 61 4

3331 19 4

3332 63 5

3341 7 5

3411 7 4

3412 47 4

3413 4 3

3421 30 4

3422 29 3

3431 8 5

3511 53 5

3512 59 3

3521 64 5

3611 18 5

4111 35 4

4112 46 2

4113 7 3

4114 3 3

4115 58 5

4121 6 1

4122 47 3

4123 24 1

4124 49 1

4131 4 2

4132 27 1

4133 10 1

4141 49 3

4142 1 2

4151 55 1

4211 40 5

4212 30 3

4213 66 1

4214 1 5

4221 29 2

4222 28 4

4223 39 2

4231 32 3

4232 12 5

4241 3 5

4311 8 4

4312 38 4

4313 37 1

4321 60 4

4322 11 5

4331 54 1

4411 46 3

4412 6 2

4421 40 1

4511 15 5

5111 15 4

5112 53 4

5113 36 1

5114 55 4

5121 59 2

5122 23 5

5123 57 2

5131 5 2

5132 65 5

5141 58 2

5211 64 4

5212 22 1

5213 17 3

5221 48 2

5222 25 3

5231 52 1

5311 51 5

5312 44 2

5321 16 1

5411 35 5

6111 18 4

6112 43 4

6113 9 3

6121 31 2

6122 50 3

6131 2 5

6211 42 5

6212 62 3

6221 34 3

6311 13 5

7111 45 3

7112 20 2

7121 41 1

7211 14 5

8111 33 4

11111 15 1

11112 33 3

11113 74 1

11114 44 5

11115 71 6

11116 32 1

11117 15 6

11121 16 2

11122 27 4

11123 58 2

11124 43 5

11125 69 6

11126 33 2

11131 8 3

11132 77 3

11133 75 3

11134 47 3

11135 74 6

11141 62 6

11142 80 6

11143 67 4

11144 44 4

11151 13 5

11152 26 6

11153 71 5

11161 66 6

11162 32 6

11171 15 5

11211 31 3

11212 14 1

11213 48 4

11214 41 2

11215 34 1

11216 16 1

11221 55 3

11222 29 3

11223 53 3

11224 39 3

11225 27 3

11231 46 4

11232 72 6

11233 36 1

11234 58 1

11241 64 6

11242 50 1

11243 43 4

11251 35 1

11252 69 5

11261 33 1

11311 17 2

11312 2 2

11313 4 2

11314 49 3

11315 8 2

11321 59 2

11322 45 6

11323 70 6

11324 77 2

11331 3 2

11332 73 6

11333 75 2

11341 60 6

11342 47 2

11351 74 5

11411 78 6

11411 78 3

11412 54 3

11413 9 3

11414 62 5

11421 79 5

11422 30 5

11423 80 5

11431 38 3

11432 67 3

11441 44 3

11511 17 5

11512 28 6

11513 13 4

11521 18 5

11522 26 5

11531 71 4

11611 31 6

11612 66 5

11621 32 5

11711 15 4

12111 66 2

12112 35 3

12113 60 2

12114 38 5

12115 18 1

12116 31 2

12121 12 1

12122 20 6

12123 10 1

12124 25 1

12125 14 6

12131 5 6

12132 40 6

12133 68 1

12134 48 3

12141 1 5

12142 24 5

12143 41 1

12151 12 5

12152 34 6

12161 16 6

12211 28 3

12212 52 2

12213 42 2

12214 65 2

12215 55 2

12221 63 6

12222 23 4

12223 21 4

12224 29 2

12231 37 2

12232 57 2

12233 53 2

12241 20 4

12242 39 2

12251 27 2

12311 54 6

12312 11 1

12312 11 4

12313 56 1

12314 46 3

12321 6 5

12322 19 5

12323 72 5

12331 10 5

12332 36 6

12341 58 6

12411 2 5

12412 52 5

12413 64 5

12421 25 5

12422 50 6

12431 43 3

12511 14 4

12512 35 6

12521 69 4

12611 33 6

13111 13 1

13112 64 2

13113 3 4

13114 79 1

13115 17 1

13121 1 1

13122 37 4

13123 6 1

13124 2 1

13131 61 6

13131 61 2

13131 61 4

13132 7 1

13133 4 1

13141 5 4

13142 49 2

13151 8 1

13211 9 6

13212 56 4

13213 76 1

13213 76 4

13214 59 1

13221 7 5

13222 22 5

13223 45 5

13231 40 4

13232 70 5

13241 77 1

13311 4 5

13312 42 5

13313 3 1

13321 68 5

13322 73 5

13331 75 1

13411 48 1

13412 60 5

13421 47 1

13511 74 4

14111 62 2

14112 46 6

14113 59 4

14114 78 5

14114 78 2

14121 5 2

14122 63 2

14123 54 2

14131 1 3

14132 9 2

14141 62 4

14211 49 6

14212 65 5

14213 79 4

14221 24 3

14222 30 4

14231 80 4

14311 41 5

14312 38 2

14321 67 2

14411 44 2

15111 8 5

15112 55 5

15113 17 4

15121 12 3

15122 28 5

15131 13 3

15211 34 4

15212 18 4

15221 26 4

15311 71 3

16111 16 4

16112 31 5

16121 66 4

16211 32 4

17111 15 3

21111 32 2

21112 69 1

21113 47 4

21114 67 5

21115 26 1

21116 66 1

21121 34 2

21122 39 4

21123 36 2

21124 50 2

21125 35 2

21131 49 4

21132 70 1

21133 73 1

21134 60 1

21141 9 4

21142 30 6

21143 38 4

21151 28 1

21152 18 6

21161 31 1

21211 18 2

21212 25 2

21213 68 2

21214 24 6

21215 12 6

21221 65 3

21222 21 5

21223 57 3

21224 20 5

21231 56 2

21232 19 6

21233 10 6

21241 52 6

21242 25 6

21251 14 5

21311 79 2

21312 6 2

21313 7 2

21314 5 5

21321 76 5

21321 76 2

21322 22 6

21323 40 5

21331 42 6

21332 68 6

21341 48 2

21411 59 5

21412 63 3

21413 1 4

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21422 24 4

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21511 55 6

21512 12 4

21521 34 5

21611 16 5

22111 26 2

22112 50 3

22113 73 2

22114 30 1

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22121 24 1

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22123 19 1

22124 52 1

22131 7 3

22132 22 1

22133 42 1

22141 63 4

22142 65 1

22151 55 1

22211 30 2

22212 19 2

22213 22 2

22214 63 5

22221 22 3

22222 51 6

22222 51 1

22222 51 5

22222 51 4

22222 51 2

22222 51 3

22223 23 3

22231 23 2

22232 21 3

22241 29 1

22311 45 3

22312 23 1

22313 37 1

22321 21 2

22322 57 1

22331 53 1

22411 29 6

22412 20 3

22421 39 1

22511 27 1

23111 80 2

23112 72 2

23113 45 2

23114 54 5

23121 40 2

23122 23 6

23123 11 3

23123 11 6

23131 37 6

23132 56 6

23141 46 2

23211 70 3

23212 21 1

23213 6 4

23221 57 6

23222 19 4

23231 72 4

23311 53 6

23312 10 4

23321 36 5

23411 58 5

24111 77 5

24112 29 5

24113 2 4

24121 20 2

24122 52 4

24131 64 4

24211 39 6

24212 25 4

24221 50 5

24311 43 2

25111 27 6

25112 14 3

25121 35 5

25211 69 3

26111 33 5

31111 71 1

31112 43 6

31113 75 4

31114 80 1

31115 13 6

31121 41 3

31122 53 4

31123 72 1

31124 64 1

31131 4 3

31132 45 1

31133 3 3

31141 54 4

31142 79 6

31151 17 6

31211 38 6

31212 10 2

31213 40 1

31214 1 6

31221 42 3

31222 23 5

31223 37 3

31231 11 2

31231 11 5

31232 6 6

31241 2 6

31311 3 5

31312 37 5

31313 61 1

31313 61 5

31313 61 3

31321 56 5

31322 7 6

31331 4 6

31411 46 1

31412 5 3

31421 49 1

31511 8 6

32111 67 6

32112 36 3

32113 70 2

32114 9 5

32121 68 3

32122 21 6

32123 56 3

32131 6 3

32132 76 6

32132 76 3

32141 59 6

32211 73 3

32212 57 5

32213 7 4

32221 19 3

32222 22 4

32231 45 4

32311 72 3

32312 40 3

32321 70 4

32411 77 6

33111 75 5

33112 53 5

33113 4 4

33121 10 3

33122 42 4

33131 3 6

33211 36 4

33212 68 4

33221 73 4

33311 75 6

34111 58 4

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34121 60 4

34211 47 6

35111 74 3

41111 44 6

41112 58 3

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41114 62 1

41121 48 5

41122 29 4

41123 46 5

41131 2 3

41132 59 3

41141 78 1

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41211 60 3

41212 20 1

41213 5 1

41221 52 3

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41311 64 3

41312 1 2

41321 9 1

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42111 47 5

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42121 25 3

42122 65 4

42131 79 3

42211 50 4

42212 24 2

42221 30 3

42311 80 3

43111 43 1

43112 41 4

43121 38 1

43211 67 1

44111 44 1

51111 74 2

51112 27 5

51113 8 4

51121 14 2

51122 55 4

51131 17 3

51211 35 4

51212 12 2

51221 28 4

51311 13 2

52111 69 2

52112 34 3

52121 18 3

52211 26 3

53111 71 2

61111 33 4

61112 16 3

61121 31 4

61211 66 3

62111 32 3

71111 15 2

111111 54 2

111112 55 3

111113 58 2

111114 65 6

111115 34 6

111116 54 1

111121 35 7

111122 17 2

111123 63 3

111124 49 2

111125 55 2

111131 29 7

111132 15 3

111133 52 3

111134 58 1

111141 30 7

111142 32 4

111143 65 5

111151 56 7

111152 34 5

111161 54 7

111211 43 7

111212 26 2

111213 53 2

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111215 35 6

111221 24 7

111222 40 3

111223 60 4

111224 17 1

111231 59 2

111232 13 4

111233 63 2

111241 10 7

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111251 55 1

111311 18 2

111312 20 6

111313 44 6

111314 29 6

111321 16 2

111322 11 2

111323 15 2

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111332 52 2

111341 58 7

111411 27 6

111412 9 6

111413 30 6

111421 41 3

111422 32 3

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111511 36 4

111512 56 6

111521 34 4

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112111 36 7

112112 25 2

112113 62 3

112114 33 5

112115 43 6

112121 64 5

112122 21 3

112123 12 5

112124 26 1

112131 28 7

112132 37 5

112133 53 1

112141 23 5

112142 47 1

112151 35 5

112211 39 2

112212 14 6

112213 38 6

112214 24 6

112221 8 2

112222 6 2

112223 40 2

112231 61 3

112232 60 3

112241 17 7

112311 31 7

112312 66 2

112313 59 1

112321 19 3

112322 13 3

112331 63 1

112411 25 6

112412 10 6

112421 49 7

112511 55 7

113111 27 2

113112 31 3

113113 45 6

113114 18 1

113121 7 2

113122 5 2

113123 20 5

113131 57 7

113132 44 5

113141 29 5

113211 45 3

113212 48 6

113213 16 1

113221 42 3

113222 11 1

113231 15 1

113311 62 7

113312 46 1

113321 52 1

113411 58 6

114111 18 5

114112 39 5

114113 27 5

114121 22 5

114122 9 5

114131 30 5

114211 33 2

114212 41 2

114221 32 2

114311 65 3

115111 43 3

115112 36 3

115121 56 5

115211 34 3

116111 54 5

121111 56 2

121112 10 2

121113 46 4

121114 41 5

121115 36 6

121121 23 7

121122 61 5

121123 19 5

121124 25 1

121131 57 2

121132 42 5

121133 62 2

121141 22 7

121142 33 4

121151 43 5

121211 22 2

121212 51 2

121213 50 2

121214 64 4

121221 4 2

121222 2 2

121223 21 2

121231 51 7

121232 12 4

121241 26 7

121311 7 6

121312 3 6

121313 28 6

121321 50 7

121322 37 4

121331 53 7

121411 64 2

121412 23 4

121421 47 7

121511 35 4

122111 9 2

122112 66 5

122113 48 2

122114 39 1

122121 3 2

122122 1 2

122123 14 5

122131 4 7

122132 38 5

122141 24 5

122211 5 6

122212 1 6

122213 8 1

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