Interval String Table for the Identification of Chords, Modes, Scales, and Melodies

copyright © 2005 by Larry J Solomon


History

Historically, the work on Interval String Theory (IST) can be traced to the early twentieth century, in 1917 Ernst Lecher Bacon published "Our Musical Idiom" in The Monist (see References). Bacon's work contains a list of interval strings. Later, in 1960, Howard Hanson used interval strings in his book, Harmonic Materials of Twentieth Century Music.

In 1973, Yale University Press published Allen Forte's The Structure of Atonal Music (SAM), which became the de facto standard for musical set theory. SAM contains a list of pitch set classes but it does not address interval strings. The title "atonal" with which SAM confines itself remains undefined. Atonality was considered nonexistent by Schoenberg and others who regarded it as a poor, derogatory term invented by critics. At the very least, "atonality" seems to be an arbitrary and subjective rubric (see the web essay Tonality, Modality, and Atonality, 2003). IST provides an alternative to SAM, especially for those who would like a theory that accounts for the difference between major and minor chords, as well as all other inversionally related chords. IST is not restricted to atonal or any other style of music. IST is comprehensive for all tone combinations in the twelve-tone equal-tempered system.

Interval Strings, Modes, and Scales

Modes and scales are different concepts, although the terms are used loosely and interchangeably in conversation. A scale is here defined as a series of pitches, notes, or pith classes arranged in ascending and/or descending order. CDEFGABC is a scale. A mode is an interval series (interval string) used to construct a class of scales. While a scale contains specific notes, pitches, or pcs, a mode does not. The major mode can be used to construct any major scale, on any tonic. IST uses a system that is similar to the traditional way of classifying modes. For instance, the major mode is TTSTTTS. The equivalent in IST notation would be 2212221, where each number in the string indicates the number of semitones. No commas or other delimiters are used between the numbers because they are all single digits. The same system is used for chords; therefore, a major chord becomes 435, i.e., 4 semitones, then 3 semitones, followed by 5 semitones. The intervals are ordered and cyclic. IST completes the octave in the same way as modes and scales. So, a major chord is not simply 4 semitones, then 3 semitones, but completes the octave by adding 5 more semitones. Therefore IS (interval string/s) are normally cyclic; 435 can become 354 or 543 (chord inversions) but not 345 (chromatic inversion), the minor triad, in IST. The  IS is distinct for major and cannot be cycled to be equivalent to minor. However, retrograding a string yields its chromatic inversion.

The "major mode" example, 2212221,considered cyclically, is actually a class of modes, because it can be cycled to become 2122212, 1222122, 2221221, 2212212, 2122122, or 1221222.  Therefore, the cyclic IS embraces the Dorian, Phrygian, Lydian, Mixolydian, Aoelian, and Locrian modes, i.e., all the medieval church modes. Combined, they are called the diatonic mode, which is actually a class of modes. A table could be constructed, which would be much larger, in which IS is not cyclic. It would separate various modes, but would still not have specific tonics. Only the more general cyclic table is shown here. IST does have a general application to modes and scales as well as chords. For a specific table of modes, see Op de Coul's list.

Some scales and modes are tonicized, while others are not. The diatonic mode is an example of a non-tonicized mode, while the Mixolydian mode implies a  variable tonic. Similarly, the chromatic scale is non-tonicized, whereas a G major scale is specifically tonicized.

Melodies

Melodies can also be expressed as an interval string. For example, P: 07020-2.2010202. (where P means pitch string) represents "Twinkle,  Twinkle, Little Star" in semitonal notation. The "0" refers to a repetition of notes (no interval motion). A minus sign indicates a change of direction downward, and a plus sign indicates a change upward (none in this tune). A period refers to a pause or cadence.

Time or rhythm can also be represented as an IS of time values. For the above tune the time string would be T: 1111112.1111112 (where T: means time string). The number "1" represents the value of quantization, the shortest time value in the tune.  The other values are relative multiples of this value. Therefore, if "1" is given the value of a quarter note, "2" is a half note, etc.

Chords

IST provides a powerful  way to identify any chord structure by using the same semitonal notation. It is not intended to displace traditional chord nomenclature systems, e.g., C#7, D+, Dm7, V7, etc. Instead, it embraces and supplements these in a more general system, but gives up some specificity in order to be comprehensive. IS are especially useful when the traditional nomenclature breaks down or does not work. For example, there is no traditional chord name for C F# B. How, then, can this chord be identified?  One way is to use SAM, although SAM does not distinguish between major and minor chords (as well as many other chords and scales) and makes an arbitrary distinction between tonal and atonal music. IST  provides a complete catalog of interval combinations, including those used in chords and non-tonicized modes. Furthermore, it distinguishes major from minor.

A major chord has no specific pitch content; i.e., it can be any major chord. E major, C major, F# major, etc. chords are recognized as major, regardless of the positions, spacings, roots, or doublings. IST accounts for the factor makes them all major. Additionally, we recognize minor chords as distinct from major, regardless of positions, spacings, doublings, or roots. Differing major chords have different notes or pitches or pcs demonstrating that the quality of major does not depend upon pitch. Therefore, a pitch or pc designation for a chord type (or set class) does not account for the essential aspect that determines its quality; i.e., major, minor, diminished, etc. However, all major chords have an IS=435. It is this that remains constant, regardless of the roots, the positions, spacings, or doublings. The ear and brain must recognize the IS to determine the quality of a chord, because it is the only constant. It seems more appropriate and logical, therefore, to designate chords by IS rather than by pc numbers.

Explanation of The Table of Interval Strings

The table contains an identification number for each set in the Catalog Number (Cat#). The next column has the Interval String notation for each set. A set may be identified as the IS, without a need for set-names, prime pitch forms, or other table dependence. Any major chord is identified as simply 435. Most of the rest of the table serves as a cross reference to other systems. The prime forms are in column 3, The SAM names are shown in column 4, interval vectors are in column 5, the common or descriptive names are shown in the last column. Chords with twelve-tone applications are shown with their hexachordal combinatorial properties in the last column. The penultimate column simply indicates mirror sets (symmetric about an axis in the cyclic set) with an "*".

The Null Set, Unison, and other interval-sets are included in the table for completeness, as are 10- to 12-note sets. The order of the sets in the table is determined by the normal form or prime of the IS within each cardinality. So, the ordering is different from that in SAM. It is also, of course, a larger table with 352 sets, because all inversionally related sets are considered distinct.

Interval String Primes from a Keyboard
The keyboard of a piano, organ, or other electronic keyboard (imagined or played), can be used as a convenient and familiar device to simplify the determination of an IS prime for any chord or pitch set. For example, a dominant seventh chord, e.g., C7, can be seen on a keyboard in four possible configurations in close position.
(1) (2) (3) (4)
These are the cyclical rotations of this set, commonly known as root, first, second, and third inversions. From these, choose the most compact form, i.e., number 2, which encompasses the smallest interval of a minor sixth or 8 semitones. Using this form, begin with the lowest key and count the semitones to the next key marked, then to the next, etc. Finally, complete the octave. The IS prime in this example is therefore 3324. This simple method can be used for any set. If two or more forms compete with the smallest span, choose the one with the smallest interval from the bass to the next note above. Finding the IS in this way eliminates the need for a computer or mathematics.

The master table provides more information besides common or descriptive names. Some sets are named with the suffix "-chord" and others with "-mode" (which actually represents a mode-class) depending on how the set is more commonly known in the literature. Generally, sets with cardinality less than 5 are called chords, whereas larger cardinalities are more often considered modes (or scales). However, this distinction is somewhat arbitrary. None of the IS in this table are restricted to a temporal order. Since modes consist of IS and scales consist of pitches or pcs, the suffix "mode" is used here for a scale-like IS.

The table uses hexadecimal number notation: 0,1,2,3,4,5,6,7,8,9,A,B,C, where A=10, B=11, and C=12 in equivalent decimals.(Hexadecimal also has D, E, and F digits, but they are not used here.) Thus it  is possible to express any IS without delimiters between the numbers; e.g., Cat#002 the designation "1B", means 1 semitone followed by 11 semitones. Cat#026, 1119, cannot be confused with 11,1,9 or any other grouping. Cat#008, 11A, could be rendered 1,1,10 in decimal, but then delimiters are required. Only a few of the IS at the beginning of the table have hexadecimal numbers beyond 9, so that almost all of the numbers in the table are equivalent to the more familiar decimal notation. The prime forms are also in hexadecimal for concision; those containing numbers beyond "9" appear only near the end of the table. The single digit string makes a more compact and elegant format. SAM does not include the first eight sets and those with cardinalities greater than 9; neither does it include any set-name with a "b" ending.


The Table of Interval Strings: a comprehensive cross-referenced list
Cat#
 Interval String
(IS)
 Prime
 Set-name
**
 IV
(vector)
 mirror *
 Common or descriptive name and
any hexachordal combinatorial properties
000
 00
 empty
 0-1
 000000
 *
 Null set
001
 0C
 0
 1-1
 000000
 *
 Unison
 002 1B 01 2-1 100000 *
 Semitone
 003 2A 02 2-2 010000 *
 Whole-tone
 004 39 03 2-3 001000 *
 Minor third
 005 48 04 2-4 000100 *
 Major third
 006 57 05 2-5 000010 *
 Perfect fourth
 007 66 06 2-6 000001 *
 Tritone
 008 11A 012 3-1 210000 *
 BACH or chromatic trimode
 009 129 013 3-2 111000
Phrygian trimode
 010 138 014 3-3 101100
Major+minor trichord.1
 011 147 015 3-4 100110
Incomplete major-seventh chord.1
 012 156 016 3-5 100011
Tritone+P4.1
 013 219 023 3-2b 111000
Minor trimode
 014
 228 024 3-6 020100 *
 Whole-tone trimode
 015 237 025 3-7 011010
Incomplete minor-seventh chord
 016 246 026 3-8 010101
Incomplete dominant-seventh chord.1/Italian-sixth
 017 255 027 3-9 010020 *
 Quartal trichord
 018 318 034 3-3b 101100 *
 Major+minor trichord.2
 019 327 035 3-7b 011010
Incomplete dominant-seventh chord.2
 020 336 036 3-10 002001
Diminished chord
 021 345 037 3-11 001110
Minor chord
 022 417 045 3-4b 100110
Incomplete major-seventh chord.2
 023 426 046 3-8b 010101
Incomplete half-dim-seventh chord
 024 435 047 3-11b 001110
Major chord
 025 444 048 3-12 000300
augmented chord
 026 516 056 3-5b 100011
Tritone+P4.2
 027 1119 0123 4-1 321000 *
 BACH or chromatic tetramode
 028 1128 0124 4-2 221100

 029 1137 0125 4-4 211110

 030 1146 0126 4-5 210111

 031 1155 0127 4-6 210021 *
 032 1218 0134 4-3 212100 *
 alternating tetramode
 033 1227 0135 4-11 121110
Phrygian tetramode
 034 1236 0136 4-13 112011

 035 1245 0137 4-Z29 111111
all-interval tetrachord.3
 036 1317 0145 4-7 201210 *
 037 1326 0146 4-Z15 111111
all-interval tetrachord.1
 038 1335 0147 4-18 102111

 039 1344 0148 4-19 101310

 040 1416 0156 4-8 200121 *
 041 1425 0157 4-16 110121
 
 042 1434 0158 4-20 101220 *
 Major-seventh chord
 043 1515 0167 4-9 200022

 044 2118 0234 4-2b 221100

 045 2127 0235 4-10 122010 *
 Minor tetramode
 046 2136 0236 4-12 112101
Harmonic-minor tetramode
 047 2145 0237 4-14 111120

 048 2217 0245 4-11b 121110
Major tetramode
 049 2226 0246 4-21 030201 *
 Whole-tone tetramode
 050 2235 0247 4-22 021120

 051 2244 0248 4-24 020301 *
 augmented seventh chord
 052 2316 0256 4-Z15b 111111
all-interval tetrachord.2
 053 2325 0257 4-23 021030 *
 Quartal tetrachord
 054 2334 0258 4-27 012111
Half-diminished seventh chord
 055 2415 0267 4-16b 110121

 056 2424 0268 4-25 020202 *
 French-sixth chord
 057 3117 0345 4-4b 211110

 058 3126 0346 4-12b 112101

 059 3135 0347 4-17 102210 *
 Major-minor tetrachord
 060 3144 0348 4-19b 101310

 061 3216 0356 4-13b 112011

 062 3225 0357 4-22b 021120

 063 3234 0358 4-26 012120 *
 Minor-seventh chord
 064 3315 0367 4-18b 102111

 065 3324 0368 4-27b 012111
Dominant-seventh/German-sixth chord
 066 3333 0369 4-28 004002 *
 Diminished-seventh chord
 067 4116 0456 4-5b 210111

 068 4125 0457 4-14b 111120

 069 4215 0467 4-Z29b 111111
all-interval tetrachord.4
 070 11118 01234 5-1 432100 *
 Chromatic pentamode
 071 11127 01235 5-2 332110

 072 11136 01236 5-4 322111
Blues pentamode
 073 11145 01237 5-5 321121

 074 11217 01245 5-2 322210

 075 11226 01246 5-9 231211

 076 11235 01247 5-Z36 222121

 077 11244 01248 5-13 221311

 078 11316 01256 5-6 311221

 079 11325 01257 5-14 221131

 080 11334 01258 5-Z38 212221
 
 081 11415 01267 5-7 310132

 082 11424 01268 5-15 220222 *
  
 083 12117 01345 5-3b 322210
 
 084 12126 01346 5-10 223111
alternating pentamode.1
 085 12135 01347 5-16 213211

 086 12144 01348 5-Z17 212320 *
 087 12216 01356 5-Z12 222121 *
 Locrian pentamode
 088 12225 01357 5-24 131221
Phrygian pentamode
 089 12234 01358 5-27 122230
Major-ninth chord
 090 12315 01367 5-19 212122

 091 12324 01368 5-29 122131

 092 12333 01369 5-31 114112

 093 12414 01378 5-20 211231
Pelog pentatonic mode
 094 13116 01456 5-6b 311221

 095 13125 01457 5-Z18 212221
Gypsy pentamode.1
 096 13134 01458 5-21 202420

 097 13215 01467 5-19b 212122
Balinese pentamode
 098 13224 01468 5-30 121321
Enigmatic pentamode.1
 099 13233 01469 5-32 113221
Neapolitan pentamode.1
 100 13314 01478 5-22 202321 *
 101 13323 01479 5-32b 113221
Neapolitan pentamode.2
 102 14115 01567 5-7b 310132

 103 14214 01578 5-20b 211231
Hirajoshi pentatonic mode
 104 21117 02345 5-2b 332110

 105 21126 02346 5-8 232201 *
 106 21135 02347 5-11 222220
 
 107 21144 02348 5-13b 221311
 
 108 21216 02356 5-10b 223111
alternating pentamode.2
 109 21225 02357 5-23 132130
Minor pentamode
 110 21234 02358 5-25 123121

 111 21315 02367 5-Z18b 212221
Gypsy pentamode.2
 112 21324 02368 5-28 122212
French-sixth pentachord.1
 113 21333 02369 5-31b 114112
Ranjani raga/Flat-ninth pentachord
 114 22116 02456 5-9b 231211

 115 22125 02457 5-23b 132130
Major pentamode
 116 22134 02458 5-26 122311

 117 22215 02467 5-24b 131221
Lydian pentamode
 118 22224 02468 5-33 040402 *
 Whole-tone pentamode
 119 22233 02469 5-34 032221 *
 Dominant-ninth/major-minor/Prometheus pentachord
 120 22314 02478 5-30b 121321
Enigmatic pentamode.2
 121 22323 02479 5-35 032140 *
 Pentatonic/Slendro/Bilahari raga/Quartal pentachord
 122 23115 02567 5-14b 221131

 123 23124 02568 5-28b 122212

 124 23214 02578 5-29b 122131
Kumoi pentamode.1
 125 31116 03456 5-4b 322111

 126 31125 03457 5-11b 222220
 
 127 31134 03458 5-Z37 212320
 
 128 31215 03467 5-16b 213211
 
 129 31224 03468 5-26b 122311
 
 130 31314 03478 5-21b 202420

 131 32115 03567 5-Z36b 222121

 132 32124 03568 5-25b 123121

 133 32214 03578 5-27b 122230
Minor-ninth chord
 134 33114 03678 5-Z38b 212221

 135 41115 04567 5-5b 321121

 136 111117 012345 6-1 543210 *
 Chromatic hexamode/1st ordall-comb.#P06/=I05+I11
 137 111126 012346 6-2 443211
Comb.#I11
 138 111135 012347 6-Z36 433221

 139 111144 012348 6-Z37 432321 *
 comb.=I04
 140 111216 012356 6-Z3 433221

 141 111225 012357 6-9 342231
comb.#I11
 142 111234 012358 6-Z40 333231

 143 111315 012367 6-5 422232
comb.I11
 144 111324 012368 6-Z41 332232

 145 111333 012369 6-Z42 324222 *
 comb.=I03
 146 111414 012378 6-Z38 421242 *
 comb.=I03
 147 112116 012456 6-Z4 432321
comb.=I06
 148 112125 012457 6-Z11 333231

 149 112134 012458 6-15 323421
comb.#I11
 150 112215 012467 6-Z12 332232

 151 112224 012468 6-22 241422
comb.#I11
 152 112233 012469 6-Z46 233331

 153 112314 012478 6-Z17 322332

 154 112323 012479 6-Z47 233241

 155 113115 012567 6-Z6 421242

 156 113124 012568 6-Z43 322332

 157 113133 012569 6-Z44 313431
Schoenberg anagram hexachord
 158 113214 012578 6-18 322242
comb.#I11
 159 113223 012579 6-Z48 232341 *
 comb.=I02
 160 113313 012589 6-Z44b 313431
Bauli raga descending mode
 161 114114 012678 6-7 420243 *
 2nd ord. all-comb.#P03+P09+I05/=P06+I02+I08
 162 121116 013456 6-Z3b 433221

 163 121125 013457 6-Z10 333321

 164 121134 013458 6-Z14 323430
comb.#P06
 165 121215 013467 6-Z13 324222 *
 alternating hexamode/comb.I07
 166 121224 013468 6-Z24 233331

 167 121233 013469 6-27 225222
comb.#I11
 168 121314 013478 6-Z19 313431

 169 121323 013479 6-Z49 224322 *
 Prometheus Neapolitan mode/comb.=I04
 170 122115 013567 6-Z12b 332232

 171 122124 013568 6-Z25 233241
Locrian hexamode/Suddha Saveri raga mode
 172 122133 013569 6-Z28 224322 *
 comb.=I06
 173 122214 013578 6-Z26 232341 *
 Phrygian hexamode/comb.=I08
 174 122223 013579 6-34 142422
Scriabin's 'mystic' or Prometheus hexachord/comb.#I11
 175 122313 013589 6-31 223431
comb.#I07
 176 123114 013678 6-18b 322242
comb.#I05
 177 123123 013679 6-30 224223
comb.=P06/#I05+I11
 178 123213 013689 6-Z29 224232
comb.=I09
 179 131115 014567 6-5b 422232
comb.#I03
 180 131124 014568 6-16 322431
comb.#I03
 181 131214 014578 6-Z19b 313431
Gypsy mode
 182 131313 014589 6-20 303630 *
 3rd ord all-comb.
 183 132114 014678 6-Z17b 322332

 184 132123 014679 6-Z50 224232 *
 comb.=I01
 185 132213 014689 6-31b 223431
comb.#I11
 186 211116 023456 6-2b 443211
Comb.#I01
 187 211125 023457 6-8 343230 *
 1st ordall-comb.#P06+I01/=I07
 188 211134 023458 6-Z39 333321

 189 211215 023467 6-Z10b 333321

 190 211224 023468 6-21 242412
comb.#I01
 191 211233 023469 6-Z45 234222 *
 comb.=I06
 192 211314 023478 6-16b 322431
Megha or "Cloud raga mode/comb.#I01
 193 211323 023479 6-Z47b 233241

 194 212115 023567 6-Z11b 333231

 195 212124 023568 6-Z23 234222 *
 Super-Locrian hexamode/comb.=I08
 196 212133 023569 6-27b 225222
comb.#I01
 197 212214 023578 6-Z25b 233241
Minor hexamode
 198 212223 023579 6-33 143241
Dorian hexamode/comb.#I01
 199 213114 023678 6-Z43b 322332

 200 213213 023689 6-30b 224223
comb.=P06/#I01+I07
 201 221115 024567 6-9b 342231
comb.#I03
 202 221124 024568 6-21b 242412
comb.#I03
 203 221133 024569 6-Z46b 233331

 204 221214 024578 6-Z24b 233331

 205 221223 024579 6-32 143250 *
 1st ord.all-comb.#P06+I03/=I09/major/quartal hexamode
 206 222114 024678 6-22b 241422
comb.#I05
 207 222123 024679 6-33b 143241
Dominant-eleventh chord/Lydian hexamode/comb.#I05
 208 222213 024689 6-34b 142422
Harmonic-overtone hexachord/aug-11th/comb.#I07
 209 222222 02468A 6-35 60603 *
 Wholetone mode/6th ord all comb.
 210 231114 025678 6-Z41b 332232

 211 311115 034567 6-Z36b 433221

 212 311124 034568 6-Z39b 333321

 213 311214 034578 6-Z14b 323430
comb.#P06
 214 312114 034678 6-15b 323421
comb.#I05
 215 321114 035678 6-Z40b 333231

 216 1111116 0123456 7-1 654321 *
 Chromatic heptamode
 217 1111125 0123457 7-2 554331

 218 1111134 0123458 7-3 544431

 219 1111215 0123467 7-4 544332

 220 1111224 0123468 7-9 453432

 221 1111233 0123469 7-10 445332

 222 1111314 0123478 7-6 533442

 223 1111323 0123479 7-Z12 444342 *
 Blues mode
 224 1112115 0123567 7-5 543342

 225 1112124 0123568 7-Z36 444342

 226 1112133 0123569 7-16 435432

 227 1112214 0123578 7-14 443352

 228 1112223 0123579 7-24 353442

 229 1112313 0123589 7-Z18 434442

 230 1113114 0123678 7-7 532353

 231 1113123 0123679 7-19 434343

 232 1113213 0123689 7-19b 434343

 233 1121115 0124567 7-5b 543342

 234 1121124 0124568 7-13 443532

 235 1121133 0124569 7-Z17 434541 *
 23 6
 1121214 0124578 7-Z38 434442

 237 1121223 0124579 7-27 344451

 238 1121313 0124589 7-21 424641

 239 1122114 0124678 7-15 442443 *
 240 1122123 0124679 7-29 344352

 241 1122213 0124689 7-30 343542
Neapolitan-minor mode
 242 1122222 012468A 7-33 262623 *
 Neapolitan-major/Major Locrian/Leading-Whole-tone mode/
 243 1123113 0124789 7-20 433452

 244 1131114 0125678 7-7b 532353

 245 1131213 0125689 7-22 424542 *
 Hungarian minor/double-harm. mode/Mayamdavagaula raga mode
 246 1132113 0125789 7-20b 433452
Pantuvarali raga mode
 247 1211115 0134567 7-4b 544332

 248 1211124 0134568 7-11 444441

 249 1211133 0134569 7-16b 435432

 250 1211214 0134578 7-Z37 434541 *
 251 1211223 0134579 7-26 344532

 252 1211313 0134589 7-21b 434641

 253 1212114 0134678 7-Z38b 434442

 254 1212123 0134679 7-31 336333
alternating heptamode.1
 255 1212213 0134689 7-32 335442
harmonic-minor heptamode
 256 1212222 013468A 7-34 254442 *
 harmonic/Super-Locrian/aug-13th heptamode
 257 1221114 0135678 7-14b 443352

 258 1221123 0135679 7-28 344433

 259 1221213 0135689 7-32b 335442
Dharmavati mode
 260 1221222 013568A 7-35 254361 *
 diatonic heptamode/Dominant-13th
 261 1222113 0135789 7-30b 343542

 262 1311114 0145678 7-6b 533442

 263 1321113 0146789 7-Z18b 434442

 264 2111115 0234567 7-2b 554331

 265 2111124 0234568 7-8 454422 *
 266 2111133 0234569 7-10b 445332

 267 2111214 0234578 7-11b 444441

 268 2111223 0234579 7-23 354351

 269 2112114 0234678 7-13b 443532

 270 2112123 0234679 7-25 345342

 271 2112213 0234689 7-28b 344433

 272 2121114 0235678 7-Z36b 444342

 273 2121123 0235679 7-25b 345342

 274 2121213 0235689 7-31b 336333
alternating heptamode.2/Hungarian major mode
 275 2122113 0235789 7-29b 344352

 276 2211114 0245678 7-9b 453432

 277 2211123 0245679 7-23b 354351

 278 2211213 0245689 7-26b 344532

 279 2212113 0245789 7-27b 344451

 280 2221113 0246789 7-24b 353442
Enigmatic heptamode
 281 3111114 0345678 7-3b 544431

 282 11111115 01234567 8-1 765442 *
 chromatic octamode
 283 11111124 01234568 8-2 665542

 284 11111133 01234569 8-3 656542 *
 285 11111214 01234578 8-4 655552

 286 11111223 01234579 8-11 565552

 287 11111313 01234589 8-7 645652 *
 288 11112114 01234678 8-5 654553

 289 11112123 01234679 8-13 445453

 290 11112213 01234689 8-Z15 555553

 291 11112222 0123468A 8-21 474643 *
 292 11113113 01234789 8-8 644563 *
 293 11121114 01235678 8-6 654463 *
 294 11121123 01235679 8-Z29 555553

 295 11121213 01235689 8-18 546553

 296 11121222 0123568A 8-22 465562

 297 11122113 01235789 8-16 554563

 298 11122122 0123578A 8-23 465472 *
 quartal octamode
 299 11122212 0123579A 8-22b 465562
Spanish octatonic mode
 300 11131113 01236789 8-9 644464 *
 301 11211114 01245678 8-5b 654553

 302 11211123 01245679 8-14 555562

 303 11211213 01245689 8-19 545752

 304 11211222 0124568A 8-24 464743 *
 305 11212113 01245789 8-20 545662 *
 306 11212122 0124578A 8-27 456553 *
 307 11212212 0124579A 8-26 456562 *
 308 11221113 01246789 8-16b 554563

 309 11221122 0124678A 8-25 464644 *
 310 11221212 0124679A 8-27b 456553

 311 12111114 01345678 8-4b 655552

 312 12111123 01345679 8-12 556543

 313 12111213 01345689 8-17 546652 *
 314 12112113 01345789 8-19b 545752

 315 12121113 01346789 8-18b 546553

 316 12121212 0134679A 8-28 448444 *
 alternating octatonic or diminished mode
 317 12211113 01356789 8-Z15b 555553

 318 21111114 02345678 8-2b 665542

 319 21111123 02345679 8-10 566452 *
 320 21111213 02345689 8-12b 556543

 321 21112113 02345789 8-14b 555562

 322 21121113 02346789 8-Z29b 555553

 323 21211113 02356789 8-13b 556453

 324 22111113 02456789 8-11b 565552

 325 111111114 012345678 9-1 876663 *
 chromatic nonamode
 326 111111123 012345679 9-2 777663

 327 111111213 012345689 9-3 767763

 328 111111222 01234568A 9-6 686763 *
 329 111112113 012345789 9-4 766773

 330 111112122 01234578A 9-7 677673
Blues mode
 331 111112212 01234579A 9-7b 677673

 332 111121113 012346789 9-5 766674

 333 111121122 01234678A 9-8 676764

 334 111121212 01234679A 9-10 668664 *
 335 111122112 01234689A 9-8b 676764

 336 111211113 012356789 9-5b 766674

 337 111211122 01235678A 9-9 676683 *
 338 111211212 01235679A 9-11 667773

 339 111212112 01235689A 9-11b 667773
Diminishing nonamode
 340 112111113 012456789 9-4b 766773

 341 112112112 01245689A 9-12 666963 *
 342 121111113 013456789 9-3b 767763

 343 211111113 023456789 9-2b 777663

 344 1111111113 0123456789 10-1 988884 *
 Chromatic decamode
 345 1111111122 012345678A 10-2 898884 *
 346 1111111212 012345679A 10-3 889884 *
 347 1111112112 012345689A 10-4 888984 *
 348 1111121112 012345789A 10-5 888894 *
 349 1111211112 012346789A 10-6 888885 *
 350 11111111112 0123456789A 11-1 AAAAA5 *
 Chromatic undecamode
 351 111111111111 0123456789AB 12-1 BBBBB6 *
 Chromatic mode/dodecamode

* the asterisk indicates that a set is a mirror, i.e., an axis of symmetry exists somewhere in the set.
** For SAM system, ignore sets with b endings and cardinalities less than 3 and greater than 9.

Conversing with Interval Strings

Conversing with, remembering, or designating interval strings can be daunting for the longer strings, but this can be simplified by using groups of numbers up to four at a time. For example,  Cat#27, 1119 can be designated in conversation as "eleven-nineteen".  Cat #313 of cardinality 8, can be rendered as "twelve-eleven, twelve-thirteen".
 313 12111213
Cat# 65, the dominant seventh, can be called  "thirty-three, twenty-four".
 065 3324
An IS of odd numbers can be grouped into three or four digits plus the remainder; e.g., the following IS is "two-twenty-two, thirty three".
 119 22233

References

Bacon, Ernst Lecher, "Our Musical Idiom", The Monist 27:1 (October 1917)
Chrisman, Richard, "Describing Structural Aspects of Pitch-Sets Using Successive Interval Arrays", Journal of Music Theory 21:1 (Spring 1977)
Forte, Allen. The Structure of Atonal Music (New Haven CT, 1973)
Hanson, Howard. Harmonic Materials of Twentieth-Century Music (New York:Appleton-Century_Crofts, 1960)
Regener, Eric, "On Allen Forte's Theory of Chords", Perspectives of New Music 13:1 (1974)
Solomon, Larry, "The List of Chords, Their Properties, and Uses", Journal of New Music Research 11:2 (1982)
Solomon, Larry. Set Theory Primer and related topics, at www.solomonsmusic.net
Solomon, Larry. Tonality, Modality, and Atonality (webessay at www.solomonsmusic.net)

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