A model for hexatonic scales

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Hexatonic scales are those who have 6 divisions . the simplest example is 6-EDO or whole-tone scale with 6 divisons  with size of 200 cent ( temperd major second ):

 

  0:             1/1                0.000 unison, perfect prime

  1:        200.000 cents     200.000

  2:        400.000 cents     400.000

  3:        600.000 cents     600.000

  4:        800.000 cents     800.000

  5:       1000.000 cents    1000.000

  6:             2/1                1200.000 octave 

                                                                     

 

I call whole tone scale one type of  6-6 hexatonic scales.This type has a pattern as 222-222.

              6-6 means that we have 2 great divisions per octave with size of tritone or 600 cent.

              222 means that there are 3 equal divisions per tritone ( or Augmented tetrachord) which have size of 200 cent or major second.

              222-222 means that there are 6 equal divisions of 200 cent making 2 tritones:                                                      

                                                                  2             2             2      -      2             2             2                   

                                                             

This way of naming tells us that:

1-           We must divide octave into 2 great divisions : equal or unequal , so we can have m-n hexatonic scales

2-           We must divide each of these 2 divisions into 3 divisions : equal or unequal , so we can have a pattern as abc in each tritone and abc-a'b'c' in each scale.( In http://www.xs4all.nl/~huygensf/doc/modename.html you can see that hexatonic scales are not considered as 2 great triple divisions but as 6 divisions.)

3-    We can have hexatonic scales in many different tuning systems.

 

m-n hexatonic scales in 12-EDO

m and n are 2 great divisions of octave in hexatonic scales . If  m=n=600 cent , we have 6-6 hexatonic scale.6 is a symbol for 600 cent in 12-EDO or its multiples.But it is also possible to have m or n with size of 300,400,500,700,800 or 900 cent. So if considering m or n equal to 3,4,5,7,8 and 9 , we will have:

3-9 hexatonic scale

4-8 hexatonic scale

5-7 hexatonic scale

7-5 hexatonic scale

8-4 hexatonic scale

9-3 hexatonic scale

Notice that :

m+n=12 semitone

a,b and c are multiples of one semitone

1 <= a,b,c =< 4 

 a+b+c = a'+b'+c' = 6

 6-6 hexatonic scale

m=600 , n=600

As wrote before , 6-6 is a kind of hexatonic scale consisting of 2 equal divisions (tritone or augmented tetrachord), each divided into 3 parts.we can consider different orders for different sizes of these 3 divisions as:

2+2+2 ~ 222  = 6

1+1+4 ~ 114  = 6

1+2+3 ~ 123  = 6

1+3+2 ~ 132  = 6

 

Look at the below picture for all the possible forms of augmenetd tetrachord in 12-EDO:

                         

and the below table shows all the 100 possible 6-6 hexatonic scales:

 

                                     

A 6-6 hexatonic scale with a pattern like 321-222 has these degrees:

C D# E# F# G# A# C'

  9-3 hexatonic scale ,   3-9 hexatonic scale

   m=900 , n=300         m=300 , n=900

                                                                                      900 + 300               300+900

                                                                                     27 scales               27 scales  

                                                                         

 

 

 8-4 hexatonic scale 4-8 hexatonic scale

m=800 , n=400                                    m=400 , n=800

                                                                     57 scales                                             57 scales

                                     

 

   7-5 hexatonic scale5-7 hexatonic scale

    m=700 , n=500                                                           m=500 , n=700

90 scales                                                                     90 scales

 

So we can have 448 possible hexatonic scales based on 12-EDO

Some examples

From : http://www.xs4all.nl/~huygensf/doc/modename.html

2 2 2- 2 2 2  Whole-tone, Messiaen mode 1, Raga Gopriya, Anhemitonic Hexatonic 
2 3 2- 2 1 2  P'yongjo: Korea, Yosen: Japan, Raga Darbar, Narayani, Suposhini, Andolika, Gorakh Kalyan, Mixolydian Hexatonic 
2 2 2- 3 1 2  Prometheus, Raga Barbara 
1 3 2 -3 1 2  Prometheus Neapolitan 
3 2 1-1 3 2  Blues scale I, Raga Nileshwari 
1 2 2- 3 2 2  Ritsu: Japan, Raga Suddha Todi 
2 2 1- 2 2 3  Arezzo Major Diatonic Hexachord, Raga Kambhoji, Devarangini, Scottish Hexatonic 
2 1 2- 2 3 2  Minor Hexatonic, Raga Manirangu, Nayaki, Palasi, Pushpalithika, Suha Sughrai, Yo: Japan, Eskimo Hexatonic 1 (Alaska: King Island) 
2 2 2- 2 3 1  Eskimo Hexatonic 2 (Alaska: Point Hope) 
2 1 4- 2 2 1  Hawaiian 
2 1 2- 1 3 3  Pyramid Hexatonic 
1 2 2- 1 3 3  Double-Phrygian Hexatonic 
2 1 3- 1 4 1  Raga Amarasenapriya 
2 1 2- 4 1 2  Raga Bagesri, Sriranjani, Kapijingla 
1 3 3- 1 3 1  Raga Bauli 
3 1 1- 2 3 2  Raga Bhanumanjari, Jog 
1 2 3- 2 2 2  Raga Bhavani 
2 3 2- 1 3 1  Raga Bhinna Pancama 
2 2 2- 1 4 1  Raga Caturangini, Ratnakanthi 
1 1 4- 1 2 3  Raga Chandrajyoti 
1 3 2- 1 1 4  Raga Dhavalangam 
2 2 1- 1 1 5  Raga Dipak 
1 2 2 -2 3 2  Raga Gandharavam 
1 3 1 -2 4 1  Raga Gaula 
2 1 2 -3 3 1  Raga Ghantana, Kaushiranjani (Kaishikiranjani) 
3 2 2- 1 2 2  Raga Gopikavasantam, Desya Todi, Phrygian Hexatonic 
1 2 3 -2 3 1  Raga Gurjari Todi 
1 3 2 -3 2 1  Raga Hamsanandi, Marva, Pancama, Puriya, Sohni 
2 2 1- 4 2 1  Raga Hamsa Vinodini 
4 1 2 -2 2 1  Raga Hari Nata, Genus secundum 
1 3 2 -2 1 3  Raga Hejjajji 
2 4 1 -1 2 2  Raga Jaganmohanam 
1 4 2 -2 2 1  Raga Jivantika 
3 3 1- 3 1 1  Raga Jivantini, Gaurikriya 
4 2 1- 1 2 2  Raga Jyoti 
1 3 3 -1 1 3  Raga Kalagada 
1 4 2 -1 1 3  Raga Kalakanthi 
1 3 1- 2 2 3  Raga Kalavati, Ragamalini 
4 1 2 -1 2 2  Raga Kamalamanohari 
1 2 4 -1 2 2  Raga Kashyapi 
4 1 2- 2 1 2  Raga Khamas, Baduhari 
2 2 3 -2 2 1  Raga Kumud, Sankara (Shankara), Lydian Hexatonic 
1 3 1 -3 3 1  Raga Lalita, Sohini, Hamsanandi, Lalit Bhairav 
2 2 3 -1 3 1  Raga Latika 
3 3 1 -2 1 2  Raga Madhukauns (hexatonic) 
2 4 1 -3 1 1  Raga Malarani (Hamsanada) 
1 3 3 -2 1 2  Raga Malayamarutam 
2 1 4 -2 1 2  Raga Manavi 
1 3 2 -1 4 1  Raga Mandari, Gamakakriya, Hamsanarayani 
3 2 2- 2 1 2  Raga Manohari 
2 2 2- 3 2 1  Raga Mruganandana 
2 3 2- 2 2 1  Raga Nagagandhari 
2 2 1 -2 4 1  Raga Nalinakanti, Kedaram, Vilasini 
2 3 2- 1 2 2  Raga Navamanohari 
2 1 3- 2 1 3  Raga Neelangi 
2 4 1- 2 2 1  Raga Nishadi 
1 4 2- 1 3 1  Raga Padi 
4 1 2- 1 3 1  Raga Paraju, Ramamanohari, Simhavahini, Sindhu Ramakriya, Kamalamanohari 
1 4 2- 1 2 2  Raga Phenadyuti, Insen, Honchoshi, Niagari: Japan 
1 2 2 -1 4 2  Honchoshi plagal form: Japan 
1 3 1- 2 1 4  Raga Purna Pancama, Malahari, Geyahejjajji, Kannadabangala 
2 2 1- 4 1 2  Raga Rageshri (Rageshwari), Nattaikurinji 
2 1 3- 3 2 1  Raga Ranjani, Rangini 
3 1 2 -1 4 1  Raga Rasamanjari 
1 4 2- 2 1 2  Raga Rasavali 
1 3 1- 4 1 2  Raga Rudra Pancama 
1 2 4 -2 1 2  Raga Salagavarali 
2 3 2 -3 1 1  Raga Brindabani Sarang, Megh (Megh Malhar) 
2 2 1- 3 3 1  Raga Sarasanana 
2 4 1- 2 1 2  Raga Sarasvati 
4 1 2- 1 1 3  Raga Saravati (Sharavati) 
2 1 3 -1 3 2  Raga Simharava (Sinharavam), Gopikatilaka 
2 1 2 -2 4 1  Raga Sindhura Kafi 
2 2 1- 2 3 2  Raga Siva Kambhoji, Vivardhini 
2 1 2- 2 2 3  Raga Suddha Bangala, Gauri Velavali 
1 1 3- 3 1 3  Raga Suddha Mukhari 
1 2 2- 2 1 4  Raga Suddha Simantini 
2 1 3- 1 1 4  Raga Syamalam 
3 2 2- 1 3 1  Raga Takka 
4 1 2- 3 1 1  Raga Tilang, Savitri, Brindabani Tilang 
2 1 4 -1 2 2  Raga Trimurti 
1 3 1 -4 2 1  Raga Vasanta, Chayavati 
1 3 1- 3 2 2  Raga Vasantabhairavi 
2 1 3- 1 2 3  Raga Vijayanagari 
1 1 4 -1 4 1  Raga Vijayasri 
4 2 1- 3 1 1  Raga Vijayavasanta 
1 2 2 -3 3 1  Raga Viyogavarali 
4 2 1- 2 1 2  Raga Vutari 
2 2 2 -1 2 3  Raga Yamuna Kalyani, Kalyani Keseri, Airavati, Ancient Chinese 
1 1 4- 1 1 4  Messiaen mode 5 
4 1 1 -4 1 1  Messiaen mode 5 inverse 
1 3 1 -3 1 3  Messiaen truncated mode 3, Prometheus (Liszt) 
3 1 3 -1 3 1  Messiaen truncated mode 3 inverse, Major Augmented, Genus tertium, Raga Devamani 
1 2 3 -1 2 3  Messiaen truncated mode 2 
1 3 2 -1 3 2  Messiaen truncated mode 2, Raga Indupriya 
1 2 1 -2 4 2  Super-Locrian hexamirror 
2 1 3- 2 3 1  Takemitsu Tree Line mode 1 
2 1 3 -2 2 2  Takemitsu Tree Line mode 2

And go to : http://www.geocities.com/scaleopia/6notes.html 

And also in : http://www.audiowish.com/downloads/presets/diatonic-shifter.html we can find these hexatonics:

 

Augmented \ Messiaen Truncated Mode 3 Inverse    Hexatonic    1 2# 3 5 6b 7           C D# E G Ab B

Blues Dorian Hexatonic                                               Hexatonic    1 2b 2# 3 5 6           C Db D# E G A

Blues Minor                                                                 Hexatonic    1 3b 4 4# 5 7b         C Eb F F# G Bb

Blues Minor Maj7                                                         Hexatonic    1 3b 4 4# 5 7           C Eb F F# G B

Double-Phrygian Hexatonic                                         Hexatonic    1 2b 3b 4 5b 6         C Db Eb F Gb A

Eskimo Hexatonic 2                                                     Hexatonic    1 2 3 4# 5# 6           C D E F# G# A

Genus Secundum                                                       Hexatonic    1 3 4 5 6 7               C E F G A B

Hawaiian                                                                     Hexatonic    1 2 3b 5 6 7             C D Eb G A B

Honchoshi Plagal Form                                                Hexatonic    1 2b 3b 4 5b 7b       C Db Db F Gb Bb

Lydian #2 Hexatonic                                                   Hexatonic    1 2# 3 5 6 7             C D# E G A B

Lydian Hexatonic                                                        Hexatonic    1 2 3 5 6 7               C D E G A B

Major Blues                                                                 Hexatonic    1 2 2# 3 5 6             C D D# E G A

Messiaen Mode 5                                                        Hexatonic    1 1# 2 4# 5 6b         C C# D F# G Ab

Messiaen Mode 5 Inverse                                           Hexatonic    1 3 4 5b 6# 7           C E F Gb A# B

Messiaen Truncated Mode 2                                       Hexatonic    1 2b 3b 4# 5 6         C Db Eb F# G A

Messiaen Truncated Mode 3 \ Half-Augmented          Hexatonic    1 2b 3 4 5# 6           C Db E F G# A

Mixolydian Hexatonic                                                  Hexatonic    1 2 4 5 6 7b             C D F G A Bb

Phrygian Hexatonic                                                     Hexatonic    1 3b 4 5 6b 7b         C Eb F G Ab Bb

Prometheus Neapolitan                                               Hexatonic    1 2b 3 5b 6 7b         C Db E Gb A Bb

Prometheus                                                                 Hexatonic    1 2 3 5b 6 7b           C D E Gb A Bb

Pyramid Hexatonic                                                      Hexatonic    1 2 3b 4 5b 6           C D Eb F Gb A

Raga Bagesri                                                              Hexatonic    1 2 3b 4 6 7b           C D Eb F A Bb

Raga Gandharavam                                                    Hexatonic    1 2b 3b 4 5 7b         C Db Eb F G Bb

Raga Jyoti                                                                   Hexatonic    1 3 4# 5 6b 7b         C E F# G Ab Bb

Raga Latika                                                                 Hexatonic    1 2 3 5 6b 7             C D E G Ab B

Raga Megh                                                                  Hexatonic    1 2 4 5 6# 7             C D F G A# B

Raga Paraju                                                                 Hexatonic    1 3 4 5 6b 7             C E F G Ab B

Raga Tilang                                                                 Hexatonic    1 3 4 5 7b 7             C E F G Bb B

Raga Trimurti                                                               Hexatonic    1 2 3b 5 6b 7b         C D Eb G Ab Bb

Raga Vasantabhairavi                                                Hexatonic    1 2b 3 4 5# 7b         C Db E F G# Bb

Raga Vasanta                                                             Hexatonic    1 2b 3 4 6 7             C Db E F A B

Raga Vijayanagari                                                       Hexatonic    1 2 3b 4# 5 6           C D Eb F# G A

Raga Vijayasri                                                             Hexatonic    1 1# 2 4# 5 7           C C# D F# G B

Raga Vijayavasanta                                                    Hexatonic    1 3 4# 5 6# 7           C E F# G A# B

Raga Viyogavarali                                                       Hexatonic    1 2b 3b 4 5# 7         C Db Eb F G# B

Raga Vutari                                                                 Hexatonic    1 3 4# 5 6 7b           C E F# G A Bb

Raga Yamuna Kalyani                                                 Hexatonic    1 2 3 4# 5 6             C D E F# G A

Ritsu                                                                            Hexatonic    1 2b 3b 4 6b 7b       C Db Eb F Ab Bb

Scottish Hexatonic                                                      Hexatonic    1 2 3 4 5 6               C D E F G A

Takemitsu Tree Line Mode 1                                       Hexatonic    1 2 3b 5b 6b 7         C D Eb Gb Ab B

Takemitsu Tree Line Mode 2                                       Hexatonic    1 2 3b 5b 6b 7b       C D Eb Gb Ab Bb

Whole-Tone \ Anhemitonic Hexatonic                         Hexatonic    1 2 3 4# 5# 7b         C D E F# G# Bb

 

                   m-n hexatonic scales in 24-EDO  

 

Like 12-EDO , m and n  are 2 great divisions of octave in 24-EDO hexatonic scales . If m=n=600 cent , we have 6-6 hexatonic scale.6 is a symbol for 600 cent in 12-EDO or itsmultiples like 24-EDO .But it is also possible to have m or n with size of 300,400,500,700,800 or 900 cent. So if considering m or n equal to 3,4,5,7,8 and 9 , we will have:

3-9 hexatonic scale

4-8 hexatonic scale

5-7 hexatonic scale

7-5 hexatonic scale

8-4 hexatonic scale

9-3 hexatonic scale

Notice that :

m+n = 24 quartertone

a,b and c are multiples of one quartertone

01 <= a,b,c =< 10

 a+b+c = a'+b'+c' = 12

6-6 hexatonic scale

m=600 , n=600

For different sizes of a,b, and c , we can consider different orders for 3 divisions of tritone or augmented tetrachord in 24-EDO:

04+04+04 ~ 040404 = 12

03+02+07 ~ 030207 = 12

02+04+06 ~ 020406 = 12

01+07+04 ~ 010704 = 12

and ......

Look at the below pictures for some of the possible forms of augmenetd tetrachord in 24-EDO:

                                              

 

Different patterns for a tritone or augmented tetrachord in 24-EDO

 

and the below table shows some of the 2970 possible 6-6 hexatonic scales in 24-EDO. For the full table click here.

                                                                  

A 6-6 hexatonic scale with a pattern like 030207-050403 has these degrees:

C Dp Ebp F# Ap Bp C'   ("p" is a sign for koron)

For an example of 5-7 hexatonic scale , we can mention C D Ep F Ap Bb C with this pattern: 040303-070304

 

m-n hexatonic scales in 96-EDO

 

Like 12-EDO , m and n  are 2 great divisions of octave in 96-EDO hexatonic scales . If m=n=600 cent , we have 6-6 hexatonic scale.6 is a symbol for 600 cent in 12-EDO or itsmultiples like 96-EDO .But it is also possible to have m or n with size of 300,400,500,700,800 or 900 cent. So if considering m or n equal to 3,4,5,7,8 and 9 , we will have:

3-9 hexatonic scale

4-8 hexatonic scale

5-7 hexatonic scale

7-5 hexatonic scale

8-4 hexatonic scale

9-3 hexatonic scale

m+n = 96  1/16-tone

a,b and c are multiples of one 1/16 tone

04 <= a,b,c =< 40

 a+b+c = a'+b'+c' = 12

6-6 hexatonic scale

m=600 , n=600

For different sizes of a,b, and c , we can consider different orders for 3 divisions of tritone or augmented tetrachord in 96-EDO:

16+16+16 ~ 161616 = 48

04+04+40~ 040440 = 48

and ......

Look at the below pictures for some of the 703 possible forms of augmenetd tetrachord in 96-EDO:

                                                         

For the full table click here. Considering 703 possible forms for augmenetd tetrachord in 96-EDO , we can have 494,209 possible 6-6 hexatonic scales in 96-EDO !!

Look at the below pictures for 9 of the 494,209 possible 6-6 hexatonic scales in 96-EDO:

                                             

 

A 6-6 hexatonic scale with a pattern like 040440-060438 has these degrees:

0  50  100  600  675  725  1200

C Dp Ebp F# Ap Bp C'   ("p" is a sign for koron)

 

 

Some definition and links

- A hexatonic scale is a musical scale with six pitches or notes per octave. Famous examples include the whole tone scale, C D E F# G# A# C, the augmented scale, C D# E G Ab B, and what some jazz theory calls the "blues scale", C Eb F F# G Bb C.

- A six-tone scale of music with varying intervalic relationships examples of which vary from period to period and culture to culture.

 

                               

- Messiaen and the Hexatonic

- Hexatonic modes

- Hexatonic Modes - Michael Capon and this one in pdf. format

- Hexatonic Scales

- A table of hexatonic modes

- whole-tone resource

- Modes of limited transposition and this one and this one and this pdf !!

- Whole-tone scale