A model for hexatonic scales
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 scale , 5-7 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 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.
- Messiaen and the Hexatonic
- Hexatonic Modes - Michael Capon and this one in pdf. format
- Modes of limited transposition and this one and this one and this pdf !!
