Sumerian and Babylonian Culture
Hurrian hymn, Ugarit, 13th century BCE, Wikimedia (public domain)
Heptachords and Scales
Hurrian hymn, Ugarit, 13th century BCE, Wikimedia (public domain)
Map of Mesopotamia, Wikimedia (CC-BY-SA Goran tek-en)
Sumerian and Babylonian Sexagesimal Arithmetic and Numbers. The divisors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
Watch, there are 60 seconds in a minute, and 60 minutes in an hour.
Star, God
Sumerian Symmetry of Opposites, Balance Beam
Symmetry, Art of Sumer, Entemena vase motif, 25th century BCE, Wikimedia (public domain)
Architecture, Reconstruction of Etemenanki, Babylon, Wikimedia (public domain), number seven
The Sumerian and Babylonian Tonal System is based on consecutive heptachords. In the mode kitmum higher pitches of the scale show flatwards shift, lower pitches sharpwards shift. Strictly speaking, the scales are not diatonic. They are not based on octaves.
The scale of kitmum is unique. The basic heptachord is palindromic. The center note is D. The scale is related to pītum and qablītum and is in a kind of center position.
This can be compared to the days of a week and the role of the sunday. The scale is cyclic like a watch.
The scale is based on multiples of a fourth 4:3 or 498 cents. In Sumerian music only just intonation was known with whole tone 204 cents and semitone 90 cents. An alternative just intonation was found in Nippur. The delta of the key signature is 2. The scale has 36 unique notes. Babylonian astronomers developed a 36 stars system.
For the cycle are needed 12 fourths. 12x498 cents = 5976 cents. This is lower, than the 5x1200 cents = 6000 cents for 5 octaves.
Cyclic scale kitmum
In total 60 semitones. The number 60 (sexagesimal) is assoziated with the highest Sumerian god An, representing the sky.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|
| Thu | Fri | Sat | Sun | Mon | Tue | Wed |
| A | B | C | D | E | F | G |
| G# | F# | E | D | C | B♭ | A♭ |
| D♯-kitmum | C♯-kitmum | B-kitmum | A-kitmum | G-kitmum | F-kitmum | E♭-kitmum |
The scales pītum and qablītum build also a pair. The key signatures of the scales are identical. Center notes are E and C, difference is a major third.
The scales išartum and embūbum are related. The center of the symmetrical scales are different. Center note (Balance point) of išartum is G, center of embūbum is A. The difference is a heptachord. The center note can be interpreted as the main finalis (f). So the Babylonian Tonal System probably knows already the concept of finalis in constructing a melody. 1 tritone per heptachord.
The preceding 5 scales all have two semitones and 4 whole tones in a heptachord. 6 heptachords build a scale. The 12 tetrachords address the 12 notes of the complete tone circle.
The last two scales nīd qablim and nīš tuḫrim are very different from the first 5 scales. Center notes are B and F. The basic heptachord contains only one semitone. 2 tritones per heptachord exist. The delta of the key signature is 5. 3 heptachords exist with 7x3=21-2=19 pitches. In Islam, Maryam, مريم is the 19th sūrah of the Qur'an, which announces Jesus. In Baháʼí Faith, a group of 19 is called a Váhid, which means unity. In total 33 semitones, not cyclic.
| God | Sumerian | Number | Mode | Center tone | Pattern |
|---|---|---|---|---|---|
| An | 𒀭 | 60 | kitmum | D | [1,½,1,1,½,1] |
| Enlil | 𒀭𒂗𒆤 | 50 | pītum | E | [½,1,1,½,1,1] |
| Enki | 𒀭𒂗𒆠 | 40 | qablītum | C | [1,1,½,1,1,½] |
| Nanna | 𒀭𒋀𒆠 | 30 | išartum | G | [1,½,1,1,1,½] |
| Utu | 𒀭𒌓 | 20 | embūbum | A | [½,1,1,1,½,1] |
| Inanna | 𒀭𒈹 | 15 | nīd qablim | B | [1,1,1,½,1,1] |
| Ninhursag | 𒀭𒎏𒄯𒊕 | 5 | nīš tuḫrim | F | [1,1,½,1,1,1] |
1, ½, 1, 1, ½, 1
Heptachordal circle (CBS 1766)
7 notes are shown, but there are only 6 unique notes. This can be compared to the days of a week and the role of the sunday.
1, ½, 1
1, ½, 1, 1, ½, 1, 1, ½, 1, 1, ½, 1, 1, ½, 1, 1, ½, 1, 1, ½, 1, 1, ½, 1, 1, ½, 1, 1, ½, 1, 1, ½, 1, 1, ½, 1, 1, ½, 1, 1, ½, 1
Heptachord Lyre, Wikimedia (public domain)
Loosening:
Tightening:
Tightening of an enneachord harp:
In 1998, Anne Draffkorn Kilmer summarised the findings of archaeomusicology: "While we have not identified an Akkadian or a Sumerian word for the octave, it is clear that the octave was known because of the substitution of the number 1 instead of 8 and 2 instead of 9 to represent the octaves of strings 1 and 2."
This means in mode kitmum: GABCDEFGA. G and A are the octaves in the enneachord (9-strings).
This is consistent with the heptatonic interpretation: DEF[GABCDEFGA]BCD
But this is also consistant with the heptachord interpretation: DEF#[GABCDEFGA]B♭CD
Fazit: Both interpretations are possible.
In Sumerian music, only one scale is perfectly symmetric. This mode represents God: kitmum. The six other modes are not perfect and asymmetric. These represent the incarnations and prophets and messengers. Asymmetry is essential for life. Four (incarnations and prophets) are nearly perfect, the two messengers are less perfect.
Analog to dodecaphony (twelve-tone technique), a heptaphony (seven-tone technique) can be developed.
12 x 6 = 72 tables x 6 = 432 columns (down) + 432 columns (up) = 864
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| E | G | B | D | F | A | C |
| D | D | D | D | D | D | D |
| B | E | A | D | G | C | F |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| E | G | B | D | F | A | C |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| B | E | A | D | G | C | F |
| G | F | E | D | C | B | A |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| E | G | B | D | F | A | C |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| B | E | A | D | G | C | F |
| F | C | G | D | A | E | B |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| E | G | B | D | F | A | C |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| B | E | A | D | G | C | F |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| E | G | B | D | F | A | C |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| B | E | A | D | G | C | F |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| E | G | B | D | F | A | C |
| D | D | D | D | D | D | D |
| B | E | A | D | G | C | F |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| A | B | C | D | E | F | G |
| E | G | B | D | F | A | C |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| B | E | A | D | G | C | F |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| A | B | C | D | E | F | G |
| E | G | B | D | F | A | C |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| B | E | A | D | G | C | F |
| G | F | E | D | C | B | A |
| A | B | C | D | E | F | G |
| E | G | B | D | F | A | C |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| B | E | A | D | G | C | F |
| F | C | G | D | A | E | B |
| A | B | C | D | E | F | G |
| E | G | B | D | F | A | C |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| B | E | A | D | G | C | F |
| A | B | C | D | E | F | G |
| E | G | B | D | F | A | C |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| B | E | A | D | G | C | F |
| A | B | C | D | E | F | G |
| E | G | B | D | F | A | C |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| B | E | A | D | G | C | F |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| E | G | B | D | F | A | C |
| D | D | D | D | D | D | D |
| B | E | A | D | G | C | F |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| E | G | B | D | F | A | C |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| B | E | A | D | G | C | F |
| G | F | E | D | C | B | A |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| E | G | B | D | F | A | C |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| B | E | A | D | G | C | F |
| F | C | G | D | A | E | B |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| E | G | B | D | F | A | C |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| B | E | A | D | G | C | F |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| E | G | B | D | F | A | C |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| B | E | A | D | G | C | F |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| E | G | B | D | F | A | C |
| D | D | D | D | D | D | D |
| B | E | A | D | G | C | F |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| C | A | F | D | B | G | E |
| E | G | B | D | F | A | C |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| B | E | A | D | G | C | F |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| C | A | F | D | B | G | E |
| E | G | B | D | F | A | C |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| B | E | A | D | G | C | F |
| G | F | E | D | C | B | A |
| C | A | F | D | B | G | E |
| E | G | B | D | F | A | C |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| B | E | A | D | G | C | F |
| F | C | G | D | A | E | B |
| C | A | F | D | B | G | E |
| E | G | B | D | F | A | C |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| B | E | A | D | G | C | F |
| C | A | F | D | B | G | E |
| E | G | B | D | F | A | C |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| B | E | A | D | G | C | F |
| C | A | F | D | B | G | E |
| E | G | B | D | F | A | C |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| B | E | A | D | G | C | F |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| E | G | B | D | F | A | C |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| B | E | A | D | G | C | F |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| E | G | B | D | F | A | C |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| B | E | A | D | G | C | F |
| G | F | E | D | C | B | A |
| E | G | B | D | F | A | C |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| B | E | A | D | G | C | F |
| F | C | G | D | A | E | B |
| E | G | B | D | F | A | C |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| B | E | A | D | G | C | F |
| E | G | B | D | F | A | C |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| B | E | A | D | G | C | F |
| E | G | B | D | F | A | C |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| B | E | A | D | G | C | F |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| E | G | B | D | F | A | C |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| B | E | A | D | G | C | F |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| E | G | B | D | F | A | C |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| B | E | A | D | G | C | F |
| G | F | E | D | C | B | A |
| E | G | B | D | F | A | C |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| B | E | A | D | G | C | F |
| F | C | G | D | A | E | B |
| E | G | B | D | F | A | C |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| B | E | A | D | G | C | F |
| E | G | B | D | F | A | C |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| B | E | A | D | G | C | F |
| E | G | B | D | F | A | C |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| B | E | A | D | G | C | F |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| A | B | C | D | E | F | G |
| B | E | A | D | G | C | F |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| E | G | B | D | F | A | C |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| A | B | C | D | E | F | G |
| B | E | A | D | G | C | F |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| E | G | B | D | F | A | C |
| G | F | E | D | C | B | A |
| A | B | C | D | E | F | G |
| B | E | A | D | G | C | F |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| E | G | B | D | F | A | C |
| F | C | G | D | A | E | B |
| A | B | C | D | E | F | G |
| B | E | A | D | G | C | F |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| E | G | B | D | F | A | C |
| A | B | C | D | E | F | G |
| B | E | A | D | G | C | F |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| E | G | B | D | F | A | C |
| A | B | C | D | E | F | G |
| B | E | A | D | G | C | F |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| E | G | B | D | F | A | C |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| B | E | A | D | G | C | F |
| D | D | D | D | D | D | D |
| E | G | B | D | F | A | C |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| B | E | A | D | G | C | F |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| E | G | B | D | F | A | C |
| G | F | E | D | C | B | A |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| B | E | A | D | G | C | F |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| E | G | B | D | F | A | C |
| F | C | G | D | A | E | B |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| B | E | A | D | G | C | F |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| E | G | B | D | F | A | C |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| B | E | A | D | G | C | F |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| E | G | B | D | F | A | C |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| B | E | A | D | G | C | F |
| D | D | D | D | D | D | D |
| E | G | B | D | F | A | C |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| B | E | A | D | G | C | F |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| E | G | B | D | F | A | C |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| B | E | A | D | G | C | F |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| E | G | B | D | F | A | C |
| G | F | E | D | C | B | A |
| B | E | A | D | G | C | F |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| E | G | B | D | F | A | C |
| F | C | G | D | A | E | B |
| B | E | A | D | G | C | F |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| E | G | B | D | F | A | C |
| B | E | A | D | G | C | F |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| E | G | B | D | F | A | C |
| B | E | A | D | G | C | F |
| A | B | C | D | E | F | G |
| C | A | F | D | B | G | E |
| D | D | D | D | D | D | D |
| E | G | B | D | F | A | C |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| B | E | A | D | G | C | F |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| E | G | B | D | F | A | C |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| B | E | A | D | G | C | F |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| E | G | B | D | F | A | C |
| G | F | E | D | C | B | A |
| B | E | A | D | G | C | F |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| E | G | B | D | F | A | C |
| F | C | G | D | A | E | B |
| B | E | A | D | G | C | F |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| E | G | B | D | F | A | C |
| B | E | A | D | G | C | F |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| E | G | B | D | F | A | C |
| B | E | A | D | G | C | F |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| E | G | B | D | F | A | C |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| B | E | A | D | G | C | F |
| D | D | D | D | D | D | D |
| E | G | B | D | F | A | C |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| B | E | A | D | G | C | F |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| E | G | B | D | F | A | C |
| G | F | E | D | C | B | A |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| B | E | A | D | G | C | F |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| E | G | B | D | F | A | C |
| F | C | G | D | A | E | B |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| B | E | A | D | G | C | F |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| E | G | B | D | F | A | C |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| B | E | A | D | G | C | F |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| E | G | B | D | F | A | C |
| C | A | F | D | B | G | E |
| A | B | C | D | E | F | G |
| B | E | A | D | G | C | F |
| D | D | D | D | D | D | D |
| E | G | B | D | F | A | C |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| C | A | F | D | B | G | E |
| B | E | A | D | G | C | F |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| E | G | B | D | F | A | C |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| C | A | F | D | B | G | E |
| B | E | A | D | G | C | F |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| E | G | B | D | F | A | C |
| G | F | E | D | C | B | A |
| C | A | F | D | B | G | E |
| B | E | A | D | G | C | F |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| E | G | B | D | F | A | C |
| F | C | G | D | A | E | B |
| C | A | F | D | B | G | E |
| B | E | A | D | G | C | F |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
| E | G | B | D | F | A | C |
| C | A | F | D | B | G | E |
| B | E | A | D | G | C | F |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| F | C | G | D | A | E | B |
| G | F | E | D | C | B | A |
| E | G | B | D | F | A | C |
| C | A | F | D | B | G | E |
| B | E | A | D | G | C | F |
| A | B | C | D | E | F | G |
| D | D | D | D | D | D | D |
| E | G | B | D | F | A | C |
| G | F | E | D | C | B | A |
| F | C | G | D | A | E | B |
In the following musical notation, the first line (bar 1-3) is the first column of I/1 (blue) (D) A C E D B F G (D):
Danish composer Per Nørgård (1932-2025) used the infinity series method.
Self-similar integer sequence: 0, 1, -1, 2, 1, 0, -2, 3, -1, 2, 0, 1, 2, -1, -3, 4, 1, 0, -2, 3, 0, 1, -1, 2, -2, 3, 1, 0, 3, -2, -4, 5, -1, 2, 0, 1, 2, -1, -3, 4, 0, 1, -1, 2, 1, 0, -2, 3, 2, -1, -3, 4, -1, 2, 0, 1, -3, 4, 2, -1, 4, -3, -5, 6, 1, 0, -2, 3, 0, 1, -1, 2, -2, 3, 1, 0, 3, -2, -4, 5, 0, 1, -1, 2, 1, 0
Tree of Nørgård's melodic infinity series
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A# | B | C# | D# | E | F# | G# | A | B | C# | D | E | F# | G | A | B | C | D | E | F | G | A | Bb | C | D | Eb | F | G | Ab | Bb | C | Db | Eb | F | Gb | Ab | Bb | Cb | Db |
| F# | G# | A# | B | C# | D# | E | F# | G# | A | B | C# | D | E | F# | G | A | B | C | D | E | F | G | A | Bb | C | D | Eb | F | G | Ab | Bb | C | Db | Eb | F | Gb | Ab | Bb |
| D# | E# | F# | G# | A# | B | C# | D# | E | F# | G# | A | B | C# | D | E | F# | G | A | B | C | D | E | F | G | A | Bb | C | D | Eb | F | G | Ab | Bb | C | Db | Eb | F | Gb |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| C# | D# | E | F# | G# | A | B | C# | D | E | F# | G | A | B | C | D | E | F | G | A | Bb | C | D | Eb | F | G | Ab | Bb | C | Db | Eb | F | Gb | Ab | Bb | Cb | Db |
| G# | A# | B | C# | D# | E | F# | G# | A | B | C# | D | E | F# | G | A | B | C | D | E | F | G | A | Bb | C | D | Eb | F | G | Ab | Bb | C | Db | Eb | F | Gb | Ab |
| D# | E# | F# | G# | A# | B | C# | D# | E | F# | G# | A | B | C# | D | E | F# | G | A | B | C | D | E | F | G | A | Bb | C | D | Eb | F | G | Ab | Bb | C | Db | Eb |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| E | F# | G# | A | B | C# | D | E | F# | G | A | B | C | D | E | F | G | A | Bb | C | D | Eb | F | G | Ab | Bb | C | Db | Eb | F | Gb | Ab | Bb | Cb | Db | Es | Fes |
| A# | B | C# | D# | E | F# | G# | A | B | C# | D | E | F# | G | A | B | C | D | E | F | G | A | Bb | C | D | Eb | F | G | Ab | Bb | C | Db | Eb | F | Gb | Ab | Bb |
| D# | E# | F# | G# | A# | B | C# | D# | E | F# | G# | A | B | C# | D | E | F# | G | A | B | C | D | E | F | G | A | Bb | C | D | Eb | F | G | Ab | Bb | C | Db | Eb |
3 modes are pure and don't have tritones. The other 4 modes have 3 consecutive full tones and contain therefore impure tritones.
Assuming equal temperament:
Lutherstophe is a seven-line strophe with the rhyme scheme ABABCCB or ABABCCX. Following seven-line schemes can be transferred from the heptachordal music to poetry: