7 turning into 12
Dia7&12Tonality â€“ a continuum
January '99 Knud Brant Nielsen
2 Table of Contents: page Preface
Introduction A Thousand Years Arnold Schonberg Came Close
4 4 5
The Find 6 From Scale to Serpent Thinking 12-Tonally, Writing 7-Tonally Transcribing from 7 to 12
7 8 8
7& 12 Codes Serpents Scales Transpositions
10 10 11 12
Practical Hints Fourth/Fifth Serpent 7-Tonal Chromatic Scale = 12-tonal Diatonic Scale Guido's Backhand Transposition/ Modulation Chromatic Semitone = Whole Tone Minus Semitone Enharmonic Interval as Disguised 12-Alteration Minor and Major Intervals of the 12-Tonality Calling Troll by Name Names and Signs of 12-Tonal Intervals The 12-tonal Intervals
13 13 14 15 16 17 18 19 20 20 21
Appendix: Degree Notation â€“ a Relevant Study Tool? Interval Tensions Degree Notation Scales transcribed into Degree Notation Some Other Examples Bartok and Bach in Degree Notation
23 23 24 26 27
3 PREFACE The topic is the dia-12-tonality (diatonic 12-tonality) as an organic consequence of the common notation and keyboard, i.e. is as a natural development of the wellknown diatonic 7-tonality: dia-7 increasing to dia-12 and in that way creating a coherent dia7&12continuum. The substance of music is the tones, its writing is the notes. 7-notation on 5 lines is the ideal writing for 7-tonal music. Over the centuries the 7-tonal notation & keyboard has developed into a perfect language for 7-tonal thinking. The common notation has rightly become the written language and in that way the basis for an enormous development. But the 7-tonal written language keyboard has been so established that it has become synonymous with tonality. 'Tonality is 7-tonality'. A fatal fallacy! The dia7&12tonality as an organic continuum is obvious: diatonic 7-tonality can he experienced as a stage of development in the history of music with diatonic 12tonality as the natural next stage. Subsequently the term 12-tonality stands for dia-12-tonality, not the octave divided into 12 equal semitones but diatonic 12-tonality constituted by minor and major steps with tonal tensions and possibilities of modulation and transposition. ... The basis of this introduction is the life's work "Ăœber TonalitĂ¤t" by the Danish chronomatic and tonal theorist Frede Schandorf. On its own this short introduction does not contain any essential news. Its content is my responsibility, marked and limited as it is by my understanding of Frede Schandorfs thoroughgoing fundamental research. I refer to Frede Schandorfs work as a most serious and weighty source of a renewing experience and understanding concerning tonality. The diatonic 12-notation is fully described and exemplified in Frede Schandorfs "Das 12-tonale Notenliniensystem". Knud Brant Nielsen.
A Thousand Years
A thousand years ago Guido of Arezzo discovered that principle for notation which – a thousand years later, in the 1970s – was deciphered by Frede Schandorf. With that, diatonic 7-tonality and diatonic 12-tonality were revealed as one organically coherent and continuous language – the dia7&12continuum/ the dia7&12tonality – as a capacious room for the unbroken development of the past one thousand years, and potentially, too, as a solid foundation for a musical creativity of a future global culture. As long ago as in the sixteen-hundreds many accidentals indicate that a larger tonality – the 12-tonality – is at work:
Michelangelo Rossi's organ toccata from 1657 (Studio Per Edizioni Scelte, Florence 1982 p.23) is written on 6 and 8 lines, neatly using 3 keys, showing one stage in the long development of 7-tonal notation.
The same section (from The New Oxford History of Music, vol VI, p. 517) in modern 7-notation on two 5-lines-systems with only two clefs. The toccata utilizes the scale 'D chromatic with E♭ F♯ G♯ B♭ and C♯, 12 different tones in all.
And here transcribed to 12-tonal notation on two 7-lines-systems using S clef and K clef ("F♯ clef and B♭ clef''). The notation uncovers that the chromatic scale is masked 12-tonality: the accidentals vanish like snow before the sun, 'D chromatic' being the 12-tonal basic scale, "12-tonal C major". Here it begins...
5 Arnold Schonberg Came Close In 1950 – at the age of 75 – Arnold Schonberg begins his radio talk "J. S. Bach" in this way: "I used to say, 'Bach is the first composer of twelve tone music'. This was a joke, of course. I did not even know whether somebody before him might not have deserved this title. But the truth on which this statement is based is that the Fugue No 24 of the first volume of the Well-Tempered Clavier, in B minor, begins with a Dux in which all twelve tones appear." Not until about 25 later Frede Schandorf found the 12-notation:
The 12-notation uncovers what had been hidden for a quarter millennium: in the third bar B♯ releases a 12-tonal modulation from 'B chromatic' to 'F♯ chromatic'. After having utilized ten different tones of 'B chromatic' Bach introduces from the dominant key 'F♯ chromatic' the accidental B♯, i.e. the third of the double dominant of chromatic'. The remaining tones A and G♯ of 'B chromatic' also belong to 'F♯ chromatic' and now become part of the cadence which establishes the 12-tonal dominant key 'F♯ chromatic'. Simultaneously G♯ brings about the 7-tonal modulation from B minor to F♯ minor. So this unique Dux has 13 different tones: by means of the thirteenth tone Bach modulates his 12-complex to the dominant, the 7-key and the 12-key here having B as the common tonic and F♯ as the common dominant. Like Bach other composers already have shown how diatonic 12-tonality can be written in 7-notation. But they had to work by intuition. By Frede Schandorfs find of the 12-notation we today have got the adequate and perfectly lucid 12-tonal written language as help for making aware of 12-tonal thinking. This awareness of diatonal 12-tonality may come about in different ways but the most natural and simple way is to transcribe from 7-notation to 12-notation. If you can 12transcribe - which is easy and plain - you can consciously write and read 12-tonality in 7-notation. On top you may get a new and deeper understanding of the essence of 7-tonality. That is the purpose of this introduction.
6 THE FIND
As long as 7-notation has existed, 12-notation has been a latent possibility – as a new answer to an old riddle: 'What is bigger than its mother when it is born? – 'An icicle!' – 'Yes, and 12-tonality!' It cost Frede Schandorf a lot of hard work to uncover the 12-notation, blurred as it is by the 7-notation – and because it is difficult to see the wood for the trees. Like other great answers the solution to this riddle is so simple that you can wonder why it has not been found long ago.
The beautiful symmetry of the 7-keyboard simply invites to write the notes 'A B C D E F G' in bass notation, the centre tone D being placed in the middle of the centre line.
Five chromatic semitones are added. G♯ being placed at the bottom as a necessary aberration, because of 12 being an even number. So the solution is a matter of course:
By simple but consequent analogy: – as 7 tones require 5 lines, so 12 tones require 7 lines; – as the centre tone of the 7-scale is written on the centre line of the 5 lines, so the centre tone of the 12-scale is written on the centre line of the 7 lines; – as the other 6 tones of the 7-scale have their own places stepwise up and down from the centre tone, so the 11 other tones of the 12-scale are placed stepwise up and down from the centre tone, every tone having a place of its own; – as the 7-scale is named from the bottom by the first letters of the alphabet 'A B C D E F G', so the 12-scale is named from the bottom by the following letters of the alphabet 'I J K L M N O P Q R S T'. Conclusion: the chromatic scale of 7-tonality is equal to the diatonic 12-scale!
7 From Scale to Serpent Separately 7-scale and 12-scale are re-formed to fourth/ fifth sequence, i.e. to a 7module and a 12-module which share 7 tones and which both of them are put in the order of fourths and fifth:
The fourth/fifth sequence is extended, the modules being repeated with accidentals:
The 7-module is repeated to the right as ♯-module (every tone with a ♯) and again as x-module (every tone with an x) ... Similarly the 7-module is repeated to the left as ♭module and as ♭♭-module
Quite similarly the 12-module is repeated to the right as ♯-module (every tone with a 4) and as x-module (every tone with an x) ... And to the left as ♭-module and ♭♭module …:
Here the 7-serpent & 12-serpent (= fourth/ fifth sequence) is written in treble clef, G clef and K clef respectively, all notes within one octave. 7-serpent & 12-serpent share a sequence of exponents (= sequence of integers), so that every tone has got two notings which are sharing one exponent: – different is notation and name, – common is sound (frequency) and number (exponent). The exponent is the civil and social registration number that indicates the individual position of every tone in the serpent order.
8 Thinking 12-Tonally, Writing 7-Tonally But who will write on 7 lines? Or play from 7 lines? Not many â€“ yet. But learning to think 12-tonally while writing and reading 7-tonally is as well realistic as relevant for musical practice â€“ and simple and manageable, too, fully worth-while. And possible without using many words, for the most natural and simple way of becoming acquainted with the diatonic 12-tonality is by transcribing from 7-notation to 12-notation â€“ and vice versa. ...
Transcribing 7 to 12 The code for transcribing 7-notation to 12-notation is simply the 7-serpent & the 12serpent, vertically coordinated. Here is a section of the code: thirteen notes with their exponents from -2 to +10:
Note by note is transcribed from 7 to 12: below every 7-note is written its 12-note translation. So every tone has one frequency and one exponent and two notations and two names (letters).
9 By this code Bach's B minor Dux is transcribed note by note:
Every note is simple transcription, except B of the first bar. In 7-notation you octavate the B (+3) of the serpent by writing it 7 steps lower in the 7-staff, using one leger line. Similarly you octavate the 12-tonal L (+3) of the serpent by writing it 12 steps lower in the 12-staff, using two leger lines. Beyond this, G clef becomes K clef, while the sign of 4/4, quaver rest, values, and bar lines are unchanged. The key signatures are analogous: B minor is two fifths apart from the basic 7scale of A minor, hence two (7-tonal) sharps; and 'B chromatic' is three fifths apart from the basic 12-scale of 'D chromatic', hence three (12-tonal) sharps. Now the difference catches one's eye: the tonally puzzling eight accidentals of the 7notation contrary to the just one accidental of the 12-notation. Here the diatonic 12notation shows its strong point, revealing a 12-tonal theme which is modulating to the dominant key, the single sharp showing the point of modulation (modulator) – quite similar to the following 7-tonal theme:
This 7-tonal theme (Bach: W.-T. Cl. I fugue no. 7) uses eight different tones: the 7 tones of E♭ major plus the modulator A, the third of F major, i.e. the dominant of the dominant key B♭ major. Similarly, the 12-tonal theme uses thirteen different tones: the 12 tones of 'Bchromatic' (= 'B-12') plus the 12-tonal modulator M♯ ('B♯'), the third of G♯ major, i.e. the double dominant of the dominant key 'F♯ chromatic'.
Frede Schandorfs 12-transcription of the opening of Bartok's 'Music for Strings, Percussion and Celesta' is a more recent example of a short tonal swing, within one bar. In the third bar the 12-notation like a seismograph registers the fine 12-tonal swing, which in the 7-notation is obscured by the many necessary accidentals.
10 7- & 12- CODES For transcription from 7 to 12 all you need is serpents, scales and transcriptions. Serpents
Serpents in 7- & 12-notation, on top with treble clefs (G-clef & K-clef), middle with alto clefs (C-clef & O-clef), and at the bottom with bass clefs (F-clef & S-clef). – That is the code for transcription from 7 to 12, and vice versa. Octavation in 12-notation is simple, following the rules: – 'once line, always line' (the notes IKMORS, including their alterations). – 'once space, always space' (the notes JLNPRT, including their alterations). That is to say that every note on line is octavated by being moved six lines up or down - and that every note in space is octavated by being moved six spaces up or down to the higher or lower sixth space. Alternatively you can note an upturned arrowhead over the note or an down-turned arrowhead below the note that are to be octavated. If more notes arc to be octavated an arrowhead is noted before the first note and a crossed out arrowhead after the last note, corresponding to the 7-tonal all'ottava with dotted line.
Scales in 7- & 12-notation, on top with treble clefs (G clef & K clef), middle with alto clefs (C clef & O clef), and at the bottom with bass clefs (F clef & S clef). (i) over the clef indicates that the basic scale of 'D chromatic ('D-12' / 'O-12') has +6 = Gâ™Ż = I. The alternative, (u) over the clef, indicates that the basic scale 'D chromatic' has -6 = Aâ™ = U. Generally Frede Schandorf has used the (u) version.
The left column is ♭-transpositions, the right column ♯-transpositions. The ordinal number (exponents) of the first and the last note indicate the number of accidentals in the key signature; for instance, the right bottom transposition has +6, stating that the key signature of 'G♯ chromatic' (G♯-12' / 'I-12') has 6 sharps.
13 PRACTICAL HINTS The codes with their letters and numbers contain all information about the diatonic 12-tonal material. While you are transcribing 7 to 12 the 12-tonality reveals its secrets, kindly informing you of its qualities. Warning! If you prefer to learn by experience you can skip the following pages, transcribe - and learn by doing! But if you would like to take a short cut the following practical hints may be helpful. Good luck! The Fourth/Fifth Serpent Notes have names and numbers: the scale is their alphabetical order, the serpent is their numerical order.
7 adjoining serpent notes make a 7-scale, 12 adjoining serpent notes make a 12-scale. The number – the exponent – is the civil registration number which states the serpent position of every single tone and with that its social relations to its surroundings, i.e. its tension value as part of an interval. Minor and diminished intervals have negative tensions, are contractive. Major and augmented intervals have positive tensions, are expansive. By calculating the tension of an interval you always have to read upwards, first the bottom tone, then the top tone. Two examples, taken from the serpent: The tone F♯ = S has the exponent +4, while the tone A = J has the exponent +1: so the interval tension is -3 degrees, from +4 to +1 being the distance of 3 serpent positions in negative direction. The interval tension may be written as '3-'. The tone E♭ = P has the exponent -5, while the tone E =Q has the exponent +2: so the interval tension is +7 degrees, from -5 to +2 being the distance of 7 serpent positions in positive direction. The interval tension may be written as '7+'.
14 7-Tonal Chromatic Scale = 12-Tonal Diatonic Scale The 7-tonal scale has 2 minor and 5 major steps (2 semitones and 5 whole tones), the 12-tonal scale has 7 minor and 5 major steps (7 semitones and 5 whole tones).
The size of the 7-tonal minor step (semitone) is identical with that of the 12-tonal one, e.g. D-E♭ = O-P or E-F = Q-R. The 7-tonal chromatic semitone is identical with the 12-tonal major step (whole-tone), e.g. E♭-E = P-Q or F-F♯ = R-S. 7-notation and 12-notation share the flaw that they do not disclose if a step is a minor or a major one. But if you read the 7-notation through 12-tonal glasses, you will see the difference: The 12-tonal whole-tone appears as a 7-tonal chromatic semitone, i.e. the two notes share a line or a space but have different accidentals – whereas the notes of the semitone in both notations are neighbours: one note is in a space, the other note is on the line over or below the first one. Through 12-tonal glasses the 7-notation may be seen as a study notation which - in certain respects - discloses what the 12-notation does not directly communicate.
15 Guido's Backhand Guido lends a helping hand when you will write a 12-scale 7-tonally. Or – as you like – when you want to construct a 7-chromatic scale. An example is the 12-scale which constitutes the material of the Fugue No 8 of the first volume of the Well-Tempered Clavier, in D♯ minor:
Guido's back of his hand shows the disposition of the 12-tonal minor and major steps, which are equal to the 7-tonal diatonic and chromatic semitones:
The hand is used once + an echo + once more. Guido found the 7-tonal notation system and with it the germ of the written language which made 7-tonal thinking possible - and which encouraged it. Schandorf decoded the chromatic scale as being identical with diatonic 12-tonality (dia-12-tonality) and as organically growing out of the 7-tonality. Who would know that Guido had the 12-tonality up his sleeve :-)
16 Transposition / Modulation The 7 central tones -3 -2 -1 0 +1 +2 +3 of the fourths/ fifths serpent form the basic 7tonal module. Rearranged they constitute the basic 7-tonal scale, D Dorian:
The 7-module may be moved for instance 3 positions to the left, i.e. transposed to -6 -5 -4 -3 -2 -1 0. Rearranged it constitutes the transposed Dorian scale with 3 flats, F Dorian:
Similarly, the 12 central tones -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 of the fourths/fifths serpent form the basic 12-tonal module. Rearranged it constitutes the basic 12-tonal scale, 'D-chromatic' ('D-12' / 'O-12'):
The 12-module may – again similarly – be moved for instance 7 positions to the left, i.e. transposed (modulated) to -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1. Rearranged it constitutes the transposed chromatic scale with 7 flats, 'D♭-chromatic' ('D♭-12' / ♭N-12'):
… (The 12-tonal scale names are used as practical technical terms in this short introduction which does not treat the twelve 12-tonal modes. The 12-tonal scale's names implicitly indicate the key signatures: the tone D / O has the exponent 0, so 'D12' / 'O-12' has no 12-tonal signature; and the tone D♭ / N♭ has the exponent -7, so 'D♭-12' / `N♭-12' has seven 12-tonal flats.)
17 Chromatic Semitone = Whole Tone Minus Semitone There are 7 fourths/ fifths between the serpent notes of the 7-tonal chromatic semitone, i.e. F-F♯:
– and there are 12 fourths/ fifths between the serpent notes of the 12-tonal chromatic semitone, e.g. P-P♯:
So the tone is written similarly in the 7-notation and the 12-notation: the note is repeated with ♯ before it.
But in 7-notation the 12-tonal chromatic semitone, the 12-chroma, is pretty blurred ... … The chromatic semitone, the chroma, is the remainder, when the diatonic semitone is subtracted from the diatonic whole tone, complying the formula: 'major minus minor equals chroma'. 7-tonality and 12-tonality share this formula. … The letter notations confirm the analogous process of formation: 7-tonally: minus equals
whole-tone: f semitone: chroma: f
12-tonally: minus equals
whole-tone: P semitone: chroma: P
(= e♭ (= (= e♭
e) e) )
18 Enharmonic Interval as Disguised 12-Alteration Traditional enharmonic change is intervallically identical with 12-tonal alteration. The interval between the two tones of the change is a 12-tonal croma(tical semitone). When you change one of the tones often is implied in the other one, i.e. not written. When you read 7-notation through 12-tonal glasses, i.e. when you think 12- tonally, the 7-tonal harmonic change is interpreted as a 12-tonal alteration, i.e. as a 12-tonal swing or modulation. In his composition 'L'Enharmonique' (bar 15-18) Jean Philippe Rameau illustrates the relation between enharmonic tones:
7-tonally the key is G minor, the key signature having two 7-tonal flats: and 12tonally the key is 'O chromatic' ('0-12' / 'T-12'), the key signature having one 12-tonal flat. The enharmonic notes which Rameau is playing off one against the other are seen in the first bar as G♯ / I and in the next bar as A♭ / I♭. The difference between the two tones appears from their melodic contexts, the first being 'A G♯ (E) A / J I (Q) J', and the other being 'G A♭ G / T I♭ T'. The two motives are each other's inversion, exposing the two enharmonic notes as complete contrasts, as Janus-faced: 'A G♯ A' showing G♯ as upward or ascending energy, and 'G A♭ G' showing A♭ as downward or descending energy. The notes illustrate the universal rule that the higher exponent (here +6 = G♯ / I) is upward or ascending energy, and the lower exponent (here -6 = A♭ / I♭) is downward or descending energy. So enharmonic notes are diametrically opposed energy. By enharmonic change – equal to 12-tonal alteration – the direction is changed: when G♯ becomes A♭, the potential upward energy is converted into potential downward energy.
19 Minor and Major Intervals of the 12-Tonality To further consistent analogy, four intervals (here) change their names: – perfect fourth becomes 'minor fourth – and – augmented fourth becomes 'major' fourth; – diminished fifth becomes 'minor' fifth – and – perfect fifth becomes 'major' fifth. Analogously to the 6 minor and 6 major intervals in 7-tonality there are 11 minor and 11 major intervals in 12-tonality. The transition from 7- to 12-tonality means change of partners: 'major' quarter and 'minor' fifth find one another and constitute a couple, the other five minor intervals choose augmented intervals and change them into major ones, the other five major intervals choose diminished intervals and change them into minor ones; altogether a total of 11 couples, sharing the 12-chroma(tic semitone) as the little difference of partners. In 7-tonality the difference is the 7-tonal chroma(tic semitone), in 12-tonality the difference is the 12-tonal chroma(tic semitone). The six 7-tonal pairs, symmetrically grouped:
Analogously, the eleven 12-tonal pairs, symmetrically grouped:
And the same eleven 12-tonal pairs, transposed into 7-notation:
Visually confusing! But that can be managed – by change of names!
20 Calling Troll by Name It is easy to translate from 7-name to 12-name – just a simple rule: 'minor and diminished becomes minor, major and augmented becomes major' For instance: 'E-G♯? "E=zero, F=one, F♯=two, G=three, G♯=four – and 'E-G♯ ' is a major third – and major and augmented become major; therefore: 'E-G♯' is a major four." Or: "'E-G♯' spans four 12-steps and therefore is a four – and major in 7 is major in 12, – therefore major: therefore 'E-G♯' is a major four." The partner is easily found: Enharmonically: "G♯ is altered to A♭: 'E-A♭' is a minor four". Check: "E=zero, F=one, F♯=two, G=three, A♭=four – and 'E-A♭' is a diminished fourth – and minor and diminished become minor; therefore: 'E-A♭' is a minor 'four". Or: "'E-A♭' spans four steps and therefore is a four – and diminished in 7 is minor in 12, therefore minor; therefore: 'E-A♭' is a minor four." Names and Notation of 12-tonal intervals The names and notation of the 12-tonal intervals (page 21) are: perfect zero minor one & major one minor two & major two minor three & major three minor four & major four minor five & major five minor six & major six minor seven & major seven minor eight & major eight minor nine & major nine minor ten & major ten minor eleven & major eleven perfect twelve
= 0 = 0 = ’1 & 1’ = ’2 & 2’ = ’3 & 3’ = ’4 & 4’ = ’5 & 5’ = ’6 & 6’ = ’7 & 7’ = ’8 & 8’ = ’9 & 9’ = ’10 & 10’ = ’11 & 11’ = 12 = 0
21 12-tonal Intervals
22 PERSPECTIVES In his composition "L'Enharmonique" the theorist and composer Jean Philippe Rameau alternates between gracieusement og hardiment, sans altĂŠrer la mesure,, between unequal and equal durations: the composer applies the contrast unequalequal as a structuring principle for his composition. Rameau applies the contrast of unequal-equal durations. Today another possibility may be unequal-equal interval sizes, i.e. opposing the unequally divided diatonic 12tonality to the equally divided dodecaphony 12-neutrality'), the thorough tonality to the consistent a-tonality. ... For the composer the awareness of the dia-12-tonality may become a point of departure for an investigation into the possibilities of a compositional interplay between unequally and equally divided intervals. For the theorist the find of the dia-12-tonal notation and thinking with its almost double number of intervals makes possible a completing counterpoint to Allen Forte's theoretical innovations. An elaborated dia-12-tonal theory may create a natural and equal - obbligato (?) - complement to Allen Forte's theory. And a co-thinking of dial2tonal theory and Allen Forte's theory may prove fruitful. ... By making aware of the complementary relation between unequally and equally divided intervals, and with that basically the relation between step and degree, a study tool is provided for solving a potential task of the coming millennium: the combining of a dia-12-tonal thinking with the traditional dodecaphonic techniques to an organic whole, displaying a vast number of latent possibilities for increased and intensified expression of a richly faceted dia-dodecaphony. Last not least, or first of all: the mere investigation into the possibilities of the dia-12tonality may open to a new, not yet heard tonal music, being not less varied and multifarious than the 7-tonal music of the passed millennium.
23 APPENDIX: DEGREE NOTATION - a Relevant Study Tool? Interval Tensions When the basic 7-serpent is supplemented by an octavated and repeated F you see – between the dotted lines – the six 7-tonal complementary intervals, on top minor, at the bottom major intervals:
The upper series of integers '7 6 5 4 3 2 1 0' indicate how many intervals you can form from the basic tones, e.g. the number 5 indicates 5 major seconds and 5 minor sevenths. The notes outside the dotted lines are identical, only g making the difference: to the left the number 7 indicates 7 available perfect primes and octaves, while to the right the number 0 indicates that the augmented prime (the chromatic semitone, the 7chroma) and the diminished octave do not occur within the framework of the basic 7-tonality. The lower series of integers '0 1 2 3 4 5 6 7' indicate – hypothetically – the tension degree of the intervals; e.g. the major seconds and the minor sevenths have the tension-degree of two, there being 2 fourth/ fifths between the notes of F and G. The perfect prime and the perfect octave with the number 0 are tension free, while the 7-chroma and the diminished octave with the number 7 have a tension of 7 degrees. When – correspondingly – the basic 12-serpent is complemented by an octavated and repeated E♭ / P you see – between the dotted lines – the eleven 12-tonal complementary intervals, on top minor, at the bottom major intervals:
Here, too, the upper series of numbers '12 11 10 9 8 7 6 5 4 3 2 1' gives the quantity and the lower series of numbers '1 2 3 4 5 6 7 8 9 10 11 12' the degree tension. While the interval-quantities are increased, the degree tensions remain unchanged, e.g. the quantity is increased into 10 major seconds and 10 minor sevenths / 10 major twos and 10 minor tens - the tension degree remaining 2.
24 As above the notes outside the dotted lines are identical, just ♯ making the difference: to the left the number 12 indicates 12 available perfect primes and octaves/ zeros and twelves, while the number 0 to the right indicates that the diminished second / augmented zero (= the 12-tonal chromatic semitone, the 12chroma) and the diminished ninth / diminished twelve do not occur within the framework of the basic 12-tonality. To the right the degree-number 12 shows that the 12-chroma surpasses the 7chroma with regards to tension. These interval surveys both of them suggest that it is the serpent motion within the framework of the octave that gradually out of the 0-tension of the perfect prime/ perfect zero generates the 7-tension of the 7-chroma and the 12-tension of the 12- chroma. This forms the basis of the subsequent hypothetical degree notation. Degree Notation The number (= exponent) of every single tone shows the tension degree of that tone, its degree number in relation to the 7-tonal D / the 12-tonal O. To make aware, the degrees may be written as a separate degree notation with an arbitrary number of lines, here 7 lines:
7- and 12-notation show the degree notation. While the 7-notation and the 12notation show steps, i.e. relative pitch, the degree notation shows the shared degree as an expression of the tonal tension. On the centre line of the degree notation, D / O are placed as 0 degrees. From here all other tones are arranged, i.e. G / T as -1 degree, having its place in the space below the centre line, that is just below the tone D / O as 0 degree; and i.e. A♯ / K♯ as +8 degrees, having its place on the first upper leger line. The degree-notation does not show the position of the octave: all 7-tonal /12-tonal A♯'s / K♯'s – from the sub-contra octave to the five-line octave – are written on the first upper leger line of the degree notation.
25 Scales Transcribed into Degree Notation The rule for the relation between step notation and degree notation reads as follows: â€“ a similar motion shows a major or an augmented interval, i.e. an expansive interval; â€“ and â€“ a contrary motion shows a minor or a diminished interval, i.e. a contractive interval. So the basic -scale (D Dorian):
The degree notation illustrates the relation between major and minor steps: the 5 ascending 7-tonal major steps are seen as ascending degree notation, showing the tension of +2 degrees, while the 2 ascending minor steps are seen as descending degree notation, showing the tension of -5 degrees. And the basic 12-tonal scale, D chromatic ('D-12 / O-12'):
Similarly the degree notation illustrates the difference between 12-tonal major and minor steps: the 5 ascending 12-tonal major steps are seen as ascending degree notation, showing the tension of +7 degrees, while the 7 ascending minor steps are seen as descending degree notation, showing a -5 degree tension.
26 Some Other Examples Some annotated examples of degree notation:
a. A repeated degree note shows a perfect prime / a perfect zero or a perfect a perfect twelve - or more perfect octaves / perfect twelves. b. Similar motion between the step notations and the degree notation indicates that the interval is an expansive interval, i. e. a major og an augmented interval; example b shows the interval as being, '2 degrees and expansive; sign: 2+. c. Contrary motion between the step notations and the degree notation indicates that the interval is a contractive interval, i.e. a minor or a diminished interval: example c shows the interval as being '5 degrees and contractive': sign: 5-. d. Complementary intervals arc degree swing within an octave / a twelve; example d shows the intervals as being '9 degrees and expansive and '9 degrees and contractive'; signs: 9+ and 9-. e. The degree tension number of the croma(tic semitone) is equal to the numerical size of the tonality; example e shows the 7-tonal chroma as being '7 degrees and expansive', sign: 7+, - and the 12-chroma as being '12 degrees and expansive'; sign: 12+. f. The (7-tonal) whole-tone scale only shows similar motions, the scale only containing major and augmented, i.e. expansive intervals: major seconds, major thirds, augmented fourths, augmented fifths, and an augmented sixth / major intervals: major twos, major fours, major six's, major eights, and a major ten; signs: 2+, 4+, 6+, 8+, and 10+. g. The (7-tonal) diminished seventh chord only shows contrary motions, the chord only containing minor and diminished, i.e. contractive intervals: minor thirds, diminished fifths, and a diminished seventh / minor threes, minor six's, a minor nine; signs: 3-, 6-, and 9-.
27 Bach and Bartok in Degree Notation Below the 7- and 12-written Bartok quotation the shared degree notation is added:
In the degree notation the 12-tonal chroma catches attention as the interval between the highest and the lowest note (the first and the last note of the third bar), the degrees +7 and -5 making clear the tension of 12 degrees between the D♯ / P♯ and Es / P of the step notation. As an upper degree-line may be read '+5 +5 +5 +7 +5 +5' (thrice the persistent C♯ / N of the first two bars, then the highest note D♯ / ♯P in the third bar, and back to C♯ / N twice in the third and fourth bars). Contrasting with this broken straight line, a lower degree line draws a fine wave motion: '-4 -4 -5 -4 -5 -4' (B♭ / K and E♭ / P).
The 7-tonal two sharps / the 12-tonal three sharps of the Bach theme cause a higher level of the degree notation. The 12-tonal chroma is seen as the lowest note -2 in the second bar and the highest note +10 in the third bar, i.e. the notes C and B♯ / N and ♯N of the step notations. In this theme it is striking to see the predominant contrary motion between the two step notations and the degree notation: the contractive intervals, i.e. the minor and diminished intervals are dominating. When Bartok only has a small predominance of contractive intervals, Bach uses fifteen contractive intervals (minor and diminished ones) and only five expansive intervals (major and augmented ones), that is three times as many contractive intervals. By this the degree notation unveils the concentrated centripetal energy which carries and impels this weighty fugue, bearing the character of Largo: broad, large, generous ...