The aim here is not just to propose a new system but also to explain how it came about, the logic behind it and the fundamental principles. With the intention of using a system that would not be too different from those already in existence, this chapter goes through the construction of a mother scale, its diatonic chords, harmonic functions, etc. Furthermore, auxiliary scales and their family of diatonic chords are introduced and explained, concluding with the analysis of melody and its influence on chords. Chapter four completes the first part with the analytical principles of the octatonic system: how are chords recognized, how to interpret them, and different harmonic situations and behaviour.
Finally, there is a comparison of the octatonic system versus a traditional system of analysis. In part two the octatonic system is used in a chronological exploration of the music, starting with 19th century influences in chapter five: spirituals, popular songs, ragtime and the blues. These constitute an intrinsic part of the repertoire and are therefore highly influential in the development of jazz harmony.
Chapter seven takes a close look at several of the major figures in jazz history and some of their most important compositions. Part three deals with some unexpected outcomes of this research. These are side ideas that are not fully developed, but that have proven to have interesting research potential. The analytical method used throughout also appears to have other applications i.
Furthermore, the principles explored have the potential of aiding the composition process, either as a way to find new tonal chord progressions or as a polytonal map. Finally, there is a suggestion that the octatonic system could be used as a musicological tool that may shed some light on why human beings organize their music so similarly. The final pages contain a glossary of terminology and abbreviations and several appendices that may facilitate the reading of this thesis.
These include chord families and key relationships that may assist the reader in following some of the more intricate analyses. To design a new theory of harmony is too large a task for one man, considering that traditional harmony took hundreds of years to evolve. By no means am I saying that this theory is in its final state. It is a work in progress which hopefully will be nourished over time by the input of musicians and educators everywhere.
Some even believe that the earliest forms of music must have been improvised and perhaps included sounds of the natural world. Although often described as the spontaneous creation of music, improvisation frequently follows a set of rules or models. For example, in the gamelan music of Indonesia, the instruments in the ensemble take turns to vary the basic melody. The South Slavonic tradition consists of combinations of themes and motifs often with a cultural or historical reference.
Indian classical music, either Hindustani or Carnatic, combines melodic Raga 6 and rhythmic Tala variations to create rich improvisations. Large arrangements or compositions with improvised sections, free form, or simply open melodies are also used as a basis for improvisation. It went from march-like rigidity to free improvisation in less than fifty years of existence. The Grove dictionary refers to improvisation as part of the definition of jazz,13 as do many articles and books in the subject of improvisation.
Therefore, we must consider that to play jazz the art of improvisation must be mastered. The techniques involved in the art of improvising jazz were not always the same. In its early development most improvisations were based around a theme that was varied or distorted. Later in its growth, we see the inclusion of harmonic embellishments based around the chord sequence largely made up of arpeggios and a few incidental passing notes.
Many famous soloists like Coleman Hawkins started basing their solos around chord structures and substitutions rather than the melody of a piece. These models are still very much in practice today and are taught in most major educational institutions and by private tutors. Quartal approaches, atonality, polytonality, symmetric patterns, intervallic concepts, etc. Naturally, as with any mathematical permutations, many of these concepts were the same in terms of the aural product.
In a random search through Google, I was able to find 8,, results for books about jazz. The problem might be traced to the fact that jazz has always developed in the hands of musicians, independent and isolated from each other. Since the very beginning in places like Memphis, St Louis, Kansas and New Orleans, just to name a few, musicians created their own names and symbols to explain music to themselves and others.
This vernacular practice brought about an extensive vocabulary of definitions and concepts that were essentially all the same or at least very similar. When the first books on jazz were published, the trend did not stop. Many authors continued using the vernacular expressions of their geographical area. Others instead tried using classical academic terms to explain a music that defied traditional classification. The rhythm25 23 Robert Witmer and Barry Kernfeld. Grove Music Online. Would it not be ideal to have one model that could serve to educate, analyse and explain the music?
In terms of rhythmic notations there seems to be a clear consensus. Quaver eighth notes in jazz are supposedly performed as a crotchet-quaver triplets quarter-eighth notes: , though many modern players would disagree. That is to say, the least rhythmic figures used for a particular rhythm the better. Melody has also achieved a certain level of convention.
Though this precision writing does not necessarily convey the actual pitch, it is accepted as an approximation to the real sonic event. Unfortunately, harmony does not enjoy the same benchmark. Nonetheless, there are very few conventions on the subject. Analysis is treated poorly and the teaching of harmony is disjointed and impractical. In this thesis, I intend to point out the problems found in most educational material whilst providing an alternative method for looking and understanding harmony in jazz.
But first, we must understand the context and evolution of harmony from the beginning of jazz to contemporary usage. This will give us a historical perspective and a chronological framework to understand the aural evolution of this music. A historical perspective The beginnings of jazz are as elusive as the definition. Officially, the story begins in when the first jazz recording appeared.
Before that, it seems to be a muddle. The word itself existed before the music and the music was not associated with the word until much later in its development. Firstly, the sound most people associate with the word jazz is far from the early traditional music that came out of New Orleans. Secondly, not all music that came from New Orleans in the early 20th century qualifies as jazz.
And thirdly, the idea that jazz is black and solely American is misleading. Whilst jazz inarguably began in New Orleans, it is clear now that it did not remain there for very long. Another important development, and what gave jazz a more exclusive characteristic, was the influence or perhaps absorption of the blues.
Exotic Scales: New Horizons For Jazz Improvisation by Joseph Befumo
Collier believes the new music quickly moved out from New Orleans into other cities35 but more important was that after the first recording of the Original Dixieland Jazz Band in the music rapidly expanded throughout the world. So what we know at this point is that jazz evolved from Dixieland, a form of music that developed from Asian and European sources. We also know that the blues, a form of music that developed from work songs and to lesser degree spirituals,36 became a part of this new music in the early part of the 20th century, thus enriching the sound and giving it its unique sonority.
But once the first jazz record, or to be more specific New Orleans jazz record, went out in to the world it became an international phenomenon in the same way ragtime had been at the end of the previous century. Taylor also mentions how quickly French composers began incorporating the new sound into their compositions. The jazz craze began as early as in Copenhagen and by the mid 20s it was the most popular music in town. When jazz reached the American audience, it also found its way to numerous musicians, many of whom later became the most famous emissaries of the jazz language.
At the time jazz became popular it had to share the attention of the audiences with other styles popular at the time besides ragtime and the blues. But Europe has much to claim as well, from the Gypsy influences of Django Reinhardt to what is now known as the Nordic Tone or Nordic jazz, which is exclusively a Scandinavian development. The global transculturation of jazz has also generated a vast diversity of new jazz styles from different countries. So this leaves us with a big question: if jazz is constantly evolving, mutating, transforming and absorbing new elements from other styles, where do we draw the line?
When does jazz stop being jazz and becomes something else? Allen, Brian,A. It also excludes major works in the area of fusion e. Therefore, in this thesis, I will mostly use examples which fall into this category as defined by Gridley, Maxham and Hoff. However, I will also include repertoire which is considered by most modern practitioners to be jazz: these are tunes played commonly in jam sessions which, though they do not swing, represent an important part of the jazz heritage, for example bossa nova.
In conclusion, jazz is heavily influenced by European music, in its native soil as well as other countries who adopted the style in early days. Though also influenced by other folkloric forms, jazz developed in the hands of people trained in European classical music, even if just the rudiments, and played mostly with western classical instruments e. Notation also played an important role in deciphering the music in European classical terms. As the music has developed over the past century, and it has continued to absorb material from other cultures, musicians still notate chord symbols representing traditional harmony.
The instruments in charge of supporting the harmony in most ensembles are still the piano and the guitar, both tempered instruments. These intervals are calculated in semitones and they are equal throughout the tempered system. Tunes are practised in every key 12 of them and they are completely interchangeable. The minor 7th is a sound and a certain amount of semitones e. C to Bb is the exact same interval as Db to B instead of Cb. Jazz musicians use less intervals than say an atonal composer from the Second Viennese School, and the intervallic reference is from the key of C major.
The smallest interval is a minor 2nd, there is no other name for this sound or distance and it is applicable to any key or combination of semitones. Below is a chart taken from Mark Levine's Jazz Theory Book56 which shows all the commonly used intervals in jazz. After the octave the intervals continue in the same fashion 9ths, 10ths, 11ths, etc. Haerle says 'An interval is simply the distance between two notes. Any semitone below the majors is called minor and the perfect intervals can be raised to augmented or lowered to diminished.
There is no mention of 2nd, 3rd, 6th or 7th being able to become diminished or augmented. A 9th, 10th, 13th and 14th can only be major or minor and 11th, 12th and 15th can be perfect, augmented or diminished. It's interesting to note that in page 33 of book one, enharmonics are introduced as the 'same physical key on the keyboard', 62 but no explanation of why enharmonics are needed. Enharmonic notation is extremely common and sometimes encouraged, like this V-I example from Hal Crook. Though notated the author says the enharmonic A major 6th is frequently used to avoid the double flat, Bbb.
The rules are simple, it must be easy to sight read. For example: This is better than this Aikin puts it nicely when writing about enharmonics, he says 'Why can't we call a given note by the same name and be done with it? Another example, in an informal query among local jazz musicians66 I asked what is the minor 7th of Db7.
This leads me to believe that jazz musicians prefer to simplify, because though it might be theoretically incorrect, the thought process is faster and therefore more accessible in midst of an improvisation. Once again, as mentioned earlier in this chapter, different authors name and conceptualize the same theory according to their background and local vernacular. Honshuku teaches them by counting the letter names,68 and Aikin the scale notes. These qualities are based on the relationship or ratio of two notes to each other rather than the exact frequencies of the notes.
It is this fact that allows us to hear relative pitch and recognize musical relationship by ear'. A competent jazz musician is able to transpose almost instantly any musical idea to any key. I believe they achieve this by the complete symmetry in which they visualize the twelve-tone system. All notes have equal distances. A chord containing a minor 9th in its structure is usually referred to as flat 9 e. The reason jazz musicians think of intervals in this manner is because everything is based around the major scale and the chords that are formed from it and since they consider all twelve keys to be equal, interval names are taken from the perspective of C.
Fig 1. If it were minor or augmented they would call it flat or sharp respectively. What is curious about this method of thinking is that if we look at chord or scales in a different key, the alteration of the notes do not coincide with the alteration of the intervals. For example F 7b9, the flattened 9th is in fact a natural G. The 'natural' 9th of Db is an E flat, the flat 13th of E is a natural C, etc. Hiroaki provides us with a possible explanation for this practice by saying that minor, augmented and diminished are used for chord tones while symbols , b for tensions.
Finally, a modern jazz musician conceptualizes almost everything through scales. I believe this is partly due to education and partly due to practicality. Almost every jazz theory book I have read and researched starts in the same manner: intervals, then scales, then triads and finally extensions. Jazz theory, whether it be modal, atonal, polytonal, etc. Or more appropriately described in jazz terms: what scale 'fits' over a given chord. Jazz musicians think of scales to know what notes are available for them inside or outside the key.
In addition there are many other referenced books such as Patterns For Jazz74 which provide the learner not only with suitable scales to match a given chord 71 Honshuku, Jazz Theory I, 4 72 Once again, slightly different from traditional harmony: a tonal centre can sometimes last for as little as two beats.
Independent of whether the key lasts for a beat or a whole piece. In conclusion, since jazz was born in the west musicians took much of their dialect and theoretical explanations from traditional classical theory. There are many names for the same aural phenomena. Simplicity is the rule.
Also, jazz musicians think in terms of chords and scales more than they do about traditional voice-leading. Any alteration to a diatonic chord sequence implies a different scale and therefore a different key, even if this 'passing' chord only lasts for a beat. This acknowledgement will serve to better understand many of the scores, transcriptions and theory books in this field. Some have speculated that this book was Slonimsky's Thesaurus2 published in , others that it was the first drafts of what would become Russell's Lydian Chromatic Concept3.
Russell was originally a drummer but life circumstances led him to become a composer and arranger, he is credited with composing the first Afro-Cuban Bebop fusion. But as Jones puts it As Berliner relates, musicians at an early stage learn to improvise mainly by imitating or copying what others play.
Later, great musicians like Coleman Hawkins developed improvisational skills based on the chords and variations of them. These included chromatic approaches, substitutions, etc. Fig 2. When comparing with the traditional major scale the claim of unity and finality is clear. His theory is long and complex, involving pages and approximately 50 years of research, but in a nutshell Russell believes that horizontal harmony and vertical harmony are the same.
This would explain why jazz musicians perceive harmony the way they do, as seen in chapter 1. Russell's theory is both fascinating and controversial, but perhaps what is far more important in the realm of this thesis is Russell's influence on nearly every jazz theory text or book published since. For example Levine redefines the concept by calling certain intervals in the scale 'avoid notes' 12 thus justifying the substitution of the Lydian instead of the Ionian, the Dorian instead of the Aeolian, etc.
First we have the issue of the 'recipe' approach. That is to say a chord symbol equals a scale. For example the brief analysis he makes in page 55, these are the first eight bars of what appears to be a tune called All The Things You Are Though puzzling at first glance this system does offer two clear advantages: 1. He calls it 'the scale which most purely conveys the sound harmonic genre of the chord.
But when every chord becomes isolated from its context and can be played with a certain scale simply because it steers clear of 'avoid' notes or because it 'works', then we lose the whole narrative and unity of the tonal context. When a chord progression is seen as isolated random bits, what is there to provide a meaningful continuous line? Which in turn creates the problem of analysis. When a chord is viewed in isolation and the only concern is what scale will fit over it, then we are not truly understanding the meaning of the piece.
The harmonic narrative is fragmented into small pieces filled with patterns or scales that perhaps have a sense of unity within the sounding chord but little or no sense of unity within the piece. The second issue is how Russell's theory seems to have altered the way musicians approach improvisation. As I will show in the analyses in chapter 7, jazz became far more scalar and pattern orientated. As Aebersold puts it: I think the old days were better in some ways, because people were forced to use their ears and really listen.
Jazz education has changed Now people use books, a visual medium -- people using their eyes instead of their ears Now, it's almost reversed, and I think we oftentimes have players who don't get lost and play the right notes, but aren't saying much. Many great teachers still encourage the aural approach, for example Ernie Watts,16 but it is perhaps perceived as too much of a risk. The final issue is the attempt of reconciliation between Russell's proposal and traditional harmony. Many of the books I will be covering in the following chapter fail to do this. Traditional harmony is understood only in its most basic elements e.
For example when a non diatonic chord appears in the middle of a progression the scale to use is the one with the most appropriate tonal gravity relation. There is no real understanding of what the chord in question is doing there or why it sounds good. It is not unusual to find even in jazz composition books suggestions like 'Until your musical taste is satisfied'17 avoiding any kind of 'rules' of voice leading.
Perhaps here we may speculate that since many of the traditional theory of harmony and voice-leading rules come from European vocal ability, jazz departed from that tradition early in its development since its average melodic line would prove challenging even to trained voices, see for example the earliest form of scat by Armstrong D Most of the terminology is borrowed from classical traditional theory but sometimes has different meaning, and finally D Russell contributed a scale syllabus that makes melodic lines fit in with chords independent of tonal context.
So now we should look at how this translates to contemporary education. Available Methodology As seen in the previous chapter, in Jazz music education harmony, is explained in terms of relations between scales and chords. Reviewing most relevant texts on the subject, such as those recommended by Professor Charles Beale 19 or the renowned Berklee Books20, we may observe a consistent method that describes the formation of chords by harmonizing the notes of a given scale. But then in page 3, when he explains minor triads or page 4 and 5 when describing augmented and diminished triads he omits the scale from the example and prefers to relate the construction of this chord to the major triad.
This is the beginning of confusion. But once again when explaining dominant seventh in page 20 he abandons the model and relates the formation of this new chord to the major seventh instead. A somewhat different method is offered by Jamey Aebersold. Furthermore, he devotes only four pages to modulation and two lines to explain it: 'A simple rule for identifying the presence of a new key is the appearance of a major chord on other than I or IV'. In this type of approach the author does not cover the origin or function of harmony at all and instead limits him or herself to providing an arbitrary association between scales and chords.
For Example, Jerry Bergonzi Fig 2. A slightly more comprehensive approach is that of Mark Levine. For example figure 2.
Jazz Scales For Guitar
Even though the methods described thus far systematically explain the construction of chords in a diatonic scale, they do not begin to explain anything about harmonic analysis. The relationship between chords and scales is left purely to the recognition of the chord symbol, for example: Xmin7 is always Dorian. If this were so how can we explain the note G in the melody? E major does not contain a G natural. The example below shows the complete version of the piece as published in the Standards Real Book recommended by Mark Levine in chapter Further on in chapter 3, the explanation of minor II-V-I is analysed as three different melodic minor keys.
In this way he standardises all minor II-V-I cadences no matter what type of music it might be, when in fact this progression is clearly only in one key, the key of C minor. The first example in which we can find inconsistency is in page 84 figure in Levine's book. The chord motion is obviously diatonic and the flat 9th of the D7 is but a chromatic passing note. If intending to sort out the appropriate scale for this progression by ear, we would conclude that it is all in G major with an added Eb during the D7. Levine on the other hand suggests the use of the diminished scale which adds strong inconsistent notes such as the Ab and F.
If we look at the original piece of music, same bars, we can see that the added notes would be nothing but disruptive to the cadence. The added Eb will be explained later, but suffice to say it does not disrupt the diatonic sound of the progression. If the sound of the cadence is seemingly diatonic and smooth, why do the scales used offer such drastic dissonance? In summary: the model undeniably has its pedagogical value when explaining the basics, but fails to explore the complexities of harmonic analysis. It is necessary for the apprentice, in improvisation or arranging, to fully understand not only the connection and smoothness between chord progressions, but also how the melodic line works within the given framework of harmony.
Consequently, an analytical model based on similar principles is needed. In other words, a system which respects the principles adopted by other educators of being accessible and easy to follow, whilst providing the absent harmonic information. Musical analysis as a requirement Nowadays we are fortunate to have many different systems of musical analysis. We have traditional methods, Schenkerian, psychological approaches, formal approaches, comparative analysis, etc.
They all serve different purposes, have different flaws and virtues and in the end they provide us with a deeper understanding of music. Cook says 'an analysis should not aim to be a carbon-copy of the listener's experience: rather it should simplify, clarify and illuminate it. What is it that I am trying to understand? Is it the harmonic structure? Is it the melodic counterpoint? Or why it brings tears to my eyes? What was quite surprising to discover during my research of analytical systems and which systems are preferred by practising musicians, is that actual players know little or nothing at all about analysis.
Popular, rock and folk musicians met me with a genuine 'what do you mean? So this makes me ask: if analysis is so fundamental to understanding music, how is it possible that the actual performers know so little about it? The answer, I believe, is that analysis is seen as a torturous, intellectual endeavour which presents little practical use to real life circumstances.
But in reality all practitioners have some kind of practical analysis system of their own. Though it is true that most orchestral players are happy just to read what is written, they all recognize if they are playing in major or minor keys, or if they are playing a scale or an arpeggio. In addition one must learn a whole new vocabulary and technical jargon just to be able to do it.
Kenny puts is nicely when saying that jazz musicians are exclusively practical and there is 'no similar concept of musica theoretica, as opposed to musica practica'. The classical approach to analysis serves us well to understand relations of pitches, direction of voices, etc. Besides, improvisation is a spontaneous form of composition in which rhythm, interaction, alterations and the element of surprise do not allow for such a global conceptualization.
Kenny finds two major flaws in Schenkerian analysis: 'It implicates a set of idealised criteria i. Since practitioners learn most of their repertoire from lead-sheets and transcriptions, it is natural to assume that the standard score format is the most comfortable. What information does the improviser require from the music then? Without a doubt the harmonic structure, if there is one. The melody or melodies if counterpoint exists.
And finally rhythm. This includes tempo, stylistic expressions e. Looking at the above we can clearly reason that a normal score would do the job. Most of the information needed can be added to the top or bottom of the staff. Here's a proposal: Fig 2. The first bracket on top of the staff indicates which scale major, minor, melodic minor, etc.
Below this bracket we may add, if they are not already provided in the score, the chord symbols which delineate the harmonic events. Underneath the staff we may add the analysis of the melody, but in reference to the chord over which it is sounding e. Since the scale is already provided on top it would be redundant to analyse it in reference to scale notes. Below the melodic analysis we may place Roman numerals to indicate harmonic movement. A few important remarks here: upper case is used for primary chords and lower case for secondary chords more on this in the next chapter.
When chords are not diatonic they should be boxed, this is to indicate that this chord needs special attention. Finally, the Roman numerals should always be referencing the tonic key or tonic centre as indicated by the main dotted bracket below. The last line of analysis is dedicated to harmonic events modulations, tonicisations, etc.
Suffice to say, this line of analysis is what most of this thesis is about. In summary, this method provides two advantages: 1. Therefore, whatever school one might be from, the material is there. If one would choose to improvise using scales, the harmonic analysis above would provide the appropriate framework. If one would choose to improvise from the melody, the intervalic relation between the melody and chords is also there. With some practice the written analysis becomes second nature as it directly relates to sound.
Now that I have explained the analytical system that will be used from here onwards, I will continue by introducing the tonal organization and harmonic concepts used when analysing pieces in the second part of this thesis. I must remind the reader at this point that the intention here is not to present scientific evidence of the origin of music or scales, but rather an explanation based on practice and with pedagogical intentions.
The theory is based on real practical tempered systems,1 hence the letters designating pitches are approximations as understood by practitioners e. Theoretical backgrounds By the late 19th century Ionian and Aeolian modes were already established as the main source for musical composition in Europe,3 to the extent they were called natural major and minor. The main and most influential work on this subject was carried out by Hugo Riemann, in which he defined the use of harmony in terms of functions.
When constructed upwards vertically the major triad contains the interval of a major third and a perfect fifth e. Although this theory became widely criticized and sometimes even ridiculed it survived the passage of time and was constantly mentioned throughout academic circles as a curious anecdote. Later works in the field of acoustics have reinforced this theory in the concept of virtual pitch. The mind perceives virtual tones below the fundamental as well as those found above in the natural harmonic sequence or harmonic series.
Bob Fink11 hypothesizes that the origin of scales and chords is actually due to the overtone series harmonic series as opposed to the traditional Pythagorean cycle of fifths. He believes that since any note produced by a flute, bamboo, or wind whistle will produce a definite 5 Hermann von Helmholtz, On the Sensations of Tone as a Physiological Basis for the Theory of Music Dover Publications Inc.
The question I ask now is: if the basic major and minor scales can be easily formed from the sum of the overtones of the three degrees I, IV and V, what would happen if we continue with the second overtones of each family group G, C and E and third overtones? In this case E, A and B. Fig 3. Though his reasoning is that C is the loudest overtone of F and G is the loudest overtone of C. Hence the relation of these three notes referred to as the trio. The second overtone, G, produces G, D and B. The third overtone E produces E, G and B, the latter B already contained in the overtones of G and the former G creating a beautiful interval of a minor sixth to the tonic or a major third to the octave.
It is interesting to consider what would result from proceeding further in the same manner, adding the third family group of the overtone G and undertone F. The latter E is already included in the overtone of C and the former C creates the dreaded flattened 9th interval from the tonic C, avoided since Pythagorean times. The overtone A is already contained in the overtones of F IV and is therefore ignored. B on the other hand generates D and F which present similar problems.
D is a minor 3rd away from the tonic which would make the C major triad ambiguous and also creates a flattened 9th to D acting as the 5th of G. Fink sees it differently, his proposition is that these intervals are too far away to have any valuable effect. He claims weaker intervals are cancelled out by the stronger ones. The same applies to B-F -D. The notes of the family group produced by the fundamental C fall in the same degree of strength as the overtone and undertone.
Thus, the family produced by the overtone and undertone fall on a weaker degree similar to the family group produced by the first overtone of the fundamental G which happen to be the same notes G-D-B. The notes in bold indicate notes that are reinforced by repetition in the spectrum. We can speculate then that the notes generated by the family groups of the second and third overtones are too far in the spectrum for any harmonic application, with the exception of E since its generated by the fundamental.
I would like to suggest an alternative way of viewing this construct based on the same phenomena of overtones: I call this the Hierarchy of Intervals. The fourth is weaker than the fifth and will therefore lose relevance when sounding together with a fifth. In a harmonic context it will always be subject to the fifth.
The same principles apply to the thirds: the major will always predominate over a minor but both are weaker than the perfect intervals. The octave is generally disregarded since it produces the same results. To avoid confusion, each level is referred to as a degree. Thus, the first degree of the hierarchy is a perfect 5th, the second degree a perfect 4th, etc.
By examining Fink's tonal scheme we can justify the addition or dismissal of our second and third overtones using the hierarchy of intervals model. The added tones G , C , F and D on the other hand are audible as extensions of the originals overtones of the overtones and therefore need to be justified in terms of harmonic relevance. To begin we must analyse these notes against the hierarchy of intervals and this we must also consider intervalic inversions.
This last interval is the third degree in the hierarchy of intervals, therefore welcomed by the ear as harmonious and relevant. C on the other hand creates an 22 The remaining overtones are omitted since there is much debate as to their relation to the tempered system. Hence, we need to consider inversions and the intervals they generate. F in either of its inversions creates a tritone which is not found in the hierarchy unless considering the interval found amid the 5th and 7th harmonic in the overtone series, which in any case is too disguised to have any relevance.
If we follow the pattern of intervals in the hierarchy we could also justify the existence conscious or unconscious of this 'octatonic' scale as part of an evolution of usage. As seen above, the fundamental C generates E and G. The major third E creates G and B which would be the next natural step to take: Fig 3.
It would be interesting to continue adding the overtones of the fourth degree to research the harmonic possibilities offered by such a system, but that will have to wait for another research project. The theories and models explained above inspired and contributed to the rationalization of this octatonic scale.
Now I intend to demonstrate how this scale may coherently provide numerous harmonic possibilities that are perceived as being diatonic from both an aural and theoretical perspective. From here onwards I will be referring to this basic organization as the Fundamental Scale and the additional note 5th will be known as Q. These were formed by combinations of 5ths, 4ths and 3rds the first four intervals in the hierarchy excluding the 8ve. This basic approach differs little from current practice, in which the basic chord still follows this organization. For example: Fig 3. Although this chord is perceived as the strongest possible polyphonic combination, it does not define the cultural inherited 26 As explained in Chapter 1 there will be occasions in which the enharmonic must be used for practical reasons.
Particularly considering that an octatonic system would require eight letters of the alphabet, hence a whole new re-conceptualization of the tonal organization which would conflict with the practical nature of this proposal. The second chord is built exclusively by 4ths and has a similar nature to the former, a lack of tonal definition. The third chord utilises a combination of 4ths and 5ths, this chord although popular in music is basically an inversion of the previous one, thus vague in its tonal designation. We can then deduce that the sixth and seventh chord, containing intervals of 3rds and 5ths from the tonic or simply 3rds between voices , offer the best balance whilst defining tonal characteristics.
Taking this into account we can proceed to harmonize the fundamental scale. The first faction of triads is formed by observing the best possible balance of the hierarchy, combinations of perfect 5th and 3rds from the root or 3rds between voices. I will refer to this group as primary triads. Although these intervals are not found in the hierarchy the distance between voices are those of major 3rds or minor 3rds. These chords are known as augmented aug and diminished dim. Our second faction observes the alternative possibilities offered by the scale i.
This chord can be used in either major or minor tonality and its purpose is usually that of ambiguity. As seen before the Q note can be either Ab or G. The rest of the chords in this figure are formed with other scale notes, and because these chords violate the hierarchy they produce tension Csus2, Csus4, Caug. These chords are often used as passing chords between primaries or internal voice movements. I will refer to these as secondary triads. Ab augmented and G diminished are essentially the same root Q , but because of the seven note system it is difficult to notate the primary and secondary using the same root.
We must nevertheless consider both options since in a piece of music we are likely to encounter Ab augmented or G diminished as part of a C major progression. We may also notice that there is no secondary chord for B, the reason will become more apparent when we begin adding upper-structures. Suffice to say, any re-harmonization of this chord would result in an inversion of another already existing chord from the scale.
Below we may observe how this works. The example is in C major beginning on the first primary chord and slowly moving away using secondary chords. The example ends again on primary I C major after passing through nearly every secondary chord. An octatonic analysis of the piece would look like this: Fig 3.
The Roman numerals below indicate the position of the chord in the scale from which they were formed. Upper case indicate primary options chords which respect the hierarchy and Lower case indicate secondary options other alternatives formed within the scale. Rp indicates Root position, inv stands for inversion. As we may observe, for all intents and purposes this piece is in one key and one scale.
All chords found here belong to C major fundamental. As mentioned before we may also observe three different chords I, V and VI becoming temporally a sus4 or sus2. What is most important at this point is the realization that the same chord symbol may appear in different places of the scale i. This begins to explain the error of assigning a chord symbol to a specific scale.
Theoretically speaking, chords sounding in the context of a key or tonal centre acquire what we could call a personality. The ear interprets the harmony creating an expectation of what will happen as a result anticipating the event towards resolution or tension. Depending on the degree of the scale from which the chord is built, and trying to link this theory with that of Riemann,35 I will refer to resolved chords as tonic chords, semi-resolved as supra-tonic,36 transition chords as sub-dominant and tension chords as dominants.
They also provide a sense of closure. Supra-tonic chords have a sense of resolution, but not of closure more on this below. Sub-dominant chords offer the feeling of transition and movement. They generate subtle tension but their direction could be towards resolution or more tension. Dominant chords generate tension from the perspective of their degree in the scale as well as their inner structure. They usually contain a tritone in either their basic form triad e. B diminished or their extension 7th, 9th. After careful experimentation and observation of the diatonic triads in the fundamental scale, it is possible to classify all of them within four functions.
For the purpose of analysis from now on the chords will be referred to in relation to the degree in the scale from which they are formed. Lower case will be used for secondary chords. Aarden, M.
Although it is debatable that the iii and VI are actual tonics they do behave as such in a diatonic cadence, as we will see below, in the sense that they generate neither tension nor the feeling of transition. This is possibly due to the fact that they share two common notes with the tonic. For the sake of clarity I will refer to these two chords iii and VI as supra-tonics, a point between sub-dominant and full Tonic resolution. Under some circumstances it acts like a sub-dominant, though it is more commonly found as a supra-tonic.
The primary II and IV can be classified as sub-dominants since their nature is that of transition, either towards resolution or further tension. The IV was the original sub-dominant in classical theory, but during the early 20th century it was more widely replaced by the II, especially in jazz. The secondary ii is rarely used and should be considered an inversion of secondary q. A more thorough explanation of this will be found below.
Arnold Schoenberg, Theory Of Harmony, In this case supra suggests tonic above the tonic, therefore not in its full state of resolution between Tonic and Subdominant. Because this intervalic combination sounds so strong to the ear possibly due to its absence in the hierarchy of intervals all chords containing the tritone interval seem to the untrained ear as the same or at least similar, consequently their function is perceived to be equivalent in terms of being inclined towards resolution, particularly to the tonic. The chords that possess this quality are primary V, VII and secondary q.
He includes the seventh of every chord as part of its normal construction, but I believe this does not affect the actual function of the chord. He describes chord I as: 'Establishes the key center, doesn't need to progress, but may go anywhere'. Schoenberg, Theory Of Harmony, Nevertheless, with the functions now defined we may proceed with harmonic movement. Basic Cadences Cadences, as traditionally understood, refer to the use of harmonic progressions not only for the use of colour but as a defining statement of tonality. When justifying a key melodically, using the scale, tonality becomes obvious due to the presence of the 7 th and 4th degree, but harmonically a C major triad for example could easily be misinterpreted as belonging to the key of F major or G major.
Thus a chord containing the defining degrees 4th and 7th, for example B diminished, would provide the most appropriate anticipating chord to the tonic C. Jaffe considers these degrees of the scale 4th and 7th as 'unstable' and consequently the functions of chords are defined by the presence of one or both of these notes. Dominants have both degrees 4th and 7th and are consequently 'most unstable'.
Historically, the most widely used cadence has been V to I. The justification of this is that the V provides a natural gravitational tendency to its root I. Seen in terms of the hierarchy of intervals the root of V is the first overtone omitting repetitions to appear; it is therefore believed to generate a natural tendency 44 Edrward Aldwell, Carl Schachter, Harmony And Voice Leading [no place of publication cited]:Hacourt Brace Jovanovich Inc. This has been mentioned from Rameau 46 to Schoenberg47 as the basic phenomenon of tonal harmony and it has been used throughout centuries as the most efficient way to establish the tonality.
However, since the notes that truly establish the key are the 4th and 7th, any chord containing these notes would work. Thus chord VII and secondary q also serve the purpose. These types of movements fall under what is known as a dominant cadence. It is considered to be an ambiguous movement since it deceives the ear to think V-I This problem is easily solved by using the extensions of the chord see below. Another solution is to use II as this is also a sub-dominant and therefore interchangeable.
These movements are known as sub-dominant cadences50 Finally the sum of the above creates what is known as the full cadence. The fundamental scale offers the same possibility by changing the tonic to the 6th degree of the scale a major 6th up from the root or a minor 3rd down thus changing the point of resolution. The basic cadences and functions work in the same fashion as those of the major fundamental scale but from its minor perspective, that is, considering A minor as the tonic chord as opposed to C major.
Since the intervals contained within the V chord of the minor scale are the same as the V chord of the major perspective, I shall use the term minor dominant for the V chord that resolves to the minor and major dominant for the V that resolves to a I major. This is mostly true for the V chord, since the addition of the 7 th creates a tritone to the third which solves the possible misinterpretation of the V being I or IV. This offers a wide range of options that can be frequently found in a wide variety of musical examples.
Also the VII degree changes its name to minor flat 5 because its diminished quality is lost with the addition of the minor 7th. These extensions, also known as upper-structures, are not simply any note in the scale above the 5th. These new notes need to respect similar principles to those of the formation of primary chords but only to the notes below it, whereas primary chords need to consider any inversion.
The upper-structure by its very definition upper needs only to concern itself with the notes that precede it. In this manner we may use the hierarchy to establish all the notes that qualify as the upper-structure of a given chord. Following the hierarchy of intervals we may add the following diatonic notes on top, taking the G as a starting point.
D The note D is a perfect 5th from G, but it is also a minor 7th from E and a major 9th from C which inverts to a major 2nd, which is not in the hierarchy. Upper-structure D Intervalic relation with 9th 2nd m7th 2nd 5th 57 chord Basic chord C E G D The note C cannot be considered an upper-structure since it is simply the octave of the root.
D B creates a major 3rd from G and a perfect 5th from E. However, it also creates a major 7th from the root C which inverts to a minor 2nd, also not in the hierarchy. D Finally the minor 3rd would have to be a Bb which is not in the scale and is therefore unavailable. Upper-structure B Intervalic relation with M7th m2nd 4th 5th M3rd chord m6th Basic chord C E G In retrospect the note B offers the best embellishment since it only conflicts with one note, whereas D conflicts with two. Now we possess a tetrad which is extremely common in jazz, C- E-G-B. Both exist in the hierarchy.
Third upper structure The next upper-structure is again defined in these terms i. D The major 3rd is not available F. Now we can confidently say that the upper-structure of chord I is major 7th, major 9th and major 13th. For a more detailed view of the upper-structures of every chord please refer to appendix 2. By the 15th century the development of the Motet in three voices brought about the existence of new chords,61 which while not belonging to the diatonic key still appeared as smooth substitutions to the actual diatonic cadence.
Interesting to observe is that this practice still appears in contemporary music, particularly that composed by song writers with 58 Kathleen Schlesinger, 'The Origin of the Major and Minor Modes Concluded ', The Musical Times, Vol. The melodic minor scale is a seven note scale which looks similar to the Aeolian mode but with the 6th and 7th raised.
Unlike the classical melodic minor which ascends in one way and descends in another, in jazz the melodic minor is the same up or down since it contains several chords which are widely used. The extensions are: Fig 3. For notation purposes it should be written out as Ab-C-E-Gb. A diatonic progression can suddenly depart from its fundamental harmony into melodic minor harmony whilst retaining the tonic centre.
Because the tonic is not lost or altered, we cannot consider it a modulation, but rather a variant or parallelism. I avoid using the word substitution here because the resulting sound is not strong enough to warrant the term as I understand it i. In the second bar it initiates a short modulation to the relative minor by the use of a full cadence. On close inspection we may notice that 1. As previously stated, the movement to the relative minor is not affected. The sense that we are in F major with a brief passage through the relative minor is also not affected.
The alteration of the chords in bar 2 are subtle enough to pass almost unnoticed in the frame of the main key. The 'parallelism' of the melodic minor is perhaps the most common non-diatonic alteration in the modern repertoire. Unlike other substitutions, the melodic minor can be found in countless popular songs. It usually appears disguised by mimicking the 'common' diatonic progressions. If we observe figure 3. A minor key may, at any point, detour to its parallel melodic minor.
Knowing the difference between the diatonic chords of the fundamental and the melodic minor will prove indispensable in recognizing when a movement is a parallelism or another type of harmonic movement more on this in the next chapter. These scales are found more typically as special effects than as tonal centres. They produce very singular types of chords and are therefore easily identified in harmonic analysis. These are commonly known as the diminished scale and the whole-tone scale.
We may notice the symmetry in the harmony as well: one diminished chord, one 7th chord. The upper-structures of these chords are quite rare in my experience but they can be found with at least one of its extensions. It also contains a perfect 11th and a natural 13th. The D7 almost appears like an altered dominant, but contains the perfect 5th and a natural 13th. Consequently, if one would want to imply 'diminished' harmony, then it would be required that either chord be used with a combination of extensions which is exclusive to this scale e.
Cdim maj9 or D7 9 with a natural 13th. Examples of this will be explored in part two of this thesis. To differentiate the scale from C diminished which starts tone, semitone, tone, etc. Whole-tone harmony is far simpler due to its construction of pure tones: Fig 3. In this scale we find that every note produces the same type of chord: a 7 th sharp 5. Its upper-structure illuminates the exclusivity of this scale: Fig 3. Unlike the diminished harmony, here one would require all three alterations 5, 9, 11 since the absence of any of these could be reinterpreted as a chord from a more traditional scale e.
Perhaps at this point it is important to remind the reader that chord symbols are intended to indicate a voicing, not an analysis. Most musicians will write a sharp 5 when analytically speaking is a flattened 13th. In other words, the required voicing will be that of C-E-G -Bb, but in the context of the of F minor this a perfectly acceptable voicing for chord V. Most of these notation problems are caused because of the jazz musician's use of enharmonic spelling. Nevertheless, it has been stated repeatedly that chord symbols should not be interpreted superficially and that it will always depend on the context not on the actual symbol.
The chromatic scale The last important resource in tonal analysis is the chromatic scale. Many examples can be found, particularly in jazz, which use the chromatic scale to parallel the diatonic harmony e. Harmonically it is important to determine when a movement is chromatic and when it is key-relation detour.
The principles of identification will be covered in the next chapter see chromatic harmony. Melodically it is much easier, but still one must understand what qualifies as a chromatic line. In most cases a chromatic line will begin and end on a chord note, most commonly the triad. Another example is what is known as a chromatic approach.
These are melodic movements that do not necessarily move in a straight direction i. These types of approaches could be described as an indirect approach to the chord tone, since the chromatic line approaches from both sides. These approaches can be applied to any note of the chord and they were common practice in the styles of Bebop or hard bop. I will explore more on these passages when analysing music from that era. This practice is particularly important to identify because it is often used as an anticipation of chords that have not yet sounded.
It would be incorrect to analyse this group of notes from the perspective of C major since the melody is heading, and thus anticipating, the next chord. This PDF book include guitar improvisation conduct. To download free ramon ricker pentatonic scales for jazz improvisation. So far this book has examined scales, arpeggios, and extensions and another device. This PDF book contain guitar improvisation guide. To download free introduction to jazz guitar improvisation sample jamie you need to Basic Jazz Booklet.
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