Analyzing and creating cadences (chord-progressions) of short music pieces using a simple mathematical model

Table of contents

Summary

Introduction

Basic assumptions for a mathematical model to calculate the degree of relatedness of different chords in a cadence

Method for the calculation of relatedness (%) of different chords

Impact of the calibration of the mathematical model

Ideas to understand, interpret and use the results of the mathematical model on harmonic relatedness of chords in a cadence

Discussion:

References

An eloquent and innovative method to analyze and create cadences (chord-progressions) of short music pieces using a simple mathematical model.

Alexander Rehm

Summary

In this paper an idea is presented which offers musicians, composers and students 1) a method to analyze the harmonic relatedness between different chords of a given cadence and 2) an opportunity to use this information to create/compose meaningful cadences. For this purpose a mathematical model is introduced where given cadences or chord progressions of a short music piece are the input. The output of the model are calculated values for the harmonic relatedness (= HR) of the chords in the cadence. Two additional parameters derived from the mathematical model - DM (= disharmonic movement) and HarmoT (= harmonic tension) - are demonstrated to be very useful for analyzing given cadences of short music pieces. Using three examples of different musical types (Pop/Beatles, Classic/Gibbons; Jazz/Coltrane) it is shown that these two parameters give new and additional insights in the harmonic concept of short music pieces compared to well know and established analysis-procedures. Details of the mathematical model as well as aspects of different calibration options of the model are presented and discussed. Further ideas are proposed how the mathematical model may help musicians and composers in their task or process of improvising or creating music. Although the introduced mathematical model has limitations (which are discussed in this paper as well) it might offer an additional and easy to use tool for anyone who works on meaningful chord progressions and harmonic concepts of short music pieces.

Introduction

Beside rhythmic components and melodic lines also cadences (defined chord progressions) play an important role in western classical and modern music. Especially in Jazz chords build the harmonic fundament of a music piece and the chord progression in a music piece is of high relevance for the “mood of the piece” but also functions as guidance for the improviser. There are numerous publications on harmony in Jazz music which deal with chords, the creation of chord progressions (cadences) and the ways to play over certain chords and cadences by the improviser (Ref. 1, 2, 3). In music theory models and concepts have been developed to describe the relationship of chords in a cadence and their meaning in a music piece. (Ref. 3-8, 11). It has also been shown that listeners of a music piece do recognize and remember chord progressions and can link those to certain musical styles (Ref. 9). The most well know and often used short chord progression in Jazz is the II-V-I cadence (Ref. 1, 2, 5, 10). Studying harmonics, chords and chord progression is a complex task, and it is not easy to understand how chords can be put in the right sequence to a) achieve a certain “sound or mood” of the music piece to be composed and to b) build a good and manageable fundament for the improviser. A well experienced Jazz musician or composer creates chord progressions very often by his intuition as she/he knows all the complex harmonic relationships of chords and sound. Laymen or students may have massive difficulties to understand and follow all these complex guidelines and may sometimes feel lost during the creative process.

In this paper an idea is presented which offers the inexperienced musician or student the opportunity to analyze the relatedness between different chords of a given cadence and further to create/compose meaningful cadences. This idea is based on the common understanding of major and minor scales and uses a simple mathematical model to calculate the relatedness of different chords. Although the model has limitations it is useful to understand why and in which way chord progressions differ among different musical styles. Further the introduced mathematical model is easy to set up and use and therefore has advantages vs. more detailed but also complex music theories on chord relationships and tension in a cadence (Ref. 6-8, 11).

Basic assumptions for a mathematical model to calculate the degree of relatedness of different chords in a cadence

The model is based on the following assumptions which also define the limitations of the model and the respective results:

1) Chords are generally based either on major scales or on minor scales.
2) Based on a given Chord a related 7-tone scale can be defined by staples of thirds under the common principle of major and minor chords
3) The first and third tone of a scale are of high importance to calculate the relatedness of different chords compared to the other tones of the scale as these two tones define the character or mood (major or minor) of the scale.
4a) The larger the difference (tone-steps) of the first or base tones (Tonika/tonic) of different scales (which are related to different chords), the smaller is the relatedness of the two different chords. (Explanation: Although the tones of the scales of Am7 and Cmaj7 are identical their relatedness will be less compared to the relatedness of Amaj7 and A7. The difference in the base tones A and C is more important concerning the relatedness as the one tone difference in the scales for Amaj7 and A7).
4b) Beside the difference in the tonic of different scales the number of “identical tones” of these two scales is also of importance. The higher the number of “identical tones” of different scales the higher is the relatedness of these scales (Explanation: The relatedness of Am7 and Cmaj7 is mainly defined on the difference of the tonic (A and C) of the two scales as the tones of both scales are identical). The relatedness of Amaj7 vs. Cmaj7 should be lower than the relatedness of Am7 and Cmaj7 as Amaj7 and Cmaj7 have (beside the difference in the tonics) in addition a lower number of “identical tones”.
5) The relatedness of different chords will be described as “percentage value”. That means that two different chords can have a theoretical relatedness of 0 – 99%. If the relatedness was 100% the chords cannot be different, they must be identical.
6) The calculated relatedness of chords is depending on: a) the definition of the basic scale and therefore the basic chord of the music piece and b) on the chronological progression of the chords. If a certain chord (= Chord-X) is the “base-chord” of a music piece a different chord within the music piece (= Chord-Y) will be rated vs. the base chord and this will result in a certain relatedness-%-value for the “Chord-Y vs. Chord-X”. A chord progression of “Chord-X-to Chord-Y” will result in the same relatedness-value. The chronological progression of chords is also important to define the relatedness of neighboring chords. So the relatedness of two chords may differ depending on its position (chronology of chord progression) or value (e.g. main- or base-chord in the music piece) of the chords. An example for this effect with the chords Cmaj7 and Ab7 will be given later in this publication.
7) The model is designed to analyze short music pieces or short parts of larger music pieces. The shortness is more defined by the number of observed chords and chord-changes than by the length or number of notes of a music piece. It is recommended to use music pieces or parts of music pieces for this analysis with a chord progression containing at least 5 and up to 30 chord changes.
8) For the calculation of relatedness simple mathematical operations are used (see below).

It is possible to argue about the above defined prerequisites & assumptions and also about the mathematical calculations which will be shown below. And it is also undisputable that this is a certain and specific definition which does not cover the entire understanding and usage of harmonic relationships of chords and scales. But those assumptions are derived from the common understanding of harmony and harmonic relationships in western music in a similar manner as words build the base and fundament of a language. So it seems acceptable and logic to use those assumptions for a model. But it must also be clear that this model and the related interpretations have limitations and give rather simplified results. An experienced musician and composer may not use it, but it may help the beginner and student. And modifications of the parameter of this model may allow new views on harmonic relationships and may induce new creative processes in music.

Method for the calculation of relatedness (%) of different chords

As described above under assumption 6, the basic scale and the related basic chord of a music piece must be known. This important chord is called “base-chord”. If the basic scale is e.g. the C-major scale, Cmaj7 is the base-chord. The first calculation reflects the harmonic relatedness (HR) of any chord in the music piece vs. the base chord and the calculated value is named “HR%-B”. The second calculation reflects the relatedness of any given chord in the music piece vs. its previous chord and the calculated value is named “HR%-P”. For the procedure of calculating the relatedness of two chords it is important to understand their position. For the calculation of HR%-B the base-chord takes position 1 and all other chords take position 2. For the calculation of HR%-P the chord looked upon (or given chord) takes position 2 and the previous chord takes position 1. Therefore, HR%-B gives the relatedness value in % of the given chord vs. the base-chord and HR%-P gives the relatedness of this chord vs. the previous chord. So for each chord in a sequence of chords (cadence) the values for HR%-B and HR%-P will be calculated except for the first chord of the cadence as a calculation of HR%-P is not applicable.

If the position of the two chords are clear the related scales should be displayed according to the example shown below, where “Cmaj7” is at Position 1 and “D7” is at position to. As both chords are major chords, their related notes could be compared.

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Table 1: Example of displaying the notes with Cmaj7 at Position1 and D7 at Position2

The base-note of the chord in Position 1 (here the C of Cmaj7) is marked with the value 3, the third note of this chord (here E of Cmaj7) is marked with the value 2 and all other notes of the related scale are marked with the value 1. For the chord in Position2 all notes of this scale are marked with the value 1. Also given is the “minimum difference in note-steps of the base-notes of the two chords”. As the base-note of Cmaj7 is C and of D7 is D this value results in 1. If the Chord in Position 2 would be Gb7 instead of D7, the minimum difference of the two base-notes would be 3.

As an example Table 2 shows the result of the calculation of the relatedness of the two chords (here Cmaj7 in Position1 and D7 in Position2) as well as some intermediate steps of the calculation process.

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Table 2: Example of the calculation of the relatedness of two chords using an Excel template

The calculation of the harmonic relatedness (HR%) is a step wise execution of several simple mathematical procedures and can therefore be easily transferred in an excel-file for proper usage.

First mathematical procedure:

The number for each note in the upper line (note of the chord in position 1) is multiplied by the number for this note in the lower line (note of the chord in position 2). The result is shown in the line named “Multiplication”.

Second math. Procedure:

For each number in the line “Multiplication” a defined value will be written in the corresponding position in the line named “Indexation”. For a number of 3 in the “Multiplication”-line the value of 1.5 will be written in the “Indexation”-line. A number of 2 in the “Multiplication”-line will turn into the value of 1.2 in the “Indexation”-line and the number of 1 and 0 in the “Multiplication”-line will result in values for 1 and 0 in the “Indexation”-line. (Remark: At this point other Indexation processes could be chosen which can be interpreted as a “different calibration method” for this calculation – see also section “Impact of the calibration of the mathematical model” below)

Third math. Procedure:

The sum of all the various values in the Indexation-line are displayed under: “Sum of Indexes”. The regular maximum value of the sum up of Indexes is 7.7 (7 identical notes plus 0.5 for a matched base-note and 0.2 for a matched third). As this calculation model is based on 7 notes per scale (defined by the chord), the calculated value for the “Sum of Indexes” has to be subtracted by the value -0.7 to give the “Comparison-Index” as shown in the line under the “Indexation”-line. In the given example in Table 2 the Sum of Indexes is 6.7 which results in a value for the “Comparison-Index” of 6.

Fourth math. Procedure:

In this procedure the “Base-tone-Factor” is calculated – this factor is of high importance for the overall calculation. The Base-tone-factor (btf) is calculated as follows:

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In the example in Table 2 “btf” = 2.5, as the minimum full tone difference between C and D is 1-full note. For base-tones of G and C of 2 different chords the “btf” would be 1 as the full-tone difference between G and C is 2.5.

Fifth math. Procedure:

A multiplication of the value for “btf” and the “Comparison-Index” results in the value for “HR” as seen in Table 2. “HR(cal)” is just a derivative of “HR” which fixes the value for “HR(cal)” at 1 in case the multiplication of: “btf” * “Comparison-Index” results in a value < 1.

Sixth math. Procedure:

Here the logarithm (for base 10) of the HR(cal) value as well as of the value for HR(Ref) = 24,5 is calculated (see Table 2). HR(Ref) can be considered as a constant value as it is simply derived by multiplication of the two highest possible values for the “Comparison Index” (maximum = 7) and the “btf” (maximum = 3,5). (Remark: At this point another mathematical calculation instead of logarithm could be chosen which could (like the Second math. Procedure – see above) be interpreted as a “different calibration method” for the HR%-calculation – see also section “Impact of the calibration of the mathematical model” below)

Final math. Procedure:

HR% is finally calculated by the division of HR(cal)log / HR(Ref)log. In the example given in Table 2 with Cmaj7 and D7 the relatedness of D7 vs. Cmaj7 is according to the above described calculation: HR% = 84.7%.

To give some more illustration of the results derived from the above described calculation various examples for the value of HR% of two chords are given in Table3.

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Table 3: Calculated values for HR% of Chord 2 (Chord in 2.position) vs. Chord 1 (Chord in 1.position) according to the calculation model described above.

According to this calculation there is no harmonic relatedness between Cmaj7 and Gbmaj7 (see HR% = 0 in Table 3). This value as all other calculated HR%-values depend on the calibration of the calculation model (see remarks at second and sixth math. Procedures as well as in section “Impact of the calibration of the mathematical model” below). As the two related scales of Cmaj7 and Gbmaj7 differ significantly (low number of identical tones and large difference in two base-tones) an HR% value of 0 seems to be rather logical for these two chords.

Looking at a typical Jazz cadence of II-V-I we would find for the base-scale of Cmaj7 (I) the following harmonic relatedness:

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Table 4: HR%-B and HR%-P values for the chords in the cadence Cmaj7-Dm7-G7-Cmaj7. Cmaj7 is the base chord of this cadence. HR%-B means the “harmonic relatedness of the given chord vs. the base-chord” (here Cmaj7). HR%-P means the “harmonic relatedness of the given chord vs. the previous chord” in the cadence. (see further explanation on HR%-B and HR%-P in the text)

The values for HR%-B (where Cmajor is the base-scale) are in line with the expectations as HR%-B for the chord (II) is 84.7% and for chord (V) is 60.8% reflecting that chord (V) generates a harmonic tension which is solved by returning to chord (I). According to the definition of harmonic relatedness (HR) a harmonic deviation (HD) could be calculated by HD% = 1 – HR%. For G7-Cmaj7 the HR%-B is 60.8% so the value for HD%-B is 39.2%.

In Table 4 a further parameter is introduced: HR%-P. HR%-P is the calculated harmonic relatedness of the given chord vs. the previous chord. In the cadence shown in Table 4 the HR%-B and HR%-P values for Dm7 (II) are identical as Cmajor is the base scale (important for calculating the HR%-B value) and the related chord Cmaj7 is the previous chord to Dm7 also. By looking at G7 at the cadence in Table 4 HR%-B and HR%-P show different values. This can be expected as Cmaj7 is the base chord which is relevant for calculating HR%-B for G7 (G7 vs. Cmaj7) but the previous chord to G7 is Dm7 so HR%-P gives the relatedness of these two chords (G7 vs. Dm7).

As already mentioned in the “Basic assumptions of the mathematical model” (see No 6 above) the position or chronology of chords in a cadence are of importance for the calculated values of HR%-P. This can be demonstrated with the chords Cmaj7 and Ab7.

For the chord progression Cmaj7-Ab7 the HR%-P value for Ab7 is: 44.9 %. For the chord-progression Ab7-Cmaj7 the HR%-P value for Cmaj7 is: 41.3%. The difference in the two HR%-P values result from the fact that the related scale to Ab7 does contain the tonic of Cmaj7 (tone C) whereas Cmaj7 does not contain the tonic of Ab7 (tone Ab). So it is obvious that the model gives a lower HR%-P value for the chord Cmaj7 in the Ab7-Cmaj7 progression vs. the HR%-P value for the chord Ab7 in the Cmaj7-Ab7 progression.

Based on the described method to calculate HR% (especially the importance of the value for btf – see above) it can be expected that the value for HR% gets smaller the larger the difference is between the base-notes of two chords. In fact there is a certain range of HR%-values for each difference in the base-notes of the two chords compared. This range of theoretically possible HR%-values for varying differences of the base-notes is listed in Table 5 and illustrated in Picture 1 below.

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Table 5: Calculated maximum and minimum HR%-values for two chords with varying differences in the base note. Further some chords are shown which have an HR% value vs. Cmaj7 representing either the minimum or maximum HR%-value at a certain base-tone difference.

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Picture 1: Theoretically possible HR%-range (blue area) as a function of the minimal difference in the base-tones of two chords (see corresponding values in Table 5).

Worth to remark, that the HR%-range is identical in both directions of the difference in the base-note. So with C as the base note in Chord 1 the chords with base notes of D or Bb (difference of 1 full tone vs the base-tone C) have the same theoretically possible HR%-range.

Further “harmonic parameters” derived from calculated HR%-values in a cadence

To get more insight and more understanding on the harmonic development in a music piece based on the various chords in a cadence, two additional parameters “Disharmonic movement” (DM) and “Harmonic Tension” (HarmoT) are proposed and can be derived from the HR- and HD-values (Remark: HD=1-HR – see details in the text) of a cadence.

The parameters are calculated according to the equations below:

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HarmoT (of a given chord) = square root of (HD%-B (of the chord) * HD%-P (of the chord))

Both newly derived parameters can be interpreted as an indication for the “Grade of disharmonic movement” (DM) or “Grade of tension” (HarmoT) which is generated by a given chord within the cadence. The related calculated values for HR%-B, HR%-P, DM and HarmoT of a typical II-V-I cadence are listed in Table 5.

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Table 5: Calculation of HR-, DM- and HarmoT-values for the given cadence with Cmaj7 as base-chord.

It is obvious that if a “V”-“I” step is interpreted as a “complete” relaxation and solution of the harmonic tension in a cadence, the calculated HarmoT-values (dropping from 41.5% to 0%) would be more in line with this view than the calculated values for DM (dropping from 41.6% to 19.6%). But both parameters meet to some extent the expectations or ideas of a Pop- or Jazz-musician or of Composers concerning the development of the tension within the often used II-V-I cadence.

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Picture 2: Display of the values for HR%-B, HR%-P, DM and HarmoT of the chords within a II-V-I cadence with Cmaj7 as base-chord. (see data in Table 5).

Impact of the calibration of the mathematical model

Although the model follows strictly defined mathematical calculations there are two points (see second and sixth mathematical procedure described above) where a change in the calibration of the model is possible and which result in different values for HR%. It is up to the idea of the user of this calculation model to use these calibration points in order to achieve results which please the user more than the results gained with the parameters used and described above. There might be unlimited options to change the calibration at these two point but for demonstration purposes only 2 variations in the second and sixth mathematical procedure will be highlighted.

1) Change in the parameter of the second mathematical procedure.

The index-values of 1.5 and 1.2 will be replaced by new index-values of 3 (instead of 1.5) and 2 (instead of 1.2). Following the logic of the model also the third mathematical procedure has to be slightly changed as instead of 0.7 with the new indexation the value of 3 has to be subtracted from the “Sum of Indexes” to get the right “Comparison-Index”.

The results of this “altered calibration” on the calculated HR%-values of the same chords listed in Table 3 for the “unaltered calibration” are displayed in Table 6. The “altered calibration” results in a different HR% value for the two chords Cm7–Dm7 whereas the HR% value for other chord combinations are not changed.

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Table 6: Values of HR% of the same chord combinations of Table 3 but calculated with an “altered calibration” in the second (and third) mathematical procedure. The “altered calibration” of the mathematical model results in a different HR% value for Cm7-Dm7 (bold letters) vs. the “unaltered calibration”.

Although this “altered calibration” shows only in one case of the examples listed in Table 6 a change in the HR%-value, the theoretically possible HR%-range is significantly changed by this altered calibration (see Table 7 and Picture 3).

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Table 7: Maximum and minimum HR%-values for two chords with varying differences in the base note calculated similar as in Table 5 but using a “altered calibration” (alteration in second math. procedure) of the mathematical model (see further details in the text).

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Picture 3: Theoretically possible HR%-range (blue area) as a function of the minimal difference in the base-tones of two chords using the “altered calibration” (alteration in second mathematical procedure) of the mathematical model as described above (see corresponding values in Table 7).

The most prominent effect is that the “altered calibration” (in the second math. procedure) allows HR%-values of “0” for chords which differ by only 0.5-notes in their base-tone. So the mathematical calculation with the “altered calibration” may give the same result of HR% = 0 for various different chord-combinations where the differences in the base-tone of the respective pairs of chords may range from 0.5 – 3 notes. The same HR%-value of 0 for such chord combinations may generate conflicts with the expectations and impressions of musicians and composers on the tension in these chord-progressions (see Discussion).

2) Change in the parameter of the sixth mathematical procedure

A simple change within the sixth mathematical procedure would be to erase the Log-function and replace it by “Multiplication of the value from the fifth math. procedure with the value 1”. It could also be understood as skipping the sixth mathematical procedure and feeding the result from the fifth mathematical procedure directly into the final mathematical procedure. In the following we name this alteration of the calibration the “Non-Log calibration” or “Non-Log alteration”.

Effects of the “Non-Log calibration” on the calculated HR%-values for the already given examples and for the HR%-range are listed in Table 8 & 9 and displayed in Picture 4.

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Table 8: Values of HR% of the same chord combinations of Table 3 but calculated with an “altered calibration” in the Sixth mathematical procedure (= “Non-Log calibration” see text). The “Non-Log calibration” of the mathematical model results in different HR% values for all chord-combinations (bold letters) vs. the “unaltered calibration”.

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Table 9: Maximum and minimum HR%-values for two chords with varying differences in the base note calculated similar as in Table 5 but using a “altered calibration” (= “Non-Log calibration”) of the mathematical model (see details in the text).

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Picture 4: Theoretically possible HR%-range (blue area) as a function of the minimal difference in the base-tones of two chords using the “Non-Log calibration” of the mathematical model as described in the text (see corresponding values in Table 9).

The “Non-Log calibration” creates significant changes in the HR%-values for different chord combinations vs. the “unaltered calibration (see Table 8 vs. Table 5). It is obvious that the HR%-values for the chord-combinations of G7-Cmaj7 and Dm7-G7 drop from 60.8% and 56% to 28.6% and 24.5% by using the “Non-Log-calibration”. This very low values in harmonic relatedness for these two chord combinations are not in line with the expectations of the author (see also Discussion below).

Ideas to understand, interpret and use the results of the mathematical model on harmonic relatedness of chords in a cadence

In this section the “unaltered calibration” is the base for all calculations. The reason for choosing this type of calibration will be explained and discussed in the “Discussion” section of this paper. A very good but also simple reason for using the “unaltered calibration” is that the results of the calculations seems to be in line with the general expectations of musicians. It is evident that this argument is already an interpretation and not solid but in combination with other arguments (see “Discussion”) it might be a meaningful approach.

Using the mathematical model on the popular music piece “Obladi-Oblada” published by the Beatles results in HR%-, DM- and HarmoT-values listed in Table 9 and displayed in Pictures 5 and 6.

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Table 9: Calculated HR%-B, HR-P%, DM and HarmoT values for the cadence of the popular song “Obladi-Oblada” using the above described mathematical procedure (unaltered calibration). The base chord of this music piece is Amaj7.

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Picture 5: The calculated values for HR%-B and HR%-P for each chord of the cadence from Obladi-Oblada are shown (for data see Table 9).

It is obvious that neither HR%-B nor HR%-P values drop below 55% during the entire cadence (see Picture 5). As a logical consequence the DM and HarmoT values reach a maximum of only 45% but drop two times to approx. 10% for DM and five-times to 0% for HarmoT (see Picture 6). The points with very low values for DM and HarmoT mark those chords where the harmonic tension is solved and a harmonic relaxation is happening. It is also worth to note that there is a constant level of highest and lowest HarmoT values during the entire cadence of Obladi-Oblada (see dashed blue lines in Picture 6). This can be understood as a sequence of multiple two step-repeats. Step1: Generating a certain intensity of harmonic tension followed by Step 2: Fully relaxation of the generated harmonic tension.

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Picture 6: The calculated values for DM (solid grey-line) and HarmoT (solid blue line) for each chord of the cadence from Obladi-Oblada are shown (for data see Table 9). The dashed lines mark the levels of the highest and lowest values for HarmoT.

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Picture 7: Calculated values for HR%-B and HR%-P for each chord of the cadence from Giant Steps by J.Coltrane. The base chord of the cadence is Bbmaj7.

In contrast to Obladi-Oblada as a typical Pop song the same calculation and analysis can be performed with the cadence published in the lead sheet of the Jazz-standard “Giant Steps” by J.Coltrane. The respective HR%-values as well as the values for DM and HarmoT of this cadence are shown in Pictures 7 and 8.

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Picture 8: The calculated values for DM (solid grey-line) and HarmoT (solid blue line) for each chord of the cadence from Giant Steps are shown. The dashed lines mark the development of the levels of the highest and lowest HamoT values.

The graphs (Pictures 5-8) clearly demonstrate significant differences in the harmonic concept of the two music pieces. During the entire cadence of Obladi-Oblada the harmonic tension drops repetitively from a certain HarmoT level (approx. 45%) to full relaxation (HarmoT=0%) and moves up to HarmoT≈45% again (see the two parallel dashed lines in Picture 6). In Giant steps the highest reached HarmoT and DM levels are much higher than the levels for these parameters in Obladi-Oblada and also the lowest level of these parameters are even higher in Giant Steps and never fall below 10%. The upper dashed line in Picture 8 demonstrates the development of the maximum of the harmonic tension level in the cadence of Giant Steps: the maximum HarmoT values move up during the cadence from 50% to nearly 80% to drop to a value around 60% at the end of the cadence. The range of the DM and HarmoT values (highest value reached vs. lowest value of this parameter in the cadence) in both songs is also different. For Obladi-Oblada the ranges are: DM-range = 41.4%; HarmoT-range = 45% whereas for Giant Steps the ranges are: DM-range = 51.9%; HarmoT-range: 61.5%. It can be assumed that both very popular songs have been seriously composed with the objective to combine the melody of the song with a well-fitting harmonic concept. According to reports J.Coltrane explained the name of the song as a description of constantly using large intervals (thirds and fourth = Giant Steps) in the baseline instead of the regular fourth and half-steps. As the baseline defines the chords, it is clear that J.Coltrane was aiming for this high harmonic tensions within Giant steps. In contrast to Giant Steps, Obladi-Oblada was composed as a Pop song. So it should generate some harmonic tension to make the song attractive, but the harmonic concept should not be to complex and should contain sections of simple harmonic repetition as this allows the listener to anticipate “what will come next” – an important feature for a so-called “Evergreen” which Obladi-Oblada definitely is.

Another example to be explored here is the short music piece “Pavane of Lord Salisbury” originally composed by Orlando Gibbons and interpreted by Glen Gould (Ref: 12, 13). As this has been originally composed by O.Gibbons as a polyphonic music piece, there are obvious differences to the clear homo- or monophonic modern songs presented above. An obvious difference is that the cadence of this “old” music piece has to be uncovered first, as in the time of O.Gibbons the modern theory of harmony was not invented yet and so the cadence has not been defined by the composer. Based on the version of G.Could certain chords could be identified and so a cadence with modern chords could be defined. The analysis of the cadence of the first part of this originally polyphonic music piece using the above described mathematical calculation gave the results displayed in Pictures 9 & 10.

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Picture 9: Calculated values for HR%-B and HR%-P for each chord of the cadence from “Pavane of Lord Salisbury” –Part 1 originally composed by O.Gibbons and interpreted by G.Gould. The identified base-chord of this music piece seems to be Am7.

From Picture 10 it can be deducted that the harmonic concept in Pavane-Part1 differs in various aspects from the harmonic concepts of the other music pieces given as examples in this publication:

1) The DM values (Disharmonic movement) stay over the entire cadence on a very similar level (see grey line in Picture 10) resulting in an average of 40.4% with an SD (standard deviation) of + 8.7% (data not shown). This can be interpreted as a “constant disharmonic impulse” at each chord-change.
2) At two distinct points in the first half of the music piece the cadence returns to the base-chord Am7 (HarmoT-values = 0) but definitely in a less repetitive manner as in Obladi-Oblada. So the listener might have the good opportunity to “recognize and learn” the base-chord and the related basic-scale (A-minor) at the beginning of this short music piece.
3) But from the middle of the music piece (= middle of the cadence) the cadence never returns or even comes close to the base-chord (see Picture 9) which generates an increasing level of tension (see development of DM & HarmoT in Picture 10) until the end of the piece.

Abbildung in dieser Leseprobe nicht enthalten

Picture 10: Calculated values for DM (solid grey-line) and HarmoT (solid blue line) for each chord of the cadence from “Pavane of Lord Salisbury” (O.Gibbons / G.Gould). The dashed blue lines mark the development of the levels of the highest and lowest HarmoT values. The dotted grey line marks the linear trend of the calculated DM values during the cadence.

Discussion:

The aim of this publication is to introduce a simple mathematical model which a) allows the inexperienced musician or student to develop an idea of the harmonic concept of a short music piece based on the respective cadence and b) to use this model as help and support in the creative process of composing and arranging. By discussing and answering the following four questions this objective might be achieved:

1) Is there a need for a mathematical model which describes harmonic concepts of small music pieces in a rather simplified way?
2) What is the right calibration of the model and does the model deliver results which are in line with the expectations of musicians and composers?
3) How can the model help to analyze, understand and interpret the harmonic concept of a short music piece based on the respective cadence?
4) How can the model help in generating meaningful chord progressions and support the process of composing or arranging music?

Discussing Question 1:

In general, a recognized or identified gap is the impetus to generate the missing. So what is missing in music theory? In the process of teaching music and also in the literature on music theory it is obvious that the idea of creating tension within the music is an important metaphor for the emotional effect of music. Creating tension in the right way and right intensity within a music piece is of great importance for the composer and musician to generate the desired emotional affection of the listeners. There are some attempts to quantify the extent of tension generated by certain chord progressions (Ref. 7, 8). In teaching lessons and in the teaching literature you find expressions like “high tension”, “increasing tension”, “low tension”, “no tension” or even “solved tension”. But you find hardly any explanations on the quantity of the tension which would allow describing “how high” or “how low” the tension is and “from which and to which level” the tension will increase”. At the moment the musician or composer is learning this idea of tension in music she/he must develop an own intrinsic scale for this “musical tension” or she/he has to recognize tension “instinctively”. There is no absolute or fixed scale which gives orientation or helps to develop this “intrinsic scale”. It can be concluded that an experienced musician and composer has such an intrinsic scale and that this is of fundamental importance for the implementation of form and structure in his creative work. Beside her/his expertise and skills in playing an instrument or creating music the intrinsic scale on musical tension is of immense importance for the individual musical output of a musician or composer. On her/his way to develop her/his intrinsic scale on musical tension the student or “younger” musician or composer is searching for something like a guidance and support in this process. Here the proposed and above demonstrated mathematical model to analyze the harmonic concept of a tonal music piece based on the cadence can be the assistance which the student or young musician/composer is searching for.

But also an experienced musician or composer may use this model to “verify” her/his ideas or just to get additional creative impulses when she/he is considering various opportunities in her/his play or compositions. Although the model does not give “ultimate statements on tension” in a cadence, it offers a tool which allows the user to quantify “tension effects” in a cadence. The opportunity to change the calibration of the mathematical model allows further to adapt the model according to the needs, expectations and impressions of the user.

Discussing Question 2:

As in any mathematical model the input-parameters (here: chords of a cadence) generate a certain output of the model (here: parameters like HR, DM and HarmoT). If the calculation processes in the model are not adequate the output will be useless as it is not in line with the expectations of the user and so it simply does not help or does not provide additional useful information. So the calibration of the model is an individual task which is depending on the expectations of the user concerning output-values of the model. In this publication 3 different types of calibrations of the introduced mathematical model have been demonstrated as well as the calibration-related effects on the output of the model (e.g. see Pictures 1, 3, 4 and Tables 3 & 5-9). It could be argued that the model will deliver any kind of output or result depending on the type of calibration – so the model itself is useless. In fact the calibration process is critical for the validity of the results – and the only way to validate the model (with a certain calibration) is to compare the results provided by the model with the expectations of the user. So if the user “expects” or “defines for himself” that a certain chord progression should generate a significant harmonic tension, the model should deliver a respective output. And if the user expects non or limited harmonic tension within a chord progression the output of the model (with a certain calibration) should deliver this expected result. Following these requirements the user will be in the position to use his “calibration-points” (calibration point = input parameter of the model which should deliver a specific and expected outcome) to develop and select a “calibration-type” for the mathematical model which forces the model to give the adequate output. The author of this publication has defined “his individual calibration type” of the model as described above in: Method of calculation of relatedness (%) of different chords.

Here are some arguments why the author favors “his individual calibration type”:

1) The range for HR% for two chords differing in their base-tone by 0-3 tones spans the entire scale from 0-100%. (see Picture1).
2) The highest values for HR% are nearly a linear function of the difference of the base tones and the lowest values for HR% exhibit the same phenomenon.
3) The difference between the highest value and the lowest value for HR% of two chords are very similar for all differences in the base-tones of the two chords.

The two altered calibration-types demonstrated in this publication do not deliver the above mentioned phenomena which result in the following striking disadvantages (according to the expectations of the author!):

1) Using the calibration-type as demonstrated in Picture 3 (“altered calibration”) can result in the same value for HR% = 0 for chords which differ by only 0.5 in the base tone but also for chords which differ by 1, 1.5, 2, 2.5 and even 3 in the base-tone. This is in contrast to the expectations of the author as an increasing difference in the base tone of two chords should correlate with a decreasing value for HR%.
2) Using the calibration-type as demonstrated in Picture 4 (“Non-Log calibration”) results in an HR%-value for the chords G7-Cmaj7 of 28.6% (see Table 8) vs 60.8% for the calibration-type preferred by the author (see Table 3). The scales of both chords contain the identical tones but differ in their base-tone. So the HR%-value should significantly differ from 100% (as Cmaj7 could be the tonic and G7 would reflect the dominant) but a value of HR% = 28.6% would be much too low (according to the expectations of the author).

In conclusion there is no correct or incorrect calibration-type of the model. The general aim is to find a calibration-type which reflects the individual expectations of the user concerning the output of the mathematical model.

Discussing Question 3:

Various methods to analyze music pieces of different types and styles (modern as well as classical music) have been introduced and used in the last decades in numerous publications. So it seems that there is no need for an additional tool or method for analyzing harmonic concepts and relationships in music pieces. But according to the knowledge of the author so far only a few attempts have been introduced or published which try to quantify harmonic relatedness and tension of the chords and used scales in short music pieces – and those concepts are rather complex. The mathematical model introduced in this publication offers an “easy to execute” approach to attribute certain and concrete values (in %) of harmonic tensions or harmonic and disharmonic movements to certain chord progressions and cadences. Therefore, a quantitative description of certain harmonic parameters within a cadence of a small music piece is possible and allows a new and different view on the harmonic concept of a music piece compared to known qualitative analysis

Looking at the example “Giant Steps” by J.Coltrane it might get clearer how the mathematical model generates a new and different view on the harmonic concept of a music piece. The idea of “large steps” in the baseline of this music piece has been responsible for the name “Giant steps” and massive changes in the values for the parameters HR%-B and HR%-P clearly demonstrate and visualize this harmonic concept (see Picture 9). By using further derived parameters (DM and HarmoT) an additional view on the harmonic concept of Giant Steps is given as demonstrated in Picture 10. Although there are massive changes in the harmonic tension (HarmoT) in the chord progression of Giant steps a clear climax in the disharmonic movement (DM) and the harmonic tension (HarmoT) is displayed (see Picture 8). Further it is obvious that the “baseline” (=lowest values) for DM and HarmoT is constant over the entire chord progression. It can be discussed whether J.Coltrance was targeting such a result which is displayed in Picture 10 or whether he just intuitively created this specific harmonic concept. The fact is that the analysis using the introduced mathematical model makes this unique harmonic concept visible. In contrast to “Giant steps” the song “Obladi Oblada” by the Beatles can be considered as a typical Pop song. The harmonic concept is totally different compared to “Giant steps” and again the DM and HarmoT values as displayed in Picture 6 of the related chord progression are a good demonstration of the harmonic concept of a successful Pop song. The DM and HarmoT values in “Obladi-Oblada” never exceed 45% in order to generate some tension but not irritating the listener by too much tension. The maximum and minimum DM and HarmoT values during the chord progression build a stable base- and top-line (see dashed lines in Picture 6) so the listener gets an idea what will happen next in terms of harmonic tension – and the listener will not be disappointed. One can argue that the chord progression in “Obladi Oblada” is so common and already understood so that there is no need for the mathematical model to display this fairly simple harmonic concept. This is true! The mathematical model is not developed to analyze and understand chord progressions of Pop or Folk songs which in most cases follow a fairly simple harmonic concept. The example just shows that the model a) is also functioning for cadences which do not need detailed analysis to understand the harmonic concept behind and b) makes the differences between harmonic concepts for music pieces of different types and styles directly and concretely visible.

The analysis of the music piece “Pavane of Lord Salisbury” by O.Gibbons and played/interpreted by G.Gould using the introduced mathematical model is another demonstration of what information the model can deliver concerning the harmonic concept of this music piece. This music piece is a good example of a “Counterpoint-type” music and therefore differs in style significantly from Giant Steps or Obladi Oblada. So it is not a surprise that displaying the DM and HarmoT values of the related chord progression (see Picture 10) offers the view on a unique harmonic concept of this music piece which differs significantly from Giant Steps and Obladi Oblada.

By looking at the given examples it can be concluded that the mathematical model introduced in this publication can be judged as a new process to analyze and visualize the harmonic concept of short music pieces and therefore delivers additional and valuable information.

Discussing Question 4:

Composing or arranging music is a creative task which also requires “craftsmanship” in such a way as the composer has to define and build the structure and form of the music piece to be composed or arranged. There are various methods, ways and guidelines (Ref. 1-5) to compose or arrange music and to define an interesting harmonic structure, but all those processes are highly individual tasks. So each musician or composer will choose her/his way to create music and may use different tools and methods to achieve the objective. When building the harmonic concept of a music piece (independently whether this is the initial composing-process or whether defining the harmonic structure is a following step) the creative person is setting the respective structure and form according to certain parameters which she/he judges as important. In this creative process the mathematical model may be helpful for the composer and musician. Especially by using and visualizing the parameters DM (= parameter for “disharmonic movement”) and HarmoT (= parameter for “harmonic tension”) for a composed chord progression the creator may get a good impression whether the harmonic concept, set by defining the cadence, is in line with her/his ideas and objectives. And if the composer recognizes deviations her/he may use the mathematical model to find alternatives in the cadence which fit better.

Giving concrete arguments to answer the question 4 is difficult as no examples for a composing process can be presented where the mathematical model was playing a role or was helpful – as the model is not known so far! So it is a theoretical discussion whether and how the model can help the composer in creating his harmonic concept. As it is the case with any “new process” or “invention” only time will tell which value the introduced mathematical model may have for composers and musicians. There are three strong benefits of the introduced mathematical model which may result in the usage of this model by musicians and composers in the near future:

1) The mathematical model is not in conflict, but in line with the existing theories on harmony of western music which build the base for various musical styles like classical music, Pop, Rock, Folk, Jazz and others.
2) The model offers various calibration opportunities which allow the user to fit the model and the related output according to her/his expectations or “calibration points” (see text).
3) Based on only a few and simple calculation procedures which can easily be transferred into an Excel sheet the model is easy to set up and use.

So far the model will not work and was not designed for atonal music structures or tonal systems or concepts which are not based on western music. But future extensions and adaptations of the model may allow to use the model for other tonal systems as well.

Remark: Anyone who is interested in the “Excel-sheet of the mathematical model” and in the “raw data” presented in this publication may ask the author by email (soundanalyse@ok.de) to provide the data.

References

1) Jazz Harmonielehre; Axel Jungbluth; Schott Verlag; 2001; ISBN: 978-3-7957-8722-6.

2) Neue Jazz-Harmonielehre; Frank Sikora, Schott Verlag; 2003; ISBN: 978-3-7957-5124-1

3) Harmonielehre; Thomas Krämer; 2009; ISBN: 978-3-7651-0261-5

4) Lehrbuch der harmonischen Analyse; Thomas Krämer; 1997; ISBN: 978-3-7651-0305-6

5) Lexikon Musiktheorie; Thomas Krämer & Manfred Dinges; 2005; ISBN: 978-3-7651-0370-4.

6) A SURVEY OF CHORD DISTANCES WITH COMPARISON FOR CHORD ANALYSIS; Thomas Rocher, Matthias Robine, Pierre Hanna, Myriam Desainte-Catherine; 2010; https://www.researchgate.net/publication/265984596

7) Analysis of Chord Progression Data; Brandt Absolu, Tao Li, and Mitsunori Ogihara; Z.W. Ra´s and A.A. Wieczorkowska (Eds.): Adv. in Music Inform. Retrieval, SCI 274, pp. 165–184.; Springer-Verlag Berlin Heidelberg 2010; https://www.researchgate.net/publication/227323875

8) Automatic estimation of harmonic tension by distributed representation of chords; Ali Nikrang, David R. W. Sears and Gerhard Widmer; 2017; https://www.researchgate.net/publication/318205203

9) Common Chord Progressions and Feelings of Remembering; Ivan Jimenez, Tuire Kuusi and Christopher Doll; 2020; Music & Science Volume 3: 1–16; DOI: 10.1177/2059204320916849

10) Transformations in Tonal Jazz: ii–V Space; Michael McClimon; MTO Society for Music Theory; Volume 23, Number 1, March 2017

11) Modeling Diatonic, Acoustic, Hexatonic, and Octatonic Harmonies and Progressions in Two- and Three-Dimensional Pitch Spaces; or Jazz Harmony after 1960; Keith J. Waters and J. Kent Williams; MTO Society for Music Theory; Volume 16, Number 3, August 2010; http://www.mtosmt.org/issues/mto.10.16.3/mto.10.16.3.waters_williams.php

12) https://www.youtube.com/watch?v=WULDLz-WUxM

13) https://www.youtube.com/watch?v=p7DJhZeC9MQ

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