jonahdempcy.com // Jazz Lessons

Overtone Series and Western Harmony

2007-07-15T11:52:00.001-07:00

Harmonics and tuning

If the first 15 harmonics are transposed into the span of one octave, they approximate some of the notes in what the West has adopted as the chromatic scale based on the fundamental tone. The Western chromatic scale has been modified into twelve equal semitones, and in relation to that scale, many of the harmonics are slightly out of tune, and the 7th, 11th, and 13th harmonics are significantly so. In the late 1930s, composer Paul Hindemith ranked musical intervals according to their relative dissonance based on these and similar harmonic relationships.

Below is a comparison between the first 20 harmonics and their equivalent frequencies in the 12-tone equal-tempered scale. Orange-tinted fields highlight differences greater than 5 cents, which is the "just noticeable difference" for the human ear. (Because physical characteristics of musical instruments cause significant variations from these theoretical values, they should not be used for tuning without adjusting for those variations.)

Harmonic	Note	Variance
1st	C1	0 cents
2nd	C2	0 cents
3rd	G2	+2 cents
4th	C3	0 cents
5th	E3	−14 cents
6th	G3	+2 cents
7th	Bb3	−31 cents

Harmonic	Note	Variance
8th	C4	0 cents
9th	D4	+4 cents
10th	E4	−14 cents
11th	G♭4	−49 cents
12th	G4	+2 cents
13th	Ab4	+41 cents
14th	Bb4	−31 cents

Harmonic	Note	Variance
15th	B4	−12 cents
16th	C5	0 cents
17th	C#5	+5 cents
18th	D5	+4 cents
19th	D#5	−2 cents
20th	E5	−14 cents

Some theorists believe that the overtone series is the basis of the experience of consonance and dissonance, for example, one writer states that

"If two different notes are played simultaneously, the composite sound includes the harmonics of both notes. In musical intervals, one or more of those harmonics are common to both notes. For example, if C3 and G3 are sounding together, their harmonic series intersect at G4 (2nd of G3, 3rd of C3) and G5 (4th of G3, 6th of C3). If these common harmonics are at the same frequency or nearly so in both notes, the composite sound will seem harmonious. If their frequencies differ significantly, we tend to hear the notes as "out of tune" with each other. Tuning involves changing the fundamental pitch of one of the notes to control the relationship between these common harmonics"

However, this theory neglects the fact that several intervals made up of electronically generated pure sine waves (with no overtone content) are still perceived as consonant or dissonant. The frequencies of the overtone series, being a range of integral multiples of the fundamental frequency, are naturally related to each other by small whole number ratios and it is these small whole number ratios that are the basis of the consonance of musical intervals, for example, a perfect fifth, say 200 and 300 Hz (cycles per second), produces a combination tone of 100 Hz (the difference between 300 Hz and 200 Hz) that is, an octave below the lower note. This 100 Hz first order combination tone then interacts with both notes of the interval to produce second order combination tones of 200 (300-100) and 100 (200-100) Hz and, of course, all further nth order combination tones are all the same, being formed from various subtraction of 100, 200, and 300. When we contrast this with a dissonant interval such as a tritone (not tempered) with a frequency ratio of 7:5 we get, for example, 700-500=200 (1st order combination tone)and 500-200=300 (2nd order). The rest of the combination tones are octaves of 100 Hz so the 7:5 interval actually contains 4 notes: 100 Hz (and its octaves), 300 Hz, 500 Hz and 700 Hz. All the intervals succumb to similar analysis as been demonstrated by Paul Hindemith in his book, The Craft of Musical Composition.

From Wikipedia

Free Jazz Mix (2001)

2007-03-27T03:13:00.001-07:00

Here is an avant-garde experimental jazz mix that I did back in 2001. I was experimenting with layering songs, for instance, taking all of the songs from an album and playing them simultaneously. This mix was done digitally using around 20 instances of WinAmp playing MP3s off my hard drive. It was done live and there was no post-production or editing after the fact. It gets pretty chaotic sometimes but I think it is interesting to hear the ebb and flow of the music when this many sounds are combined. Sometimes it is utter chaos but other times there is a sort of "serendipity" that occurs. Listen and tell me what you think!

Free Jazz Mix
MP3 23 MB

Great Info on Jazz Standards

2007-02-27T16:01:00.001-08:00

This site has a lot of great information on jazz standards: www.jazzstandards.com

This is useful both for knowing the legality of playing/recording/releasing a song (i.e. whether the song is in the public domain or not) as well as for historical and research purposes.

There is also a ranking system which ostensibly ranks a jazz standard by popularity. I'm not sure how this is determined, whether it is by looking at sheer number of recordings of that song or if other factors are taken into account. But, it is interesting and seems at least generally accurate&emdash; that is, songs in the top 100 are obviously more popular than those ranked 800 or 900 &emdash; but I'm not sure if I agree that Body and Soul is more popular than All the Things You Are. Here are the top 10 according to jazzstandards.com:

Rank

Year

Title <- Sort contents by clicking on heading

1	1930	Body and Soul	C		J	R	S	A	O
2	1939	All the Things You Are	C	M	J	R	S	A	O
3	1935	Summertime	C		J	R	S	A	O
4	1944	'Round Midnight	C	M	J	R	S	A	O
5	1935	I Can't Get Started (with You)	C	M	J	R	S	A	O
6	1937	My Funny Valentine	C	M	J	R	S	A	O
7	1942	Lover Man (Oh, Where Can You Be)	C	M	J	R	S	A	O
8	1930	What Is This Thing Called Love?	C	M	J	R	S	A	O
9	1933	Yesterdays	C	M	J	R	S	A	O
10	1946	Stella By Starlight	C		J	R	S	A	O

Source: www.jazzstandards.com

Different Lands

2007-02-26T16:01:00.000-08:00

Different Lands is an interesting independent record label primarily focusing on world fusion, but releasing music ranging from acoustic improvisation to electronic music (curiously referred to as "computer music" on their site). There are MP3 downloads so you can hear samples of the tracks.

The Different Lands links page has some good sites for more info about world music, electronic music, independent record labels and other topics.

Press Release Submission on All About Jazz

2007-02-25T13:35:00.001-08:00

For all you jazz musicians out there, next time you have a release, make sure to post it on AllAboutJazz.com. You can use this handy form:

All About Jazz press release submission form

Listen to the Real Book:

2007-02-23T16:01:00.001-08:00

This is an excellent idea: Someone has gone through and taken the index of tunes from the "bootleg" Real Book we have all grown to know and love, and <em>linked the songs to MP3 samples</em> via Amazon.com. Great idea and a highly useful tool for learning new tracks. I used to check Amazon (well, CDNow.com which is currently an Amazon property) but this page now has direct links to the audio sample pages. Very useful and very cool.<br><br> Real Book Listen | Tune List

1 A Call For All Demons Sun Ra
2 A Child Is Born Thad Jones
3 A Fine Romance Jerome Kern
4 A Family Joy Michael Gibbs
6 A Foggy Day George Gershwin
7 A Night In Tunisia Dizzy Gillespie / Frank Paparelli
8 African Flower Duke Ellington
9 Afro Blue Mongo Santa Maria
10 Afternoon In Paris John Lewis
11 Airegin Sonny Rollins
etc...

___________________________
Sent by Jonah Dempcy via the free Email Scheduler service.
Register now at http://www.emailschedule.com

Expected vs. Unexpected

2007-02-10T22:09:00.000-08:00

I believe that a crucial part of making good, entertaining and enjoyable music is striking the balance between expected and unexpected. Regardless of other music content, if this balance is not struck, the audience will either be bored or annoyed. True exhilaration can be elicited through building excitement through repetition, building expectation and then surprising the listener. But, it is very difficult to get this just right. I believe that this is one of the key factors to "great" music -- that it is stable, predictable and repetitive enough to build energy but erratic and spontaneous enough to excite the listener.

Jazz Picante

2007-01-30T08:01:00.001-08:00

Nice jazz podcast/DJ mixes: Jazz Picante

Here's this week's episode: MP3

Rune Grammofon

2007-01-29T00:00:00.001-08:00

Description from the site www.runegrammofon.com:

Rune Grammofon is a record label dedicated to releasing work by the most adventurous and creative Norwegian artists and composers.
Being music enthusiasts almost to the point of absurdity, we don't want to limit ourselves to certain genres, as long as there's real heart and personality. With an increasing number of indifferent records bombarding the market place, we modestly aim to recapture the magic connected to the discovery of new artists and buying their records. Our aim is to put love and care into each release, giving it the best possible design and packaging. No plastic, ever. All releases come in the digipak format with exclusive design by brilliant designer Kim Hiorthøy.

Jeff Ballard

2007-01-28T05:00:00.001-08:00

Jeff Ballard is one of my favorite contemporary jazz drummers. He is spot-on and even when playing sparse and syncopated rhythms, tends to keep the energy level high and the rhythm driving. Of course, I don't want to over-simplify &emdash; Ballard is a versatile musician &emdash; but my favorite recordings of his are the ones that have a driving rhythm, either in a upper tempos or if they are a slower tune, by filling at double-time. Check out his website here:

www.jeffballard.com

Roy Hargrove - Nothing Serious

2007-01-27T15:59:00.000-08:00

I just picked up the 2006 release Nothing Serious by Roy Hargrove. It's a very articulate and professional post-bop recording which, although not entirely innovative, is still a solid album and recommended listening. I feel it's important to listen to both experimental "outside" jazz as well as recordings which are more typical of the idiom.

In any case, if you want to hear a solid modern jazz recording, check out Roy Hargrove - Nothing Serious.

Subs for V7 - I-vi-ii-V7 practice

2006-09-23T04:30:00.000-07:00

Subs for V7 in the key of C

Play straight through or mix and match ...

| C | A-7 | D-7 | G7sus4 | C | A-7 | D-7 | Bb7sus4 |
| C | A-7 | D-7 | Db7sus4 | C | A-7 | D-7 | E7sus4 |

| C | A-7 | D-7 | G7sus4(9) | C | A-7 | D-7 | Bb7sus4(9) |
| C | A-7 | D-7 | Db7sus4(9) | C | A-7 | D-7 | E7sus4(9) |

| C | A-7 | D-7 | G7sus4^b9 | C | A-7 | D-7 | Bb7sus4^b9 |
| C | A-7 | D-7 | Db7sus4^b9 | C | A-7 | D-7 | E7sus4^b9 |

| C | A-7 | D-7 | G7Alt. | C | A-7 | D-7 | Bb7Alt. |
| C | A-7 | D-7 | Db7Alt. | C | A-7 | D-7 | E7Alt. |

| C | A-7 | D-7 | G13b9 | C | A-7 | D-7 | Bb7sus4b9 |
| C | A-7 | D-7 | Db13b9 | C | A-7 | D-7 | E7sus4b9 |

| C | A-7 | D-7 | G7#11 | C | A-7 | D-7 | Bb7#11 |
| C | A-7 | D-7 | Db7#11 | C | A-7 | D-7 | E7#11 |

Here is a chart for all of the above chord qualities showing possible chord voicings and scales:

G7sus4

Chord voicings
Shell: G-C-F
...... 1-4-b7

Root: G-C-D-F
..... 1-4-5-b7

Rootless Shell: D--F--C
............... 5--b7-4

Scales
5th mode of C Major - G A B C# D E F
..................... 1 2 3 4. 5 6 b7

G7sus4(9)

Chord voicings
F Maj. Triad: A-C-F
............. 2-4-b7

Rootless: D--F--A--C
......... 5--b7-9--11*
.. (*i.e. 5--b7-9--4)

Scales
5th mode of C Major - G A B C# D E F
..................... 1 2 3 4. 5 6 b7

G7#11

Chord voicings
Rootless 1: D--F--A--C#
........... 5--b7-9--#11

Rootless 1 Shell: D--F--C#
................. 5--b7-#11

Rootless 2: F--A--C#--E
........... b7-9--#11-13

Rootless 2 Shell: F--A--E
................. b7-9--13
or
Rootless 2 Shell: F---C#---E
................. b7--#11--13

Rootless 3: F--B--C#--E
........... b7-3--#11-13

Scales:
5th mode of D Melodic Minor - G A B C# D E F
............................. 1 2 3 #4 5 6 b7

Coming soon:

G7sus4b9, G7Alt., G13b9 - Chord voicings and scales

Kurt Rosenwinkel - Zhivago

2006-09-06T01:56:00.000-07:00

Kurt Rosenwinkel (gtr) playing live ~2000-01 with Mark Turner (sax), Ben Street (bass) and Jeff Ballard (drums). Rosenwinkel is one of my favorite contemporary jazz recording artists and I'm always eager to hear his new material. He began making forays into electronic jazz with 2003's Heartcore but returned to a more traditional instrumentation with his latest album, Deep Song (2005). Kurt Rosenwinkel is seemingly always open to experimentation, however, as his recent collaboration with Q-Tip (A Tribe Called Quest) evinces. Check out this great song from my favorite album of his, The Next Step (2001).

Rhythm Changes Piano Solo

2006-09-05T20:09:00.000-07:00

Jazz piano solo on rhythm changes. Performed using Native Instruments' Akoustik Piano playing along to a backing track by Jamey Aebersold. The solo was slightly sped up (~15%) after it was recorded, giving the tune more of a bebop feel.

Tags: acoustic piano, aebersold, jazz, jazz solo, piano, rhythm changes, solo

The backing track is copyright Jamey Aebersold and excerpted here for educational purposes only, under the belief that this constitutes fair use. However, all rights are reserved by the original copyright holder. Please support Jamey Aebersold by buying his great play-alongs! www.aebersold.com

Shell Voicings on Piano

2006-08-28T00:13:00.000-07:00

One of the habits that some students of jazz have is to play 4-note voicings all the time, leading to a full sound that can be clunky or out of place at times. While it is certainly appropriate to play full voicings at times, it is necessary to have both sparse and full voicings in one's arsenal. With that, I offer a simple method of creating sparse shell voicings on piano. You can adapt this to other instruments quite easily, but for the sake of this discussion, we will be focusing on keyboard instruments.

To start with, everyone learns the basic 4-note voicings for a given chord, e.g. C-E-G-B for C major 7. Most people will be familiar with this in at least the root position and 2nd inversion (in this case, G-B-C-E) and perhaps also 3rd inversion (B-C-E-G). If you aren't familiar with basic 4-note voicings for chords, then I recommend brushing up before continuing this lesson.

Root Position Shell Voicings

The first shell voicings to learn are root position voicings for all chord types. For all chord types except Minor7 b5 (half-diminished), the shell voicing consists of the 1-3-7 or 1-3-6 of the chord. For Minor7 b5, you can think of it as an Eb-6 and voice it accordingly. (1-3-6 = Eb Gb C)

Here are a few examples:

C Major 7 = C-E-B

C Major 6 = C-E-A

D-7 = D-F-C

D-7b5 = F-Ab-C

G7 = G-B-F

So, we can call these root position shell voicings. (Although the D-7b5 is technically a first inversion shell voicing, it could also be considered a root position shell voicing for F-6).

2nd Inversion Shell Voicings

Next to learn are 2nd inversion shell voicings. These begin from the 5th of a chord and contain the 5-7-3 of the chord, except for 6 chords which contain 6-1-5 of a chord (3rd inversion).

Here are some examples:

C Major 7 = G-B-E

C Major 6 = A-C-G

D-7 = A-C-F

D-7b5 = Ab-C-F

G7 = D-F-B

The reason that 6 chords have a special rule is because you wouldn't want to play 2nd inversion (5-6-3) because of the major 2nd interval between the 5 and 6. It's preferable to play 6-1-5 which is the 3rd inversion. (You could also look at this as root position of the relative minor, e.g. the notes A-C-G are root position A minor 7, as well as being 3rd inversion C major 6).

Here's an example of a practice you could do that mixes and matches these shell voicings:

C: C-E-B
A-7: A-C-G
D-7: A-C-F
G7: G-B-F
C: G-B-E
A-7: E-G-C
D-7: D-F-C
G7: D-F-B
C: C-E-B

(repeats)

You can practice this in all keys, or adapt the voice leading technique to a song or chord progression of your choosing.

3-7 Voicings

Another useful sparse voicing which might be considered a shell voicing is to simply play the 3 and 7 of a chord.

Here's the same progression as above, playing only 3-7 or 7-3 for chord voicings.

C: E-B
A-7: C-G
D-7: C-F
G7: B-F
C: B-E
A-7: G-C
D-7: F-C
G7: F-B
C: E-B

These shell voicings can be quite useful, either for accompanying someone without stepping on their toes or for adding contrast to full voicings.

Adapting Shell Voicings For Extended and Altered Chords

While not technically shell voicings (as the name shell voicing generally implies the rudimentary notes of the chord), you can also adapt these voicings for 9, 13 and altered chords.

This will be explored more in-depth in another post, but for now, suffice it to say that any of these shell voicings can be played as a rootless voicing in another key. For example, the root position shell voicing for C Major 7 (C-E-B) can be played as a rootless voicing for A-9. Played as an A minor 9, the notes C-E-B correspond to the 3-7-9 of the chord.

There are too many correlations to enumerate here, but I recommend trying all shell voicings in other contexts (over other root notes) and exploring to find interesting combinations. You can simply move the root note through all steps of the scale and see if anything works. For instance, taking the root position C Major 7 shell voicing, play C-E-B in the right hand and test it against all 7 possible root notes in the key of C. Many of the voicings won't work, but you may find interesting combinations that do work. In this example, C-E-B could also be the 5-7-#11 of an F Major 7 #11 chord, or it could be the 7-9-13 of a D7sus4(13) chord.

I will go into depth with some of the more interesting combinations in a later post, but for now, go ahead and familiarize yourself with these shell voicings, or if you are already familiar with them, try combining them with different root notes to generate rootless voicings with extensions and altered notes.

Giant Steps Played by Robot

2006-08-23T16:09:00.000-07:00

I came across this humorous and intriguing video clip while reading David Valdez' blog, Casa Valdez Studios:

You won't believe this video that Dan Gaynor just turned me on to. I've
heard tons of guys play Giant Steps like this at Berklee but this guy
takes the cake. Here is the next generation of young unswinging tenor
players. Things are looking pretty grim for the future, but at least
we'll have live Jazz in space.

Link: Giant Robot Steps

Barry Harris Subs - Clarification

2006-08-20T10:21:00.000-07:00

[reposted from comments on "Barry Harris Substitutions"]

Bobby said...: I see how all the final chords resolve to Cmaj except the last. G7 to C, V7 to I; Bb7 to C , fine bVII7 to I; Db7 to C , ok bII7 to I; and I'm fine with adding a II chord before each dominant, but E7 resolving to C? This I don't really get, either theoretically or aurally.
I responded...: You can think of E7 resolving to C in a number of ways.

For one thing, E7 regularly resolves to A-7 which is an inversion of C6. Also, the chord E7 can be played along with a shell G7 to yield the G13b9 chord.

You could play the 5th mode of A Harmonic Minor over this chord (E F G# A B C D) and it would sound quite fine.

But, I'm guessing you don't have a problem this kind of E7b9 -> C transition, but rather E7#11 -> C. E7#11, to which you would play 4th mode of B Melodic Minor as a scale (E F# G# A# B C# D), is somewhat difficult to make work. E7#11 to C is "out there" in the same way that E7#11 to A-7 sounds a bit odd.

However, you can still make it work. Here's an example of a lick in 8th-notes that uses this substitution:

Chord changes:
E7#11 | C

Descending 8th notes:
A# F# D C# B A# G# F# | G

Another way to look at it would be as an Ab-7b5 or Bb7sus4b9 chord, since both of those share the key of B Melodic Minor with E7#11. Bb7->C is quite common and sus4b9 chords are often substituted for unaltered dominant7 chords so it's not unheard of ...

On the topic of why this could work theoretically, it's because E7#11 shares 2 of the 4 notes comprising the diminished 7th "backbone" of the C major key. According to Barry Harris, a major key is comprised of a major 6 chord from the root and a diminished 7 chord from the leading tone. Hence, the key of C is comprised of C6 and Bdim7. This yields an 8-note scale, C D E F G G# A B, which Barry Harris considers superior to the 7-note scale taught in mainstream music theory.

It also accounts for a second tritone that resolves. Whereas conventional music theory only examines the resolution of the tritone of the 4-7, Barry Harris' theory includes the resolution of the tritone of the 2-b6 as well.

In conventional music theory, it is said that the 4 resolves to the 3 while the 7 resolves to the 1. Barry Harris' theory simply states that any of the 4 tones of the diminished backbone of a key will resolve to a neighboring tone from the major 6 chord from the tonic. Here's a table showing this:

Dim7 from 7th | Maj6 from root

B | A, C

D | C, E

F | E, G

G# | G, A

This table is for the key of C. The left column consists of the notes of the Bdim7 chord and the right column has notes from the C6 chord. This table illustrates how notes from the Bdim7 chord resolve to neighboring tones from the C6 chord.

To tie this back in to the discussion of E7#11 resolving to C, we simply say that it is not as strong of a resolution as E7b9 to C, because only 3 of the 4 notes from the diminished backbone are present. In the case of E7b9 (or G7b9, et al.) the notes B, D, F and G# are present, so it resolves strongly to C. In the case of E7#11, only B, D and G# are present, so the resolution is weaker. But, it is still possible to make it work. Just experiment at home first before trying it on the fly at a gig!

Also, of course, you would probably not want to use this reharmonization while accompanying someone. It would clash if they played F or G, both of which are commonly played over G7. So, I would only play E7#11 as a sub for G7 resolving to C when playing a solo.

I intend to upload audio examples of these subs shortly. In the meantime, check out the previous post on the topic ("Barry Harris Substitutions") and experiment with the various methods of reharmonization.

Dick Hyman Lesson on Art Tatum

2006-07-27T17:55:00.000-07:00

These are excerpts of Dick Hyman teaching some of the techniques and runs used by Art Tatum. There are some nice patterns and motifs featured, often surprisingly easy to play (though certainly difficult at the speed Tatum played them at!).

Art Tatum's lessons (introduction)
00:57

Art Tatum's lessons (G turn)
00:57

Art Tatum's lessons (descending pentatonic scales)
01:31

Art Tatum's lessons (displaced pentatonic scales)
00:44

Art Tatum's lessons (neighbor tone devices in Db)
02:48

Art Tatum's lessons (diatonic devices)
03:31

Portland Jazz Jams

2006-07-27T17:08:00.001-07:00

This is a good resource for NW jazz artists, as well as having a Jazz Knowledge Wiki that aims to provide a repository for jazz education. Check it out!

Art Tatum - Humouresque (video)

2006-07-27T17:08:00.000-07:00

Art Tatum- Humouresque
02:45

Art Tatum - Yesterdays (video)

2006-07-27T17:07:00.000-07:00

Art Tatum - Yesterdays (1954)
01:59

Barry Harris Substitutions

2006-07-09T22:30:00.000-07:00

Here is a table of substitutions using Barry Harris' method of transposition by minor 3rds:

*iii*	*VI7*	ii	V7
E-7	A7	D-7	G7
G-7	C7	F-7	Bb7
Bb-7	Eb7	Ab-7	Db7
Db-7	Gb7	B-7	E7

Shown here are iii-VI7-ii-V7 progressions in 4 keys. (C, Eb, Gb and A, respectively). You can freely substitute any of the chords from the vertical rows, e.g. you can substitute F-7 for D-7.

Here are some example chord progressions using this technique:

Mixed ii-V7s

|| E-7 | A7 | F-7 | Bb7 ||

|| Bb-7 | Eb7 | B-7 | E7 ||

|| G-7 | C7 | Ab-7 | Db7 ||

|| Db-7 | Gb7 | B-7 | E7 ||

Chromatic ii-V7s

|| E-7 | A7 | Ab-7 | Db7 ||

|| G-7 | C7 | B-7 | E-7 ||

|| Bb-7 | Eb7 | D-7 | G7 ||

|| Db-7 | Gb7 | F-7 | Bb7 ||

Minor7 to Dom.7 a Whole Step Up

|| E-7 | Gb7 | F-7 | G7 ||

|| G-7 | A7 | Ab-7 | Bb7 ||

|| Bb-7 | C7 | B-7 | Db7 ||

|| Db-7 | F7 | D-7 | E7 ||

For the sake of simplicity, we can consider these as being in the key of C, though in actuality these progressions can be used as substitutions over iii-VI7-ii-V7 progressions in the keys of C, Eb, Gb and A.

This is perfect for turnarounds-- there are 256 possible chord progressions generated by the above table. So, dig in and explore the possibilities!

The Geometry of Harmony

2006-07-07T19:29:00.000-07:00

Excerpt from THE GEOMETRY OF MUSICAL CHORDS by Dmitri Tymoczko:

Musical chords have a non-Euclidean geometry that has been exploited by Western composers in many different styles. A musical chord can be represented as a point in a geometrical space called an orbifold. Line segments represent mappings from the notes of one chord to those of another. Composers in a wide range of styles have exploited the non-Euclidean geometry of these spaces, typically by utilizing short line segments between structurally similar chords. Such line segments exist only when chords are nearly symmetrical under translation, reflection, or permutation. Paradigmatically consonant and dissonant chords possess different near-symmetries, and suggest different musical uses.

Excerpt from Princeton Univ. page:

Composer reveals musical chords' hidden geometry

by Chad Boutin · Posted July 6, 2006; 02:00 p.m.

Composers often speak of fitting chords and melodies together, as though sounds were physical objects with geometric shape -- and now a Princeton University musician has shown that advanced geometry actually does offer a tool for understanding musical structure.

In an attempt to answer age-old questions about how basic musical elements work together, Dmitri Tymoczko has journeyed far into the land of topology and non-Euclidean geometry, and has returned with a new -- and comparatively simple -- way of understanding how music is constructed. His findings have resulted in the first paper on music theory that the journal Science has printed in its 127-year history, and may provide an additional theoretical tool for composers searching for that elusive next chord...

Making graphical representations of musical ideas is not itself a new idea. Even most nonmusicians are familiar with the five-line musical staff, on which the notes that appear physically higher represent sounds that have higher pitch. Other common representations include the circle of fifths, which illustrates the relationships between the 12 notes in the chromatic scale as though they were the 12 hours on a clock's face.

"Tools like these have helped people understand music with both their ears and their eyes for generations," Tymoczko said. "But music has expanded a great deal in the past hundred years. We are interested in a much broader range of harmonies and melodies than previous composers were. With all these new musical developments, I thought it would be useful to search for a framework that could help us understand music regardless of style."

Traditional music theory required that harmonically acceptable chords be constructed from notes separated by a couple of scale steps -- such as the major chord, whose three notes comprise the first, third and fifth elements in the major scale, forming a familiar harmony that most audiences find easy to enjoy. Many 20th-century composers abandoned this requirement, however. Modern chords are often constructed of notes that sit right next to one another on the keyboard, forming "clusters" -- dissonant by traditional standards -- that to this day often challenge listeners' ears.

"Western music theory has developed impressive tools for thinking about traditional harmonies, but it doesn’t have the same sophisticated tools for thinking about these newer chords," Tymoczko said. "This led me to want to develop a general geometrical model in which every conceivable chord is represented by a point in space. That way, if you hear any sequence of chords, no matter how unfamiliar, you can still represent it as a series of points in the space. To understand the melodic relationship between these chords, you connect the points with lines that represent how you have to change their notes to get from one chord to the next."

Here's a video sample of Tymoczko's system in action: Link

I wonder if software will be released to the general public that will be able to MIDI files and display geometric shapes as in the video sample. Seeing a visual representation of the music like this with perfect mathematical accuracy is highly preferable to seeing music visualizations purely based on amplitude or frequency response.

Patterns Using Stowell's Dom.7 Reharmonizations

2006-06-23T21:19:00.000-07:00

This is a follow-up to the previous post on John Stowell's dominant 7 reharmonizations. I recommend reading it first.

Here are some stock patterns that utilize Stowell's dom.7 reharmonizations. These are substitutions for a standard C7 resolving to F Major7.

The first example begins with a chord on the first downbeat. The other examples begin on the first upbeat, which is the "correct" way to play this lick. I simply began the first example with a chord on the downbeat to assist the ear in hearing the proper rhythm.

C7#11 to FMaj7

C7b6 to FMaj7

C7sus4b9 to FMaj7

C-7b5 to FMaj7

C7Alt. to FMaj7

Here are the correlations between the above chord substitutions and their respective melodic minor scales:

G melodic minor- one tension (#11) = C7#11
F melodic minor- one tension (b13) = C7b6
Bb melodic minor- two tensions (b9, #9) = C7sus4b9
Eb melodic minor- three tensions (#9, #11, b13) = C-7b5
C# melodic minor- four tensions (b9, #9, #11, b13) = C7Alt.

John Stowell Dominant 7 Harmonizations

2006-04-17T16:42:00.000-07:00

David Valdez: John Stowell's Jazz guitar mastery book

Over a C7 chord you can play:
G melodic minor- one tension (#11)
F melodic minor- one tension (b13)
Bb melodic minor- two tensions (b9, #9)
Eb melodic minor- three tensions (#9, #11, b13)
C# melodic minor- four tensions (b9, #9, #11, b13)
He arpeggiates these scales to obtain his signature open, modern and often vertical sound. John also explains in his book/DVD how to use the melodic minor scale (a third down) over Major chords. This gives us the Augmented Major scale, which has a #11 and a #5. John also uses the Melodic Minor scale a whole step up to obtain a Major scale with a b9. These are obviously very spicy scale choices over a Major 7th chord but John really makes them work in a way that isn't jarring to the ears.

G melodic minor over C7 is what Mark Levine calls the Lydian Dominant scale, which is, numerically: 1 2 3 #4 5 6 b7.

F melodic minor (along with F harmonic minor) is one of the default choices over a Dom.7 chord resolving to minor, or, for slightly more interest, resolving to major.

Bb melodic minor is essentially reharmonizing the C7 to C7sus4b9, yielding the Phrygian Nat.6 scale: 1 b2 b3 4 5 6 b7. This could also be used over C-7 iv chords, eg: the first chord in the sequence C-7 F7 Bb-7 Eb7 Ab.

Sus4b9 chords are ubiquitous throughout modern jazz, with players like Herbie Hancock and McCoy Tyner using them heavily. Often, they are used over pedal basslines, such as in the chord progression:

|| G7sus4b9 | Gsus13 | Gsus4b9 | G13b9 | C6/9 ||

Eb melodic minor reharmonizes the C7 to C-7b5, a less-common substitution but not unheard-of. Incidentally, when I attended a John Stowell / Kurt Rosenwinkel master class in Seattle a couple of years ago, Rosenwinkel said that he freely substitutes Dom.7, minor7 and minor7b5 chords for each other. So, definitely try throwing in chords of different qualities with the same root, though I wouldn't try this with Major7 chords.

Here's an example of reharmonizing a basic iii-vi-ii-V7 progression in the key of C:

|| E-7 A-7 | D-7 G7 | E-7b5 A7 | D-7b5 G7 | E7 A-7b5 | D7 G7 | E7 A7 D-7b5 G7 | C6/9 ||

C# melodic minor is commonly known as the altered scale. It is ubiquitous in modern jazz and has perhaps achieved over-use as a "go-to" scale for altered chords. Nevertheless, it is a useful scale for using altered harmony. The scale degrees are 1 b2 b3 3 #4 #5 b7. In terms of which extensions of the chord this translates to, b2 and b3 correspond to b9 and #9 of a Dom.7 chord, and #4 and #5 are usually notated as #11 and b13.

Augmented Major is the scale Mark Levine calls the Lydian #5 scale. It contains the 1 2 3 #4 #5 6 7 of a scale, and can be derived by playing the third degree of the melodic minor scale. (eg: C Lydian #5 is the third degree of A melodic minor).

This is an interesting scale that has received some use in modern jazz and fusion. Kurt Rosenwinkel uses it extensively on his 2001 album, The Next Step.

Major add b9 is the other scale choice discussed in the above post. There's no real name for it, and Major add b9 is a little misleading because there isn't even a root! But we can't exactly call it a Major #1 scale, or can we?

In any case, the scale is derived by playing the melodic minor scale a whole tone above the root, eg: D melodic minor over a C major chord. The resulting scale yields the following notes: b2 2 3 4 5 6 7.

When I was on the just-jazz email list some years ago, I put forth the idea that melodic minor harmony correlates to major scale harmony by starting the melodic minor scale a whole step above the major scale. For example:


Major mode       |  Mel. minor mode

D Dorian         |         D Mel. minor
E Phrygian       |         E Phrygian Nat.6
F Lydian         |         F Lydian #5
G Mixolydian     |         G Lydian Dominant (eg: the chord G7#11)
A Aeolian        |         A Mixolydian b6
B Locrian        |         B Locrian Nat.2
C Ionian         |         C# Altered

In each case, the mel. minor scale harmony will be only one note different from the major scale harmony.

John Stowell's use of the D melodic minor scale over C major chords fits in with the above chart, and it makes for a pretty interesting harmonic substitution. The only song I can think of that uses this scale is Herbie Hancock's Tell me a Bedtime Story, which uses the A melodic minor scale over a G major chord in the intro.

Source Blog: Casa Valdez Studios