Last week's lecture in my jazz theory course was on a number of esoteric, derived, 7- and 8-note scales. The professor demonstrated each at the piano and took us through various harmonic applications. Several of these came from Nicolas Slonimsky's Thesaurus of Scales and Melodic Patterns.

Written in the lecture notes was a claim that there are over 23000 possible 7-note scales, increasing exponentially for 8-note scales. I was skeptical, so I did a quick calculation in my head:

The total number of collections of notes within an octave, assuming there's a root, would allow each of the 11 remaining notes to be included in the collection or not. That's 2 to the 11th power (2^11), which is only 2048. So there must be fewer 7-note scales than that. And I remember from my probability classes that there will be even fewer 8-note scales.

This line of thinking got me asking and answering a multitude of questions about the number of scales of different types. I will assume for the rest of this post that every scale includes a root note and is confined to an octave.

How many n-note scales are there?

I established above that there are 2048 possible collections of any number of notes, so whatever I come up with here should total 2048. To get started, there's only one 1-note scale (the root, alone) and 11 2-note scales (the root plus one of 11 possible other notes). Similarly, there's only one 12-note scale (the whole chromatic scale) and 11 11-note scales (exclude one of the 11 non-root notes).

For everything in between, I'll need to calculate how many ways n - 1 notes can be chosen from 11 non-root notes. This calls for a binomial expansion! Instead of just looking it up, I chose to derive it while I was trying to sleep the other night.

If note order mattered, the first one could take any of the 11 available slots, the second would have 10 available, and so on. That's 11 * 10 * 9 * etc. for as many notes as you have beyond the root. More generally, that's 11! / (11 - (n - 1))! For a 3-note scale, you have the root, 11 possibilities for the next note, 10 possibilities for the third note. That's 11! / (11 - (3 - 1))! = 11! / 9! = 11 * 10, as above.

But order doesn't matter, so each of these calculations must be divided by the number of ways to order the collection. Luckily, that's easy. Any collection of n elements can be ordered n! different ways. (The first can take any of n slots, the second any of n - 1 slots, and so on.)

Now I have a formula. There are 11! / ((11 - (n - 1))! * (n - 1)!) possible n-note scales. The shorthand for this is 11C(n - 1), as in "11 choose (n - 1)". (See the Wikipedia entry if you're totally lost but still care.)

1-note scales: 11C0 = 11! / (11! * 0!) = 1
2-note: 11C1 = 11! / (10! * 1!) = 11
3-note: 11C2 = 11! / (9! * 2!) = 11 * 10 / 2 = 55
4-note: 11C3 = 11! / (8! * 3!) = 11 * 10 * 9 / (3 * 2) = 165
5-note: 11C4 = 11! / (7! * 4!) = 11 * 10 * 9 * 8 / (4 * 3 * 2) = 330
6-note: 11C5 = 11! / (6! * 5!) = 11 * 10 * 9 * 8 * 7 / (5 * 4 * 3 * 2) = 462
7-note: 11C6 = 11! / (5! * 6!) = 11C5 = 462
8-note: 11C7 = 11! / (4! * 7!) = 11C4 = 330
9-note: 11C8 = 11! / (3! * 8!) = 11C3 = 165
10-note: 11C9 = 11! / (2! * 9!) = 11C2 = 55
11-note: 11C10 = 11! / (1! * 10!) = 11C1 = 11
12-note: 11C11 = 11! / (0! * 11!) = 11C0 = 1

Add those all up, and it's the same total from before, 2048 possible scales of any length.

So now I know that there are only 462 possible 7-note scales. How else can I limit this?

How many 7-note scales are there with no more than three half steps between adjacent notes?

This can be approached backwards, as in how many are there with an interval of at least four half steps between adjacent notes? To answer, place the big interval at the beginning, and multiply by seven modes for each resulting possibility.

The root starts the scale again, and the first three chromatic notes are off limits, because that's where I want the big interval. Now there are eight remaining notes to choose from, and six notes left in the scale.

8C6 = 8! / (2! * 6!) = 8 * 7 / 2 = 28

Each of those 28 scales has six other modes unaccounted for, with the big interval somewhere other than directly above the root. That's 28 * 7 = 196 scales with an interval of at least four half steps between adjacent notes. Now, to answer the question, subtract that from the total 7-note scales: 462 - 196 = 266.

That's cool. That amounts to only 38 generative scales with 7 modes each. I wonder how many are usable and how many are rubbish?

How many 7-note scales are there with no more than two half steps between adjacent notes and no consecutive half steps?

I'm now limited to 1- and 2-step intervals. I'll call them h for half step, w for whole step. I need seven of these intervals for seven notes, and they must add up to 12 half steps. So there must be two half steps and five whole steps.

There are only 14 ways to order these intervals without adjacent half steps. And they happen to correspond exactly to the modes of the major and (Jazz) Melodic Minor scales:

hwhwwww = Altered Dominant
hwwhwww = Locrian
hwwwhww = Phrygian
hwwwwhw = Dorian b2
whwhwww = Locrian #2
whwwhww = Aeolian
whwwwhw = Dorian
whwwwwh = Melodic Minor
wwhwhww = Mixolydian b6
wwhwwhw = Mixolydian
wwhwwwh = Ionian (major)
wwwhwhw = Lydian Dominant
wwwhwwh = Lydian
wwwwhwh = Lydian Augmented

That's just two scales along with their modes. This is suddenly not so overwhelming.

How many 7-note scales are there with only major and minor thirds?

This will take some trickery. I'll extend the scale to two octaves, using thirds instead of seconds. (This orders the notes as 1 3 5 7 2 4 6.) There must be three major thirds and four minor thirds. Further, the scale needs to collapse back to one octave with no duplicate notes. Three major thirds add up to an octave, as do four minor thirds, so I'll keep them from all sticking together.

I'll spare you the long story on these possibilities. There are 28. That's four generative scales with seven modes each. Using m for minor third, M for major third, h for half step, w for whole step, a for augmented second, they are:

MmMmmMm = wwhwwwh = Ionian (major)
mMMmmMm = whwwwwh = Melodic Minor
mMMmmmM = whwwhah = Harmonic Minor
MmMmmmM = wwhwhah = Harmonic Major

That's fantastic! That's a very digestible chunk of knowledge.

How many 6-note scales are there with no half steps?

Only one: the Whole Tone Scale. Cool!

How many 6-note scales are there with no intervals greater than two half steps between adjacent notes?

Only one: the Whole Tone Scale. Cool!!

How many 5-note scales are there using only whole steps and minor thirds between adjacent notes?

Using w for whole step and m for minor third:

wwwmm
wwmwm
wwmmw
wmwwm
wmwmw
wmmww
mwwwm
mwwmw
mwmww
mmwww

These are all modes of two pentatonic scales: the major/minor pentatonic everyone knows and the other pentatonic scale no one knows.

The theme in all of this is that scalar possibilities are not endless. The numbers may be daunting, but they are finite. When the scope is limited, as in the questions above, the resulting possibilities aren't so mind-boggling. It makes mastery very achievable. Then you get into applications of all these scales, and how they relate within the progression of time, and the possibilities truly are endless.

What other useful questions can you come up with? Leave some in the comments, with a solution if you're as nerdy as I am.

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I started lessons with bassist Bob Magnusson last week. He's played with Sarah Vaughan and Joe Pass and just about everyone else.

I'm still working on repertoire and guitar-specific concepts with Bob Boss, so Magnusson is taking me back to fundamentals. On the first lesson, he gave me a monster arpeggio workout that he's been doing since he was in Sarah Vaughan's band in the 70s.

It goes like this. Starting with a Cmaj7 arpeggio, C E G B, play the lowest root note on your instrument. Then play the lowest chord tone, then the next two chord tones. That's the first four eighth notes. (On guitar, that's C on 5th string, open 6th E, G on 6th, B on 5th.) Then play the four-note arpeggio ascending from that lowest chord tone, in eighth notes. Then the next inversion ascending from the next-lowest chord tone. Repeat until you reach the top of your instrument's range. Then reverse with descending arpeggios. When you reach the bottom of your range again, play the low root on the final downbeat. The variations on the pattern at the beginning and end of the exercise are to keep the root firmly in your head as you hear what you play.

Now do it with C7, Cm7, Cm7b5, and Cdim7. Now with all twelve roots. That's 60 different arpeggios.

As I plunk through these, I look for efficient and logical places to shift positions. I find that I know the fretboard well enough to navigate through them all, but I end up getting stuck in awkward, avoidable hand positions. So I made a chart.

These are all the easy fingerings for each inversion of each arpeggio. I define easy as not requiring any finger stretches. These are the fingerings I will try favor when I do the arpeggio exercises above. The numbers refer to the strings used for each note. "6655" in 1st inversion means play the 3 on the 6th string, 5 on the 6th string, 7 on the 5th string, root on the 5th string. Fingerings in parentheses require a slight hand shift, but no stretching.

maj7

• root position: 6554 (6543) 5443 5432 4332 4321 3221 (3322)
• 1st inversion: 6655 6544 5544 5433 4433 4322 3211 2211
• 2nd inversion: 6554 5443 4332 3221
• 3rd inversion: 6655 6654 5544 5543 4433 4432 (3322) 3321 2211

7

• root position: 6554 6544 5443 5433 4332 4322 3221 3211
• 1st inversion: 6655 6554 5544 5443 4433 4332 4322 3322 3221 3211 2211
• 2nd inversion: 6554 5443 4432 4332 3321 3221
• 3rd inversion: 6655 6654 5544 5543 4433 4432 (3322) 3321 3211 2211

m7

• root position: 6655 6654 6544 5544 5543 5433 4433 4432 4322 (3322) 3321 3211 2211
• 1st inversion: 6554 6544 5443 5433 4332 4322 3221 3211
• 2nd inversion: 6655 6554 5544 5443 4433 4332 3322 3221
• 3rd inversion: 6654 5543 4432 3321 3221

m7b5

• root position: 6654 6554 5543 5443 4432 4332 3321 3221
• 1st inversion: 6655 6544 5544 5433 4433 4322 (3322) 3321 3211 2211
• 2nd inversion: 6554 5443 4332 4322 3221 3211
• 3rd inversion: 6655 5544 4433 4432 3322 3321 2211

dim7

• root position: 6655 6554 5544 5443 4433 4432 4332 4322 3322 3321 3221 3211
• inversions: symmetrical, all same as root

I spent the weekend working on this. I used a script I wrote long ago to give me a random root and random arpeggio so I didn't have to systematically go through all 60. The next step is to apply this to real tunes, starting with Autumn Leaves, switching arpeggios in time with each chord change.

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I rearranged my desk at home last week. I set up all my fake books for instant access as I continue expanding my repertoire. So far, I have the Hal Leonard Real Books, Volumes 1, 2, and 3, and Chuck Sher's New Real Book series, Volumes 1, 2, and 3. I have these all lined up in a small bookshelf on my desk, ready for grabbing.

This presented a problem. When I was after a specific tune, I had to search each book's index one by one. Problems don't last long around me. I consulted the Internet and found The Fake Book Index from Seventh String Software, the makers of Transcribe!. The index searches through any selection of over 70 fake books, including the six on my desk. So when I wanted to learn Ellington's Isfahan, I searched all my books at once and found it was in Hal Leonard's Volume 3 and Chuck Sher's Volume 2. I got them both out and compared.

Further, I have a professional tip for Mozilla Firefox users. Go to The Fake Book Index and check all the books you want to search. (I include the Charlie Parker Omnibook and Aebersold Play-A-Long Series as well as the six mentioned above.) Right-click in the text field at the top of the page, where you would type the title of the song you seek. Select "Add a Keyword for this Search..." This creates a bookmark that you can access from the URL bar. Name it "Fake Book Index" or whatever you like. Use a short keyword like "fb" and save it. Enter "fb body and soul" in your URL bar, and it takes you directly to your custom search results.

Now learn.

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I am one week into my second semester in SDSU's Master's program in jazz studies. I'm still loving it, and I'm on track to finish a semester early. If all goes as planned, I could be done by Christmas.

I'm most excited for a class I'm auditing this semester, Rick Helzer's final undergrad jazz theory course. I'll be in his graduate theory seminar in the fall, so this should ensure that my knowledge is where it should be. In the first two lectures, we ripped through Bill Evans's "Very Early" and the harmonic applications of every mode of Harmonic Minor and Harmonic Major. I had never touched Harmonic Major before. I knew it was 1 2 3 4 5 b6 7, but never used it, so I ran all five positions on guitar after class the other night.

I'll have private lessons with Bob Boss again. He's a repertoire machine, and helping me become one too. (See my post from last semester, Learning Standards Again.) I also signed up with bassist Bob Magnusson. My classmate, guitarist Travis Daudert, has said great things about lessons with Magnusson. I recall renting An Evening with Joe Pass last year and discovering Magnusson right up there with Pass on the Musicians Institute stage.

In addition to the above, I'm in a combo and two graduate seminars: one on classical theory and one on music research and writing. I bet the latter will give me another surplus of material to post here.

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Earlier this month, I was fortunate to spend an afternoon at SDSU for a jazz guitar clinic given by Howard Paul, president and CEO of Benedetto Guitars. Howard gave an excellent presentation on the evolution of jazz guitar building practices, complete with demos of seven Benedetto models.

We first looked at the construction of flat top guitars, durable for treks across the country and steel strings for volume. Because the bridge and tailpiece are placed together on the top of the guitar, right in the middle, an incredible amount of bracing is required. The top of any acoustic guitar needs to be delicate enough to vibrate with the strings and project sound like a speaker cone. The tight steel strings of a flat top would rip that delicate top right off without extra bracing underneath it. Unfortunately, that extra bracing reduces the acoustic qualities of the guitar, so it couldn't be heard well in a jazz band setting.

Developments by Gibson and others in the early 1900s brought more traditional stringed instrument designs to the guitar. Like a violin or cello, the top was carved into an arch with f-holes next to the bridge instead of a single round hole under the strings. The tailpiece was attached to the side of the guitar, called a violin or trapeze tailpiece, to provide a downward force on the bridge. Thus, there was no more danger of ripping off the top, so that extra bracing was no longer necessary, which allowed the top to resonate more freely.

Ideally, no holes were drilled anywhere in the top. The tailpiece was attached to the side. The pickguard was attached to the neck, supported by an additional attachment to the side if necessary. When pickups came to prominence in the 30s, they were attached to the pickguard or neck. The volume knob was accessible on the pickguard, and the output jack on the side. In fact, nothing even touched the top of the guitar except the bridge, which was essential in transferring the strings' vibrations to the top.

Built-in pickups were also developed as an alternative to the floating pickup. This required a change in the bracing, which traditionally runs in a big "X" underneath the top. Cutting a big hole or two in the top for pickups requires parallel bracing, running along the sides of the pickups, parallel to the strings. Installing pickups allows for more tonal flexibility while maintaining most, but not the optimum, acoustic qualities. So, with occasional exceptions, a built-in pickup means parallel bracing, and floating or no pickup means x-bracing.

Of course, with pickups and amplification comes feedback, especially with hollow, highly resonant bodies. This leads us to solid bodies, Telecasters, Les Pauls, Stratocasters, and away from traditional jazz guitar.

The finest woods used in jazz guitars come from Europe, specifically spruce and maple. According to Howard, they generally grow on hillsides in harsh environments, leading to tight, stable growth rings. Spruce happens to be optimal for the resonant top, and maple for the reflecting back and sides.

Tops are traditionally hand-carved from two adjacent pie-slices of European spruce, glued together down the center of the resulting guitar. This takes many man-hours and wastes most of the wood purchased for the process. A more cost-effective means of constructing the top comes from laminated sheets of wood, cut from tree trunks in thin spirals. Different wood types can be laminated together, with perpendicular grain directions for stability, in a variety of thicknesses. They are then machine-pressed into the proper shape for a given guitar top. The laminated method results in less expensive guitars, less feedback, but less acoustic quality. The ultimate in jazz guitar construction remains the hand-carved top.

Howard summarized his presentation with a continuum of jazz guitar designs, ranging from the traditional ideal to the more modern sounds of fusion and pop.

On the traditional end:

• expensive
• designed for acoustic qualities
• hand-carved top
• all European woods, usually spruce and maple
• x-bracing
• floating or no pickup
• floating bridge
• trapeze tailpiece
• no holes besides f-holes in the top, nothing touching except bridge
• 17" wide, measured at the widest point of the lower bout

On the modern end:

• inexpensive
• laminated top
• 2 pickups, built-in
• parallel bracing or a center block (semi-hollow)
• sometimes no sound holes
• Tune-o-matic bridge
• stoptail bridge
• smaller body

Of course, jazz guitars can fall anywhere on this spectrum, combining features from both ends. My own Epiphone Sheraton II falls squarely on the cheap side, but I still love the sound I get from it. And I love having money to buy food.

For the ultimate detail in jazz guitar construction, see the definitive book on the subject, written by Robert Benedetto himself, Making an Archtop Guitar.

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