James Tauber

...and just because pianos (keys) and non-slide guitars (frets) are discrete doesn't mean that vocals or other (continuous) instruments won't find Bb and A# to be in close, yet distinguishable, places.

How does the topology of equal temperament relate to the topology of the diatonic key torus?

It is, though, a subtle distinction, and in my experience the distinction only matters in a minority of cases. It can be confusing to people if you try to make a strong distinction about something that usually doesn't affect them. At least, it usually isn't worth making such distinctions unless you are discussing a particular context where the difference really does matter in some practical sense. That's my approach, anyway.
Cheers, Tony.

Hi James,
re. "If you are in Gm and you write A# you are conveying something completely different than if you write Bb."

Can you please clarify :

If this were piano music, would the player hit a different key? (i.e. would that note sound different?)

Would you expect the two to be written differently in staff notation?

On an instrument with a continuous scale, should these be played at the same or different pitches (frequencies)?

The A, B, C... letter notation looks somewhat Western to me :-) Does it actually make much sense to use it for anything other that Western tonal music?

I think I'm talking about the Tonnetz, or at least a close relative. Take the circle of fifths, which is trivially a circle. Now take the diatonic scale built on each root, and identify pitches at the octave -- so we forget about differing inversions, and only pay attention to the roles (root, third, fifth, seventh, etc.) each pitch plays in the chord. That's another circle, and multiplying the two circles yields a torus.

If the relative minors are included as intermediate steps on the circle of fifths, it's possible to make changes one voice at a time. (conversely, moving all the voices in parallel jumps a few steps: IV-V)

[what does a Lie Bracket sound like? I-IV-V-II-I? probably better the other way around: I-VII-IV-V-I]

Finally, continuing with the phsyical inspiration, there's an uncertainty principle for assonance. How long does an accord need to be held before any unwanted beats become audible?

See http://math.ucr.edu/home/baez/week234.html for way more mathematical-physics/music crossover discussion. (the Baez family being known for indulging in a bit of both)

I don't care so much for the final model {T,I,P,R,L} that he presents, though. It is somewhat interesting that a typical I IV V I progression would be RL TT RL (on a guitar, play E A B E to get an obvious slide at the TT), and maybe there's even some interest in using the identity [L,R] = TT to rewrite that as RL [L,R] RL -- but those generators are far too productive. It's possible to get all over the chromatic keyboard with only a few operations.

On the other hand, it's difficult to go wrong by moving around by fifths, using 7ths or runs on the changes, and taking advantage of related minors either for leading or for variation. Maybe there's something in between, more restrictive than the Baez model, yet freer than the 3-chords-plus-ornaments model, that still tends to wind up suggesting good lines?

On the gripping hand, it's certainly possible to go overboard on worrying about throwing in extra changes. I once read over the end of a Ludwig van piece, and discovered that what sounded like a fairly complex passage was picking up its texture from someplace other than the harmony; it went for a page or two straight in variations on F.

James,

I've just begun to learn to play some Bossa Nova and it's littered with the E# and A# notation. It's driving me batty. I suppose I'll just have to get use to it

James,

I'm what you would call a "Wannabe" guitarist.. I've played for years, but with no training. I've played live shows, but not with any true skill..
I can't read "Real" music, rather chord diagrams, and Tab..
(Crappy I know.)
My point is I always knew there was a difference, but your post here (I know I just read it and it's old) Finally made me understand.
I'm not big on theory, but it made sense. (I know enough to get me in trouble.. LOL)

Thanks for the "Simple Terms"
Just thought you'd like to know even the simpletons can hear the knowlege your trying to pass on.
AND Thanks...

Joe

I've written before inquiring about a piece of music that has b note with a double flat. What note is that? I can read music but I don't know everything about accidentals. I admist I didn't do the math question last time. Thank you for your help.
Annie

Over here, they teach you first the general and then move on to the specific.

So I have been taught this: an octave is defined by the most simple relationship between 2 frequencies, 2/1. Accordingly, an octave is defined by the nest most relation, 3/2. Which, according to Fourier is equivalent to the interval between any base frequency and its harmonic2. A quint is the interval between harmonic2 and harmonic3. Then you can define a tone to being equal to 2 quints minus 1 octave, which is a ratio of 9/8 or the interval between F9 and F8.

Now if one wants to define a scale, one has to consider how many times a quint fits into an octave. Which is S=log(2)
/log(3/2), which is an irrational number. So you do something called "continued fraction", and you get:
1st approximation: S=5/3 aka the pentatonic scale.
2nd approx: S=12/7 aka the tempered or chromatic scale.
The next interesting one is the 4th, S=53/31, which gives you 1 quint = 3 tones + D, and 1 octave = 5 tones + 2D = 1 quint + 2 tones + D.

D is then the diatonic semitone, tone-d= chromatic semitone, and chromatic semitone - diatonic semitone = 1 comma. (9 commas = 1 tone). That's called Holder's comma, as opposed to a whole lot of other commas.

The usefulness of that is that it is said that the difference of 1 comma two apparently is claimed to be enough to cause beating (audible interference) when playing two apparently enharmonic notes.

Which is called a meantone tuning.

The 12-tone equal tempered is the only meantone tuning where both semitones are equal, and thus the only one having an unbroken cycle of fifths.

Which, you have to admit, is more interesting in class then being told "An octave=12 semitones, a quint is 7 semitones, and a semitone is 100 cent. That's how is is, always ways, and always will be." Yes, that's a phrase I've actually heard in real live.

By the way, "b note with a double flat" is a challenging concept for me. Sometimes B=H and B-flat=B? Anyway, two flats should be 2 semitones lower than whatever b is. If that b = Si, then then the answer should be La. Which is a, I think.

JoeK

Oh, I forgot relevance:

Violins are tuned in perfect fifths. By bowing strings in pairs. Or that's what the dictionary says.

Which would be a just intonation tuning, which is the opposite of the equal tempered tuning. In equal tempered, your 12 subdivisions of the octave have the same interval, whereas in just tuning, there intervals are whole number fractions fractions, and they are not uniform.

If you google a bit, you find out that
Leopold Mozart, in his violin method of 1756 -- (..) tells us that keyboard instruments of his time were played with some form of tempered [i.e., well-tempered] tuning, but that in the "right ratio" [i.e., meantone] tuning that he recommends for the violin, flats are higher by a comma than enharmonically equivalent sharps."

Which should be enough to convince everybody...?

i AM STUPID

As noted above, all Indian notes except Sa and Pa, can move a microtone or two depending on the raaga. This practice can be traced back to the ancient concept of Shruthi, where an octave is divided into 22 shruthis. Each of the 10 notes (i.e. excluding Sa and Pa) have two variants. For Eg. Komal Rishab (r) can be either the normal r or r+. The pitch difference between these two notes is small, but the use of one instead of the other, guarantees a simple ratio with the other notes used in that Raaga. That is the reason, why r is used in some Raagas and r+ is some others. It has been observed that the shruthis are such that, between two adjacent saptasvara notes there is a ratio of 256/243 and between shudh and komal notes the ratio is 25/24. It is this simple ratio that makes the music pleasing, natural and sweet. The 22 shruthis, their frequencies and equivalents are given in the table 3. An octave can be divided into 1200 cents. Scales are usually defined in cents so that for any base pitch, the scale can be calculated. The difference in cents between two frequencies, f1 and f2, is given as log(f1/f2) * 1200 * log(2) or approximately log(f1/f2) * 3986.3137

Table 3 - Shruthi
Shruthi Names Western Eq. Abbreviation Hz. Adjucent Ratio Cents Ratio
Shadja C S 261.63 0 1
Ekasruti Rishabha r 273.38 256/243 76.03 256/245
Dvisruti Rishabha Db r+ 279.07 111.73 16/15
Trisruti Rishabha R- 290.7 25/24 182.40 10/9
Chatusruti Rishabha D R 294.33 203.91 9/8
Shudha Gandhara g 310.08 256/243 294.13 32/27
Sadharana Gandhara Eb g+ 313.96 315.64 6/5
Antara Gandhara E G 327.03 25/24 386.31 5/4
Chyuta Madhyama Gandhara G+ 331.12 407.82 81/64
Shudha Madhyama F m 348.84 256/243 498.04 4/3
Tivra Sudha Madhyama m+ 353.20 519.55 27/20
Prati Madhyama F# M 367.92 25/24 590.22 45/32
Chyuta Panchama Madhyama M+ 372.52 611.73 729/512
Panchama G P 392.45 256/243 701.95 3/2
Ekasruti Dhaivata d 413.44 256/243 792.18 128/81
Dvisruti Dhaivata Ab d+ 418.61 813.68 8/5
Trisruti Dhaivata A D- 436.05 25/24 884.36 5/3
Chatusruti Dhaivata D 441.50 905.86 27/16
Shudha Nishada n 465.12 25/24 996.09 16/9
Kaishiki Nishada Bb n+ 470.93 1017.60 9/5
Kakali Nishada B N 490.56 256/243 1088.29 15/8
Chyuta Shadja Nishada N+ 496.69 1109.78 243/128
Tara Shadja C S' 523.26 25/24 1200 2
Carnatic Scales

For screwing around purposes, is A# the same as Bb on a guitar tuner?!?!

Can someone give a concrete numeric example of A# and Bb frequencies that are not the same? What is the difference in Hz that we are talking about here, say for a guitar?

A# and Bb may serve different functions to the music scholar in the same way as "hear" and "here" do to the written word but I'm not sure that music is composed solely with the intention of impressing the academics.

Maybe we should just close our eyes and enjoy what we here (see you knew what I meant).

ps. Stumbled on this site through Monty Python Word Association sketch amazing how many MP fans are into music theory

Hey,
I am an absolute newb to music theory.

If I have understood it correctly, it is because the modern scale system, 12-tet, only divides an octave into 12 semitones and thus there are no room for separation of flats and sharps.

In another scale system with more intervals witin an octave there would be room for a separation in frequency(pitch) between say A sharp and B flat. Is this correct?

No Jesper,

The Flat-Sharp model in this context is specific to the 12 tone system. So A-sharp and B-flat are the exact same frequency, audibly indistinguishable; at least that is how I understand it.

I beleive the point the author was making is that while they would SOUND the same, it is bad "musical grammar" to call them incorrectly... so a G-minor third is correctly described as G and B-flat (not A-sharp)

Anybody know where I can see a list or chart of the diatonic ratios for sharps and flats?