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#945227 - 09/29/08 05:19 PM 3+3=5
Gary D. Offline
4000 Post Club Member

Registered: 08/30/08
Posts: 4649
Loc: South Florida
Musical math. ;\)

I love showing this to students (beginners), especially small kids. It makes them giggle.

I show them that we call C-G a 5th because the total number of notes is 5. Then we stick an E in the middle, count the thirds and learn that a third plus a third equals a fifth.

They always get caught for a moment until we explore the fact that the middle note gets counted twice.

Since we had another thread about how to teach theory, this is something I like to do. I like to point out the inconsistencies about the way we notate music, to make it clear that analysis is never more than a tool!
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#945228 - 09/29/08 07:22 PM Re: 3+3=5
btb Offline
4000 Post Club Member

Registered: 01/21/04
Posts: 4261
Loc: Pretoria South Africa
Hi Gary D,

Here’s a diagram showing the accurate relationship
of THIRD intervals ... in terms of semitones, the
lower third occupies 4 semitones, whereas the upper third only has 3 semitones.

The confusion of 3+3=5 is purely because of the antiquated numbering of the notes of the major scale ... but what should be heard is that the lower 4 semitone interval is a much "fuller"
sound than the upper 3 semitone interval.

Thanks for your e-mail ... glad you liked blowing the bogey of theory out of the water ... music should always be a delight.

third plus third diagram

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#945229 - 09/29/08 09:51 PM Re: 3+3=5
John v.d.Brook Online   content
7000 Post Club Member

Registered: 03/18/06
Posts: 7200
Loc: Olympia, Washington, USA
What's really interesting is that basic music notation pre-dates the acceptance of our modern counting system. The invention of the printing press forced the issue, and by the mid-1500s, most of Europe had changed over (from Roman to Arabic). It's really fun to watch the student's expression when you tell them that there was no zero, which is why we start counting from the 1st note to the 3rd tone (or what ever)!
_________________________
"Those who dare to teach must never cease to learn." -- Richard Henry Dann
Full-time Private Piano Teacher offering Piano Lessons in Olympia, WA. www.mypianoteacher.com
Certified by the American College of Musicians; member NGPT, MTNA, WSMTA, OMTA

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#945230 - 09/30/08 01:51 AM Re: 3+3=5
Gary D. Offline
4000 Post Club Member

Registered: 08/30/08
Posts: 4649
Loc: South Florida
 Quote:
Originally posted by btb:
Hi Gary D,

Here’s a diagram showing the accurate relationship
of THIRD intervals ... in terms of semitones, the
lower third occupies 4 semitones, whereas the upper third only has 3 semitones.
Of course, but it is hard to get beginners to think this way. They don't think:

0-4-7, major
0-3-7, minor
0-3-6, diminished (three of four notes)

\:\)

Actually, I was just having a bit of fun when I started this thread. I love showing children a G7 chord and then telling them that 3+3+3=7.

It's another one of those little things that makes sense to them once they've thought through the whole idea of "double counting" notes.
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#945231 - 09/30/08 01:55 AM Re: 3+3=5
Gary D. Offline
4000 Post Club Member

Registered: 08/30/08
Posts: 4649
Loc: South Florida
 Quote:
Originally posted by John v.d.Brook:
What's really interesting is that basic music notation pre-dates the acceptance of our modern counting system. The invention of the printing press forced the issue, and by the mid-1500s, most of Europe had changed over (from Roman to Arabic). It's really fun to watch the student's expression when you tell them that there was no zero, which is why we start counting from the 1st note to the 3rd tone (or what ever)! [/b]
Exactly!

I also tell people that it is useful to think of the piano as having been invented with only white notes. (Of course this is a monumental distortion of what actually took place, but for people starting out it gives them an idea of how our notational system sort of evolved from a 7 note system into a 12 note system. ;\)
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#945232 - 09/30/08 02:42 AM Re: 3+3=5
Pete the bean Offline
Full Member

Registered: 12/06/04
Posts: 450
Loc: Canada
I tell my students the keyboard layout is a code invented by some very smart people to show the user how music is put together in black and white. In white, all the modes, and in black the pentatonic scale.
Then I show them the beautiful geometry of the keyboard using tetrachords of the major scale (ww1/2). I describe how the upper tetra chord can become the bottom of the scale a 5th above and adds sharp to each new key. By going down by 5ths from C adds a flat to the key signature. The geometry is beautiful.

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#945233 - 09/30/08 11:37 AM Re: 3+3=5
Betty Patnude Offline
4000 Post Club Member

Registered: 06/11/07
Posts: 4896
Loc: Puyallup, Washington
Yes, it's really fun to perplex a student into a bright idea. From confusion comes understanding.

This is right up there when you sing "The square of the hypoteneuse of the right triangle is equal to the sum of the square of the two adjacent sides"....a song from "Merry Andrew", I believe. (Correct me, please.)

I can sing it, I can say it, but it doesn't mean a thing to me.

In testing, I was once told that I could have a doctorate in anything as long as it wasn't math or science. What a whoop of laughter that generated from me! I already knew that!

So things that help us compute musically are so much fun for me - I really liked this topic!

Betty

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#945234 - 09/30/08 11:45 AM Re: 3+3=5
Karisofia Offline
Full Member

Registered: 02/13/08
Posts: 201
Loc: Wisconsin
 Quote:
Originally posted by John v.d.Brook:
What's really interesting is that basic music notation pre-dates the acceptance of our modern counting system. The invention of the printing press forced the issue, and by the mid-1500s, most of Europe had changed over (from Roman to Arabic). It's really fun to watch the student's expression when you tell them that there was no zero, which is why we start counting from the 1st note to the 3rd tone (or what ever)! [/b]
This is very interesting. Makes me want to read up on the subject. Any recommended resources?
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#945235 - 09/30/08 12:09 PM Re: 3+3=5
keystring Online   content
Yikes! 10000 Post Club Member

Registered: 12/11/07
Posts: 11190
Loc: Canada
 Quote:
This is right up there when you sing "The square of the hypoteneuse of the right triangle is equal to the sum of the square of the two adjacent sides"....a song from "Merry Andrew", I believe. (Correct me, please.)

I can sing it, I can say it, but it doesn't mean a thing to me.
Would you believe that there is a link between that and some deep philosophies affecting perceptions of music? \:\) (I'm reading an interesting book)

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#945236 - 09/30/08 12:18 PM Re: 3+3=5
lilylady Offline
4000 Post Club Member

Registered: 03/17/05
Posts: 4974
Loc: boston north
It is stuff like that that has always bothered me about math and logic.

Mon - Tues
one day

But C - G equals 5? In other words we count the first thing said as ONE.

With that logic, M - T is two.

UGH...

And are there 7 or 8 days in a week?

And how old am I exactly? What is that first year, nothing? We start at age 1 when we have already lived for one year.

And don't get me started on the 2000 year celebration!!!

Thank goodness piano stuff made more sense to me!

I would - let's take steps. Step one is C, etc and we would arrive at step five being G.
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#945237 - 10/01/08 01:21 AM Re: 3+3=5
btb Offline
4000 Post Club Member

Registered: 01/21/04
Posts: 4261
Loc: Pretoria South Africa
Pete goes into raptures about the TONE, TONE, semitone patterning within the Major scale ... but even better is the point picked up by GaryD that we now have a keyboard palette of a basic 12 notes instead of the original stagnant 7.

By adding the 5 black notes our cunning forebears opened up a magical world of endless modulation through the various major and minor keys ... sadly however, most pianists become slaves to the alphabetic 7 white notes ... and bravely grasp the nettle as they venture up the sharp and flat-rich keys.

But 12 is superb number (as championed by history) because of it's flexible ability to be divided by 6, 4, 3 and 2 ... and the antique
musical term of "octave" should more correctly be called a 6-TONE because of it's makeup up of 6 WHOLE tones ( 12 semitones).

Here's the earlier diagram ... but showing 6 WHOLE tones (6T).

6 whole tones

Thus, in the 3+3=5 analogy, the intervals should read

major third = 2T
minor third = 1+T (+ representing an additional semitone)

the logical 6T interval sequence should then read

6T = OCTAVE
5+ = leading note
5T
4+ = submediant
4T
3+ = DOMINANT
3 T
2+ = SUBDOMINANT
2T = mediant
1+
1T = supertonic
+
0= TONIC

I’m sure the girls are good at some other things (me chauvinist dog)but mathematics doesn’t seem to be one of them.

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#945238 - 10/01/08 09:38 AM Re: 3+3=5
John v.d.Brook Online   content
7000 Post Club Member

Registered: 03/18/06
Posts: 7200
Loc: Olympia, Washington, USA
Karisofia, Ask Dr. Math had some interesting resources.
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Full-time Private Piano Teacher offering Piano Lessons in Olympia, WA. www.mypianoteacher.com
Certified by the American College of Musicians; member NGPT, MTNA, WSMTA, OMTA

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#945239 - 10/01/08 10:50 AM Re: 3+3=5
Kreisler Offline


Registered: 11/27/02
Posts: 13706
Loc: Iowa City, IA
The problem is that we're equating cardinal with ordinal numbers.

Cardinal numbers identify quantity - 4 apples, or 15 years old.

Ordinal numbers identify order or rank - the 6th page, or the 2nd track

Intervals are ordinal. When we speak of the interval between C and G, we mean the "Fifth note over from C" not "5 notes away from C." That's why intervals are named with ordinal numbers, not cardinal numbers.

Keep this in mind, and there's no confusion. The confusion that people have with the numbering of intervals is manufactured.

3 + 3 does NOT equal five in mathematics OR music, and the statement 3 + 3 = 5 ignores the fact that intervals use ordinal numbers.

Part of the problem is that interval is often defined as the distance between two notes. This is untrue. An interval indicates position within the scale. A chord is made up of the 1st, 3rd, and 5th notes of the scale; not the notes that are 3 and 5 away from the root.


(I minored in mathematics in college.)

\:D
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#945240 - 10/01/08 11:05 AM Re: 3+3=5
lilylady Offline
4000 Post Club Member

Registered: 03/17/05
Posts: 4974
Loc: boston north
Thanks Kreisler,

I had never heard of ordinal or cardinal numbers.

But I still am having difficulting with the 6th page (in my mine, page 6 or 6 pages with this being the last one). Are they both the same or am I mixing them up as ordinal then cardinal?

I like this:

"Part of the problem is that interval is often defined as the distance between two notes. This is untrue. An interval indicates position within the scale. A chord is made up of the 1st, 3rd, and 5th notes of the scale; not the notes that are 3 and 5 away from the root."

I was one of the thousands that were taught that an interval was the distance between two notes as a definition. And have even repeated that teaching. (she slaps her hand)

I think that an interval indicates the position within a scale sounds good.

Except I can see getting caught with the question - in the key of C, the interval F to B (a 4th) wouldn't be the notes of the F scale (Bb needed) and that they are the 4th and 7th tones of the C scale so that might as well, be confusing. ???

A TRIAD is made up of the 1st, 3rd and 5th notes of a scale (a chord in its root position) but I also teach it as the root name, 4 hs, 3 hs (half steps) etc for major, minor, Aug, dim, each accordingly. Ordinal, correct?
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"Music gives a soul to the universe, wings to the mind, flight to the imagination, and life to everything."

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#945241 - 10/01/08 01:13 PM Re: 3+3=5
Kreisler Offline


Registered: 11/27/02
Posts: 13706
Loc: Iowa City, IA
Going with the page analogy:

Three pages + Three pages only equals Six pages if all the pages are different.

If you have Three pages (pg. 1, 2, and 3) and Three pages (pg. 3, 4, and 5), then you have Five pages, because the third page belongs to both sets.

In the case of 4hs, 3hs, you're using cardinal numbers because you're counting things. 4 half steps is the number of half steps between the notes.
_________________________
"If we continually try to force a child to do what he is afraid to do, he will become more timid, and will use his brains and energy, not to explore the unknown, but to find ways to avoid the pressures we put on him." (John Holt)

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#945242 - 10/01/08 01:37 PM Re: 3+3=5
lilylady Offline
4000 Post Club Member

Registered: 03/17/05
Posts: 4974
Loc: boston north
Have patience with me, Kreisler!

'4 hs, and 3 hs' always seems logical to me since to take a step (physically) you move from A to B.

But, I am also dyslexic. And my mind mixes up things, or twists thing, even though in general, I know that I understand.

So counting is cardinal?

I guess it is amazing that I 'think' in piano! But don't seem to think in terms of days of the week and dates (as mentioned above). So, please don't ask how old I am!

I just don't know 'how' I am thinking!

You have been helpful. And I am trying to digest the math. I have been questioning this stuff all my life.

One last question here...

"4 half steps is the number of half steps between the notes."

To me it is not the 'between the notes' because the last note IS one of the steps counted (count 4 - so not inbetween two notes). An example of where I get so confused with the numbers game.

But...I do know what intervals are named correctly, and how to show it and recognize it as a pianist. It is just the wording that sometimes confuses me.
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"Music gives a soul to the universe, wings to the mind, flight to the imagination, and life to everything."

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#945243 - 10/01/08 05:59 PM Re: 3+3=5
Kreisler Offline


Registered: 11/27/02
Posts: 13706
Loc: Iowa City, IA
A note and a step are two different things. If the last note was a step, then you'd be able to answer the question: "How many half steps is F#?"

Steps and half steps are distances between notes that you can count.
_________________________
"If we continually try to force a child to do what he is afraid to do, he will become more timid, and will use his brains and energy, not to explore the unknown, but to find ways to avoid the pressures we put on him." (John Holt)

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#945244 - 10/01/08 06:49 PM Re: 3+3=5
Gary D. Offline
4000 Post Club Member

Registered: 08/30/08
Posts: 4649
Loc: South Florida
 Quote:
Originally posted by btb:
But 12 is superb number (as championed by history) because of it's flexible ability to be divided by 6, 4, 3 and 2 ... and the antique
musical term of "octave" should more correctly be called a 6-TONE because of it's makeup up of 6 WHOLE tones ( 12 semitones).
This always leads to fights between advocates of equal-temperament and those who view equal-temperamant as nothing short of an invention of the Devil. ;\)

It has always fascinated me that:

440*3/2=660 (what we have in nature, assuming a fundamental of 440)
AND
440*2^(5/12)=659.255114 (tempered 5th)

Are SO close.

I don't think most people have any idea that our piano 5ths are so close to perfect (less then 2 cents flat), or how unlikely that most people can hear this tiny distance unless both notes in the 5ths are struck together. \:\)
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#945245 - 10/01/08 06:59 PM Re: 3+3=5
Gary D. Offline
4000 Post Club Member

Registered: 08/30/08
Posts: 4649
Loc: South Florida
 Quote:
Originally posted by lilylady:
But...I do know what intervals are named correctly, and how to show it and recognize it as a pianist. It is just the wording that sometimes confuses me.
Please ignore anything that confuses you here. I started this thread with a twinkle in my eye.

If you know intervals, that's all that is important.

If you think of a major third as four half steps up or down from a starting point and a minor third as three away, that's all you need to know.

Or if you know all your scales, if you know which thirds are "big" and which are "small", that works too. \:\)
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#945246 - 10/01/08 10:30 PM Re: 3+3=5
Kreisler Offline


Registered: 11/27/02
Posts: 13706
Loc: Iowa City, IA
Very true!

I don't actually think about cardinal numbers or set theory when I'm composing, analyzing, or improvising.
_________________________
"If we continually try to force a child to do what he is afraid to do, he will become more timid, and will use his brains and energy, not to explore the unknown, but to find ways to avoid the pressures we put on him." (John Holt)

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#945247 - 10/01/08 11:33 PM Re: 3+3=5
btb Offline
4000 Post Club Member

Registered: 01/21/04
Posts: 4261
Loc: Pretoria South Africa
According to my reference
"By an interval in music is meant the difference in pitch between any two notes."

In my book this difference is a cardinal number ... C to G is 7 semitones.

It is therefore total rot to suggest that "Intervals are ordinal."

Shame on you prof for not majoring in mathematics like some of us.

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#945248 - 10/02/08 12:23 AM Re: 3+3=5
Gary D. Offline
4000 Post Club Member

Registered: 08/30/08
Posts: 4649
Loc: South Florida
 Quote:
Originally posted by btb:
According to my reference
"By an interval in music is meant the difference in pitch between any two notes."

In my book this difference is a cardinal number ... C to G is 7 semitones.

It is therefore total rot to suggest that "Intervals are ordinal."

Shame on you prof for not majoring in mathematics like some of us. [/b]
Forget about labels for a moment. How do you think invervals?

For example, for me augemented is:

x-4-4-x, meaning that the octave just takes you back to where you started and it is divided evenly.

diminished:

x-3-3-3-x

major scale:

x-2-2-1-2-2-2-x(1)

etc.
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#945249 - 10/02/08 04:29 AM Re: 3+3=5
btb Offline
4000 Post Club Member

Registered: 01/21/04
Posts: 4261
Loc: Pretoria South Africa
Hi GaryD

Here’s a schedule of the absurd interval names with which we burden our students ... the nonsensical fine line drawn between the "enharmonically equal intervals" will never be
understood by our Computer generation ... only old fuddy-duddies like us (count me out!) can possibly defend calling the same single semitone interval in the one case AUGMENTED PRIME (#) and on the other MINOR 2ND (b).

intervalnames

Diminished and augmented is just another way of saying add or subtract a semitone.

JS Bach’s monumental WTC (Well-tempered Clavichord) is founded on the acceptance of a keyboard octave of 12 equally spaced notes (thus well-tempered) to gain free modulation into 12 different major and minor keys ... at the cost of a slight tweaking of certain notes to achieve the equal semitone spread ... but we have got so used to this "off-key" tweak that we no longer notice ... violinists on the other hand, with their absolute pitch range, might well wince in adjusting to some of the sounds coming from the well-tempered piano.

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#945250 - 10/02/08 06:32 AM Re: 3+3=5
ROMagister Offline
500 Post Club Member

Registered: 04/26/08
Posts: 518
Loc: Bucuresti, Romania
If everyone would just add semitones in their heads, (and label the names modulo 12), everything would be EASY ! That's what computer MIDI does, and what I thought as a kid decades ago. But doing that in real time is for computers, not humans...

Now I also think 'intervallic reading' is a great help to us fallible humans... only it gets even more complicated in non-C key signatures - to keep track whether the # is in the key signature or not etc. Or, easier back to addition modulo 12 ? What's easier, what pro musicians (not us amateur geeks) do ?

e.g.
A 5th above G# is D# (and they are NOT in the key signature of G Major)
A 5th above F# is C# (and the first is, and second is NOT in G Major)
OR
G# 8+7=15 -12=3 D#
F# 6+7=13 -12=1 C#

And... I'd still want to understand in what way pre-well-tempered intervals differed - in what way the Circle of Fifths didn't close. I understand the basic definition there was 1 tone = 2 fifths - 1 octave ? and the major third was 2 such tones, or the harmonic 5/4 (quite far from such approximations) ?

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#945251 - 10/04/08 12:07 AM Re: 3+3=5
Ferdinand Online   content
500 Post Club Member

Registered: 04/23/07
Posts: 926
Loc: California
 Quote:
Originally posted by btb:
According to my reference
"By an interval in music is meant the difference in pitch between any two notes."

In my book this difference is a cardinal number ... C to G is 7 semitones.

It is therefore total rot to suggest that "Intervals are ordinal."

Shame on you prof for not majoring in mathematics like some of us. [/b]
btb, I can't tell whether you are just trying to provoke an argument here, but if so I'll take the bait.

As Kreisler said, C to G is a 5th because G is the 5th note in the letter sequence beginning with C. In the preceding statement, "C", "G" and "5th" are meant in the general sense, not specifically C natural or perfect 5th. It holds for Cb to G#, Cx to G natural, etc. The 5 is an ordinal number.

As you said, C natural to G natural is 7 semitones. This 7 is a cardinal number.

These are two different, equally valid ways of thinking about intervals. Each has its appropriate use, and the two are not in conflict.

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#945252 - 10/04/08 12:51 AM Re: 3+3=5
btb Offline
4000 Post Club Member

Registered: 01/21/04
Posts: 4261
Loc: Pretoria South Africa
Mustn’t twist your tail Ferdinand ... at least you concede that an interval of 7 semitones (C-G) is a cardinal number ... sorry if I insist that an interval by definition can’t be anything but a cardinal number.

Use of the number series (minor 2nd etc. ) is a ill-judged dodge by musicologists, who had run out of suitable number formats for their musical lingo ... only for the dusty blighters to add the heavy-going I, II, III, IV, V, VI, VII, VIII format to differentiate the triads ...
(not forgetting so-called "minors" with the lower case i, ii, iii, iv, v, vi, vii, viii.)

Under the moggy blundering legacy of our musicologist forebears we must forgive the learned Professor for saying "Intervals are ordinal".

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#945253 - 10/06/08 01:56 AM Re: 3+3=5
Coolkid70 Offline
Full Member

Registered: 03/27/08
Posts: 378
Loc: Irvine, CA
 Quote:
Originally posted by Gary D.:
 Quote:
Originally posted by btb:
But 12 is superb number (as championed by history) because of it's flexible ability to be divided by 6, 4, 3 and 2 ... and the antique
musical term of "octave" should more correctly be called a 6-TONE because of it's makeup up of 6 WHOLE tones ( 12 semitones).
This always leads to fights between advocates of equal-temperament and those who view equal-temperamant as nothing short of an invention of the Devil. ;\)

It has always fascinated me that:

440*3/2=660 (what we have in nature, assuming a fundamental of 440)
AND
440*2^(5/12)=659.255114 (tempered 5th)

Are SO close.

I don't think most people have any idea that our piano 5ths are so close to perfect (less then 2 cents flat), or how unlikely that most people can hear this tiny distance unless both notes in the 5ths are struck together. \:\) [/b]
On a side note, we can prove mathematically that this approximation is the absolute best we can do.

We've come a long way!
_________________________
Kawai K-3 (2008)

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#945254 - 10/07/08 01:22 AM Re: 3+3=5
btb Offline
4000 Post Club Member

Registered: 01/21/04
Posts: 4261
Loc: Pretoria South Africa
Real cool ... here’s a scientific analysis of the major scale notes . It will be noted that the 4th and 5th are so close to being true that we regard them as "consonant" notes ... it’s interesting to be reminded that the 6th and 3rd are most out of whack.

well-tempered

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New Topics - Multiple Forums
Noodling board
by Maarkr
Yesterday at 10:20 PM
New Movement Composed "To Rule" 4/20/2014
by hsheck
Yesterday at 10:17 PM
Understanding Sharps
by imustlearn
Yesterday at 08:18 PM
When a beginner is not a beginner anymore?
by Eight Octaves
Yesterday at 08:09 PM
DEBUSSY-"Serenade for the Doll" from 'Children's Corner'
by Hal Freedman
Yesterday at 07:14 PM
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