> If you choose the frequency 440hz, jump down one 7-semitone step and up 5, you have an A major scale
Also I have one more question about - when making chords using the 3rd and 5th step, for a note on the right boundary of this n <-> 2n interval, I would jump into 2n <-> 4n interval, right? But I'd I were not using chords then is "all" (simplified) music made in only only one x <-> 2x range and it's not crossed?
Another question- why do pianos have different black and white keys? Do you plan on doing a blog post that explains how "your " explanation of music theory fits various instruments? I've always wanted to understand music from a mathematical perspective and your post was eye opening!
I am going to try out your tool once I get hold of a computer.
Each doubling of frequencies is divided into twelve somewhat evenly-spaced semitones. If you go up seven out of twelve (using index 0 for your initial frequency), that's essentially half way to the doubling of the frequency with a ratio of 3:2. E.g., if your starting frequency is 440, going up seven steps will give you a note with a frequency of 660.
Another way of dividing up the space between a frequency and it's doubling uses eight of the twelve semi-tone divisions . The eight notes form an 'OCTive'. The octive has a 1-based index for the starting frequency. You can see seven out of twelve highlighted notes in the dashboard circles . The eighth note has the same name as the first one as you complete the circle but it would either be double or half the frequency. Going up seven semi-tones using the twelve division 'chromatic' scale is the same as going up five steps in the more selective eight note octive scale.
Yes, the circle continues from 2n<->4n and 4n<->8n and n/2<->n and n/4<->n/8 etc. The boundaries are those of human hearing from about 20hz to 20khz.
The white keys on a piano are the key of C Major. The black keys are the 'accidental' notes--the five notes of the twelve division chromatic scale that are discarded when selecting the notes for the eight (7 + the 1st/8th note that is counted twice to complete the circle) note octive.
Thank you, that was a detailed answer that gave me a lot of new things and I have a new question and some old questions-
You are saying there are 8 notes in an octave, but we only selected 7 notes after selecting 7th semi tone 7 times. I think you said the 8 note is 2x the original. Is that correct understanding?
So I got the point about 2^(1/12), what I didn't get is the staring point for A major and the jump structure. I thought all jumps are selecting 7th semi tone. So what is meant by "jump down one 7-semitone step and up 5"?
You partially answered my second question I think- saying that for chords it's okay to go from n <-> 2n scale to 2n <-> 4n scale. My question is for non chord music is this a common occurrence?
The music tradition sees the notes rather as intervals between two notes.
C to the same C is "unison" (You may need to instruments to play two identical C notes at the same time.)
C to C#/Db (this note has two names) is "small second"
C to D is "large second". And so on.
C to the next higher C is "perfect octave".
So if we take the C major scale, it has 7 different notes. But if we also include the next higher up C, you can pair the base C with 8 different notes, when we include pairing with itself, and pairing with the higher C. When you have two instruments playing, these are the pairings you can make when you play two notes at the same time.
> C to the same C is "unison" (You may need to instruments to play two identical C notes at the same time.)
> C to C#/Db (this note has two names) is "small second"
While C# and Db are the same note (in equal temperament [1]), the intervals C/C# and C/Db have different names: C# is called 'augmented unison' [2]. For the name, you start from the basic interval (e.g. C/C) and apply the accidentals (# or b) [3].
This is an example of why the traditional approach to music theory can be cryptic for a beginner. After the Western music moved to well and equally tempered scales (starting from the early 1700's), the context in which there is a difference between C# and Db has disappeared. But we still use terminology and notation from 500-800 years ago.
I actually didn't realize that the notes of the major scale and it's modes were contiguous on the circle of fifths until seeing guitardashboard.com last night. That's what the down one, up five is saying.
A fifth is seven semi-tones above the root. It has a 3:2 frequency ratio to the initial note. It so happens that if you jump up by seven semi-tones five times, you've got most of the major scale, albeit strewn over 3 doublings of the initial frequency. Usually, the major scale is thought of as occurring within a single doubling so in the key of C you'd have to divide the frequencies D and A by 2; E and B by 4 to get back into the original n<->2n range. Any time you double or halve the frequency you get a note with the same mood/purpose/name but one octave higher or lower. The frequency ratios end up something like this:
C = 1
G = 3/2
D = (3/2)(3/2) (/2 to get back to the initial range)
A = (3/2)(3/2)(3/2) /2
E = (3/2)(3/2)(3/2)(3/2) /2/2
B = (3/2)^5 /2/2
There's one note missing, though--the one with a 4:3 frequency ratio: F. To get that one, we invert the 3/2 relationship and take the frequency that's 2/3 of our initial frequency. That's in the halved range, n/2<->n, though so we've gotta double to get back into our starting range. This is the one time we go down seven semi-tones instead of the five times that we go up.
I'm not sure how useful it is to think in these terms but it does show that you can derive the major scale by using simple ratios which, psycho-acoustically speaking, are generally considered more pleasant than complex ratios when played at the same time. The relationship between B and C, (3/2)^5 == 243/32 is already pretty tense. E.g., you could alternate between two notes with that ratio to make it sound like Jaws is lurking somewhere nearby:
Even nonmusical people can almost always tell that, for example, 110 Hz, 220 Hz, 440 Hz, 880 Hz etc. sound like the same note.
So if you take an A major chord:
A (220.00 Hz), C# (277.18 Hz), E (329.63 Hz)
and then switch to
C# (277.18 Hz), E (329.63 Hz), A (440.00 Hz)
or also
E (329.63 Hz), A (440.00 Hz), C# (554.37 Hz)
and also something like
A (110.00 Hz), A (220.00 Hx), E (329.63 Hz), A (440.00 Hz), C# (554.37 Hz)
it makes pretty much no difference. People will hear and feel it as the same chord, the same notes, the same musical meaning.
There are some nuances in color, especially about which note is the lowest. And other concerns, like at which frequency ranges the other instruments (or your other hand on a piano) are playing at the same time, and if you want to spread out to avoid the others, or make the frequencies more crowded to put more emphasis. And also purely mechanistic concerns, how you are able to reach the notes with your fingers, for example, on guitar fretboard or piano keyboard.
> why do pianos have different black and white keys?
The 7 white keys match the C major (same as A minor) scale: C D E F G A B (and C).
I don't know why the ancient musicians thought to make C major as kind of the default scale, but that's how we now tend to see it. You can start a scale from any of the 12 semitones, but piano, and some other instruments, and the traditional musical writing notation are build so that C major is the default, and the remaining 5 semitones (the piano black keys) are written as how they deviate from the default scale.
I can see the construction from D onwards - jumps of 7, but how did you get to D? The instruction was 7)semi tones down, which from A is D "in previous scale"), but what about the 5 semi tones up?
One new question that popped in my mind- is this series of instructions (7down, 5up) a reverse engineering of the A major sequence to fit the 7- jump rule or is there some theory about it? I have no clue what different scales even mean or what major and minor means. Feel free to ignore the new questions (or the old one!)
The blog means two different sequences of jumps. The sequence of down-jumps begins from A. Then the sequence of up-jumps begins from A.
{jumps down: 1, jumps up: 5}
First, you start from A, and execute the down-jumps. In this case, there is only one. Then you start again from A (returning to A is not counted as a jump), and execute the up-jumps.
> Up two 7-semitone steps and down four gives you A minor.
Up-jump sequence: A, up to E, up to B.
Down-jump sequence: A, down to D, down to G, down to C, down to F.
Sort: A B C D E F G
I don't know what's behind this idea, or how useful it is. It seems to work, though. But I think there are simpler ways to construct the scales. Or just memorize them.
