The major scale is almost uniform (and 42) | Maths and Music | N J Wildberger
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- čas přidán 27. 07. 2024
- Building from the discussion of uniform scales in our last video in this series, we show that the major scale, which is the central framework of modern western music, actually has a somewhat curious property of being "almost uniform". This feature of the major scale turns out to be intimately connected with our current staff notation involving sharps and flats to give "key signatures" to all 12 major, and minor, scales.
Again we lean heavily on our arithmetical, mod 12 approach to the chromatic 12 tone scale. There is an interesting mathematical question which arises from this discussion.
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The question that I raise here about finding all the 7 note scales which have this almost uniform property has been answered by Federico Rocca, a frequent contributor to this channel and also the Wild Egg Maths channel. In his comment, pinned below, you can find the following list of interval sequences (in lexographic order) along with the list of 12 overlaps generated by sequential translations. He used a Python program to generate this: perhaps others might check it? [This is a "mathematical / musical classification": can you think of any others?]
Interval [1, 1, 1, 1, 1, 1, 6]: (7, 6, 5, 4, 3, 2, 2, 2, 3, 4, 5, 6)
Interval [1, 1, 1, 3, 1, 1, 4]: (7, 5, 3, 2, 3, 5, 6, 5, 3, 2, 3, 5)
Interval [1, 1, 1, 3, 1, 2, 3]: (7, 4, 3, 4, 3, 4, 6, 4, 3, 4, 3, 4)
Interval [1, 1, 1, 3, 2, 1, 3]: (7, 4, 3, 4, 3, 4, 6, 4, 3, 4, 3, 4)
Interval [1, 1, 1, 4, 1, 1, 3]: (7, 5, 3, 2, 3, 5, 6, 5, 3, 2, 3, 5)
Interval [1, 1, 2, 1, 3, 1, 3]: (7, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4)
Interval [1, 1, 2, 2, 1, 1, 4]: (7, 4, 4, 2, 4, 4, 6, 4, 4, 2, 4, 4)
Interval [1, 1, 2, 2, 1, 3, 2]: (7, 3, 4, 4, 4, 3, 6, 3, 4, 4, 4, 3)
Interval [1, 1, 2, 2, 2, 2, 2]: (7, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2)
Interval [1, 1, 2, 3, 1, 2, 2]: (7, 3, 4, 4, 4, 3, 6, 3, 4, 4, 4, 3)
Interval [1, 1, 3, 1, 3, 1, 2]: (7, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4)
Interval [1, 2, 1, 2, 1, 2, 3]: (7, 3, 3, 6, 3, 3, 6, 3, 3, 6, 3, 3)
Interval [1, 2, 1, 2, 1, 3, 2]: (7, 3, 3, 6, 3, 3, 6, 3, 3, 6, 3, 3)
Interval [1, 2, 2, 1, 2, 2, 2]: (7, 2, 5, 4, 3, 6, 2, 6, 3, 4, 5, 2) --- The major
There are some interesting features about this list, which I will discuss in a future video. Note for example that only four of the fourteen overlap sequences does not have a 6 centrally placed. Note also that the major scale is the only one on the list with interval sequence consisting of just 1's and 2's.
Video Contents:
00:00 Moving towards ley signatures and staff notation
00:10 Understanding and analysing musical notes
00:34 The key of D and a Prelude of J S Bach
02:09 Diminished and uniform scales
06:09 Major scale is almost uniform! (and 42)
10:10 Translates of minor scale I(s)=[2,1,2,2,1,3,1]
12:24 Translates of another scale I(s)=[2,1,2,1,2,2,2]
***********************
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With the help of a bit of python code, I have found these intervals regarding having a 7 note scale (up to cyclic rotations) with a 6 note transposition overlap:
Interval [1, 1, 1, 1, 1, 1, 6]: (7, 6, 5, 4, 3, 2, 2, 2, 3, 4, 5, 6)
Interval [1, 1, 1, 3, 1, 1, 4]: (7, 5, 3, 2, 3, 5, 6, 5, 3, 2, 3, 5)
Interval [1, 1, 1, 3, 1, 2, 3]: (7, 4, 3, 4, 3, 4, 6, 4, 3, 4, 3, 4)
Interval [1, 1, 1, 3, 2, 1, 3]: (7, 4, 3, 4, 3, 4, 6, 4, 3, 4, 3, 4)
Interval [1, 1, 1, 4, 1, 1, 3]: (7, 5, 3, 2, 3, 5, 6, 5, 3, 2, 3, 5)
Interval [1, 1, 2, 1, 3, 1, 3]: (7, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4)
Interval [1, 1, 2, 2, 1, 1, 4]: (7, 4, 4, 2, 4, 4, 6, 4, 4, 2, 4, 4)
Interval [1, 1, 2, 2, 1, 3, 2]: (7, 3, 4, 4, 4, 3, 6, 3, 4, 4, 4, 3)
Interval [1, 1, 2, 2, 2, 2, 2]: (7, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2)
Interval [1, 1, 2, 3, 1, 2, 2]: (7, 3, 4, 4, 4, 3, 6, 3, 4, 4, 4, 3)
Interval [1, 1, 3, 1, 3, 1, 2]: (7, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4)
Interval [1, 2, 1, 2, 1, 2, 3]: (7, 3, 3, 6, 3, 3, 6, 3, 3, 6, 3, 3)
Interval [1, 2, 1, 2, 1, 3, 2]: (7, 3, 3, 6, 3, 3, 6, 3, 3, 6, 3, 3)
Interval [1, 2, 2, 1, 2, 2, 2]: (7, 2, 5, 4, 3, 6, 2, 6, 3, 4, 5, 2) - -> The major
Regarding the number 42, if we have n intervals in the scale, we will find (n-1) matching notes in each of the n rows (6x7).
The matching notes for the pentatonic scale have a similar shape to the major, except that we need to subtract 2 from each number:
(7, 2, 5, 4, 3, 6, 2, 6, 3, 4, 5, 2) - (2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2) =
(5, 0, 3, 2, 1, 4, 0, 4, 1, 2, 3, 0)
[2, 2, 3, 2, 3]
Something pretty similar happens with other intervals
(7, 2, 5, 4, 3, 6, 2, 6, 3, 4, 5, 2) + (1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1) =
(8, 4, 6, 5, 4, 7, 4, 7, 4, 5, 6, 4)
[2, 2, 1, 2, 2, 1, 1, 1]
(7, 2, 5, 4, 3, 6, 2, 6, 3, 4, 5, 2) - (1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1) =
(6, 1, 4, 3, 2, 5, 0, 5, 2, 3, 4, 1)
[2, 2, 1, 2, 2, 3]
(7, 2, 5, 4, 3, 6, 2, 6, 3, 4, 5, 2) - (3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2) =
(4, 0, 2, 1, 0, 3, 0, 3, 0, 1, 2, 0)
[3, 2, 5, 2]
(7, 2, 5, 4, 3, 6, 2, 6, 3, 4, 5, 2) - (4, 2, 4, 4, 3, 4, 2, 4, 3, 4, 4, 2)
(3, 0, 1, 0, 0, 2, 0, 2, 0, 0, 1, 0)
[5, 2, 5]
Thanks Federico! I have taken your list and also copied it to the video description.
Interesting to note that of all the almost uniform scales that you found, only the major scale and the “degenerate” 7-note scale [1,1,1,1,1,1,6] both transpose so that all overlap counts (except 1) are accounted for.
@@mdreid Hi, Mark! What do you mean with 'both transpose'? A vertical scale transposed with an horizontal one?
What I do find special about both scales plus two more ones is that there is a kind of unique center of symmetry:
(7, 6, 5, 4, 3, 2, 2, 2, 3, 4, 5, 6): [1, 1, 1, 6, 1, 1, 1] --> the "degenerate"
(7, 2, 5, 4, 3, 6, 2, 6, 3, 4, 5, 2): [2, 1, 2, 2, 2, 1, 2]--> the major
(7, 4, 4, 2, 4, 4, 6, 4, 4, 2, 4, 4): [2, 1, 1, 4, 1, 1, 2]
(7, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2): [1, 2, 2, 2, 2, 2, 1]
Regarding the other intervals, we need to take them by pairs to find some symmetry:
(7, 3, 3, 6, 3, 3, 6, 3, 3, 6, 3, 3): [1, 2, 1, 2, 1, 2, 3], [3, 2, 1, 2, 1, 2, 1]
(7, 4, 3, 4, 3, 4, 6, 4, 3, 4, 3, 4): [1, 1, 1, 3, 1, 2, 3], [3, 2, 1, 3, 1, 1, 1]
(7, 5, 3, 2, 3, 5, 6, 5, 3, 2, 3, 5): [1, 1, 1, 3, 1, 1, 4], [4, 1, 1, 3, 1, 1, 1]
(7, 3, 4, 4, 4, 3, 6, 3, 4, 4, 4, 3): [1, 1, 2, 2, 1, 3, 2], [2, 3, 1, 2, 2, 1, 1]
(7, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4): [1, 1, 2, 1, 3, 1, 3], [3, 1, 3, 1, 2, 1, 1]
@@MrRoccmanI was making the observation that both the degenerate and major scales have the largest number of unique overlap counts as you transpose them through all 12 steps. Both those scales have transpositions with overlap counts of 6, 5, 4, 3, and 2 notes with the original scale. None of the other scales you found seem to have that many distinct overlap counts.
Put another way, for these two scales I can always find a transposition that introduces 1, 2, 3, 4, 5 or 6 new notes that are not in the original scale. The other 7 notes scales do not have this property.
@@mdreid Got it!
How does a mathematician play a piano?
then says "et cetera" and stops.
Not an answer to the question posed but maybe still relevant: The major scale of C is composed of the tones of the chords F, C, G which are adjacent in the circle of fifths. In the same way the major scale of G is composed of the tones of the chords C, G, D. There is a tautological overlap C, G for a total of 5 tones. The observation that the C scale is almost uniform says in other words that this 5 can be increased to 6 due to an overlap of 1 additional tone between the chords of F and D, namely A. So maybe the real question is where this A comes from. It has 5 times the frequency of F and 3 times the frequency of D, while D has 3^3 times the frequency of F... approximately! Combining these relations in a cycle we obtain that the frequency of A is 3^4 / 5 = 16.2 times itself, which modulo a multiplicative 2 is the same as 1.0125 which is close to 1. It appears that the overlapping A comes from this numerical coincidence.
I've played piano for over 60 years and never saw the symmetry until you showed me the numerical way of writing the scales. I'm also struck by the palindromic line along the bottom showing the overlaps. If you add the original again at the end, it reads the same forward and backward! The minor scale is even more interesting when looking at that pattern. Thanks, Norman, this is truly enlightening!
This music and math series reminds me (in spirit) of Leonard Bernstein's 1973 Norton Lecture series "The Unanswered Question", which I just discovered is available on CZcams.
The scale [1,2,2,2,2,2,1] is almost uniform. Start from C and then from D to see this.
I've been thinking lately I need to look into link between math and music theory. I mean, if I can understand physically what is happening it will give more to understanding music.
WELL DONE
Here is an interesting video on the geometry of music.
czcams.com/video/eJL3cYun_A4/video.html
and a related video on the symmetry of music
czcams.com/video/rNxAyI3idlc/video.html
where he shows that music theory has 3 kinds of symmetry.
Love your videos!
Could you delve into rhythm eventually?
A lot of interesting math there but the mayor challenge for me is to find some way of "wrapping my head around it".
Experimenting with Indian konnakol a bit, seams easier (somehow like adding blocks of pulses up to a sum in contrast to dividing some fixed sum??) but can't point to what if any real difference there is to western thinking around rhythm, the whole thing gives me a headache :)
Hi ralph12d Yes great direction, and I hope to be able to say something decent about rhythm . Not only Indian but also a lot of African styles that one can study and that give fascinating alternatives to our admittedly boring approach generally.
The property does not correspond so well to uniformity since 7 consecutive notes of the chromatic scale is changed by 1 note when we transpose by 1 semi-tone.
Black and white keys corrspond to the Beatty-like sequences (integer part of 12n/5 for n integers), and its complement (strict integer part of 12m/7 for m integers). Maybe this fact is more related to uniformity.
Do you have any ideas around why specific chords sound nice to us?
I am pretty sure our brain doesn't actually calculate the frequency changes etc. So I guess it has to be something about how these sounds occur in nature idk.
HI Dracony, That is such a hard question to answer as it has obviously a very large cultural aspect, and also within any culture a strong historical dependence. However there are also some physical reasons: an obvious one is that the harmonics of a note (overtones and undertones) have a closer physical connection with a note than others and this is reflected to some extent in chordal structures. I will have to talk about harmonics and overtones also in this series at some later point. But if we look at the history say of 20th century jazz music, we find that the notion of what sounds "good" harmonically has shifted quite a bit over the decades. Similar statements could be made about classical music say in the 20th century, and no doubt also about pop, where chord progressions seem to ebb and flow in popularity somewhat.
There’s a good book by William Sethares called “Tuning, Timbre, Spectrum, Scale” that digs into this sort of question. He starts by considering the overtone series and how the interactions between two different notes’ overtones “rub” to form a set of dissonances of varying amplitudes. Scales can be derived by finding sets of notes with low dissonance.
He goes on to consider timbres that arise from different overtone series (i.e., with non-integer frequency relationships). Each different timbre gives rise to different sets of notes with low dissonance for that timbre.
Obviously all the modes of major scale have the property, you ask about, but I think, You want some less trivial answer.