If you start at any note of the piano and keep going up fifths until you’ve got a total of 7 notes, you get the Lydian mode. I explained this last time.
Above I show how it works starting with the note C. If we take these 7 notes and list them in increasing order starting with C, we get
which is C Lydian. Great! But please remember that C is an arbitrary choice of starting note. If we’d started at any other note we’d still get a Lydian scale.
Lydian is mostly made from notes of the major scale—with one exception, which shows up as the F♯ in the above example. What if we want the major scale, also called the Ionian mode? Then we should go up by fifths until we get 6 notes, but also go down a fifth to get one more:
C is still in red, which means the scale starts there! If we start there and write the 7 notes of the scale in increasing order, we get
which is C Ionian. Again, there’s nothing special about C here. The whole story I’m telling would work just as well with any other note.
It’s fun to keep playing this game. Let’s start at C and go up by fifths until we get 5 notes, but also go down by fifths to get 2 more:
Now we get C Mixolydian:
Next let’s start at C and go up by fifths until we get 4 notes, but also go down by fifths to get 3 more:
Now we get C Dorian:
Next let’s start at C and go up by fifths until we get 3 notes, but also go down by fifths to get 4 more:
Now we get C Aeolian, also called C natural minor:
Next let’s start at C and go up by fifths until we get 2 notes, but also go down by fifths to get 5 more:
Now we get C Phrygian:
Finally we can start at C and just go down by fifths to get 6 more notes:
Now we get C Locrian:
Now we’ve created all 7 modes of the major scale starting at C. If we list them in the order they were created, we get this chart:
We can see some interesting things here. As we work our way down the chart, each new mode differs from the previous one by having one of its notes lowered a half-step! I show this by having the note turn blue. We say each mode is ‘darker’ than the previous one.
There’s an interesting pattern in how the notes get lowered. Let’s understand it! Here’s the order in which notes get lowered as we move down the chart:
Each of these notes is a fifth below the one before!
It’s easy to see why this happens from all the pictures I drew. In each new mode we added a new note that’s a fifth below the last one we added. We can see them all in the very last picture, showing the darkest of our modes:
See? Working down from C, which is the note that appears in all 7 modes we’re discussing, we get
as we go around, each note being a fifth below the previous one!
I hope that’s clear. Now for another question: what if we extend this chart by lowering the one tone that hasn’t been lowered yet, the C?
We get another mode of the major scale: C♭ Lydian!
And the pattern of blue bars is beautifully continued! That’s because C♭ is a fifth below the previous lowered note, G♭.
It’s really cool how by lowering the one tone that hadn’t been lowered—the root of the scale, the so-called ‘tonic’—we suddenly pop from the darkest mode, Locrian, back to the brightest mode, Lydian. Why is that? It’s easiest to see using our diagrams. Let’s compare C Locrian to C♭ Lydian:
In terms of the notes these scales contain, the only difference is that we’ve replaced C by C♭. But this lowers the very bottom note of the scale. And this means that instead of the notes being bunched up near the bottom of the scale, making the scale very dark, they are now bunched up near the top, making it very bright. By lowering the bottom note, the other notes become higher by comparison!
Now, you may have been wanting to complain that C♭ isn’t a thing:
You would be wrong. C♭ really does exist: in various contexts, like listing the notes in a scale one letter at a time, musicians do have good reasons to call one of those notes C♭. This is actually a rather deep topic:
But in my series on modes I’m only talking about the tones an ordinary modern piano can play, tuned in equal temperament, so some of the nuances Adam Neely discusses are not relevant here. For our discussion now, C♭ is just another name for B.
In short, while C♭ Lydian does exist, when played on a modern piano it sounds just like B Lydian. So let’s redraw our chart, calling it B Lydian:
A bunch of notes in the last row now get new names, with sharps rather than flats, to make sure that each letter from A to G appears once. But they still sound the same.
More importantly: now we are back to Lydian, so we can play the game all over again!
We can lower one tone at a time, just as we did before, and go through the 7 modes from B Lydian to B Locrian. And then we can lower that B a half-tone, and so on:
This chart could go on forever! In each row we lower one note by a half-step. It’s always a fifth below the note we lowered in the previous row. The chart eventually repeats. But it repeats only after we’ve covered all 7 modes of the major scale starting on all 12 notes in the chromatic scale—a total of 84 modes!
(If you have the ability to create a beautiful long chart like this without dying of boredom, please send it to me.)
It’s fun to think about exactly why when we lower the first note in C Locrian we get B Lydian—why the darkest mode suddenly transforms into the brightest one. I think I’ll let you ponder that.
All the ideas in this post emerged from a short comment by Mark Reid on Mastodon:
Furthermore, which note is flatted cycles through the circle of fifths. If you continue the pattern past the bottom row the next note to be flattened would be the C down to a B, creating a B Lydian scale. So the whole pattern through the modes repeats a half step down.
I am apparently much less efficient at transmitting information! But I had to imagine those wheel-shaped charts to really appreciate why things work as they do. And it may pay to ponder those charts, since they reveal a lot of interesting patterns.
• Part 1: modes of the major scale.
• Part 2: minor scales, and a cube of modes.
• Part 3: all 7-note scales drawn from the 12-tone chromatic scale with at most a whole tone between consecutive notes.
• Part 4: modes of the major and melodic minor scale.
• Part 5: modes of the Neapolitan major scale.
• Part 6: the special role of the Lydian mode, and the circle of fifths.
• Part 7: cycling through all 84 modes of the major scale in all keys.
• Part 8: how the group 
Azimuth 
Nice! Did you explain where does the 7 come from though? I mean why take 7 notes and not any other number <=12?
No, I’m not attempting to explain the 12-tone chromatic scale, the 7-tone major scale, or why we pick 7 tones out of twelve in the particular pattern we use.
People have written a lot about this! It’s fascinating and controversial topic. But it’s a different game. Right now I’m taking this stuff as given and simply studying patterns in the 7 modes of the major scale.
Short answer: It's because 3/2 is the simplest number between 1 and 2, and 2^(7/12) is the closest approximation of 3/2 by a power of 2 that isn't more complicated.
That’s a good short answer to Tali’s question! Indeed, if you keep multiplying by 3/2 you get a pattern that resembles the ones I’ve drawn above, but doesn’t quite close up because (3/2)12 is not quite equal to 27:
The little gap is called the Pythagorean comma. But in equal temperament, which I’m using in all my posts on modes, we approximate 3/2 by 27/12 to make that gap go away.
27/12 ≈ 1.49830707688…
Tali wrote:
Toby wrote:
Actually now I don’t see how this answers Tali’s question.
It explains something nice about the 12-tone equal-tempered scale: a perfect fifth has a frequency ration of 3/2, which sounds nice and simple, and if we go up 12 perfect fifths we almost go up 7 octaves, since 3/2 is close to 27/12. So, we can build a equal-tempered scale where each note has a note almost a perfect fifth above it if that scale has 12 notes.
But I don’t see it explaining why the 12-tone equal-tempered scale has nice 7-tone ‘subscales’, namely the modes of the major scale.
I think for that perhaps we should notice some other nice math fact. Going up a fifth in the 12-tone equal-tempered scale seven times almost gets us back to the same note, four octaves up! And that’s one of the lessons of Russell’s Lydian chromatic theory: he notes that if we do this we get
so we’re almost but not quite back to C.
Thus we need to make a compromise if we’re trying to pick a 7-tone subscale of the 12-tone chromatic scale that contains a lot of fifths. One compromise is Lydian:
where the last step is a bit less than a fifth. Another compromise is the major scale, also called Ionian:
where the second to last step is a bit less than a fifth. There are actually 7 such compromises, since we can put the too-short step anywhere we want. And these are Lydian, Ionian, Mixolydian, Dorian, Aeolian, Phrygian and Locrian!
Notice I’m claiming 7 fifths in the 12-tone equal-tempered scale is almost 4 octaves. Since a fifth in 12-tone equal-tempered scale is 27/12, this means mathematically that
249/12 ≈ 24
or
16.9514095… ≈ 16
which is not great but not too bad. 16.9514095… is bigger than 16 so we need to squash one of our fifths.
Alternatively we could claim that 7 perfect fifths is almost 4 octaves. In our story we’re already committed to the 12-tone equal-tempered scale and looking for 7-tone subscales, but quite likely in reality a 7-tone scale was invented first. Maybe people did it because 7 perfect fifths is almost 4 octaves. Let’s see how close it comes. We’re saying
(3/2)7 ≈ 24
or
17.0859375 ≈ 16
This is a tiny bit worse than our previous approximation, but basically about the same: we’re basically saying 17 is close to 16.
So I think we’ve seen the number 7 show up for at least two different reasons! One way to put it is this. You could invent a 7-tone equal-tempered scale based on
(3/2)7 ≈ 24
and you could invent a 12-tone equal-tempered scale based on
(3/2)12 ≈ 27
Our current musical system goes for the 12-tone equal-tempered scale—but there happen to be halfway decent approximations of a 7-tone equal-tempered scale inside it. I don’t think that’s a further coincidence: I think it follows from the two approximations above.
And finally: these ‘further approximations’ are not unique, i.e. there’s not a unique best way to try to jam a 7-tone equal-tempered scale inside the 12-tone equal-tempered scale. Even if we pick the starting note, there are 7 different approximations, the modes, where we stick the too-short fifth in different places.
Very nice!
So my short answer should have used 2^(4/7) rather than 2^(7/12).
(There are multiple solutions to the problem of approximating 3/2 by a power of 2 in a way that can’t be made both simpler[*] and more precise[†]. Compared to 4/7, 7/12 is more precise but also more complicated, while 3/5 (which gives us the pentatonic scale) is simpler but also less precise.)
[*] (defined as having a smaller denominator, or equivalently a smaller numerator)
[†] (defined by having a ratio close to 1, rather than a difference close to 0, although none of these examples depends on that subtlety)
I think you would really like Jacob Gran’s YouTube channel, especially his series on Tonal Voice Leading. It’s not exactly related to modes (except insofar as the voice leading rules he covers were originally used for modal music) but if we think of your blog posts on modes as being about explaining how the western musical alphabet works, Jacob Gran’s videos are about explaining the western musical grammar.
Thanks, I’ll check it out! When it comes to music I’m interested in a lot more than just modes. I’ve been enjoying the YouTube videos of various music theorists and music theorists: while I hate videos for learning math they seem like a perfect way to learn about music, since they let you hear the music, see scores or someone playing an instrument, and hear them talk. But I haven’t run into Jacob Gran, whose YouTube channel is here.
I wrote:
Owen Lynch took some time on the weekend to do this! He put it here. I will feature it in my next post on modes, along with some more math. His chart is amazing! But it’s circular, and I wouldn’t mind a linear one, an extension of this to a chart with 84 rows, since that will fit more easily into the skinny-column format of this blog:
I see that Owen chose G-sharp over A-flat.
Very cool! And you inspired me to play around with some Python code to create the 84 rows. (I’m thinking it should produce text, HTML, and PNG output…)
Well, FWIW:
I think I got filed in your spam folder. I tried to post the HTML of the 84-row chart, and I think WP didn’t like all that HTML.
It not only put that comment in spam, it got rid of the HTML. So I guess WordPress really dislikes comments with lots of HTML. But don’t worry: I liked it!
Thanks for creating that chart! I’m going to use it in Part 8, and credit you. Part 8 will be a kind of climax of my story of modes, or least a local maximum.
I have a lot of complaints about WP software… 🙄
Glad you liked the chart! It was a fun little project. A very small payback for all I’ve gotten from your posts over the years. I don’t have the background to understand a good chunk of it, but I’m always trying to learn more, and what I do understand fascinates me!
FWIW: This first version used the pattern of the notes in the columns to generate each 84-row column. Quick and easy. I’m thinking of doing a second version that uses the logic discussed in your post and generates full rows — all the notes of each scale. But I need to study that logic a bit more to code it right.
[…] John Baez has been putting out an excellent series of posts about music theory on his blog. The most recent, the seventh, is about how you can generate scales by picking out piano notes in intervals of fifths. What’s interesting is that you can generate all seven major scale modes in each of the twelve […]
Mathematician and educator John Baez has an excellent series of blog posts about music theory. The seventh concerns generating scales by using notes separated by fifths. Shifting the start point generates the seven major scale modes. Shifting the root key generates those seven modes in the twelve keys (a total of 7×12=84 scales).
John asked if any of his readers would be interested in creating that table of all 84 rows. It sounded like — and turned out to be — a fun exercise. This post explores in detail the Python solution I came up with.
It’s actually the third solution I came up with. The first and second used different approaches to the problem — the second wasn’t entirely successful. Exploring those seems like a good illustration of different programming approaches to the same problem, but also topics for another post. I’ll stick here to the third (and final) solution.
First, here’s the table the code is to create…
the link for Part 8 at the end is broken, has wrong date in the url path, instead of 08/24 should write 08/21