Ideas which I'm not sure about: maybe a I becoming a I7 and then going to it's IV (the IV becoming the new I)? Or another mode? Or an immediate modulation afterwards? Or a I which is not the end of a cadence ? Or a I going to a IIIb ?

You could find examples there: https://www.hooktheory.com/theorytab/index

What does your teacher mean exactly by Intermediary Chord ?

Look at that : https://www.hooktheory.com/chordProgressionNode#key=C&scale=major&path=1.4.1

I made a C F C progression and then there are still a lot of songs going to different chords, so the second I is used in an intermediary position. Is that right?

I'm with Mark, you are going to have to define what you mean by intermediary chord.

There's one example that comes to mind: 'Reach Out I'll Be There' by the Four Tops. I haven't ever seen any transcriptions or original sheets, but whichever is the 'official' key, I've always considered the most stable chord (I) to be the first one in the pre-chorus, under the 'Reach out' lyrics. I believe that's quite an 'intermediary position'... What do you think?

The teacher means: using I in the middle of a progression, neither as the first chord not the last.
Cuantas your answer is the right one.......How do I search for songs based on a certain progression in Hook theory. Just cannot find a place to type the progression(s) in.

I'm afraid that was just me: a brain full of songs and still enough memory to access them...but I can't help you any further, I really don't know where or how a song search by chords, or progressions, can be done.

I have never personally used hook theory until today, but looking at the page @Mark Tomato Alley linked, you do not type them, instead you clear the chords he imputed with the refresh button on the top middle of the webpage, then you can choose any chord progression by clicking on the chord names and see the songs in their database that has that progression.

I believe that most music out there will have chord I as an intermediary chord. Just check for the chords of yours favorite songs and It will be there. There are many ways which we can use the I in the middle of a song without taking its strenght as a conclusion chord... You can use inversions, add dissonances, or do not use it after a V chord, or, if you do, not making this V chord a dominant chord by inverting it... If you think that we start harmonizing songs using only chords I, IV e V, you can see that it it isn't rare to find I chord anywhere in a piece of music. Chords built over scale degrees and harmonic functions are different things, although related.

A lot of the most popular western music is strongly based on repetition of different sections, so it's so easy to find examples of I chords playing a certain amount of times through a song, I think that's the case you were talking about, @igor reis . That's why I was more inclined to consider the question a bit more tricky, meaning that the song should not show any I chord in the first part of it, but I might be wrong...

Yep! I never heard of the concept intermediary chord. So, can be some use for the I chord that I never heard of, or it can be a simple thing :D Specially because of our tonal-centered approach to music. If the question was upon the II, IV and V degree, I would give an aswer with the same idea: their functions, specially from the cadential point of view. If he asked for examples of intermediary III, VI and VII... Then I would be confused!

But fuctional harmony is just one artistic choice we can have in the process of music making, and there are numerous ways to move inside this realm. We can intend to get to a certain tonal center, but, before we prepare our cadence, we flow through realms of chords in a sequence chosen by other criteria instead of harmonic functions. We stay suspended before being pulled by gravity.

Anyways, functional harmony is one of my favourite tools in sound, and it is deeply primordial, since is about the natural inclination of us to fix a sound in memory and then relate all other subsequent sounds with it. So, from the simply fact that our brain can solidily store a sound memory through a certain time, it comes a whole sophiscated and omnipresent musical system. (... probably we don't hear the city noise as musical dissonance because there are sounds occurring all the time and very fast, we never having enough time to fix a tonal center as it would be the first note played by an orchestra in an empty hall. That might be the reason with an ambulance siren can be so annyoing... some of them are simple fast intervals, maybe with a slide - at least here where I live -, but because it establishes a tonal center, all city noise come together with the siren in a massive dissonance. I don't know, I just thought about that :D)

But, yeah, if I play a C and 5 seconds later I play a B, automatically our brains will wait another sound in five seconds (natural spectation due the little pattern of two sounds that just happen), and our brains will desire to come back to C, because the information of these two sonic inputs are in a near memory and, therefore, making a dissonance. But if I play a C, in a silent room, and after a minute I play a B, the B would naturally sound as a tonic established out of nothing, from silence, and our brains wouldn't tie such a wide gap of time in a way it would sound minimally rythmical.

Personally, I always admired composers who wrote long chord progressions, with light and seameless modulations, making us eager for a tonic center, delaying it but always keep our interest in what is going on and increasing the tension, giving us hints that it is about to get there... but it doesn't! and again! then to release it in the most beautiful way.

Try out Shostakovich's 5th Symphony's last movement...



Or Wagner's Lohengrin overture...



Ok, I think I might have gone a bit further again XD

😳Ok let me digest it before I...😬