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PowerChromatic, Inverted Bebop, Diminished and all their relatives. A complete list!

Posted: Thu May 21, 2020 3:43 pm
by EdvinW
When Brendan talks about his PowerChromatic tuning for diatonic half-valved harmonicas, there is a set of properties he often mentions as advantages:
* Each octave is the same
* Each draw two half steps above its blow note
* It has complete major scales through it's range, possibly with bends

(Of course there are many modes that are useful, but whenever I write "major scale" the other modes are implied.)

There are many tunings which are very useful without meeting these criteria, Easy 3rd for instance doesn't meet any of them, but for general purpose single-note playing I think it's a reasonable list.

Last summer, Patrick (IaNerd) Proposed the Inverted Bebop tuning, and Brendan noted that the IBB share all these properties of the PC. A while ago I started to ponder: What other such tunings are there?

I started a systematic search, and when considering shifted or transposed tunings as equivalent I ended up with what I found to be a surprisingly short list of tunings.

I'll limit the list to tunings with period four, as a shorter period can't meet the criteria, and a longer period reduces the range of the instrument.

So here we go!

Diminished: (Top row contains blow notes, bottom row draw notes. )

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c  d# f# a  c  d# f# a
d  f  g# b  d  f  g# b
This well known tuning has all the requested properties. In fact, one could say it's sort of optimal since it can play ANY major scale.

I won't elaborate more on the dimi here, but for to point out what's probably its greatest weakness: Every key requires bends!

Let's add an additional criterion to the list: As many complete major scales as possible should be playable without bends.

As it turns out, no more than two tunings exist that meet this criterion with at least one complete scale without bends. They are:

Inverted Bebop:

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g  b  c  e  g  b  c  e  
a  c# d  f# a  c# d  f# 

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g  a  c  e  g  a  c  e  
a  b  d  f# a  b  d  f#
That's it! There are no other four-periodic tunings where draws are consistently one full step above their blows, which contain a complete major scale!

So far I've listed known tunings, but I have some news as well :)

Going through all possible combinations (henceforth accepting we'll have to use bends), I found that the PowerChromatic has two "sisters". I like to call them that, as the intervals between consecutive blow (or draw) notes are the same as for the PC (i.e. 2, 3, 4 and 3 half steps) but in different order; the three tunings are made from the only three possible permutations. In particular, the sisters are like PowerChromatic in that they have one note occurring twice.

One of them is obtained from raising both notes in either hole 2 or 3 (in the PowerChromatic tuning above) and the other by lowering the notes of hole 1 or 4, repeating this in every octave. I thus refer to them as the Sharp Sister and the Flat Sister, respectively.

Sharp Sister:

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g  bb c  e  g  bb c  e     (or  g  a  c# e  g  a  c# e
a  c  d  f# a  c  d  f#	        a  b  d# f# a  b  d# f#)
Flat Sister:

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f# a  c  e  f# a  c  e     (or  g  a  c  eb g  a  c  eb
g# b  d  f# g# b  d  f#         a  b  d  f  a  b  d  f)
With bending, both PC sisters have the same set of playable notes as the PC and thus the same scales. The leftmost variations contain the same notes as the above out-laid PowerChromatic in G, so lacking any scale to name them for that's how I've named them for myself. (Any suggestions for another system are welcome)

I've seen snippets of the pattern of the Sharp Sister appearing as elements in some of Patrick's tunings, but never repeated for two periods or more:
viewtopic.php?f=9&t=212&start=40#p1591 (almost two periods!)

The pattern of the Flat sister I believe to be new.

Finally, there are three more repeating tunings that technically meet the criteria, but which probably are not very useful. At least not as general-purpose tunings. I include them for completeness. I call them the 1AKB, or the 1 Awkward Key Bastards, as each is capable of playing one single complete major scale using bends. They are capable of playing some pentatonic scales and blues scales, but much fewer than the previously listed tunings.

Here are the three tunings: (To highlight the awkwardness, I give them all in the key of C!)

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b  c# d# g  b  c# d# g
c# d# f  a  c# d# f  a

b  d  eb g  b  d  eb g  
c# e  f  a  c# e  f  a

c  eb g  b  c  eb g  b  
d  f  a  c# d  f  a  c#
That completes the list!

Summing up, there has been three known four periodic tunings with each draw note one full step above its blow note: the Diminished, the PowerChromatic and the Inverted Bebop. The new tunings I've introduced are the two sisters to the PowerChromatic, which might be useful, and the three 1 Awkward Key Bastards which might be less so. The complete lit thus spans exactly 8 tunings, or a mere 5 if we exclude the bastards.

After stumbling over this I found it interesting, so I thought some of you might find it so as well :)

(I hope to elaborate a little on the new tunings in later posts, but as always there should be some considerable stress on the word 'hope' ;) )

Re: PowerChromatic, Inverted Bebop, Diminished and all their relatives. The complete list!

Posted: Thu May 21, 2020 4:45 pm
by triona
:idea: Image

dear geetings

Re: PowerChromatic, Inverted Bebop, Diminished and all their relatives. A complete list!

Posted: Fri May 22, 2020 9:36 am
by Brendan
Thanks Edvin! I love your combination of curiosity about harmonica tunings with a sophisticated analytical approach to researching and categorising them. It throws up interesting new layouts. Within your criteria (repeating scale, 4 hole period, draw always 2 semitones above the blow) you seem to have found them all! That's cool in itself, even if the new ones you found seem to have flaws from a practical playing perspective.

As you say, Diminished is really the ultimate tuning of this type. I think the fact that all major scales need bends is not such a drawback, considering its other advantages. You could even say it adds a nice character to each scale :)

I recently made a Dimi harp for a friend wanting to play jazz on diatonic, who was frustrated with the Richter overblow approach. While testing it, I was struck once again what a great tuning it is! A part of me would like to drop out for 6 months and woodshed this tuning exclusively, it has so much potential.

Using the C-based scales you showed, PowerChromatic is easy and flowing in keys like G, A, C, D, F and their relative minors. I have in it most keys and usually play the D major/Bm position by choice, because of the way they have all draws on their primary notes, 1-3-5 of the scale. Inverted Bebop has a similar shape in this respect.

The big disadvantage of PowerChromatic (shared by all the non Diminished tunings you show) is that a full chromatic scale can't be played with all draw bends alone. That D# note is missing, requiring an overblow or valved bend, which is a drag! Using half-valved chromatics gets over this issue easily, and I normally choose them for playing jazz type tunes.

But, sticking to diatonics, one variation I tried was this (it breaks Edvin's rules, but what the heck!):

GA. AB. CD. D#F#. GA. AB. CD. D#F#. GA.....

This is fully chromatic using bends, but you have a two semitone bend to get the E note. That can sound nice actually! But getting a clear chromatic transition between E and F is not easy, so I prefer standard PowerChromatic for general playing. However this variation is a good tuning for Balkan/Middle Eastern music if you play based in D. I use it that way for my tune 'Sweet Bulgarity'.

It would be interesting if Edvin (assuming he has the time and inclination) could map out slide half-valved harps for his new layouts, using the draw bending criterion. Maybe more interesting discoveries await!

If you do that it opens up more options too. The best of those is probably Wholetone, which has a 3 hole period. That's an advantage in terms of range and getting around tunes with big jumps, but the slider has to be used a lot.

Thanks for starting this interesting discussion Edvin.

Re: PowerChromatic, Inverted Bebop, Diminished and all their relatives. A complete list!

Posted: Fri May 22, 2020 10:22 am
by Lizzy
I think the dimi is the most versatile tuning but the conformity to law makes it, at least to me not so forgiving to improvise with like the powerchromatic.
It would be good to have an alternative 10 hole powerchromatic tuning to switch with but i dont have the theory mind like you guys :D And bends can give so much expression to a tuning so i dont mind them either :D

Re: PowerChromatic, Inverted Bebop, Diminished and all their relatives. A complete list!

Posted: Fri May 22, 2020 10:55 am
by triona
Lizzy wrote:
Fri May 22, 2020 10:22 am
I ... dont have the theory mind like you guys :D
It is me too struggling hard with that. :D

dear greetings

Re: PowerChromatic, Inverted Bebop, Diminished and all their relatives. A complete list!

Posted: Mon May 25, 2020 11:11 am
by EdvinW
Thank you for the appreciation!

The main take-home message from my post, I think, isn't necessarily about the cool new tunings. Rather, it's that if we want these properties, we have relatively few options. Our two choices are thus to either work with one of the tunings I present, or look outside this class; we don't need to spend time wondering if it contains more tunings! There's a saying I like among mathematicians, which can be applied to many other endeavours, and that's to "work very hard on being lazy". Getting things properly sorted out means less work when we think about them in the future :)

Among the tunings I listed, I think the three previously known ones all stand out in their own way, maximising different things. Among the new ones, the one I find most interesting is probably the Sharp Sister. For instance, if you're playing blues in D you get the low 7th (the C) as a bendable draw note sitting right next to the root for easy trills (or whatever ornaments are called in blues music :P). It does this while retaining many nice features from the PC.

When I talk of the need for bends as a drawback of the Dimi, I don't mean to suggest it's not a great tuning. I just think that property might be what keeps most people from using it, and why, outside the Richter-like tunings, it seems less popular than spiral tuning, variations on the Solo tuning, and others.

That's a good point about slideharps Brendan! With two seminotes span in each hole, they are the only option if we want to keep the period below 4. Making a similar analysis for slideharps would be much more involved though. If we, for instance, dictate that slide in and slide out both should be periodical and each posses some scale playable with bends, and even if we force the slide in notes to be no further than two seminotes from the slide out ones, there's about a thousand combinations!

I'm not sure if you had any specific kinds of chromatic layouts in mind, but I've been looking at three approaches since reading your answer:

1: Use a regular slider, that always changes the pitch by some fixed amount, like a traditional chromatic.
2: Take a step back from entire tunings, look at the slide-in and slide-out tunings separately and learn about how they interact.
3: Look at tunings of period three, as this drastically reduces the range of possibilities.

They've all been more or less fruitful I think, but I'm afraid writing up what I've found will have to wait a little. I should try to get some work down today, and despite trying I've found out that tuning tinkering in the same hours I'm trying to work is hard. (Who would had thought? :P)

I'll hopefully post again in a few days. Hopefully..

Re: PowerChromatic, Inverted Bebop, Diminished and all their relatives. A complete list!

Posted: Mon May 25, 2020 1:31 pm
by IaNerd
Bravo, Edvin!