A late hello :)

Hi,

I joined these yahoo groups about half a year ago, but I never introduced myself. I wanted to wait until I have my 31-EDO guitar, but there were some problems which I didn't expect that delayed the process. Well, anyway...

--------------------------------------------------

I'm an autodidact from Germany who's into microtonal music for a long time.

My interest more or less started when I still went to school, when I realized that (in common practice) there were only 12 pitches per octave, but an infinite number of ways to note them via classical notation. I was convinced that there was some deeper meaning behind the general agreement that it depends on the musical context whether one should write 'D#' or 'Eb'.

This (among other things) led me to pythagorean tuning, 5-limit just intonation, 53-EDO, and later meantone temperament. I also realized that classical notation can be extended using linear tunings with a fifth generator and an octave period, and that 5, 7 or 12 notes work especially well, since even after modulation the ordering of the notes remains.
It took me a lot of time to realize that this is the same concept Erv Wilson described as "moments of symmetry", though.

After graduation from school, I started studying at a German university with a computer science major (which includes a lot of mathematics) and a musicology minor. I learned a lot of useful and interesting things related or applicable to music theory. Also, a professor recommended to me a book by Martin Vogel, who has become one of my major inspirations.

A very fascinating and hard-to-describe moment was when I discovered scala. For me, microtonality was just an abstract concept that existed in my mind until then, but now I could actually hear what a just 5:4, 7:4 or 7:6 sounds like!

Discussions about microtonal music in regular music forums tended to be more tiring than helpful, and since I didn't know of any German microtonal music forum, I eventually gave up on them, and continued to study microtonal music on my own.

Later, I stumbled upon Ron Sword's microtonal guitarist forum where I learned some interesting and useful basics, and which was basically my gate to the English speaking xenharmonic community. When I wanted to join nonoctave.com, I realized that it didn't exist anymore. In October 2012, I joined Ig's forum, and when it went offline again after less than a month due to a lack of general interest, I found the yahoo groups to be the only other remaining discussion platform that was halfway acceptable to me, so I registered here, and I have to say I learned a lot! :)

In an earlier discussion, I was asked here about my real name, and why I'm using a pseudonym. My reason why I use a pseudonym is that this is an open platform, and I want to be able to freely express myself without having to worry that anyone who knows my real name can google it, and read about my "weird" interests, or personal opinions. I hope that's understandable. However, it's not that I'm someone who was active here before, and simply wants to hide their previous identity (I didn't achieve anything noteworthy related to microtonal music that is publicly connected to my real name, or other pseudonyms).
Also, I have to apologize to you, Carl; I think my reaction to your criticism back then was a little harsh, and I shouldn't have called another opinion 'dogmatic', just because it didn't coincide with mine.

--------------------------------------------------

You can find further info about me in my SoundCloud profile, though there are hardly any audio files yet, since I'm still waiting for my guitar:

https://soundcloud.com/gedankenwelt

If you have questions, just ask, though I'm going to create some new topics anyway - I hope for some interesting discussions. ;)

Best
- Gedankenwelt

P.S.: Sorry in case of bad English

Well, I should apologize too. We've had great contributions from pseudonymous members over the years. I still do believe things work best when everyone uses their real names, but I understand your reasons. I never suspected you weren't a native English speaker.

-Carl

--- In tuning@yahoogroups.com, "gedankenwelt94" <gedankenwelt94@...> wrote:
>
> Hi,
>
> I joined these yahoo groups about half a year ago, but I never introduced myself. I wanted to wait until I have my 31-EDO guitar, but there were some problems which I didn't expect that delayed the process. Well, anyway...

https://soundcloud.com/gedankenwelt/bach-fugue-in-d-minor-5-limit

I like the performance values. Really disappointing to hear it
fade out instead of finishing!

-Carl

--- In tuning@yahoogroups.com, "gedankenwelt94" <gedankenwelt94@...> wrote:
>

> You can find further info about me in my SoundCloud profile,
> though there are hardly any audio files yet, since I'm still
> waiting for my guitar:
>
> https://soundcloud.com/gedankenwelt

Dear Carl,

I'm glad you replied! I don't know you personally, but I think you're a very nice person, and I had a bad conscience about the development of the discussion.

Anyway, I'm happy about your feedback back than, because I didn't think too much about the "distance" the use of pseudonyms can create between people, even if it is "just" on the internet.

I'll reply to your other post later, I'm a little short on time...

- Gedankenwelt

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Well, I should apologize too. We've had great contributions from pseudonymous members over the years. I still do believe things work best when everyone uses their real names, but I understand your reasons. I never suspected you weren't a native English speaker.
>
> -Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> https://soundcloud.com/gedankenwelt/bach-fugue-in-d-minor-5-limit
>
> I like the performance values. Really disappointing to hear it
> fade out instead of finishing!
>
> -Carl

I'm glad to hear that you like the adaptation! :)
I was surprised myself how well the notes in meantone notation translated to 5-limit JI here.

I could have finished the adaptation, but I planned to revise the score format in the future, so I felt that it was not worth working with a format that will become obsolete.

I also started an adaptation of "The Art of Fugue" / Contrapunctus I, where I couldn't avoid a very early upward shift by a syntonic comma.
To me (who doesn't have absolute pitch) it didn't sound bad, but the notation became a little ugly in the current / limited score format, so I didn't finish that adaptation either.

As mentioned in the description, my plan was to revise the score format and write a parser that converts this microtonal score format to the Csound score format, but I can't say when I will find the time for this, since currently my focus lies on other projects.

For example, I realized that the Vocaloid 3 editor seems to be well suited for making microtonal vocal music:
I started to write a so-called "job plugin" that converts all notes in a tune to pitches based on a 12-tone maximally even scale from a specified EDO, and additionally applies 1-step accidentals if a certain short character sequence (like "!^" or "!v") is found in the lyrics. So, everything one has to do is to input some notes, put accidentals in the lyrics where necessary, and run the plugin with the specified EDO. :)

"gedankenwelt94" wrote:

> As mentioned in the description, my plan was to revise the score
> format and write a parser that converts this microtonal score
> format to the Csound score format, but I can't say when I will
> find the time for this, since currently my focus lies on other
> projects.

You might consider striking up a conversation with Chuckk,
whose tool "Rationale" does something similar.

> I also started an adaptation of "The Art of Fugue" /
> Contrapunctus I,

Excellent!

> For example, I realized that the Vocaloid 3 editor seems to be
> well suited for making microtonal vocal music:
> I started to write a so-called "job plugin" that converts all
> notes in a tune to pitches based on a 12-tone maximally even
> scale from a specified EDO, and additionally applies 1-step
> accidentals if a certain short character sequence (like "!^" or
> "!v") is found in the lyrics. So, everything one has to do is
> to input some notes, put accidentals in the lyrics where
> necessary, and run the plugin with the specified EDO. :)

That sounds awesome. Vocaloid is a really cool tool.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> You might consider striking up a conversation with Chuckk,
> whose tool "Rationale" does something similar.

Thanks, this definitely looks interesting, and Chukk has my sympathy for supporting open-source / Linux, and OSC (as an alternative to MIDI)! I'll take a closer look when I return to my synthesizer project.

> > For example, I realized that the Vocaloid 3 editor seems to be
> > well suited for making microtonal vocal music:
> > I started to write a so-called "job plugin" that converts all
> > notes in a tune to pitches based on a 12-tone maximally even
> > scale from a specified EDO, and additionally applies 1-step
> > accidentals if a certain short character sequence (like "!^" or
> > "!v") is found in the lyrics. So, everything one has to do is
> > to input some notes, put accidentals in the lyrics where
> > necessary, and run the plugin with the specified EDO. :)
>
> That sounds awesome. Vocaloid is a really cool tool.

Yeah, I'm pretty excited, too! :)
My tool isn't very reliable yet, but when it did what I wanted it to do, the results were very satisfying. I was stunned when I first heard how realistic IA's voice can sound (a Japanese vocaloid), but when I tried out melody lines based on septimal harmonies, neutral thirds and very small steps, I was deeply impressed!

Of course one doesn't have to limit oneself to EDOs or a notation based on 12-note ME scales, that was just the most convenient implementation method I could think of. I'll also implement extended just intonation, by converting the 12-EDO pitches to the 12-tone pythagorean MOS scale, and adding accidentals for the pythagorean, syntonic and septimal comma, for the undecimal quartertone and so on.

Very cool, g94! It sounds like your background is very similar to mine
- eerily so, in fact. Great to have you around!

I'm curious, are you a guitarist by trade then?

The forums you found are kind of peripheral splinter groups. The three
forums you should definitely be on are:
1) This
2) tuning-math
3) XA on Facebook

That's where all of the serious discussion takes place.

Unfortunately, you're joining about a year after one of the largest
sustained spurts of activity that we've ever had. Like I said, things
are in a lull right now, though they'll pick up again. I plan on
continuing my previous practice of posting here like a madman as soon
as my current ridiculously time-consuming job stops being so
ridiculously time-consuming. But, in the meantime, there's a
ridiculous amount of high-quality literature on this list, since this
is where the whole thing started (before I joined, unfortunately), so
feel free to keep learning stuff.

You're going to want to join the Yahoo! tuning-math group if you
aren't there yet. That's where the serious mathematical discussion is
taking place, and from the looks of it, you're interested in that. I
strongly encourage you to get on there!

If you want to get up to speed quickly, then if you haven't already, I
strongly recommend taking the time to learn as much of the
"Mathematical Theory" pages on the wiki as you possibly can. It's very
dry and technical, so feel free to ask on tuning-math and we'll answer
for you.

Mike

On Fri, May 17, 2013 at 2:08 AM, gedankenwelt94
<gedankenwelt94@...> wrote:
>
> Hi,
>
> I joined these yahoo groups about half a year ago, but I never introduced myself. I wanted to wait until I have my 31-EDO guitar, but there were some problems which I didn't expect that delayed the process. Well, anyway...

Hi Mike! :)

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Very cool, g94! It sounds like your background is very similar to mine
> - eerily so, in fact. Great to have you around!
>
> I'm curious, are you a guitarist by trade then?

Yeah, I also realized that our main interests (free modulation / generalization of MOS / flexible notation) seem to overlap! If you find the time, I'd love to hear how you discovered microtonal music, but this isn't urgent, and you mentioned that you have a very time-consuming job. ;)

Yes, I'm mainly a guitarist, and I wait with composing music until I have my 31-EDO guitar. I also plan to add 94-EDO / Garibaldi[53] fret markers to my fretless guitar as a future project, here's the design:

/makemicromusic/topicId_29123.html#29123

...which makes me wonder: Why do I see none mentioning interval notations based on MOS scales? I think they can be very useful, especially for generalized guitar tabs that aren't limited to a specific equal division, or in case of larger EDOs where something like 8/ instead of "36th fret" is much more readable.

> You're going to want to join the Yahoo! tuning-math group if you
> aren't there yet. That's where the serious mathematical discussion is
> taking place, and from the looks of it, you're interested in that. I
> strongly encourage you to get on there!

Thanks! I already joined simultaneously with the tuning list, but first there's some not-so math heavy stuff I want to discuss ... or, what I personally think is not so math heavy. I often have a hard time estimating what is complicated for people who already disliked math in school.

You won't see me at Facebook though, since I don't want to support that platform. I liked the idea of an independent forum, but apparently there wasn't much interest.

Looking forward to future discussions ;)
- Gedankenwelt

--- In tuning@yahoogroups.com, "gedankenwelt94" <gedankenwelt94@...> wrote:
> ...which makes me wonder: Why do I see none mentioning interval notations based on MOS scales? I think they can be very useful, especially for generalized guitar tabs that aren't limited to a specific equal division, or in case of larger EDOs where something like 8/ instead of "36th fret" is much more readable.

This notation method comes to mind:

/tuning/topicId_105842.html#105842

Ryan Avella

--- "Ryan Avella" <domeofatonement@...> wrote:
>
> --- "gedankenwelt94" <gedankenwelt94@> wrote:
>> ...which makes me wonder: Why do I see none mentioning interval
>> notations based on MOS scales?
>
> This notation method comes to mind:
> /tuning/topicId_105842.html#105842

As do

http://lumma.org/music/theory/notation/OnLinearNotations.pdf

and

/tuning-math/message/11629

and many others.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- "Ryan Avella" <domeofatonement@> wrote:
> >
> > --- "gedankenwelt94" <gedankenwelt94@> wrote:
> >> ...which makes me wonder: Why do I see none mentioning interval
> >> notations based on MOS scales?
> >
> > This notation method comes to mind:
> > /tuning/topicId_105842.html#105842
>
> As do
>
> http://lumma.org/music/theory/notation/OnLinearNotations.pdf
>
> and
>
> /tuning-math/message/11629
>
> and many others.

Thanks for your replies!

Yes, those linear notations can be easily applied to intervals as well. But each time I saw someone mentioning them, they were exclusively applied to notes.
And that made me wonder if other people aren't aware of the possibility, or if they just don't think it's useful.

Mike's notation indeed doesn't need any modifications to apply it directly to intervals. Personally, I strongly prefer to notate the unison as a '0' instead of a '1' though, but that's for everyone to decide on their own.

- Gedankenwelt

--- In tuning@yahoogroups.com, "gedankenwelt94" <gedankenwelt94@...> wrote:

> Thanks for your replies!
>
> Yes, those linear notations can be easily applied to intervals as
> well. But each time I saw someone mentioning them, they were
> exclusively applied to notes.
> And that made me wonder if other people aren't aware of the
> possibility, or if they just don't think it's useful.
>
> Mike's notation indeed doesn't need any modifications to apply it
> directly to intervals. Personally, I strongly prefer to notate the
> unison as a '0' instead of a '1' though, but that's for everyone to
> decide on their own.
>
> - Gedankenwelt

Maybe I misunderstood your idea. Linear notations of the kind I
mentioned all assign note names to scale degrees, and accidental
symbols to intervals.

Can you clarify what you meant? Byzantine notation works on
intervals instead of scale degrees
http://en.wikipedia.org/wiki/Musical_notation#Byzantine_Empire

As for instruments, the Samchillian Tip Tip Tip Cheeepeeeee is
based on intervals...
http://www.samchillian.com

-Carl

On Saturday 25 May 2013 1:41:01 gedankenwelt94 wrote:

> Mike's notation indeed doesn't need any modifications to
> apply it directly to intervals. Personally, I strongly
> prefer to notate the unison as a '0' instead of a '1'
> though, but that's for everyone to decide on their own.

There are existing standards we don't get to decide.
Jianpu, which is a "movable doh" system for diatonic scales,
starts with '1' and uses '0' for rests. Guitar tablature
and musical set theory, both for chromatic scales, start
with 0.

Graham

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Maybe I misunderstood your idea. Linear notations of the kind I
> mentioned all assign note names to scale degrees, and accidental
> symbols to intervals.
>
> Can you clarify what you meant? Byzantine notation works on
> intervals instead of scale degrees
> http://en.wikipedia.org/wiki/Musical_notation#Byzantine_Empire
>
> As for instruments, the Samchillian Tip Tip Tip Cheeepeeeee is
> based on intervals...
> http://www.samchillian.com

Sorry, I should have been more specific.

Typically, a diatonic C major scale is notated as following: C D E F G A B C. In order to modulate, notate MODMOS scales or scales with more than 7 notes etc., # or b can be used to indicate chroma alterations.

However, it's possible to notate a non-specific major scale using an interval notation: 1 2 3 4 5 6 maj7 (or simply "7") 8. Here again, we can use # and b to indicate chroma alterations. For example, 1 2 b3 4 5 b6 b7 8 describes a natural minor scale.

Likewise, we could notate a 12-tone schismatic scale (6 fifths up and 5 fifths down) like following 0 1 2 3 4 5 6 7 8 9 10 11 12. If we use / and \ for a chroma (= pythagorean or syntonic comma) up and down resp., we can notate a 5-limit major scale like following: 0 2 4\ 5 7 9\ 11\ 12. The natural minor scale would be 0 2 3/ 5 7 8/ 10/ 12.
(you may want to notate the chroma (or other accidentals) before the basic intervals, instead of after them)

An advantage of starting with 0 instead of 1 is that the notation can be used as a step notation for an EDO where the chroma is tempered out. For example, we can apply the above Schismatic[12] notation to 12-EDO by simply ignoring the chroma, and we get an ordinary step notation.
I think this is especially useful in the case of quasi-equal MOS scales, which are approximations of a smaller EDO within a larger one.

Likewise, we could apply the diatonic notation to 7-EDO by ignoring any chroma. However, since the unison is a '1' (and not a '0'), 1 2 3 4 5 6 7 and 8 would refer to 0 1 2 3 4 5 6 and 7 steps, octaves would be 8, 15, 22 and 29 instead of 7, 14, 21 and 28, and so on.

If I had to suggest a standard, I'd say the unison should be '0' instead of '1', except for the diatonic and antidiatonic scale, where it would be counterintuitive for musicians with a (western) classical background.

- Gedankenwelt

Hi Graham,

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> There are existing standards we don't get to decide.
> Jianpu, which is a "movable doh" system for diatonic scales,
> starts with '1' and uses '0' for rests. Guitar tablature
> and musical set theory, both for chromatic scales, start
> with 0.

I don't think it makes much sense to differ between interval notation applied to guitar tablature, or not. Guitar tablature (or tablature for other string instruments) works with any kind of interval notation, as long as the unison is associated with the open string.

For example, it's no problem to use tablatures for violins based on a pythagorean diatonic interval notation (you can add accidentals to access higher limits, like syntonic comma accidentals / and \ for the 5-limit). Here's the C major scale on a typical violin:

E|---------------------
A|----------------2\-b3
D|---1-2\-b3-4-5\------
G|-4-------------------

I think it's best if standards are non-mandatory, but I agree that it's important not to ignore already established standards.

Best
-Gedankenwelt

"gedankenwelt94" <gedankenwelt94@...> wrote:

> Sorry, I should have been more specific.
>
> Typically, a diatonic C major scale is notated as following:
> C D E F G A B C. In order to modulate, notate MODMOS scales
> or scales with more than 7 notes etc., # or b can be used to
> indicate chroma alterations.
>
> However, it's possible to notate a non-specific major scale
> using an interval notation: 1 2 3 4 5 6 maj7 (or simply "7") 8.
> Here again, we can use # and b to indicate chroma alterations.
> For example, 1 2 b3 4 5 b6 b7 8 describes a natural minor scale.

Hm, I think that's the same thing. By writing 1 2 3... N, you
just mean 'the interval between the tonic and Nth degree of
the major scale. Same with the letters.

-Carl

Hi~
One of the great advantages to MOS scales as a master set is their ability to fit onto a generalized keyboard. In fact the two are tied quite closely. Since many of them will occur their more than anywhere else, a notation based on how scales lay out on the keyboard is quite logical and advantageous. For instance have build instruments and mounting them in generalized patterns such sign of a slant up or down keyboard has made the reading of notes on them extremely easier.

-- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, "gedankenwelt9
4" <gedankenwelt94@...> wrote:
> ...which makes me wonder: Why do I see none mentioning interval notations based on MOS scales? I think they can be very useful, especially for generalized guitar tabs that aren't limited to a specific equal division, or in case of larger EDOs where something like 8/ instead of "36th fret" is much more readable.

--
signature file

/^_
_ /

/^_,',',',_ //^/Kraig Grady_^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hm, I think that's the same thing. By writing 1 2 3... N, you
> just mean 'the interval between the tonic and Nth degree of
> the major scale. Same with the letters.

I think we're slightly talking past one another, though we probably mean the same - or at least something similar.

You're correct that the principle behind an MOS-based notation for notes and for intervals is basically the same. If that's what you meant, we probably agree.

However, if I use an interval notation (based on numbers), I'm refering to the distance between two notes, while when I use letters for note names, I'm refering to absolute pitches, which makes a difference.

And I think it's a good idea to use different notations for notes and for intervals, just like mathematicians distinguish between points (absolute) and vectors (relative). Well, at least sometimes they do. ;)

Best
- Gedankenwelt

"gedankenwelt94" <gedankenwelt94@...> wrote:

> if I use an interval notation (based on numbers), I'm refering to
> the distance between two notes, while when I use letters for note
> names, I'm refering to absolute pitches, which makes a difference.

Can you show an example? Writing C D E etc. doesn't refer to
pitches. One also needs to know the concert pitch. In normal
practice A=440 is often assumed, but I don't think that's a
safe assumption in the microtonal world. Even performers of
early music often use A=415 or another standard.

In the regular temperament practice, one would also need to
specify a tuning. A B C D E F G H I may be used to notate
negri[9], but one would also have to say whether one is using
the TE tuning, or the POTE tuning, etc. (see the wiki for
more information).

The point is that these notations really refer to offsets
into a scale, just like the numerals 1 2 3 etc. might do.

Differential notation (like Byzantine notation) really is
different though. An example is writing the ascending major
scale like this:

L L s L L L s

Just trying to clarify,

-Carl

Hi Kraig! :)

--- In tuning@yahoogroups.com, kraiggrady <kraiggrady@...> wrote:
>
> Hi~
> One of the great advantages to MOS scales as a master set is their
> ability to fit onto a generalized keyboard. In fact the two are tied
> quite closely. Since many of them will occur their more than anywhere
> else, a notation based on how scales lay out on the keyboard is quite
> logical and advantageous. For instance have build instruments and
> mounting them in generalized patterns such sign of a slant up or down
> keyboard has made the reading of notes on them extremely easier.

That's definitely an important relationship, though I'm looking at it the other way round:

MOS scales are exceptionally well-suited to notate music in the underlying linear tuning. If one is mainly interested in a certain linear tuning, it makes sense to choose an instrument that is build around such MOS scales.

For me that's the case in 94-EDO, where I'm mainly interested in garibaldi temperament; hence my fretboard design which is built around garibaldi MOS scales.

However, in 31-EDO I want to freely think in and modulate between different linear tunings. I think that different MOS-based notations can still be advantageous, but that an instrument with a strong focus on a specific linear tuning (like a generalized keyboard) is rather disadvantageous when trying to think in other linear tunings.

Best
- Gedankenwelt

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> "gedankenwelt94" <gedankenwelt94@> wrote:
>
> > if I use an interval notation (based on numbers), I'm refering to
> > the distance between two notes, while when I use letters for note
> > names, I'm refering to absolute pitches, which makes a difference.
>
> Can you show an example? Writing C D E etc. doesn't refer to
> pitches. One also needs to know the concert pitch. In normal
> practice A=440 is often assumed, but I don't think that's a
> safe assumption in the microtonal world. Even performers of
> early music often use A=415 or another standard.
>
> In the regular temperament practice, one would also need to
> specify a tuning. A B C D E F G H I may be used to notate
> negri[9], but one would also have to say whether one is using
> the TE tuning, or the POTE tuning, etc. (see the wiki for
> more information).

You're correct, one has to set some parameters first. However, once a reference pitch and a tuning is specified, "A B C D E F G H I A" will refer to a clearly defined scale starting at a specific pitch, while something like "0 1 2 3 4 5 6 7 8 9" would be a generalization of the scale where no pitch is specified. It could refer to "A B C D E F G H I A" as well as to "B C D E Fb G H I A B" or "I A B C D E# F G H I", or to any other transposition of the scale.
(# and b represent a chroma up and down)

> The point is that these notations really refer to offsets
> into a scale, just like the numerals 1 2 3 etc. might do.
>
> Differential notation (like Byzantine notation) really is
> different though. An example is writing the ascending major
> scale like this:
>
> L L s L L L s

By using the diatonic notation I mentioned in my previous post, it's possible to notate the above scale as 2 2 b2 2 2 2 b2. Likewise, a pentatonic scale L s s L s could be written in diatonic notation as b3 2 2 b3 2.

And I think I see now where our misunderstanding lies:
My focus was on the meaning of the single symbols (2 = major second, b3 = minor third etc.), while your focus was whether scales are notated in relation to a (given or abstract) reference note, or as subsequent steps. Thanks for your help to clear things up! :)

Best
- Gedankenwelt

"gedankenwelt94" <gedankenwelt94@...> wrote:

> You're correct, one has to set some parameters first. However,
> once a reference pitch and a tuning is specified, "A B C D E F
> G H I A" will refer to a clearly defined scale starting at a
> specific pitch, while something like "0 1 2 3 4 5 6 7 8 9" would
> be a generalization of the scale where no pitch is specified.
> It could refer to "A B C D E F G H I A" as well as to "B C D E
> Fb G H I A B" or "I A B C D E# F G H I", or to any other
> transposition of the scale.

We're advocating the same thing then. I do not think of the
letters as pitches.

> > Differential notation (like Byzantine notation) really is
> > different though. An example is writing the ascending major
> > scale like this:
> >
> > L L s L L L s
>
> By using the diatonic notation I mentioned in my previous post,
> it's possible to notate the above scale as 2 2 b2 2 2 2 b2.
> Likewise, a pentatonic scale L s s L s could be written in
> diatonic notation as b3 2 2 b3 2.

Have you made two proposals then? Clearly this is incompatible
with the above.

-Carl

I wrote:
> > You're correct, one has to set some parameters first. However,
> > once a reference pitch and a tuning is specified, "A B C D E F
> > G H I A" will refer to a clearly defined scale starting at a
> > specific pitch, while something like "0 1 2 3 4 5 6 7 8 9" would
> > be a generalization of the scale where no pitch is specified.
> > It could refer to "A B C D E F G H I A" as well as to "B C D E
> > Fb G H I A B" or "I A B C D E# F G H I", or to any other
> > transposition of the scale.
>
> We're advocating the same thing then. I do not think of the
> letters as pitches.

That is, like Mike I think, I intend to use a specialized
'key signature' marking to specify transpositions. -C.

What I'm doing is basically a generalization of the diatonic
notation used in Jazz notation (which is derived/adopted
from classical notation). Letters refer to notes, while
numbers refer to intervals.

If I write "C D E F G A B" in diatonic notation, I'm refering
to a specific major scale (i.e. the C major scale).
The letters represent notes that can be used independently
from the scale, and can be identified with specific pitches
if a fixed tuning and a concert pitch are chosen.

The intervals 1,2,3,4,5,6,(maj)7 are defined as the distance
between the root of the diatonic major scale, and the
1st,2nd,3rd,4th,5th,6th,7th scale degree, respectively.

Notes and intervals can be modified by a chroma # or b.
So "2" refers to the (diatonic) major second, "b2" to the
minor second, and "#2" to the augmented second.

There are (at least) two ways to represent a scale structure
using intervals:

1) as a set of intervals that represent the distance between
the root and the scale degrees.
2) as a list of intervals between subsequent scale degrees.

For example, the harmonic minor scale can be represented as
"1 2 b3 4 5 b6 maj7 (8)" by using method 1,
or as "2 b2 2 2 b2 #2 b2" by using method 2.

However, this is independent from the basic notation idea.
I only described how notes and intervals are represented,
not in which context they should or should not be used.

-Gedankenwelt

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> "gedankenwelt94" <gedankenwelt94@> wrote:
>
> > You're correct, one has to set some parameters first. However,
> > once a reference pitch and a tuning is specified, "A B C D E F
> > G H I A" will refer to a clearly defined scale starting at a
> > specific pitch, while something like "0 1 2 3 4 5 6 7 8 9" would
> > be a generalization of the scale where no pitch is specified.
> > It could refer to "A B C D E F G H I A" as well as to "B C D E
> > Fb G H I A B" or "I A B C D E# F G H I", or to any other
> > transposition of the scale.
>
> We're advocating the same thing then. I do not think of the
> letters as pitches.
>
> > > Differential notation (like Byzantine notation) really is
> > > different though. An example is writing the ascending major
> > > scale like this:
> > >
> > > L L s L L L s
> >
> > By using the diatonic notation I mentioned in my previous post,
> > it's possible to notate the above scale as 2 2 b2 2 2 2 b2.
> > Likewise, a pentatonic scale L s s L s could be written in
> > diatonic notation as b3 2 2 b3 2.
>
> Have you made two proposals then? Clearly this is incompatible
> with the above.
>
> -Carl

"gedankenwelt94" <gedankenwelt94@...> wrote:

> However, this is independent from the basic notation idea.
> I only described how notes and intervals are represented,
> not in which context they should or should not be used.

Aha, now I understand the scope of your proposal. Sorry to
belabor it. -Carl

Hi Gedankenwelt,

Am 28.05.2013 04:52, schrieb gedankenwelt94:
> What I'm doing is basically a generalization of the diatonic
> notation used in Jazz notation (which is derived/adopted
> from classical notation). Letters refer to notes, while
> numbers refer to intervals.
You mean Figured bass notation (DE: Generalbass)?
The difference between these both is that Generalbass keeps always in
the scale so that 2 can be major or minor whereas Jazz refers to the
distances in the C major Key from G upwards (call it Mixolydian).

>
> If I write "C D E F G A B" in diatonic notation, I'm refering
> to a specific major scale (i.e. the C major scale).
> The letters represent notes that can be used independently
> from the scale, and can be identified with specific pitches
> if a fixed tuning and a concert pitch are chosen.
>
> The intervals 1,2,3,4,5,6,(maj)7 are defined as the distance
> between the root of the diatonic major scale, and the
> 1st,2nd,3rd,4th,5th,6th,7th scale degree, respectively.
I'm a little confused, do you mean 1 9/8 5/4 4/3 3/2 5/3 15/8?
And ...

>
> Notes and intervals can be modified by a chroma # or b.
> So "2" refers to the (diatonic) major second, "b2" to the
> minor second, and "#2" to the augmented second.
... what exactly is this chroma? 25/24?
I think, these two questions are closely related.

>
>
> There are (at least) two ways to represent a scale structure
> using intervals:
>
> 1) as a set of intervals that represent the distance between
> the root and the scale degrees.
> 2) as a list of intervals between subsequent scale degrees.
>
> For example, the harmonic minor scale can be represented as
> "1 2 b3 4 5 b6 maj7 (8)" by using method 1,
> or as "2 b2 2 2 b2 #2 b2" by using method 2.
If I'm getting this right, you base all scales on the major scale?

I hope my questions don't throw us to far back.
Maybe I'd better re-read the whole thread ;)

Best,
Wolf

Hi Wolf,

--- In tuning@yahoogroups.com, Wolf Peuker <wolfpeuker@...> wrote:
>
> Am 28.05.2013 04:52, schrieb gedankenwelt94:
> > What I'm doing is basically a generalization of the diatonic
> > notation used in Jazz notation (which is derived/adopted
> > from classical notation). Letters refer to notes, while
> > numbers refer to intervals.
> You mean Figured bass notation (DE: Generalbass)?
> The difference between these both is that Generalbass keeps always in
> the scale so that 2 can be major or minor whereas Jazz refers to the
> distances in the C major Key from G upwards (call it Mixolydian).

What I meant is basically the latter, i.e. "2" refers to the major second, but not to the minor second (which would be "b2").
As far as I know, Jazz notation is a little inconsistent about the notation of the seventh. In chord notation, a minor seventh is usually notated as "7", while in scales, it's not unusual to notate it as "b7".

I went with the definition that "maj7" or "7" refer to the major seventh, while "b7" refers to the minor seventh, "#7" to the augmented seventh and bb7 to the diminished seventh.
It would also have been possible to notate the major seventh as "maj7" or "#7", the minor seventh as "7", the augmented seventh as "##7" or "x7", and the diminished seventh as "b7".

In Jazz you usually see "+" for augmented, and there's "°" for diminished. However, what I want is a flexible and simple system where you have seven basic intervals, and all other intervals in the linear tuning can be derived by applying one or more chromas # or b.

> > If I write "C D E F G A B" in diatonic notation, I'm refering
> > to a specific major scale (i.e. the C major scale).
> > The letters represent notes that can be used independently
> > from the scale, and can be identified with specific pitches
> > if a fixed tuning and a concert pitch are chosen.
> >
> > The intervals 1,2,3,4,5,6,(maj)7 are defined as the distance
> > between the root of the diatonic major scale, and the
> > 1st,2nd,3rd,4th,5th,6th,7th scale degree, respectively.
> I'm a little confused, do you mean 1 9/8 5/4 4/3 3/2 5/3 15/8?
> And ...
>
> >
> > Notes and intervals can be modified by a chroma # or b.
> > So "2" refers to the (diatonic) major second, "b2" to the
> > minor second, and "#2" to the augmented second.
> ... what exactly is this chroma? 25/24?
> I think, these two questions are closely related.

The notation can be used with any linear tuning that is generated by stacking "fifths" with a size between 4\7 and 3\5 (~685.7 to 720 cents). That way, you get a diatonic major scale with the form LLsLLLs when stacking 5 fifths upwards and 1 fifth downwards (modulo octave).

This scale is a MOS, which means the chroma is defined as the difference between the large step L and the small step s. You can also view the chroma as the difference between 7 fifths and 4 octaves.
If we think of notes in terms of octave equivalency, applying a chroma to notes in the diatonic scale leads either to transposed versions of the scale, or to so-called MODMOS scales, like the harmonic or melodic minor scale.

Unless a specific linear temperament is applied to the tuning (like meantone), its intervals remain "unmapped", i.e. no just ratios are associated with the intervals.

However, most temperaments associated with the generator range will associate the fifth with the ratio 3/2, and in those cases we can associate our scale with the pythagorean diatonic scale. So one valid interpretation of the diatonic major scale would be 1/1 9/8 81/64 4/3 3/2 27/16 243/128, and the chroma can be associated with the apotome or pythagorean chromatic semitone 2187/2048.

If we have a specific temperament, also interpretations that vary by one of the "unison vectors" (= tempered-out commas) become available. For example, in meantone temperament, the syntonic comma 81/80 is tempered out. So in that case, we can associate the major third with 5/4, which is 81/64 * 80/81 (the pythagorean major third divided by a syntonic comma). Likewise, we can use 5/3 instead of 27/16, 15/8 instead of 243/128, and for our chroma 25/24 or 135/128 instead of 2187/2048.
In this case, we can call this notation "meantone notation" (or Meantone[7] notation), which is a special case of this diatonic notation. But unless we apply a specific temperament, it's just a diatonic notation, and if we're really strict, we cannot even be sure our "fifth" is associated with 3/2.

We could as well have chosen superpyth temperament, in which the septimal comma 64/63 is tempered out. Here, we could associate the major third with 9/7, but not with 5/4.

Also, the notation can be directly applied to an abstract temperament (no need for a specific linear tuning), which means things like adaptive JI are possible. Or, you could even use the notation with a generator that doesn't produce a diatonic scale, if the definition of basic intervals / notes and the "chroma" is slightly adapted.

> > There are (at least) two ways to represent a scale structure
> > using intervals:
> >
> > 1) as a set of intervals that represent the distance between
> > the root and the scale degrees.
> > 2) as a list of intervals between subsequent scale degrees.
> >
> > For example, the harmonic minor scale can be represented as
> > "1 2 b3 4 5 b6 maj7 (8)" by using method 1,
> > or as "2 b2 2 2 b2 #2 b2" by using method 2.
> If I'm getting this right, you base all scales on the major scale?

Not necessarily; I could also write "1 b2 2 b3 3 4 #4 5 b6 6 b7 maj7 (8)" to notate the chromatic scale that is generated by stacking 6 fifths up and 5 fifths down. Or any other scale whose notes can be generated by stacking fifths.

Just imagine there's a sequence of fifths that extends in both directions infinitely, like this:

... Fbb Cbb Gbb Dbb Abb Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# F## C## G## D## A## E## B## ...

or for intervals:

... bb4 bb1 bb5 bb2 bb6 bb3 bb7 b4 b1 b5 b2 b6 b3 b7 4 1 5 2 6 3 maj7 #4 #1 #5 #2 #6 #3 #7 ##4 ##1 ##5 ##2 ##6 ##3 ##7 ...

You can use any of those for a scale, for example 1 2 #2 4 5 #5 #6 (8), or with explicit note names: A B B# D E E# F## (A). This may look a little weird, but if we assume (septimal) meantone temperament, then #2, #5 and #6 can be interpreted as 7/6, 14/9 and 7/4 respectively, which gives us a septimal minor scale.

> I hope my questions don't throw us to far back.
> Maybe I'd better re-read the whole thread ;)

No, I think it's good that you asked; I bet there are other people who had similar questions, but where to shy to ask. ;)
Plus, I'm often uncertain how much of what I'm writing is understandable for other people, and feedback helps me.

Best
-Gedankenwelt

Hi Gedankenwelt,

Am 28.05.2013 17:37, schrieb gedankenwelt94:
> Hi Wolf,
> [...]
>>> If I write "C D E F G A B" in diatonic notation, I'm refering
>>> to a specific major scale (i.e. the C major scale).
>>> The letters represent notes that can be used independently
>>> from the scale, and can be identified with specific pitches
>>> if a fixed tuning and a concert pitch are chosen.
>>>
>>> The intervals 1,2,3,4,5,6,(maj)7 are defined as the distance
>>> between the root of the diatonic major scale, and the
>>> 1st,2nd,3rd,4th,5th,6th,7th scale degree, respectively.
>> I'm a little confused, do you mean 1 9/8 5/4 4/3 3/2 5/3 15/8?
>> And ...
>>
>>>
>>> Notes and intervals can be modified by a chroma # or b.
>>> So "2" refers to the (diatonic) major second, "b2" to the
>>> minor second, and "#2" to the augmented second.
>> ... what exactly is this chroma? 25/24?
>> I think, these two questions are closely related.
>
> The notation can be used with any linear tuning that is generated by stacking "fifths" with a size between 4\7 and 3\5 (~685.7 to 720 cents). That way, you get a diatonic major scale with the form LLsLLLs when stacking 5 fifths upwards and 1 fifth downwards (modulo octave).
Why exactly these borders?
>
> This scale is a MOS, which means the chroma is defined as the difference between the large step L and the small step s. You can also view the chroma as the difference between 7 fifths and 4 octaves.
> If we think of notes in terms of octave equivalency, applying a chroma to notes in the diatonic scale leads either to transposed versions of the scale, or to so-called MODMOS scales, like the harmonic or melodic minor scale.
I see: *the* chroma is only defined for MOS scales.
>
> Unless a specific linear temperament is applied to the tuning (like meantone), its intervals remain "unmapped", i.e. no just ratios are associated with the intervals.
Is it right, that linear temperament means a infinite amount of tones
like the following?

-INF ..., t(-1), t(0), t(1), ... +INF

..whereas EDOs are cyclic?

>
> However, most temperaments associated with the generator range will associate the fifth with the ratio 3/2, and in those cases we can associate our scale with the pythagorean diatonic scale. So one valid interpretation of the diatonic major scale would be 1/1 9/8 81/64 4/3 3/2 27/16 243/128, and the chroma can be associated with the apotome or pythagorean chromatic semitone 2187/2048.
>
> If we have a specific temperament, also interpretations that vary by one of the "unison vectors" (= tempered-out commas) become available. For example, in meantone temperament, the syntonic comma 81/80 is tempered out. So in that case, we can associate the major third with 5/4, which is 81/64 * 80/81 (the pythagorean major third divided by a syntonic comma). Likewise, we can use 5/3 instead of 27/16, 15/8 instead of 243/128, and for our chroma 25/24 or 135/128 instead of 2187/2048.
> In this case, we can call this notation "meantone notation" (or Meantone[7] notation), which is a special case of this diatonic notation. But unless we apply a specific temperament, it's just a diatonic notation, and if we're really strict, we cannot even be sure our "fifth" is associated with 3/2.
>
> We could as well have chosen superpyth temperament, in which the septimal comma 64/63 is tempered out. Here, we could associate the major third with 9/7, but not with 5/4.
>
> Also, the notation can be directly applied to an abstract temperament (no need for a specific linear tuning), which means things like adaptive JI are possible. Or, you could even use the notation with a generator that doesn't produce a diatonic scale, if the definition of basic intervals / notes and the "chroma" is slightly adapted.
What do you mean with abstract temperament?

Thanks in advance!

Best,
Wolf

H Wolf,

--- In tuning@yahoogroups.com, Wolf Peuker <wolfpeuker@...> wrote:
>
> Am 28.05.2013 17:37, schrieb gedankenwelt94:
> > [...]
> > The notation can be used with any linear tuning that is generated by stacking "fifths" with a size between 4\7 and 3\5 (~685.7 to 720 cents). That way, you get a diatonic major scale with the form LLsLLLs when stacking 5 fifths upwards and 1 fifth downwards (modulo octave).
> Why exactly these borders?

Do you mean (a) why those borders lead to a diatonic scale with form LLsLLLs, or (b) why I used borders to limit this diatonic notation to the diatonic scale?

An answer to (a):
If the generator fifth becomes larger, also the large step L increases, while the small step s decreases. If the fifth becomes smaller, it's of course vice-versa.

One extreme cases where the pattern is preserved is when the small steps almost vanish, which means our scale converges to 5-EDO, and is the case if we choose a larger generator that is close to 3\5.

The other extreme case is when s becomes so large that it's almost identical to L, which means our scale converges to 7-EDO, and is the case if we choose a smaller generator that is close to 4\7.

An answer to (b):
These are borders for the primary use of the notation that guarantee that there's "no strange behaviour".

You can use a generator fifth that lies between 1\2 (600 cents) and 4\7 (~685.7 cents), which means you'll get an antidiatonic scale where "major intervals" are smaller than "minor intervals", and b increases the pitch by a chroma, while # decreases it.
Otherwise, it will work perfectly fine if you choose a temperament that makes sense.

However, if you choose a generator slightly above 3\5, you won't even have a 7-note MOS. You could still define something similar to a chroma that allows you to modulate, build something similar to MODMOS scales or notate every note / interval in the tuning. But the "chroma" will be so large that in the case of modulations or "pseudo-MODMOS scales", the nominal's order usually isn't preserved, which leads to weird results.
You *can* do this, but in this case a notation based on something like a 5- or 8-note MOS will probably work better.

> > Unless a specific linear temperament is applied to the tuning (like meantone), its intervals remain "unmapped", i.e. no just ratios are associated with the intervals.
> Is it right, that linear temperament means a infinite amount of tones
> like the following?
>
> -INF ..., t(-1), t(0), t(1), ... +INF
>
> ..whereas EDOs are cyclic?

I'll try to answer that as far as I can, but I don't know if got everything right:

Linear temperaments can be abstract, which means it's only about which intervals are associated with which just ratios, but there is no specification about the actual tuning of the intervals. This is how I used the term in the passage you quoted.

If an (abstract) linear temperament is applied to a linear tuning, the result is called a (non-abstract) linear temperament.

A linear temperament always has an infinite number of notes / intervals when we look at it as an abstract temperament, or if we refer to the abstract temperament that was applied to the tuning. In that sense they're always non-cyclic.

However, you asked about the number of "tones", which I'd usually associate with actual pitches. In that sense, an abstract linear temperament doesn't have tones (because no pitches are specified), so the question doesn't apply there.

In a (non-abstract) linear temperament, there are tones with specific pitches in the underlying linear tuning, and if that tuning is based on an EDO, it's cyclic.

So if we talk about the (non-abstract) linear temperament which is meantone temperament applied to 19-EDO (via patent val mapping)*, then I'd say it has infinitely many notes from a temperament perspective, but from a tuning perspective it only has a finite number of tones (octave equivalency assumed) and is cyclic.

That's at least how I see it, but I'm not sure if that's the correct way.

* Note that 19-EDO itself is only a rank-1 tuning, because it can be generated with 1\19 and no period. However, if we use 11\19 as a generator and an octave period (which would happen in the above example), we have a rank-2 (or linear) tuning.

> > Also, the notation can be directly applied to an abstract temperament (no need for a specific linear tuning), which means things like adaptive JI are possible. Or, you could even use the notation with a generator that doesn't produce a diatonic scale, if the definition of basic intervals / notes and the "chroma" is slightly adapted.
> What do you mean with abstract temperament?

See above, or here:
http://xenharmonic.wikispaces.com/Abstract+regular+temperament

In meantone temperament, it means that for example a major second can represent both 9/8 and 10/9, but an actual interval size is not specified.
If adaptive JI is used with an abstract temperament, the same interval can have a different size in different contexts.

Best
-Gedankenwelt