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= ARCHIVED WIKISPACES DISCUSSION BELOW =
{{WSArchiveLink}}
'''All discussion below is archived from the Wikispaces export in its original unaltered form.'''
 
== Note names in the diatonic scale ==
 
The Examples section currently contains the following table:
 
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== **to have** v.s **to be** a constant structure ==
Interval matrix as note names:
the introduction uses <em>to have</em> which I find better to understand, should we decide for a more consistent terminology for the verb?
 
{| class="wikitable center-all"
!
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! (8)
|-
! C
| C
| D
| E
| F
| G
| A
| B
| C
|-
! D
| C
| D
| Eb
| F
| G
| A
| Bb
| C
|-
! E
| C
| Db
| Eb
| F
| G
| Ab
| Bb
| C
|-
! F
| C
| D
| E
| <span style="background-color: #ffcc44;">F#</span>
| G
| A
| B
| C
|-
! G
| C
| D
| E
| F
| G
| A
| Bb
| C
|-
! A
| C
| D
| Eb
| F
| G
| Ab
| Bb
| C
|-
! B
| C
| Db
| Eb
| F
| <span style="background-color: #ffcc44;">Gb</span>
| Ab
| Bb
| C
|}


- '''xenwolf''' September 04, 2017, 12:15:04 AM UTC-0700
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== Construction of Constant Structures ==
This usage seems incoherent to me: if the scale in the example is the diatonic scale containing C, D, E, F, G, A, and B, then ''the scale in question doesn't contain any notes with sharps or flats,'' and it's nonsensical to talk about those notes. Instead, the table should describe the notes from a single scale, and the paragraph that follows it should also refer to the notes within that same scale.
Could some material or resources on -how- to construct these scales be added? It seems to get exponentially harder the more notes you want and the brute force approach isn't getting me anywhere.


I'm particularly interested in CS Scales because they seem to be the best option for bringing Just Intonation to isomorphic keyboards.
I suggest something like the following instead:


- '''kai.lugheidh''' March 21, 2017, 07:33:10 PM UTC-0700
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I've noticed that when constructing regular scales for certain limits in a way that extends the concept of well-formedness – that is, having no more different scale steps than generating primes, and always breaking up any extraneous scale steps in the same way – Scala always reports them as being constant structures. Could I be on to something here?


- '''kai.lugheidh''' June 01, 2017, 08:15:01 PM UTC-0700
Interval matrix as steps of 12edo:
 
{| class="wikitable center-all"
!
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! (8)
|-
! 0\12
| 0\12
| 2\12
| 4\12
| 5\12
| 7\12
| 9\12
| 11\12
| 12\12
|-
! 2\12
| 0\12
| 2\12
| 3\12
| 5\12
| 7\12
| 9\12
| 10\12
| 12\12
|-
! 4\12
| 0\12
| 1\12
| 3\12
| 5\12
| 7\12
| 8\12
| 10\12
| 12\12
|-
! 5\12
| 0\12
| 2\12
| 4\12
| <span style="background-color: #ffcc44;">6\12</span>
| 7\12
| 9\12
| 11\12
| 12\12
|-
! 7\12
| 0\12
| 2\12
| 4\12
| 5\12
| 7\12
| 9\12
| 10\12
| 12\12
|-
! 9\12
| 0\12
| 2\12
| 3\12
| 5\12
| 7\12
| 8\12
| 10\12
| 12\12
|-
! 11\12
| 0\12
| 1\12
| 3\12
| 5\12
| <span style="background-color: #ffcc44;">6\12</span>
| 8\12
| 10\12
| 12\12
|}
 
Interval matrix as note names:
 
{| class="wikitable center-all"
!
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! (8)
|-
! C
| C
| D
| E
| F
| G
| A
| B
| C
|-
! D
| D
| E
| F
| G
| A
| B
| C
| D
|-
! E
| E
| F
| G
| A
| B
| C
| D
| E
|-
! F
| F
| G
| A
| <span style="background-color: #ffcc44;">B</span>
| C
| D
| E
| F
|-
! G
| G
| A
| B
| C
| D
| E
| F
| G
|-
! A
| A
| B
| C
| D
| E
| F
| G
| A
|-
! B
| B
| C
| D
| E
| <span style="background-color: #ffcc44;">F</span>
| G
| A
| B
|}
 
In 12edo, the intervals from F to B and from B to F are the same size: 6\12, or 600 cents. From F to B, this interval spans four steps of our diatonic scale; but from B to F it spans five. Since the same (600¢) interval spans different numbers of scale steps at different points in the scale, this scale is not a constant structure.
 
However, in tunings that assign different interval sizes for F–B and B–F — such as meantone and superpyth — the diatonic scale ''is'' a constant structure. For example, 31edo (meantone) tunes F–B and B–F to 15\31 (581¢) and 16\31 (619¢) respectively, so the four-scale-step interval is distinct from the five-scale-step one:
 
{| class="wikitable center-all"
!
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! (8)
|-
! 0\31
| 0\31
| 5\31
| 10\31
| 13\31
| 18\31
| 23\31
| 28\31
| 31\31
|-
! 5\31
| 0\31
| 5\31
| 8\31
| 13\31
| 18\31
| 23\31
| 26\31
| 31\31
|-
! 10\31
| 0\31
| 3\31
| 8\31
| 13\31
| 18\31
| 21\31
| 26\31
| 31\31
|-
! 13\31
| 0\31
| 5\31
| 10\31
| <span style="background-color: #ffcc44;">15\31</span>
| 18\31
| 23\31
| 28\31
| 31\31
|-
! 18\31
| 0\31
| 5\31
| 10\31
| 13\31
| 18\31
| 23\31
| 26\31
| 31\31
|-
! 23\31
| 0\31
| 5\31
| 8\31
| 13\31
| 18\31
| 21\31
| 26\31
| 31\31
|-
! 28\31
| 0\31
| 3\31
| 8\31
| 13\31
| <span style="background-color: #ffcc44;">16\31</span>
| 21\31
| 26\31
| 31\31
|}
 
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== Density of CS Scales in EDO's  ==
--[[User:Bcmills|Bcmills]] ([[User talk:Bcmills|talk]]) 12:18, 2 May 2024 (UTC)
What is the definition of density?


- '''xenwolf''' June 12, 2015, 01:21:15 AM UTC-0700
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I guess I was thinking of something akin to this definition, but for scales instead of numbers:
I like this one better:
{| class="wikitable center-all"
!
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! (8)
|-
! C
| P1
| M2
| M3
| P4
| P5
| M6
| M7
| P8
|-
! D
| P1
| M2
| m3
| P4
| P5
| M6
| m7
| P8
|-
! E
| P1
| m2
| m3
| P4
| P5
| m6
| m7
| P8
|-
! F
| P1
| M2
| M3
| <span style="background-color: #ffcc44;">A4</span>
| P5
| M6
| M7
| P8
|-
! G
| P1
| M2
| M3
| P4
| P5
| M6
| m7
| P8
|-
! A
| P1
| M2
| m3
| P4
| P5
| m6
| m7
| P8
|-
! B
| P1
| m2
| m3
| P4
| <span style="background-color: #ffcc44;">d5</span>
| m6
| m7
| P8
|}
Because now the CS violation can be called out as A4 = d5 when L = 2s.
 
--[[User:Frostburn|Frostburn]] ([[User talk:Frostburn|talk]]) 15:24, 3 May 2024 (UTC)
 
== “has” vs. “is” ==
 
The current version of the page uses two variations: “have/has constant structure” and “is a constant structure”. (See the [[Talk:Constant_structure/WikispacesArchive#.2A.2Ato_have.2A.2A_v.s_.2A.2Ato_be.2A.2A_a_constant_structure|previous discussion]] from the Wikispaces archive.)
 
Erv Wilson's original papers tended to mention only “constant structures” (plural noun), and the Yahoo tuning list entries cited in the [[http://www.tonalsoft.com/enc/c/constant-structure.aspx|Tonalsoft entry]] on constant structures consistently use “is”, with or without an article (“is/was a constant structure” or “is CS”). As far as I can tell, “has constant structure” is a neologism created in the Wikispaces entry.


https://en.wikipedia.org/wiki/Natural_density
The only Google results I see for [https://www.google.com/search?q=%22has+constant+structure%22 "has constant structure"] and [https://www.google.com/search?q=%22have+constant+structure%22 "have constant structure"] are a couple of pages on this wiki, the phrase “you can have constant-structure 17 and 19-limit otonal scales” in the Yahoo tuning list [https://robertinventor.com/tuning-math/frame.html?s___9/msg_8250-8274.html&8271 message 8271], and a bunch of mathematical and chemical contexts that have nothing to do with the use of the phrase in the context of Xenharmonic music.


Basically, it is a way to ask the question of "how many CS scales are there compared to non-CS scales?"
So I'm going to resolve this inconsistency by changing this article to consistently use “is”.


- '''Sarzadoce''' June 12, 2015, 09:01:02 PM UTC-0700
--[[User:Bcmills|Bcmills]] ([[User talk:Bcmills|talk]]) 13:18, 6 May 2024 (UTC)
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