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m it's very annoying to have to constantly change from a section of the page to the very bottom just to read a single note; the disorientation makes me forget what the note was even for; furthermore, *all* of the notes are about that one table, so the separation is even more indefensible
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→Step ratio ranges: fixed irrelevant and duplicate content. Feel free to revert if there is a clarity issue.
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== Step ratio spectrum ==
== Step ratio spectrum ==
{{Main| Step ratio }}
{{Main| Step ratio }}TAMNAMS names nine specific simple [[Blackwood's R|L:s ratios]] tabulated below, which correspond to the simplest edos that have the mos scale. The two extremes, equalized and collapsed, are degenerate cases and define the boundaries for valid tuning ranges. An equalized mos has large and small steps be the same size ({{nowrap|L {{=}} s}}), so the mos pattern is no longer apparent. A collapsed mos has small steps shrunken down to zero ({{nowrap|s {{=}} 0}}), merging adjacent tones s apart into a single tone. In both cases, the mos structure is no longer valid.


=== Simple step ratios ===
TAMNAMS names nine specific simple [[Blackwood's R|L:s ratios]]. These correspond to the simplest edos that have the mos scale.
{| class="wikitable"
|+ style="font-size: 105%;" | Step ratio names
|-
! TAMNAMS Name
! Ratio
! Hardness
! Diatonic example
|-
| Equalized
| L:s = 1:1
| 1.000
| [[7edo]]
|-
| Supersoft
| L:s = 4:3
| 1.333
| [[26edo]]
|-
| Soft (or monosoft)
| L:s = 3:2
| 1.500
| [[19edo]]
|-
| Semisoft
| L:s = 5:3
| 1.667
| [[31edo]]
|-
| Basic
| L:s = 2:1
| 2.000
| [[12edo]]
|-
| Semihard
| L:s = 5:2
| 2.500
| [[29edo]]
|-
| Hard (or monohard)
| L:s = 3:1
| 3.000
| [[17edo]]
|-
| Superhard
| L:s = 4:1
| 4.000
| [[22edo]]
|-
| Collapsed
| L:s = 1:0
| ∞ (infinity)
| [[5edo]]
|}
For example, the 5L 2s (diatonic) scale of 19edo has a step ratio of 3:2, which is ''soft'', and is thus called ''soft diatonic''. Tunings of a mos with L:s larger than that ratio are ''harder'', and tunings with L:s smaller than that are ''softer''.
The two extremes, equalized and collapsed, are degenerate cases and define the boundaries for valid tuning ranges. An equalized mos has large and small steps be the same size ({{nowrap|L {{=}} s}}), so the mos pattern is no longer apparent. A collapsed mos has small steps shrunken down to zero ({{nowrap|s {{=}} 0}}), merging adjacent tones s apart into a single tone. In both cases, the mos structure is no longer valid.
=== Step ratio ranges ===
In between the nine specific ratios there are eight named intermediate ranges of step ratios. These names are useful for classifying mos tunings which don't match any of the nine simple step ratios. There are also two additional terms for broader ranges: the term ''hyposoft'' describes step ratios that are ''soft-of-basic'' but not as soft as 3:2; similarly, the term ''hypohard'' describes step ratios that are ''hard-of-basic'' but not as hard as 3:1.
In between the nine specific ratios there are eight named intermediate ranges of step ratios. These names are useful for classifying mos tunings which don't match any of the nine simple step ratios. There are also two additional terms for broader ranges: the term ''hyposoft'' describes step ratios that are ''soft-of-basic'' but not as soft as 3:2; similarly, the term ''hypohard'' describes step ratios that are ''hard-of-basic'' but not as hard as 3:1.


By default, all ranges include their endpoints. For example, a hard tuning is considered a quasihard tuning. To exclude endpoints, the modifier ''strict'' can be used, for example ''strict hyposoft''.
By default, all ranges include their endpoints. For example, a hard tuning is considered a quasihard tuning. To exclude endpoints, the modifier ''strict'' can be used, for example ''strict hyposoft''.


Note that mosses with soft-of-basic step ratios always exhibit [[Rothenberg propriety]], or are ''proper'', whereas mosses with hard-of-basic step ratios do not, or are ''not proper'', with one exception: mosses with only one small step per period are always proper, regardless of the step ratio. It has been argued, however, that this is not a particularly important property, both because "improper" MOSSes still admit an ordering if you allow "off-by-one" errors and because larger moses tend to sound more distinct when {{nowrap|L/s > 1}}, which is in some sense the more vast/varied side of the tuning spectrum, because as L/s becomes larger, the scale becomes increasingly close to the [[equalized]] tuning, which is usually radically different from most "proper" tunings while softer tunings don't have much room to be different compared to the basic tuning. (This is explained in more detail in [[TAMNAMS/Appendix#Extending the spectrum's edges]].)
In some cases it can be clearer to name step ratio ranges by their ranges in hardness (for example, 1-1.33 for ultrasoft) or by their boundary step ratios (for example, equalized-to-supersoft for ultrasoft) than by the step ratio ranges tabulated here.
 
{| class="wikitable"
|+ style="font-size: 105%;" | Intermediate ranges
|-
! TAMNAMS Name
! Ratio range
! Hardness
|-
| Hyposoft
| 3:2 ≤ L:s ≤ 2:1
| 1.500 ≤ L/s ≤ 2.000
|-
| Ultrasoft
| 1:1 ≤ L:s ≤ 4:3
| 1.000 ≤ L/s ≤ 1.333
|-
| Parasoft
| 4:3 ≤ L:s ≤ 3:2
| 1.333 ≤ L/s ≤ 1.500
|-
| Quasisoft
| 3:2 ≤ L:s ≤ 5:3
| 1.500 ≤ L/s ≤ 1.667
|-
| Minisoft
| 5:3 ≤ L:s ≤ 2:1
| 1.667 ≤ L/s ≤ 2.000
|-
| Minihard
| 2:1 ≤ L:s ≤ 5:2
| 2.000 ≤ L/s ≤ 2.500
|-
| Quasihard
| 5:2 ≤ L:s ≤ 3:1
| 2.500 ≤ L/s ≤ 3.000
|-
| Parahard
| 3:1 ≤ L:s ≤ 4:1
| 3.000 ≤ L/s ≤ 4.000
|-
| Ultrahard
| 4:1 ≤ L:s ≤ 1:0
| 4.000 ≤ L/s ≤ ∞
|-
| Hypohard
| 2:1 ≤ L:s ≤ 3:1
| 2.000 ≤ L/s ≤ 3.000
|}
 
One may ask "what about [[hypersoft]] and [[hyperhard]], given you have [[hyposoft]] and [[hypohard]]?" and they would be right: see [[TAMNAMS/Appendix#Extended spectrum]] which details a more complete glossary that this set of terms is a subset of.
 
=== Central spectrum ===
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Central spectrum of step ratio ranges and specific step ratios
|+ style="font-size: 105%;" | Spectrum of step ratio ranges and specific step ratios
|-
|-
! colspan="3" | Step ratio ranges
! colspan="3" | Step ratio ranges
! Specific<br />step ratios
! Specific<br />step ratios
!Hardness
! Notes
! Notes
|-
|-
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|
|
| '''1:1<br />(equalized)'''
| '''1:1<br />(equalized)'''
|1
| Trivial/pathological
| Trivial/pathological
|-
|-
| rowspan="7" | 1:1 to 2:1<br />(soft-of-basic)
| rowspan="7" | 1:1 to 2:1<br />(soft-of-basic)
| colspan="2" | 1:1 to 4:3<br />(ultrasoft)
| colspan="2" | 1:1 to 4:3<br />(ultrasoft)
|
|
|
| Step ratios especially close to 1:1 may be called pseudoequalized
| Step ratios especially close to 1:1 may be called pseudoequalized
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|
|
| '''4:3<br />(supersoft)'''
| '''4:3<br />(supersoft)'''
|1.33
|
|
|-
|-
| colspan="2" | 4:3 to 3:2<br />(parasoft)
| colspan="2" | 4:3 to 3:2<br />(parasoft)
|
|
|
|
|
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|
|
| '''3:2<br />(soft)'''
| '''3:2<br />(soft)'''
|1.5
| Also called monosoft
| Also called monosoft
|-
|-
| rowspan="3" | 3:2 to 2:1<br />(hyposoft)
| rowspan="3" | 3:2 to 2:1<br />(hyposoft)
| 3:2 to 5:3<br />(quasisoft)
| 3:2 to 5:3<br />(quasisoft)
|
|
|
|
|
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|
|
| '''5:3<br />(semisoft)'''
| '''5:3<br />(semisoft)'''
|1.67
|
|
|-
|-
| 5:3 to 2:1<br />(minisoft)
| 5:3 to 2:1<br />(minisoft)
|
|
|
|
|
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|
|
| '''2:1<br />(basic)'''
| '''2:1<br />(basic)'''
|2
|
|
|-
|-
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| rowspan="3" | 2:1 to 3:1<br />(hypohard)
| rowspan="3" | 2:1 to 3:1<br />(hypohard)
| 2:1 to 5:2<br />(minihard)
| 2:1 to 5:2<br />(minihard)
|
|
|
|
|
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|
|
| '''5:2<br />(semihard)'''
| '''5:2<br />(semihard)'''
|2.5
|
|
|-
|-
| 5:2 to 3:1<br />(quasihard)
| 5:2 to 3:1<br />(quasihard)
|
|
|
|
|
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|
|
| '''3:1<br />(hard)'''
| '''3:1<br />(hard)'''
|3
| Also called monohard
| Also called monohard
|-
|-
| colspan="2" | 3:1 to 4:1<br />(parahard)
| colspan="2" | 3:1 to 4:1<br />(parahard)
|
|
|
|
|
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|
|
| '''4:1<br />(superhard)'''
| '''4:1<br />(superhard)'''
|4
|
|
|-
|-
| colspan="2" | 4:1 to 1:0<br />(ultrahard)
| colspan="2" | 4:1 to 1:0<br />(ultrahard)
|
|
|
| Step ratios especially close to 1:0 may be called pseudocollapsed
| Step ratios especially close to 1:0 may be called pseudocollapsed
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|
|
| '''1:0<br />(collapsed)'''
| '''1:0<br />(collapsed)'''
|infinity
| Trivial/pathological
| Trivial/pathological
|}
|}
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